Companion document to the 2018 Mathematics Content Standards
2021 WYOMING MATH
PERFORMANCE STANDARDS WITH 2018 CONTENT STANDARDS
AND PERFORMANCE LEVEL DESCRIPTORS (PLDS)
Wyoming State Board of Education Members
Chairman Ryan Fuhrman, Vice Chair Max Mickelson, Treasurer Bill Lambert,
Debbie Bovee, Dan McGlade, Robin Schamber, Forrest Smith,
Ken Clouston, Ellen Creagar, Mark Mathern, and Amy Pierson
Ex Officio Members: Sandra Caldwell and Scott Thomas
Jillian Balow, Superintendent of Public Instruction
WDE Facilitators - Laurie Hernandez, Director of Standards & Assessment, Barb Marquer, Supervisor, and
Consultants: Jill Stringer, Cat Palmer, Alicia Wilson, Rob Black, Lori Pusateri-Lane, & Shannon Wachowski
Content Standards Effective - July 12, 2018
TO BE FULLY IMPLEMENTED IN DISTRICTS BY THE BEGINNING OF SCHOOL YEAR 2021-22
Performance Standards Effective – July 1, 2021
TO BE FULLY IMPLEMENTED IN DISTRICTS BY THE BEGINNING OF SCHOOL YEAR 2024-25
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2021 Math Wyoming Content & Performance Standards & PLDs
Companion document to the 2018 Mathematics Content Standards
MATH WYOMING 2018 CONTENT AND 2021 PERFORMANCE STANDARDS
ACKNOWLEDGMENT: The Wyoming State Board of Education would
like to thank the Wyoming Department of Education, as well as educators,
parents and community members, business and industry representatives,
community college representatives, and the University of Wyoming
representatives for their work on the development of these Math Content
Standards (CS), Performance Standards (PS), and Performance Level
Descriptors (PLDs).
INTRODUCTION:
The Wyoming Math Content and Performance
Standards (WYCPS) were last reviewed and approved in 2012 in accordance
with Wyoming State Statute W.S. 21-2-304(c). The 2018 Wyoming Math
Content Standards were developed collaboratively through the
contributions of Math Standard Review Committee (MSRC) members from
across the state. In 2020, a new committee was convened with members
from the original MSRC to identify the Performance Standards. For both
reviews, the committee’s work was informed and guided by initial public
input through community forums, as well as input solicited from
stakeholder groups.
RATIONALE: Mathematics is the language that defines the blueprint of
the universe. Mathematics is woven into all parts of our lives and is more
than a list of skills to be mastered. The essence of mathematics is the ability
to employ critical thinking and reasoning to solve problems. To be
successful in mathematics, one must see mathematics as sensible, useful,
and worthwhile. The 2018 Wyoming Mathematics Content and
Performance Standards address two kinds of knowledge: mathematical
content and mathematical practice.
WHY DO WE HAVE STANDARDS FOR MATHEMATICS?
Uniform and consistent mathematical education is necessary as it ensures
that all students in Wyoming are prepared for success in and out of the
classroom. Therefore, the 2018 Wyoming Mathematics Content and
Performance Standards:
Provide students, parents, and educators focus and coherence through
application including understanding of mathematical concepts and
processes.
• Align K-12 with clearly defined goals and outcomes for learning.
• Emphasize conceptual understanding.
• Encourage multiple models, representations, and strategies.
• Use technology to optimize mathematical understanding.
Develop students’ mathematical thinking.
• Develop reasoning, solving, representing, proving, communicating, and
connecting across contexts and applications.
• Recognize and identify mathematics in the world around us.
• Engage students in making sense, building conceptual understanding,
developing procedural fluency, and employing adaptive reasoning.
• Build constructive attitudes to see mathematics as sensible, useful and
worthwhile, and to increase confidence in one’s own ability to do
mathematics.
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2021 Math Wyoming Content & Performance Standards & PLDs
ORGANIZATION OF THE STANDARDS: (with terminology)
Standard Code: Grade. Math Domain. Cluster Code. Standard #
Key: K.G.I.4 = Grade K. Domain Geometry (G). Cluster I. Standard 4
DOMAINS are the core concepts to be studied in math. The Math Standards
usually consist of 5-6 domains in each grade level. The math domains are
listed below.
• Kindergarten – Counting & Cardinality (CC)
• K-5 – Operations & Algebraic Thinking (OA)
• K-5 – Number & Operations in Base Ten (NBT)
• K-5 – Measurement & Data (MD)
• K-HS – Geometry (G)
• 3-5 – Number & Operations – Fractions (NF)
• 6-7 – Ratios & Proportional Relationships (RP)
• 8-12 – Functions (F)
• 6-8 – Expressions & Equations (EE)
• 6-8 – The Number System (NS)
• 6-12 – Statistics & Probability (SP)
• 9-12 – Number & Quantity (N)
• 9-12 –Algebra (A)
CONTENT STANDARDS define the content and skills students are expected
to know and be able to do by the end-of-the-grade level or grade band.
They are built foundationally and then in learning progressions. They do not
dictate what methodology or instructional materials should be used, nor
how the material is delivered. In this standards document, you will find
these are broken out into individual grades for Kindergarten through 8th
grade and then banded by domains for high school (9-12).
STANDARDS FOR MATHEMATICAL PRACTICES
1. Make sense of problems and persevere in solving them.
2. Reason abstractly and quantitatively.
3. Construct viable arguments and critique the reasoning of others.
4. Model with mathematics.
5. Use appropriate tools strategically.
6. Attend to precision.
7. Look for and make use of structure.
8. Look for and express regularity in repeated reasoning.
ADVANCED STANDARDS (+)
The high school standards specify the mathematics that all students should
study to be college and career ready. Each standard without a (+) symbol
should be in the common mathematics curriculum for all students.
Advanced mathematics standards, those designated with a (+) sign, are
integrated into the higher level math courses after Algebra II. These
standards encourage student experiences in higher level mathematical
thinking and/or STEM pathways.
(Adapted from CCSS https://edu.wyoming.gov/downloads/standards/final-2012-math-standards.pdf)
PERFORMANCE LEVEL DESCRIPTORS (PLDs) describe the performance
expectations of students for each of the four (4) performance level
categories: Advanced, Proficient, Basic, and Below Basic. These are a
description of what students within each performance level are expected to
know and be able to do.
PERFORMANCE STANDARDS (PS) are the standards all students are
expected to learn and be assessed on through the district assessment
system by the end-of-the grade band. They specify the degree of
understanding or demonstration of the knowledge and/or skill for a
particular content standard. As such, they employ clear action verbs and
describe “how good is good enough.”
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For those designated as PS, the content standard is marked with the code
in blue highlight and an asterisk (*) and the Proficient PLD Statement is
the PS expectation and is highlighted in blue.
Districts are expected to give students multiple opportunities to
demonstrate proficiency on the Performance Standards through the District
Assessment System (DAS) and provide appropriate supports for student
success. Teachers should provide extra focus, targeted supports, and offer
multiple opportunities to demonstrate student understanding (mastery). In
the secondary level, only students electing to take a course aligned to these
standards need to be assessed in the DAS.
A snapshot of the Math Performance Standards can be found on page 8.
MATHEMATICAL LITERACY
“Mathematical literacy is an individual’s capacity to formulate, employ, and
interpret mathematics in a variety of contexts. It includes reasoning
mathematically and using mathematical concepts, procedures, facts, and
tools to describe, explain, and predict phenomena. It assists individuals to
recognize the role that mathematics plays in the world and to make the
well-founded judgements and decisions needed by constructive, engaged,
and reflective citizens.” https://www.achieve.org/files/StrongStandards.pdf
WHY DO WE HAVE THE STANDARDS FOR MATHEMATICAL
PRACTICE?
Procedural knowledge alone will not prepare our 21st Century students to
be globally competitive. Mathematical thinkers also visualize problems and
recognize that multiple strategies may lead to a single solution. They realize
mathematics is applicable outside of the classroom and are confident in
their ability to apply mathematical concepts to all aspects of life. The
Standards for Mathematical Practice cultivate mathematically literate and
informed citizens. Using mathematics as a means of synthesizing complex
concepts and making informed decisions is paramount to college and career
success. The Standards for Mathematical Practice develop skills that serve
students beyond the math classroom.
http://www.corestandards.org/Math/Practice/
COMPUTATIONAL THINKING
Computational thinking is necessary and meaningful in mathematics.
Computational thinking has developed into competencies in problem
solving, critical thinking, productivity, and creativity. Over time, engaging in
computational thought builds a student’s capacity to persevere, work
efficiently, gain confidence, tolerate ambiguity, generalize concepts, and
communicate effectively. In order to adapt to global advancements in
technology, students will need to use their computational thinking skills to
formulate, articulate, and discuss solutions in a meaningful manner.
MODELING
Modeling links classroom mathematics and statistics to everyday life, work,
and decision making. Modeling is the process of choosing and using
appropriate mathematics and statistics to analyze empirical situations, to
understand them better, and to improve decisions. Quantities and their
relationships in physical, economic, public policy, social, and everyday
situations can be modeled using mathematical and statistical methods.
When making mathematical models, technology is valuable for varying
assumptions, exploring consequences, and comparing predictions with data.
The basic modeling cycle involves: (1) identifying variables in the situation
and selecting those that represent essential features, (2) formulating a
model by creating and selecting geometric, graphical, tabular, algebraic, or
statistical representations that describe relationships between the variables,
(3) analyzing and performing operations on these relationships to draw
conclusions, (4) interpreting the results of the mathematics in terms of the
original situation, (5) validating the conclusions by comparing them with the
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situation, and then either improving the model or, if it is acceptable, (6) http://www.corestandards.org/Math/Content/HSM/
reporting on the conclusions and the reasoning behind them.
Mathematics | Standards for Mathematical Practice
“The Standards for Mathematical Practice describe varieties of expertise that mathematics educators at all levels should seek to develop in their students. These
practices rest on important “processes and proficiencies” with longstanding importance in mathematics education. The first of these are the National Council of
Teachers of Mathematics (NCTM) process standards of problem solving, reasoning and proof, communication, representation, and connections. The second are
the strands of mathematical proficiency specified in the National Research Council’s report Adding It Up: adaptive reasoning, strategic competence, conceptual
understanding (comprehension of mathematical concepts, operations and relations), procedural fluency (skill in carrying out procedures flexibly, accurately,
efficiently and appropriately), and productive disposition (habitual inclination to see mathematics as sensible, useful, and worthwhile, coupled with a belief in
diligence and one’s own efficacy). “
Source: http://www.corestandards.org/Math/Practice/
1. Make sense of problems and persevere in solving them.
problems using a different method, and they continually ask themselves,
"Does this make sense?" They can understand the approaches of others to
Mathematically proficient students start by explaining to themselves the
solving complex problems and identify correspondences between different
meaning of a problem and looking for entry points to its solution. They
approaches.
analyze givens, constraints, relationships, and goals. They make conjectures
about the form and meaning of the solution and plan a solution pathway
2. Reason abstractly and quantitatively.
rather than simply jumping into a solution attempt. They consider
Mathematically proficient students make sense of quantities and their
analogous problems, and try special cases and simpler forms of the original
relationships in problem situations. They bring two complementary abilities
problem in order to gain insight into its solution. They monitor and evaluate
to bear on problems involving quantitative relationships: the ability
their progress and change course if necessary. Older students might,
to decontextualize—to abstract a given situation and represent it
depending on the context of the problem, transform algebraic expressions
symbolically and manipulate the representing symbols as if they have a life
or change the viewing window on their graphing calculator to get the
of their own, without necessarily attending to their referents—and the
information they need. Mathematically proficient students can explain
ability to contextualize, to pause as needed during the manipulation process
correspondences between equations, verbal descriptions, tables, and
in order to probe into the referents for the symbols involved. Quantitative
graphs, or draw diagrams of important features and relationships, graph
reasoning entails habits of creating a coherent representation of the
data, and search for regularity or trends. Younger students might rely on
problem at hand; considering the units involved; attending to the meaning
using concrete objects or pictures to help conceptualize and solve a
of quantities, not just how to compute them; and knowing and flexibly using
problem. Mathematically proficient students check their answers to
different properties of operations and objects.
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3. Construct viable arguments and critique the reasoning of others.
Mathematically proficient students understand and use stated assumptions,
definitions, and previously established results in constructing arguments.
They make conjectures and build a logical progression of statements to
explore the truth of their conjectures. They are able to analyze situations by
breaking them into cases and can recognize and use counterexamples. They
justify their conclusions, communicate them to others, and respond to the
arguments of others. They reason inductively about data, making plausible
arguments that take into account the context from which the data arose.
Mathematically proficient students are also able to compare the
effectiveness of two plausible arguments, distinguish correct logic or
reasoning from that which is flawed, and—if there is a flaw in an
argument—explain what it is. Elementary students can construct arguments
using concrete referents such as objects, drawings, diagrams, and actions.
Such arguments can make sense and be correct, even though they are not
generalized or made formal until later grades. Later, students learn to
determine domains to which an argument applies. Students at all grades
can listen or read the arguments of others, decide whether they make
sense, and ask useful questions to clarify or improve the arguments.
4. Model with mathematics.
Mathematically proficient students can apply the mathematics they know to
solve problems arising in everyday life, society, and the workplace. In early
grades, this might be as simple as writing an addition equation to describe a
situation. In middle grades, a student might apply proportional reasoning to
plan a school event or analyze a problem in the community. By high school,
a student might use geometry to solve a design problem or use a function to
describe how one quantity of interest depends on another. Mathematically
proficient students who can apply what they know are comfortable making
assumptions and approximations to simplify a complicated situation,
realizing that these may need revision later. They are able to identify
important quantities in a practical situation and map their relationships
using such tools as diagrams, two-way tables, graphs, flowcharts and
formulas. They can analyze those relationships mathematically to draw
conclusions. They routinely interpret their mathematical results in the
context of the situation and reflect on whether the results make sense,
possibly improving the model if it has not served its purpose.
5. Use appropriate tools strategically.
Mathematically proficient students consider the available tools when
solving a mathematical problem. These tools might include pencil and
paper, concrete models, a ruler, a protractor, a calculator, a spreadsheet, a
computer algebra system, a statistical package, or dynamic geometry
software. Proficient students are sufficiently familiar with tools appropriate
for their grade or course to make sound decisions about when each of these
tools might be helpful, recognizing both the insight to be gained and their
limitations. For example, mathematically proficient high school students
analyze graphs of functions and solutions generated using a graphing
calculator. They detect possible errors by strategically using estimation and
other mathematical knowledge. When making mathematical models, they
know that technology can enable them to visualize the results of varying
assumptions, explore consequences, and compare predictions with data.
Mathematically proficient students at various grade levels are able to
identify relevant external mathematical resources, such as digital content
located on a website, and use them to pose or solve problems. They are
able to use technological tools to explore and deepen their understanding
of concepts.
6. Attend to precision.
Mathematically proficient students try to communicate precisely to others.
They try to use clear definitions in discussion with others and in their own
reasoning. They state the meaning of the symbols they choose, including
using the equal sign consistently and appropriately. They are careful about
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specifying units of measure, and labeling axes to clarify the correspondence
with quantities in a problem. They calculate accurately and efficiently,
expressing numerical answers with a degree of precision appropriate for the
problem context. In the elementary grades, students give carefully
formulated explanations to each other. By the time they reach high school
they have learned to examine claims and make explicit use of definitions.
7. Look for and make use of structure.
Mathematically proficient students look closely to discern a pattern or
structure. Young students, for example, might notice that three and seven
more is the same amount as seven and three more, or they may sort a
collection of shapes according to how many sides the shapes have. Later,
students will see 7 × 8 equals the well-remembered 7 × 5 + 7 × 3, in
preparation for learning about the distributive property. In the
expression x
2
+ 9x + 14, older students can see the 14 as 2 × 7 and the 9 as 2
+ 7. They recognize the significance of an existing line in a geometric figure
and can use the strategy of drawing an auxiliary line for solving problems.
They also can step back for an overview and shift perspective. They can see
complicated things, such as some algebraic expressions, as single objects or
as being composed of several objects. For example, they can see 5 -
3(x - y)
2
as 5 minus a positive number times a square and use that to realize
that its value cannot be more than 5 for any real numbers x and y.
8. Look for and express regularity in repeated reasoning.
Mathematically proficient students notice if calculations are repeated, and
look both for general methods and for shortcuts. Upper elementary
students might notice when dividing 25 by 11 that they are repeating the
same calculations over and over again, and conclude they have a repeating
decimal. By paying attention to the calculation of slope as they repeatedly
check whether points are on the line through (1, 2) with slope 3, middle
school students might abstract the equation (y - 2)/(x - 1) = 3. Noticing the
regularity in the way terms cancel when expanding (x - 1)(x + 1), (x -
1)(x
2
+ x + 1), and (x - 1)(x
3
+ x2 + x + 1) might lead them to the general
formula for the sum of a geometric series. As they work to solve a problem,
mathematically proficient students maintain oversight of the process, while
attending to the details. They continually evaluate the reasonableness of
their intermediate results.
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2021 Math Wyoming Content & Performance Standards & PLDs
PERFORMANCE STANDARDS (PS) SNAPSHOT (# OF CONTENT STANDARDS TIED TO PS)
The PS are also fully listed at the end of this document, starting on pg. 162.
Elementary School Math Performance Standards (by individual grade level)
Kindergarten
Grade 1
Grade 2
Grade 3
Grade 4
Grade 5
Grade 6
(5 out of 22) (5 out of 21) (6 out of 26) (6 out of 25) (6 out of 28) (5 out of 26) (8 out of 29)
K.CC.A.1
1.OA.A.1
2.OA.A.1
3.OA.C.7
4.OA.A.3
5.OA.A.1
6.RP.A.3
K.CC.B.4
K.OA.D.2
K.OA.D.3
1.OA.C.6
1.NBT.E.1
1.NBT.G.4
2.OA.B.2
2.NBT.D.1
2.NBT.E.7
3.OA.D.8
3.NBT.E.2
3.NF.F.2
4.NBT.E.5
4.NBT.E.6
4.NF.F.2
5.NBT.D.7
5.NF.E.2
5.NF.F.6
6.NS.B.1
6.NS.C.3
6.NS.D.8
K.MD.G.3
1.MD.I.3 2.MD.H.8
2.G.J.3
3.NF.F.3
3.MD.I.7
4.NF.G.3
4.MD.I.3
5.MD.I.5 6.EE.E2
6.EE.F.7
6.G.H.1
6.SP.J.5
Middle School (by grade level) & High School Math Performance Standards (by grade band)
Grade 7
(8 out of 24)
Grade 8
(7 out of 28)
HS - Number
and Quantity
(7 out of 27)
HS - Algebra
(11 out of 27)
HS - Function
(9 out of 28)
HS -
Geometry
(11 out of 43)
HS - Statistics
& Probability
(11 out of 31)
7.RP.A.2
8.NS.A.1
N.RN.A.2
A.SSE.B.3
F.IF.A.1
G.CO.A.1
S.ID.A.2
7.RP.A.3
7.NS.B.3
7.EE.D.4
8.EE.C.5
8.EE.D.7
8.EE.D.8
N.Q.C.1
N.Q.C.2
N.CN.F.7
A.APR.C.1
A.APR.D.3
A.CED.G.1
F.IF.C.7
F.BF.D.1
F.BF.E.3
G.CO.A.5
G.CO.C.10
G.SRT.F.5
S.ID.B.6
S.ID.C.7
S.IC.E.4
7.G.F.4
7.G.F.6
8.F.E.2
8.F.F.4
N.VM.G.3
N.VM.H.4
A.CED.G.2
A.CED.G.3
F.LE.F.1
F.LE.F.2
G.SRT.G.8
G.C.I.2
S.IC.E.6
S.CP.F.1
7.SP.G.1
7.SP.H.4
8.G.H.7 N.VM.I.6 A.REI.H.2
A.REI.I.3
A.REI.I.4
F.TF.H.1
F.TF.H.3
F.TF.I.5
G.GPE.L.5
G.GPE.L.7
G.GMD.M.3
S.CP.F.2
S.CP.F.5
S.MD.H.3
A.REI.J.6
A.REI.J.7
G.MG.O.2
G.MG.O.3
S.MD.I.5
S.MD.I.7
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Table of Contents
Performance Standards (PS) Snapshot (# of Content Standards tied to PS) ........................................................................................................................................... 8
How to Read This Document………………………………………………………………………………………………………………………………………………………………………………………………………….10
Kindergarten Math Content & Performance Standards & PLDs............................................................................................................................................................ 11
Grade 1 Math Content & Performance Standards & PLDs .................................................................................................................................................................... 18
Grade 2 Math Content & Performance Standards & PLDs .................................................................................................................................................................... 26
Grade 3 Math Content & Performance Standards & PLDs .................................................................................................................................................................... 35
Grade 4 Math Content & Performance Standards & PLDs .................................................................................................................................................................... 46
Grade 5 Math Content & Performance Standards & PLDs .................................................................................................................................................................... 59
Grade 6 Math Content & Performance Standards & PLDs .................................................................................................................................................................... 72
Grade 7 Math Content & Performance Standards & PLDs .................................................................................................................................................................... 85
Grade 8 Math Content & Performance Standards & PLDs .................................................................................................................................................................... 96
High School Math Content & Performance Standards & PLDs ............................................................................................................................................................ 109
NUMBER AND QUANTITY (NQ) ........................................................................................................................................................................................................ 110
ALGEBRA (A)..................................................................................................................................................................................................................................... 118
FUNCTIONS (F) ................................................................................................................................................................................................................................. 127
GEOMETRY (G) ................................................................................................................................................................................................................................. 139
STATISTICS AND PROBABILITY (SP) .................................................................................................................................................................................................. 154
Snapshot of the set of Math Performance Standards (PS). ................................................................................................................................................................. 163
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HOW TO READ THIS DOCUMENT
The Math Standards have 4 main sections:
1) Domain (in black bold) with the domain-specific learning expectations in blue, aka cluster headings.
2) Standard Code and Content Standard (CS) (in black bold).
3) Performance Level Descriptors (PLDs) describe the performance expectations within each of the 4 levels (Advanced, Proficient, Basic,
and Below Basic) to assist with teacher judgments on student proficiency.
4) Performance Standard (PS) – For the targeted subset of the Proficient PLD statements identified as the PS, the CS code are denoted
with an asterisk (*) and highlighted in blue and the Proficient PLD (aka the PS) is highlighted in a lighter blue.
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2021 Math Wyoming Content & Performance Standards & PLDs
Companion document to the 2018 Mathematics Content Standards
Kindergarten Math Content & Performance Standards &
PLDs
GRADE K MATH PRACTICES
MP1 Make sense of problems and persevere in solving
them.
K.MP.1 In Kindergarten, students begin to build the understanding that doing
mathematics involves solving problems and discussing how they solved
them. Students can explain the meaning of a problem and look for ways to
solve it. Students check their thinking by using concrete objects or pictures to
help them conceptualize and solve problems. Students are also working on
increasing stamina as they work on problems.
MP2 Reason abstractly and quantitatively.
K.MP.2 Students begin to recognize what a number is and that it also
represents a specific quantity. Then, they connect the quantity to written
symbols. Students make meaning of word problems and use manipulatives
to express and solve their thinking. Students are also working on increasing
stamina as they work on problems.
MP3 Construct viable arguments and critique the
reasoning of others.
K.MP.3 Students construct arguments using concrete illustrations, such as
objects, pictures, drawings, and actions. They also begin to develop their
mathematical communication skills as they participate in mathematical
discussions involving questions such as, “How did you get that? and Why is
that true?” They explain their thinking to others and respond to others’
thinking by making connections. Students are also working on increasing
stamina as they work on problems.
MP4 Model with mathematics.
K.MP.4 Students experiment with representing problem situations in multiple
ways including using objects, acting out, drawing pictures, numbers, words
(mathematical language), making a chart or list, creating equations, etc.
Students need opportunities to connect the different representations and
explain the connections. They should be able to use all of these
representations as needed.
MP5 Use appropriate tools strategically.
K.MP.5 Students begin to explore the different available tools when thinking
about the concepts of numbers. They begin to learn which tools help
strengthen their understanding of concepts. For instance, kindergarteners
may decide that it might be advantageous to use linking cubes to represent
two quantities and then compare the two representations side-by-side.
MP6 Attend to precision.
K.MP.6 As kindergarteners begin to develop their mathematical
communication skills, they try to use clear and precise mathematical
vocabulary in their discussions with others and in their own reasoning.
Students learn to attend to the shapes of numbers, quickly recognize
quantities (subitizing), and simple drawings to show their work.
MP7 Look for and make use of structure.
K.MP.7 Students begin to notice a number pattern or structure. For instance,
students recognize the pattern that exists in the teen numbers; every teen
number is written with a 1 (representing one ten) and ends with the digit that
is first stated, and the pattern of numbers 0-9 repeat in the following numbers
of 20, 30, etc. They also recognize that 3 + 2 = 5 and 2 + 3 = 5.
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MP8 Look for and express regularity in repeated reasoning.
K.MP.8 Students notice repetitive actions in counting and computation, etc. For example, they may notice that the next number in a counting sequence is one
more. When counting by tens, the next number in the sequence is ten more (or one more group of ten). Students also notice that when adding two numbers, order
of adding doesn’t affect the sum (Commutative Property).
COUNTING AND CARDINALITY
Know number names and the count sequence.
*K.CC.A.1
K.CC.A.1A Count to 100 by ones and by tens.
K.CC.A.1B Count backwards by ones from 20.
In addition to Proficient, the Advanced student is able to:
A1. Count to 100 by ones from a given number.
A2. Count to 100 by tens from a given multiple of ten.
B. Count backward by ones from a given number within 100.
The Proficient student is able to:
A1. Count to 100 by ones, starting at one.
A2. Count to 100 by multiples of ten, starting at ten.
B. Count backwards by ones from 20.
The Basic student is able to:
A. Count to 100 by ones, with prompting.
B. Count backwards from 10 by ones.
The Below Basic student does not meet the Basic performance level.
K.CC.A.2 Count forward beginning from a given number within the known sequence (instead of having to begin at 1).
In addition to Proficient, the Advanced student is able to count forward a sequence of numbers, crossing two decades/tens starting at a number between
30 and 90.
The Proficient student is able to count forward a sequence of 10 numbers, crossing a decade/ten starting at a number between 1 and 30.
The Basic student is able to count forward a sequence of 10 numbers, crossing a decade/ten starting at a number between 1 and 30, with prompting.
The Below Basic student does not meet the Basic performance level.
K.CC.A.3 Write numbers from 0 to 20. Represent a number of objects with a written numeral 0-20 (with a 0 representing a count of no objects).
In addition to Proficient, the Advanced student is able to:
A. Write any two digit number above 20.
B. Represent a number of objects with a written numeral between 20 and 40.
The Proficient student is able to:
A. Write numbers from 0 to 20.
B. Represent a number of objects with a written numeral 0-20 (with 0 (zero) representing a count of no objects).
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The Basic student is able to:
A. Write numbers from 0 to 10.
B. Represent a number of objects with a written numeral 0-10 (with 0 (zero) representing a count of no objects).
The Below Basic student does not meet the Basic performance level.
Count to tell the number of objects.
*K.CC.B.4 Understand the relationship between numbers and quantities; connect counting to cardinality.
K.CC.B.4A Use one-to-one correspondence when counting objects.
K.CC.B.4B Understand that the last number name said, tells the number of objects counted regardless of their arrangement.
K.CC.B.4C Understand that each successive number name refers to a quantity that is one more, and each previous number name refers to a quantity that is
one less.
In addition to Proficient, the Advanced student is able to count up or count back starting at an initial quantity when given an additive or removed item task.
The Proficient student is able to count and tell the number of objects in a range from 10 to 39.
A. Use one-to-one correspondence when counting objects.
B. Understand that the last number name said, tells the number of objects counted regardless of their arrangement.
C. Understand that each successive number name refers to a quantity that is one more, and each previous number name refers to a quantity that
is one less.
The Basic student is able to count and tell the number of objects in a range from 1-9.
A. Use one-to-one correspondence when counting objects.
B. Understand that the last number name said, tells the number of objects counted.
The Below Basic student does not meet the Basic performance level.
K.CC.B.5
K.CC.B.5A When counting, answer the question "how many?" by counting up to 20 objects arranged in a line, a rectangular array, a circle, or as
many as 10 objects in a scattered configuration.
K.CC.B.5B When counting, given a number from 1-20, count out that many objects.
In addition to Proficient, the Advanced student is able to, when counting:
A. Answer the question "how many?" by counting beyond 20 in a scattered configuration.
B. Count out the number of objects given a number 25-35.
The Proficient student is able to, when counting:
A. Answer the question "how many?" by counting up to 20 objects arranged in a line, a rectangular array, a circle, or as many as 10 objects in a
scattered configuration.
B. Count out the number of objects given a number from 1-20.
The Basic student is able to, when counting:
A. Answer the question "how many?" by counting up to 20 objects arranged in a line, a rectangular array, a circle, or as many as 10 objects in a
scattered configuration. OR
B. Count out the number of objects given a number from 1-20.
The Below Basic student does not meet the Basic performance level.
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Compare numbers.
K.CC.C.6 Identify whether the number of objects in one group is greater than, less than, or equal to the number of objects in another group, e.g., by
using matching and counting strategies. (Include groups with up to ten objects.)
In addition to Proficient, the Advanced student is able to order 3 or more groups of objects (Include groups with up to ten objects.) from greatest to least or
least to greatest.
The Proficient student is able to identify whether the number of objects in one group is greater than, less than, or equal to the number of objects in
another group. (Include groups with up to ten objects.)
The Basic student is able to identify whether the number of objects in one group is equal to or not equal to the number of objects in another group.
(Include groups with up to five objects.)
The Below Basic student does not meet the Basic performance level.
K.CC.C.7 Compare two numbers between 1 and 10 presented as written numerals.
In addition to Proficient, the Advanced student is able to compare 3 or more non-consecutive numbers between 1 and 20 presented as written numerals.
The Proficient student is able to compare two numbers between 1 and 10 presented as written numerals.
The Basic student is able to compare two numbers between 1 and 5 presented as written numerals.
The Below Basic student does not meet the Basic performance level.
OPERATIONS AND ALGEBRAIC THINKING
Understand addition as putting together and adding to, and understand subtraction as taking apart and taking from.
K.OA.D1 Model situations that involve representing addition and subtraction with objects, fingers, mental images, drawings, sounds (e.g., claps),
acting out situations, verbal explanations, expressions, or equations.
In addition to Proficient, the Advanced student is able to model situations that involve representing addition and subtraction with objects, fingers, mental
images, drawings, sounds (e.g., claps), acting out situations, verbal explanations, and write the corresponding expression, or equation.
The Proficient student is able to model situations that involve representing addition and subtraction with objects, fingers, mental images, drawings,
sounds (e.g., claps), acting out situations, verbal explanations, expressions, or equations.
The Basic student is able to model situations that involve representing addition with objects, fingers, mental images, drawings, sounds (e.g., claps), acting
out situations, verbal explanations, expressions, or equations.
The Below Basic student does not meet the Basic performance level.
*K.OA.D.2 Solve word problems using objects and drawings to find sums up to 10 and differences within 10.
In addition to Proficient, the Advanced student is able to create a word problem to find sums up to 10 and differences within 10.
The Proficient student is able to solve word problems using objects and drawings to find sums up to 10 and differences within 10.
The Basic student is able to solve word problems using objects and drawings to find sums up to 5 and differences within 5.
The Below Basic student does not meet the Basic performance level.
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*K.OA.D.3 Decompose numbers less than or equal to 10 in more than one way.
In addition to Proficient, the Advanced student is able to decompose numbers with parts that are less than or equal to 10 in more than one way identifying
patterns.
The Proficient student is able to decompose numbers less than or equal to 10 in more than one way.
The Basic student is able to decompose numbers less than or equal to 5 in more than one way.
The Below Basic student does not meet the Basic performance level.
K.OA.D.4 For any number from 1 to 9, find the number that makes 10 when added to the given number.
In addition to Proficient, the Advanced student is able to, for any number from 10-19, find the number that makes 20 when added to the given number.
The Proficient student is able to for any number from 1 to 9, find the number that makes 10 when added to the given number.
The Basic student is able to for any number from 1 to 4, find the number that makes 5 when added to the given number.
The Below Basic student does not meet the Basic performance level.
K.OA.D.5 Fluently add and subtract within 5.
In addition to Proficient, the Advanced student is able to fluently add and subtract within 5 including missing addend problems.
The Proficient student is able to fluently add and subtract within 5.
The Basic student is able to fluently add within 5.
The Below Basic student does not meet the Basic performance level.
NUMBERS AND OPERATIONS IN BASE TEN
Work with numbers 11-19 to gain foundations for place value.
K.NBT.E.1
K.NBT.E.1A Describe, explore, and explain how the counting numbers 11 to 19 are composed of ten ones and more ones.
K.NBT.E.1B Describe, explore, and explain how the counting numbers 11 to 19 are decomposed into ten ones and more ones.
In addition to Proficient, the Advanced student is able to describe, explore, and explain how the counting numbers 11 to 19 are:
A. Composed of one unit of ten and more ones.
B. Decomposed into one unit of ten and more ones.
The Proficient student is able to describe, explore, and explain how the counting numbers 11 to 19 are:
A. Composed of ten ones and more ones.
B. Decomposed into ten ones and more ones.
The Basic student is able to describe, explore, and explain how the counting numbers 11 to 19 are composed of ten ones and more ones.
The Below Basic student does not meet the Basic performance level.
MEASUREMENT AND DATA
Describe and compare measurable attributes.
K.MD.F.1 Describe several measurable attributes of one or more objects.
In addition to Proficient, the Advanced student is able to describe several measurable and non-measurable attributes of one or more real-world object(s).
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The Proficient student is able to describe several measurable attributes of one or more objects.
The Basic student is able to describe at least two attributes of one or more objects.
The Below Basic student does not meet the Basic performance level.
K.MD.F.2 Make direct comparisons of the length, capacity, weight, and temperature of objects, and recognize which object is shorter/longer, taller,
lighter/heavier, warmer/cooler, and which holds more/less.
In addition to Proficient, the Advanced student is able to make a direct comparison of three or more objects based on the length, capacity, weight, and
temperature of objects, and order them shorter/longer, taller, lighter/heavier, warmer/cooler, and which holds more/less and vice versa.
The Proficient student is able to make direct comparisons of the length, capacity, weight, and temperature of objects, and recognize which object is
shorter/longer, taller, lighter/heavier, warmer/cooler, and which holds more/less.
The Basic student is able to make direct comparisons of the length of objects, and recognize which object is shorter/longer.
The Below Basic student does not meet the Basic performance level.
Classify objects and count the number of objects in each category.
*K.MD.G.3 Classify objects into given categories; count the numbers of objects in each category and sort the categories by count. (Limit category
counts to be less than or equal to 10.)
In addition to Proficient, the Advanced student is able to construct categories from the given objects and justify reasoning for each category.
The Proficient student is able to classify objects into given categories; count the numbers of objects in each category and sort the categories by count.
(Limit category counts to be less than or equal to 10.)
The Basic student is able to identify objects from given categories; count the numbers of objects in each category. (Limit category counts to be less than
or equal to 10.)
The Below Basic student does not meet the Basic performance level.
K.MD.G.4 Identify U.S. coins by name (pennies, nickels, dimes, and quarters).
In addition to Proficient, the Advanced student is able to identify the value of at least two U.S. coins (pennies, nickels, dimes, and quarters).
The Proficient student is able to identify U.S. coins by name (pennies, nickels, dimes, and quarters).
The Basic student is able to identify at least two U.S. coins by name (pennies, nickels, dimes, and quarters).
The Below Basic student does not meet the Basic performance level.
GEOMETRY
Identify and describe shapes (squares, circles, triangles, rectangles, hexagons, cubes, cones, cylinders, and spheres).
K.G.H.1 Describe objects in the environment using the names of shapes, and describe the relative positions of these objects using terms such as
above, below, beside, in front of, behind, and next to.
In addition to Proficient, the Advanced student is able to using the names of shapes describe the position of one object relative to two or more objects.
The Proficient student is able to describe objects in the environment using the names of shapes, and describe the relative positions of these objects using
terms such as above, below, beside, in front of, behind, and next to.
The Basic student is able to describe the relative positions of objects using terms such as above, below, beside, in front of, behind, and next to.
The Below Basic student does not meet the Basic performance level.
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K.G.H.2 Correctly name shapes regardless of their orientations or overall size.
In addition to Proficient, the Advanced student is able to correctly identify shapes within a compound figure.
The Proficient student is able to correctly name shapes regardless of their orientations or overall size.
The Basic student is able to correctly name shapes: squares, circles, triangles, rectangles, cones, and cubes.
Assessment Boundary: does not include hexagon, cylinder, or sphere.
The Below Basic student does not meet the Basic performance level.
K.G.H.3 Identify shapes as two-dimensional or three-dimensional.
In addition to Proficient, the Advanced student is able to justify the difference between two-dimensional and three-dimensional shapes.
The Proficient student is able to identify shapes as two-dimensional or three-dimensional.
The Basic student is able to identify shapes as two-dimensional.
The Below Basic student does not meet the Basic performance level.
Analyze, compare, create, and compose shapes.
K.G.I.4 Analyze and compare two- and three-dimensional shapes, using informal language to describe their similarities, differences, and attributes.
In addition to Proficient, the Advanced student is able to group shapes based on attributes and justify reasoning.
The Proficient student is able to analyze and compare two- and three-dimensional shapes, using informal language to describe their similarities,
differences, and attributes.
The Basic student is able to use informal language to describe the similarities, differences, and attributes of corresponding two- or three-dimensional
shapes (comparing two-dimensional to two-dimensional and three-dimensional to three-dimensional).
The Below Basic student does not meet the Basic performance level.
K.G.I.5 Model shapes in the world by building shapes from components (e.g., sticks and clay balls) and drawing shapes.
In addition to Proficient, the Advanced student is able to predict and test what components are needed to change a two-dimensional shape to a three-
dimensional shape (e.g., square to cube).
The Proficient student is able to model shapes in the world by building shapes from components (e.g., sticks and clay balls) and drawing shapes.
The Basic student is able to model two-dimensional shapes in the world by building shapes from components (e.g., sticks and clay balls) or drawing two-
dimensional shapes.
The Below Basic student does not meet the Basic performance level.
K.G.I.6 Use simple shapes to compose squares, rectangles, and hexagons.
In addition to Proficient, the Advanced student is able to decompose a two-dimensional figure to determine the shapes that were used to build it.
The Proficient student is able to use simple shapes to compose squares, rectangles, and hexagons.
The Basic student is able to use given simple shapes to compose squares and rectangles.
The Below Basic student does not meet the Basic performance level.
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2021 Math Wyoming Content & Performance Standards & PLDs
Companion document to the 2018 Mathematics Content Standards
Grade 1 Math Content & Performance Standards & PLDs
GRADE 1 MATH PRACTICES
MP1 Make sense of problems and persevere in solving
them.
1.MP.1 In first grade, students realize that doing mathematics
involve
solving
problems and discussing how they solved them. Students explain the
meaning of a problem and look for ways to solve it. Students may use
concrete objects or pictures to help them conceptualize and solve problems.
They may check their thinking by revisiting their work and asking themselves,
“Does this make sense?" or, “Should I try another strategy? Students are
also working on increasing stamina as they work on problems.
MP2 Reason abstractly and quantitatively.
1.MP.2 Students recognize that a number represents a specific quantity.
They connect the quantity to written symbols. Quantitative reasoning means
being able to explain through manipulatives or drawings what a problem
means while attending to the meanings of the quantities. Students make
meaning of a problem situation and translate into a number sentence.
MP3 Construct viable arguments and critique the
reasoning of others.
1.MP.3 First graders construct arguments using concrete illustrations
referents, such as objects, pictures, drawings, and actions. They also
practice their mathematical communication skills as they participate in
mathematical discussions involving questions like, “How did you get that?”
Explain your thinking, “Why is that true?” They not only explain their own
thinking, but listen to others’ explanations. They decide if the explanations
make sense and ask questions for clarity.
MP4 Model with mathematics.
1.MP.4 Students experiment with representing problem situations in multiple
ways including using objects, acting out, drawing pictures, numbers, words
(mathematical language), making a chart or list, creating equations, etc.
Students need opportunities to connect the different representations and
explain the connections. They should be able to use all of these
representations as needed.
MP5 Use appropriate tools strategically.
1.MP.5 Students begin to consider the different tools available when thinking
about the concepts of number. They evaluate the available tools (including
concrete manipulatives, drawings, estimation, and applications) when solving
a mathematical problem and decide when certain tools might be helpful and
give a reason for using the tool to solve the problem. For instance, first
graders decide it might be best to use colored chips to model an addition
problem.
MP6 Attend to precision.
1.MP.6 Students begin to develop their mathematical communication skills.
They try to use clear and precise mathematical vocabulary in their
discussions with others and in their own reasoning. Students learn to express
their work with mathematical language and symbols.
MP7 Look for and make use of structure.
1.MP.7 First graders begin to discern a number pattern or structure. For
instance, if students recognize12 + 3 = 15, then they also know 3 + 12 =
15. (Commutative Property of addition.) To add 4 + 6 + 4, the first two
numbers can be added to make a ten, so 4 + 6 + 4 = 10 + 4 = 14.
Students continue to develop their understanding of patterns in our number
system.
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MP8 Look for and express regularity in repeated reasoning.
1.MP.8 Students notice repetitive actions in counting and computation, etc. When children have multiple opportunities to add and subtract ten, including multiples
of ten, then they notice the pattern and gain a better understanding of place value. Students also notice that when adding two numbers, order of adding doesn’t
affect the sum (Commutative Property). They also notice that three numbers create a family when adding or subtracting (2 + 3 = 5 and 5 2 = 3).
OPERATIONS AND ALGEBRAIC THINKING
Represent and solve problems involving addition and subtraction.
*1.OA.A.1 Use addition and subtraction within 20 to solve word problems involving situations of adding to, taking from, putting together, taking apart,
and comparing, with unknowns in all positions, by using objects, drawings, or equations with a symbol for the unknown number to represent the
problem.
In addition to Proficient, the Advanced student is able to create and solve an addition or subtraction word problem within 20.
The Proficient student is able to use addition and subtraction within 20 to solve word problems involving situations of adding to, taking from, putting
together, taking apart, and comparing, with unknowns in all positions, by using objects, drawings, or equations with a symbol for the unknown number to
represent the problem.
The Basic student is able to use addition and subtraction within 10 to solve word problems involving situations of adding to, taking from, putting together,
taking apart, and comparing, with unknowns in all positions, by using objects, drawings, or equations with a symbol for the unknown number to represent
the problem.
The Below Basic student does not meet the Basic performance level.
1.OA.A.2 Solve word problems that call for the addition of three whole numbers whose sum is less than or equal to 20, by using objects, drawings, or
equations.
In addition to Proficient, the Advanced student is able to solve a missing addend word problem that calls for the addition of three whole numbers whose
sum is less than or equal to 20, by using objects, drawings, or equations.
The Proficient student is able to solve word problems that call for the addition of three whole numbers whose sum is less than or equal to 20, by using
objects, drawings, or equations.
The Basic student is able to add three whole numbers whose sum is less than or equal to 20, by using objects, drawings, or equations.
The Below Basic student does not meet the Basic performance level.
Understand and apply properties of operations and the relationship between addition and subtraction.
1.OA.B.3 Apply Commutative and Associative Properties of addition as strategies to add and subtract.
In addition to Proficient, the Advanced student is able to describe the relationships when applying the properties of addition.
The Proficient student is able to apply Commutative and Associative properties of addition as strategies to add and subtract.
The Basic student is able to apply Commutative Property of addition.
The Below Basic student does not meet the Basic performance level.
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1.OA.B.4 Understand subtraction as an unknown-addend problem.
In addition to Proficient, the Advanced student is able to write all equivalent addition or subtraction equations relating three whole numbers within 20.
The Proficient student is able to understand subtraction as an unknown-addend problem.
The Basic student is able to understand subtraction as an unknown-addend problem within 10.
The Below Basic student does not meet the Basic performance level.
Add and subtract within 20.
1.OA.C.5 Relate counting to addition and subtraction using strategies, such as, by counting on and back.
In addition to Proficient, the Advanced student is able to, given a counting on or counting back situation, write related addition or subtraction equations
(e.g., 42, 41, 40, 39. 42 3 = 39, or 42 39 = 3).
The Proficient student is able to relate counting to addition and subtraction using strategies, such as, by counting on and back.
The Basic student is able to relate counting to addition by counting on.
The Below Basic student does not meet the Basic performance level.
*1.OA.C.6 Add and subtract within 20, demonstrating fluency in addition and subtraction within 10. Use strategies such as counting on; making ten
using the relationship between addition and subtraction.
In addition to Proficient, the Advanced student is able to add and subtract within 20, demonstrating fluency in addition and subtraction within 10 by
utilizing multiple strategies, such as: counting on; making ten using the relationship between addition and subtraction.
The Proficient student is able to add and subtract within 20, demonstrating fluency in addition and subtraction within 10. Use strategies such as counting
on; making ten using the relationship between addition and subtraction.
The Basic student is able to add and subtract within 10 using strategies such as counting on; making ten using the relationship between addition and
subtraction.
The Below Basic student does not meet the Basic performance level.
Work with addition and subtraction equations.
1.OA.D.7 Understand the meaning of the equal sign, and determine if equations involving addition and subtraction are true or false.
In addition to Proficient, the Advanced student is able to determine if equations involving addition and subtraction are true or false and rewrite false
equations to make them true.
The Proficient student is able to understand the meaning of the equal sign, and determine if equations involving addition and subtraction are true or false.
The Basic student is able to use the equal sign to make a true addition or subtraction equation given a visual representation of a situation.
The Below Basic student does not meet the Basic performance level.
1.OA.D.8 Determine the unknown whole number in an addition or subtraction equation relating three whole numbers.
In addition to Proficient, the Advanced student is able to describe multiple strategies for determining the unknown whole number in an addition or
subtraction equation relating three whole numbers.
The Proficient student is able to determine the unknown whole number in an addition or subtraction equation relating three whole numbers.
The Basic student is able to determine the unknown whole number in an addition or subtraction equation relating three whole numbers less than 10.
The Below Basic student does not meet the Basic performance level.
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NUMBER AND OPERATIONS IN BASE TEN
Extend the counting sequence.
*1.NBT.E.1 Extend the number sequences to 120. In this range:
1.NBT.E.1A Count forward and backward, starting at any number less than 120.
1.NBT.E.1B Read numerals.
1.NBT.E.1C Write numerals.
1.NBT.E.1D Represent a number of objects with a written numeral.
In addition to Proficient, the Advanced student is able to:
• Count forward and backward by 10, starting at any number less than 120. OR
• Count forward and backward by 2, starting at any even number less than 120. OR
• Count forward and backward by 5, starting at any multiple of 5 less than 120.
The Proficient student is able to extend the number sequences to 120. In this range:
A. Count forward and backward, starting at any number less than 120.
B. Read numerals.
C. Write numerals.
D. Represent a number of objects with a written numeral.
The Basic student is able to extend the number sequences to 120 with guidance. In this range:
A. Count forward starting at any number less than 120.
B. Read numerals.
C. Write numerals.
D. Represent a number of objects with a written numeral.
The Below Basic student does not meet the Basic performance level.
Understand place value.
1.NBT.F.2 Understand that the two digits of a two-digit number represent amounts of tens and ones. Understand the following as special cases:
1.NBT.F.2A 10 can be thought of as a bundle of ten ones — called a “ten.”
1.NBT.F.2B The numbers from 11 to 19 are composed of a ten and one, two, three, four, five, six, seven, eight, or nine ones.
1.NBT.F.2C The numbers 10, 20, 30, 40, 50, 60, 70, 80, 90 refer to one, two, three, four, five, six, seven, eight, or nine tens (and 0 ones).
In addition to Proficient, the Advanced student is able to represent any two digit number in multiple ways using tens and ones (e.g., 67 is 6 tens and 7
ones, or 4 tens and 27 ones, etc.).
The Proficient student is able to understand that the two digits of a two-digit number represent amounts of tens and ones. Understand the following as
special cases:
A. 10 can be thought of as a bundle of ten ones — called a “ten.”
B. The numbers from 11 to 19 are composed of a ten and one, two, three, four, five, six, seven, eight, or nine ones.
C. The numbers 10, 20, 30, 40, 50, 60, 70, 80, 90 refer to one, two, three, four, five, six, seven, eight, or nine tens (and 0 ones).
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The Basic student is able to build any two digit number using manipulatives to represent amounts of tens and ones and show understanding of the
following special cases:
A. 10 can be thought of as a bundle of ten ones — called a “ten.”
B. The numbers from 11 to 19 are composed of a ten and one, two, three, four, five, six, seven, eight, or nine ones.
The Below Basic student does not meet the Basic performance level.
1.NBT.F.3 Compare pairs of two-digit numbers based on the values of the tens digit and the ones digits, recording the results of comparisons with the
words "is greater than," "is equal to," "is less than," and with the symbols >, =, and <.
In addition to Proficient, the Advanced student is able to write comparisons recording the results using both the greater than and the less than words and
symbols (e.g.,27 > 21 and 21 < 27).
The Proficient student is able to compare pairs of two-digit numbers based on the values of the tens digit and the ones digit, recording the results of
comparisons with the words "is greater than," "is equal to," "is less than," and with the symbols >, =, and <.
The Basic student is able to compare pairs of two-digit numbers based on the values of the tens digit and the ones digit, stating the results of comparisons
with the words "is greater than," "is equal to," "is less than."
The Below Basic student does not meet the Basic performance level.
Use place value understanding and properties of operations to add and subtract.
*1.NBT.G.4 Add within 100, using concrete models or drawings and strategies based on place value:
1.NBT.G.4A Including adding a two-digit number and a one-digit number.
1.NBT.G.4B Adding a two-digit number and a multiple of 10.
1.NBT.G.4C Understand that in adding two-digit numbers, adds tens and tens, ones and ones; and sometimes it is necessary to compose a ten.
1.NBT.G.4D Relate the strategy to a written method and explain the reasoning used.
In addition to Proficient, the Advanced student is able to subtract within 100, using concrete models or drawings and strategies based on place value:
A. Including subtracting a one-digit number from a two-digit number.
B. Subtracting a multiple of 10 from a two-digit number.
The Proficient student is able to add within 100, using concrete models or drawings and strategies based on place value:
A. Including adding a two-digit number and a one-digit number.
B. Adding a two-digit number and a multiple of 10.
C. Understand that in adding two-digit numbers, adds tens and tens, ones and ones; and sometimes it is necessary to compose a ten.
D. Relate the strategy to a written method and explain the reasoning used.
The Basic student is able to add within 100, using concrete models, manipulatives, or drawings and strategies based on place value:
A. Including adding a two-digit number and a one-digit number.
B. Adding a two-digit number and a multiple of 10.
The Below Basic student does not meet the Basic performance level.
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1.NBT.G.5 Given a two-digit number, mentally find 10 more or 10 less than the number, without having to count; explain the reasoning used.
In addition to Proficient, the Advanced student is able to given a two-digit number, mentally find multiples of 10 more or multiples of 10 less than the
number, without having to count; explain the reasoning used.
The Proficient student is able to given a two-digit number, mentally find 10 more or 10 less than the number, without having to count; explain the
reasoning used.
The Basic student is able to mentally find 10 more or 10 less than a multiple of 10 less than 100, without having to count; explain the reasoning used.
The Below Basic student does not meet the Basic performance level.
1.NBT.G.6 Subtract multiples of 10 from an equal or larger multiple of 10 both in the range 10-90, using concrete models, drawings, and strategies
based on place value.
In addition to Proficient, the Advanced student is able to, without counting, subtract multiples of ten and explain the reasoning used.
The Proficient student is able to subtract multiples of 10 from an equal or larger multiple of 10 both in the range 10-90, using concrete models, drawings,
and strategies based on place value.
The Basic student is able to, starting at any multiple of ten in the range 10-90, verbally count backwards by tens.
The Below Basic student does not meet the Basic performance level.
MEASUREMENT AND DATA
Measure lengths indirectly and by iterating length units.
1.MD.H.1 Order three objects by length; compare the lengths of two objects indirectly by using a third object.
In addition to Proficient, the Advanced student is able to, given different organizations of more than three objects, justify an arrangement of the objects by
length.
The Proficient student is able to order three objects by length; compare the lengths of two objects indirectly by using a third object.
The Basic student is able to compare two objects by length.
The Below Basic student does not meet the Basic performance level.
1.MD.H.2 Use nonstandard units to show the length of an object as the number of same size units of length with no gaps or overlaps.
In addition to Proficient, the Advanced student is able to measure the same object using a non-standard unit of one length and then a non-standard unit of
a different length. Explain how the two measurements relate to the size of the unit chosen.
The Proficient student is able to use nonstandard units to show the length of an object as the number of same size units of length with no gaps or
overlaps.
The Basic student is able to use nonstandard units to show the length of an object as the number of same size units of length with no gaps or overlaps
with guidance.
The Below Basic student does not meet the Basic performance level.
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Work with time and money.
*1.MD.I.3A Tell and write time in hours and half-hours using analog and digital clocks.
In addition to Proficient, the Advanced student is able to approximate time to the nearest hour or half hour on an analog clock based only on the hour
hand.
The Proficient student is able to tell and write time in hours and half-hours using analog and digital clocks.
The Basic student is able to tell and write time in hours using analog and digital clocks.
The Below Basic student does not meet the Basic performance level.
*1.MD.I.3B Identify U.S. coins by value (pennies, nickels, dimes, quarters).
In addition to Proficient, the Advanced student is able to find equivalent values of coins up to and including quarters using coins of lesser value.
The Proficient student is able to identify U.S. coins by value (pennies, nickels, dimes, quarters).
The Basic student is able to identify at least two U.S. coins by value (pennies, nickels, dimes, quarters).
The Below Basic student does not meet the Basic performance level.
Represent and interpret data.
1.MD.J.4 Organize, represent, and interpret data with up to three categories; ask and answer questions about the total number of data points, how
many in each category, and how many more or less are in one category than in another.
In addition to Proficient, the Advanced student is able to compare two different data sets with at least three categories to ask and answer questions.
The Proficient student is able to organize, represent, and interpret data with up to three categories; ask and answer questions about the total number of
data points, how many in each category, and how many more or less are in one category than in another.
The Basic student is able to ask and answer questions about the total number of data points, how many in each category, and how many more or less are
in one category than in another when given an organized set of data.
The Below Basic student does not meet the Basic performance level.
GEOMETRY
Reason with shapes and their attributes.
1.G.K.1 Distinguish between defining attributes (e.g., triangles are closed and three-sided) versus non-defining attributes (e.g., color, orientation,
overall size); for a wide variety of shapes; build and draw shapes to possess defining attributes.
In addition to Proficient, the Advanced student is able to compare and contrast defining attributes from given shapes and use the comparison to change
one shape into the other shape.
The Proficient student is able to distinguish between defining attributes (e.g., triangles are closed and three-sided) versus non-defining attributes (e.g.,
color, orientation, overall size); for a wide variety of shapes; build and draw shapes to possess defining attributes.
The Basic student is able to identify attributes (both defining and non-defining) for a wide variety of shapes.
The Below Basic student does not meet the Basic performance level.
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1.G.K.2 Use two-dimensional shapes (rectangles, squares, trapezoids, rhombuses, and triangles) or three-dimensional shapes (cubes, rectangular
prisms, cones, and cylinders) to create a composite figure, and create new figures from the composite figure.
In addition to Proficient, the Advanced student is able to decompose a composite figure (made of two and three dimensional shapes) and then compose
to the original figure and create a new composite figure.
The Proficient student is able to use two-dimensional shapes (rectangles, squares, trapezoids, rhombuses, and triangles) or three-dimensional shapes
(cubes, rectangular prisms, cones, and cylinders) to create a composite figure, and create new figures from the composite figure.
The Basic student is able to use two-dimensional shapes (rectangles, squares, trapezoids, rhombuses, and triangles) to create a composite figure.
The Below Basic student does not meet the Basic performance level.
1.G.K.3 Partition circles and rectangles into two and four equal shares and:
1.G.K.3A Describe the shares using the words halves, fourths, and quarters, and use the phrases half of, fourth of, and quarter of.
1.G.K.3B Describe the whole as two of, or four of the shares.
1.G.K.3C Recognize that decomposing into more equal shares creates smaller shares.
In addition to Proficient, the Advanced student is able to partition multiple circles and rectangles into two and four equal shares and:
A. Describe the shares using the words halves, fourths, and quarters, and use the phrases half of, fourth of, and quarter of.
B. Describe the whole as two of, or four of the shares.
C. Recognize that decomposing into more equal shares creates smaller shares.
The Proficient student is able to partition circles and rectangles into two and four equal shares and:
A. Describe the shares using the words halves, fourths, and quarters, and use the phrases half of, fourth of, and quarter of.
B. Describe the whole as two of, or four of the shares.
C. Recognize that decomposing into more equal shares creates smaller shares.
The Basic student is able to partition circles and rectangles into two equal shares and:
A. Describe the shares using the word halves.
B. Describe the whole as two of the shares.
The Below Basic student does not meet the Basic performance level.
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Companion document to the 2018 Mathematics Content Standards
Grade 2 Math Content & Performance Standards & PLDs
GRADE 2 MATH PRACTICES
MP1 Make sense of problems and persevere in
solving them.
2.MP.1 In second grade, students realize that doing mathematics
involves solving problems and discussing how they solved them.
Students explain the meaning of a problem and look for ways to
solve it. They may use concrete objects or pictures to help them
conceptualize and solve problems. They may check their thinking
by asking themselves, “Does this make sense?” They make
conjectures about the solution and plan out a problem-solving
approach. Students work on increasing stamina.
MP2 Reason abstractly and quantitatively.
2.MP.2 Students recognize that a number represents a specific
quantity and connect the quantity to written symbols. Quantitative
reasoning entails being able to explain through manipulatives or
drawings what a problem means, while attending to the meanings
of the quantities. Students make meaning of a problem situation
and translate into a number sentence. Second graders begin to
know and use different properties of operations and relate addition
and subtraction.
MP3 Construct viable arguments and critique the
reasoning of others.
2.MP.3 Second graders may construct arguments using concrete
illustrations, such as objects, pictures, drawings, and actions. They
practice their mathematical communication skills as they
participate in mathematical discussions involving questions like,
“How did you get that?” Explain your thinking, “Why is that true?”
They not only explain their own thinking, but listen to others’
explanations and compare strategies. They decide if the
explanations make sense and ask appropriate questions for clarity.
MP4 Model with mathematics.
2.MP.4 Students experiment with representing problem situations in multiple
ways including numbers, words (mathematical language), drawing pictures, using
objects, making a chart or list, creating equations, etc. Students need
opportunities to connect the different representations and explain the
connections. They should be able to use all of these representations as needed.
MP5 Use appropriate tools strategically.
2.MP.5 Students decide how and when to use the available tools appropriately
and efficiently when solving a mathematical problem. Students reason whether or
not a tool was helpful in solving the problem. For instance, second graders may
decide to solve a problem by drawing a picture rather than writing an equation.
MP6 Attend to precision.
2.MP.6 Students begin to develop their mathematical communication skills,
(orally and written). They use clear and precise mathematical language and
symbols when explaining their own reasoning.
MP7 Look for and make use of structure.
2.MP.7 Second graders look for patterns. For instance, they adopt mental math
strategies based on patterns (making ten, fact families, doubles, adding and
subtracting numbers by place, and equal shares). Their understanding of the
number system develops into 3- and 4- digit numbers.
MP8 Look for and express regularity in repeated reasoning.
2.MP.8 Second grade students notice repetitive actions in counting and
computation, etc. When children have multiple opportunities to add and subtract,
they look for shortcuts, such as tens are added to tens, ones are added to ones,
and sometimes the ones make a new ten. They also notice when a whole is
shared into equal groups, the size of the share gets smaller the more shares.
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OPERATIONS AND ALGEBRAIC THINKING
Represent and solve problems involving addition and subtraction.
*2.OA.A.1 Use addition and subtraction within 100 to solve one- and two-step word problems involving situations of adding to, taking from, putting
together, taking apart, and comparing, with unknowns in all positions, by using drawings and equations with a symbol for the unknown number to
represent the problem.
In addition to Proficient, the Advanced student is able to express solutions to one- and two-step word problems using multiple representations (e.g., 32 +
= 50 and = 50 32).
The Proficient student is able to use addition and subtraction within 100 to solve one- and two-step word problems involving situations of adding to, taking
from, putting together, and taking apart, and comparing with unknowns in all positions, by using drawings and equations with a symbol for the unknown
number to represent the problem.
The Basic student is able to use addition within 100 to solve one- and two-step word problems involving situations of adding to and putting together, by
using drawings and equations with a symbol for the unknown number to represent the problem.
The Below Basic student does not meet the Basic performance level.
Add and subtract within 20.
*2.OA.B.2 Fluently add and subtract within 20 using mental strategies. By end of Grade 2, know automatically all sums of two one-digit numbers based
on strategies.
In addition to Proficient, the Advanced student is able to fluently add and subtract within 100 using mental strategies.
The Proficient student is able to fluently add and subtract within 20 using mental strategies. By end of Grade 2, know automatically all sums of two one-
digit numbers based on strategies.
The Basic student is able to fluently add and subtract within 20 using mental strategies.
The Below Basic student does not meet the Basic performance level.
Work with equal groups of objects to gain foundations for multiplication.
2.OA.C.3 Determine whether a group (up to 20) has an odd or even number of objects (i.e. by pairing objects or counting them by 2s).
2.OA.C.3A If the number of objects is even, then write an equation to express this as the sum of two equal addends.
2.OA.C.3B If the number of objects group is odd, then write an equation to express this as a sum of a near double (double plus 1).
In addition to Proficient, the Advanced student is able to determine whether a group (up to 100) has an odd or even number of objects.
A. If the number of objects is even, then write an equation to express this as the sum of two equal addends.
B. If the number of objects in a group is odd, then write an equation to express this as a sum of a near double (double plus 1).
The Proficient student is able to determine whether a group (up to 20) has an odd or even number of objects (i.e. by pairing objects or counting them by
2s).
A. If the number of objects is even, then write an equation to express this as the sum of two equal addends.
B. If the number of objects in a group is odd, then write an equation to express this as a sum of a near double (double plus 1).
The Basic student is able to determine whether a group (up to 20) has an odd or even number of objects (i.e. by pairing objects or counting them by 2s).
The Below Basic student does not meet the Basic performance level.
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2.OA.C.4 Use addition to find the total number of objects arranged in rectangular arrays with up to 5 rows and up to 5 columns; write an equation to
express the total as a sum of equal addends.
In addition to Proficient, the Advanced student is able to write at least two equations to express the total as the sum of equal addends (e.g., an 5 + 5 +
5 + 5 array is the same as 10 + 10 array).
The Proficient student is able to use addition to find the total number of objects arranged in rectangular arrays with up to 5 rows and up to 5 columns;
write an equation to express the total as a sum of equal addends.
The Basic student is able to use addition to find the total number of objects arranged in rectangular arrays with no more than 10 objects; write an equation
to express the total as a sum of equal addends.
The Below Basic student does not meet the Basic performance level.
NUMBER AND OPERATIONS IN BASE TEN
Understand place value.
*2.NBT.D.1 Understand that the three digits of a three-digit number represent amounts of hundreds, tens, and ones; and demonstrate that cases: a. 100
can be thought of as a bundle of ten tens — called a “hundred.” b. The numbers 100, 200, 300, 400, 500, 600, 700, 800, 900 refer to one, two, three, four,
five, six, seven, eight, or nine hundreds (and 0 tens and 0 ones).
2.NBT.D.1A 100 can be thought of as a bundle of ten tens — called a “hundred.”
2.NBT.D.1B The numbers 100, 200, 300, 400, 500, 600, 700, 800, 900 refer to one, two, three, four, five, six, seven, eight, or nine hundreds (and 0 tens and 0
ones).
2.NBT.D.1C Three-digit numbers can be decomposed in multiple ways (e.g., 524 can be decomposed as 5 hundreds, 2 tens and 4 ones or 4 hundreds, 12
tens, and 4 ones, etc.)
In addition to Proficient, the Advanced student is able to understand that the four digits of a four-digit number represent amounts of thousands, hundreds,
tens, and ones; and demonstrate that:
A. 1000 can be thought of as a bundle of ten hundreds -- called a "thousand."
B. Multiples of one hundred larger than 1000 can be referred to both as a count of hundreds or by place value with thousands and hundreds (e.g.,
4200 is forty-two hundred and four thousand, two hundred).
C. Four-digit numbers can be decomposed in multiple ways (e.g., 2524 can be decomposed as 2 thousands, 5 hundreds, 2 tens and 4 ones or 25
hundreds, 2 tens, and 4 ones, etc.)
The Proficient student is able to understand that the three digits of a three-digit number represent amounts of hundreds, tens, and ones; and demonstrate
that:
A. 100 can be thought of as a bundle of ten tens — called a “hundred.”
B. The numbers 100, 200, 300, 400, 500, 600, 700, 800, 900 refer to one, two, three, four, five, six, seven, eight, or nine hundreds (and 0 tens and
0 ones).
C. Three-digit numbers can be decomposed in multiple ways (e.g., 524 can be decomposed as 5 hundreds, 2 tens and 4 ones or 4 hundreds, 12
tens, and 4 ones, etc.)
The Basic student is able to understand that the three digits of a three-digit number represent amounts of hundreds, tens, and ones; and demonstrate
that:
A. 100 can be thought of as a bundle of ten tens — called a “hundred.”
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B. The numbers 100, 200, 300, 400, 500, 600, 700, 800, 900 refer to one, two, three, four, five, six, seven, eight, or nine hundreds (and 0 tens and
0 ones).
C. Three-digit numbers can be decomposed into place value (e.g., 524 can be decomposed as 5 hundreds, 2 tens, and 4 ones, etc.)
The Below Basic student does not meet the Basic performance level.
2.NBT.D.2 Skip-count by 10s and 100s within 1000 starting at any given number.
In addition to Proficient, the Advanced student is able to skip-count by multiple units of 10s and 100s within1000 starting at any given number.
The Proficient student is able to skip-count by 10s and 100s within 1000 starting at any given number.
The Basic student is able to skip-count by 10s and 100s within 1000 starting at any multiple of 10 or multiple of 100.
The Below Basic student does not meet the Basic performance level.
2.NBT.D.3 Read and write numbers to 1000 using base-ten numerals, number names, and expanded form.
In addition to Proficient, the Advanced student is able to read and write numbers to 10000 using base-ten numerals, number names, and expanded form.
OR Represent a three digit number in multiple ways using base 10 numerals or expanded form (e.g., 674 could be: 600 + 70 + 4 or 500 + 170 + 4 or
600 + 60 + 14).
The Proficient student is able to read and write numbers to 1000 using base-ten numerals, number names, and expanded form.
The Basic student is able to read and write numbers to 1000 using base-ten numerals and number names and write numbers to 100 using expanded
form.
The Below Basic student does not meet the Basic performance level.
2.NBT.D.4 Compare pairs of three-digit numbers based on meanings of the hundreds, tens, and ones digits, using the words "is greater than," "is equal
to," "is less than," and with the symbols >, =, and < to record the results of comparisons.
In addition to Proficient, the Advanced student is able to write comparisons recording the results using both the greater than and the less than words and
symbols (e.g., 127 > 121, and 121 < 127).
The Proficient student is able to compare pairs of three-digit numbers based on meanings of the hundreds, tens, and ones digits, using the words "is
greater than," "is equal to," "is less than" and with the symbols >, =, and < to record the results of comparisons.
The Basic student is able to compare pairs of three-digit numbers from a pictorial representation based on meanings of the hundreds, tens, and ones
digits, using the words "is greater than," "is equal to," "is less than" and with the symbols >, =, and < to record the results of comparisons.
The Below Basic student does not meet the Basic performance level.
Use place value understanding and properties of operations to add and subtract.
2.NBT.E.5 Add and subtract within 100 using strategies based on place value, properties of addition, and/or the relationship between addition and
subtraction.
In addition to Proficient, the Advanced student is able to add and subtract within 100 using multiple strategies based on place value, properties of
addition, and the relationship between addition and subtraction.
The Proficient student is able to add and subtract within 100 using strategies based on place value, properties of addition, and/or the relationship between
addition and subtraction.
The Basic student is able to add and subtract within 100 in problems that do not require regrouping using strategies based on place value, properties of
addition, and/or the relationship between addition and subtraction.
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The Below Basic student does not meet the Basic performance level.
2.NBT.E.6 Add up to four two-digit numbers using strategies based on place value and/or properties of addition.
In addition to Proficient, the Advanced student is able to add up at least four two-digit numbers using multiple strategies based on place value and
properties of addition.
The Proficient student is able to add up to four two-digit numbers using strategies based on place value and/or properties of addition.
The Basic student is able to add up to four one-digit numbers with sums greater than 20 using strategies based on place value and/or properties of
addition.
The Below Basic student does not meet the Basic performance level.
*2.NBT.E.7 Add and subtract within 1000, using concrete models or drawings and strategies based on place value, properties of addition, and/or the
relationship between addition and subtraction:
2.NBT.E.7A Relate the strategy to a written method and explain the reasoning used.
2.NBT.E.7B Understand that in adding or subtracting three-digit numbers, add or subtract hundreds and hundreds, tens and tens, ones and ones.
2.NBT.E.7C Understand that sometimes it is necessary to compose or decompose tens or hundreds.
In addition to Proficient, the Advanced student is able to add and subtract within 1000 using multiple strategies based on place value, properties of
addition, and the relationship between addition and subtraction. Relate the strategies used to written methods and explain the reasoning.
The Proficient student is able to add and subtract within 1000, using concrete models or drawings and strategies based on place value, properties of
addition, and/or the relationship between addition and subtraction:
A. Relate the strategy to a written method and explain the reasoning used.
B. Understand that in adding or subtracting three-digit numbers, add or subtract hundreds and hundreds, tens and tens, ones and ones.
C. Understand that sometimes it is necessary to compose or decompose tens or hundreds.
The Basic student is able to add and subtract within 1000 in problems that do not require regrouping, using concrete models or drawings and strategies
based on place value, properties of addition, and/or the relationship between addition and subtraction.
The Below Basic student does not meet the Basic performance level.
2.NBT.E.8
2.NBT.E.8A Mentally, add 10 or 100 to a given number 100-900.
2.NBT.E.8B Mentally, subtract 10 or 100 from a given number 100-900.
In addition to Proficient, the Advanced student is able to, mentally:
A. Add multiple units of 10s or 100s to a given number 100-900, and
B. Subtract multiple units of 10s or 100s from a given number 100-900.
The Proficient student is able to, mentally:
A. Add 10 or 100 to a given number 100-900, and
B. Subtract 10 or 100 from a given number 100-900.
The Basic student is able to, mentally:
A. Add 10 or 100 to a given multiple of 10 or multiple of 100 in the range 100-900, and
B. Subtract 10 or 100 from a given multiple of 10 or multiple of 100 in the range 100-900.
The Below Basic student does not meet the Basic performance level.
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2.NBT.E.9 Explain why addition and subtraction strategies work, using place value and the properties of addition. (Explanations may be supported by
drawings, objects, or written form.)
In addition to Proficient, the Advanced student is able to evaluate different addition and subtraction strategies for a given situation, determine the most
efficient strategy, and justify reasoning.
The Proficient student is able to explain why addition and subtraction strategies work, using place value and the properties of addition. (Explanations may
be supported by drawings, objects, or written form.)
The Basic student is able to explain why addition strategies work, using place value and the properties of addition. (Explanations may be supported by
drawings, objects, or written form.)
The Below Basic student does not meet the Basic performance level.
MEASUREMENT AND DATA
Measure and estimate lengths in standard units.
2.MD.F.1 Measure the length of an object by selecting and using appropriate tools such as rulers, yardsticks, meter sticks, and measuring tapes.
In addition to Proficient, the Advanced student is able to use the most appropriate measurement tool and provide justification for the selection.
The Proficient student is able to measure the length of an object by selecting and using appropriate tools such as rulers, yardsticks, meter sticks, and
measuring tapes.
The Basic student is able to measure the length of an object with a given tool.
The Below Basic student does not meet the Basic performance level.
2.MD.F.2 Measure the same object or distance using a standard unit of one length and then a standard unit of a different length. Explain how the two
measurements relate to the size of the unit chosen.
In addition to Proficient, the Advanced student is able to measure an object or distance using a standard unit of length and describe other measurements
that could be used to measure the length in another unit of the same system. Explain the relationship of the units. Students are not required to know direct
unit conversions.
The Proficient student is able to measure the same object or distance using a standard unit of one length and then a standard unit of a different length.
Explain how the two measurements relate to the size of the unit chosen.
The Basic student is able to measure an object or distance using a standard unit.
The Below Basic student does not meet the Basic performance level.
2.MD.F.3 Estimate lengths using units of inches, feet, centimeters, and meters.
In addition to Proficient, the Advanced student is able to give an estimation and provide justification for the unit selection, when given a real-world
scenario.
The Proficient student is able to estimate lengths using units of inches, feet, centimeters, and meters.
The Basic student is able to estimate lengths using units of inches, feet, centimeters, and meters when given a scaled pictorial comparison of one unit and
a given object (e.g., the student is asked to estimate the length of a car in meters, when given a pictorial representation of a meter stick in relation to the
car)
The Below Basic student does not meet the Basic performance level.
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2.MD.F.4 Measure in standard length units to determine how much longer one object is than another.
In addition to Proficient, the Advanced student is able to measure three or more lengths and order or compare the measurements.
The Proficient student is able to measure in standard length units to determine how much longer one object is than another.
The Basic student is able to identify longer or shorter given two lengths.
The Below Basic student does not meet the Basic performance level.
Relate addition and subtraction to length.
2.MD.G.5 Use addition and subtraction within 100 to solve word problems involving lengths that are given in the same units.
In addition to Proficient, the Advanced student is able to express multiple representations of solutions to addition and subtraction word problems involving
lengths (e.g., 32 + = 50 and = 50 32).
The Proficient student is able to use addition and subtraction within 100 to solve word problems involving lengths that are given in the same units.
The Basic student is able to use addition within 100 to solve word problems involving lengths that are given in the same units.
The Below Basic student does not meet the Basic performance level.
2.MD.G.6 Use a number line diagram with equally spaced points to:
2.MD.G.6A Represent whole-number sums and differences within 100 on a number line diagram.
2.MD.G.6B Locate the multiple of 10 before and after a given number within 100.
In addition to Proficient, the Advanced student is able to use multiple strategies to solve whole-number sums and differences within 100 on a number line
diagram.
The Proficient student is able to use a number line diagram with equally spaced points to:
A. Represent whole-number sums and differences within 100 on a number line diagram.
B. Locate the multiple of 10 before and after a given number within 100.
The Basic student is able to use a number line diagram with equally spaced points to locate the multiple of 10 before and after a given number within 100.
The Below Basic student does not meet the Basic performance level.
Work with time and money.
2.MD.H.7 Tell and write time from analog and digital clocks in five minute increments using a.m. and p.m.
In addition to Proficient, the Advanced student is able to determine the amount of time until the next hour.
The Proficient student is able to tell and write time from analog and digital clocks in five minute increments using a.m. and p.m.
The Basic student is able to match corresponding times in five minute increments on analog and digital clocks.
The Below Basic student does not meet the Basic performance level.
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*2.MD.H.8 Solve word problems up to $10 involving dollar bills, quarters, dimes, nickels, and pennies, using $ (dollars) and ¢ (cents) symbols
appropriately.
In addition to Proficient, the Advanced student is able to count up from an amount (using the least amount of bills and coins) to determine the change
from a 10 dollar bill.
The Proficient student is able to solve word problems up to $10 involving dollar bills, quarters, dimes, nickels, and pennies, using $ (dollars) and ¢ (cents)
symbols appropriately.
The Basic student is able to solve word problems up to $10 involving dollar bills and dimes using $ (dollars) and ¢ (cents) symbols appropriately.
The Below Basic student does not meet the Basic performance level.
Represent and interpret data.
2.MD.I.9 Generate measurement data based on whole units and show data by making a line plot.
In addition to Proficient, the Advanced student is able to answer addition and subtraction questions that compare measurements using information
presented in a line plot.
The Proficient student is able to generate measurement data based on whole units and show data by making a line plot.
The Basic student is able to place given measurement data on a line plot.
The Below Basic student does not meet the Basic performance level.
2.MD.I.10
2.MD.I.10A Use data to draw a picture graph and a bar graph (with single-unit scale) to represent a data set with up to four categories.
2.MD.I.10B Use data to solve simple put-together, take-apart, and compare problems using information presented in a bar graph.
In addition to Proficient, the Advanced student is able to solve put-together, take-apart, and compare problems using information presented in multiple bar
and/or picture graphs.
The Proficient student is able to use data to:
A. Draw a picture graph and a bar graph (with single-unit scale) to represent a data set with up to four categories.
B. Solve simple put-together, take-apart, and compare problems using information presented in a bar graph.
The Basic student is able to use data to:
A. Draw a picture graph or a bar graph (with single-unit scale) to represent a data set with up to four categories.
B. Solve simple put-together problems using information presented in a bar graph.
The Below Basic student does not meet the Basic performance level.
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GEOMETRY
Reason with shapes and their attributes.
2.G.J.1 Identify triangles, quadrilaterals, pentagons, hexagons, and cubes. Recognize and draw shapes having specified attributes, such as a given
number of angles or a given number of equal faces. (Sizes are compared directly or visually, not compared by measuring.)
In addition to Proficient, the Advanced student is able to compare and contrast sets of defining attributes from given shapes and use the comparison to
change one shape into the other shape.
The Proficient student is able to identify triangles, quadrilaterals, pentagons, hexagons, and cubes. Recognize and draw shapes having specified
attributes, such as a given number of angles or a given number of equal faces. (Sizes are compared directly or visually, not compared by measuring.)
The Basic student is able to identify triangles, quadrilaterals, pentagons, hexagons, and cubes.
The Below Basic student does not meet the Basic performance level.
2.G.J.2 Partition a rectangle into rows and columns of same-size squares and count to find the total number of them.
In addition to Proficient, the Advanced student is able to manipulate a given amount of same-size squares (up to 25) into all of the different possible
combinations of rectangles and represent the rectangle combinations with repeated addition equations or expressions.
The Proficient student is able to partition a rectangle into rows and columns of same-size squares and count to find the total number of them.
The Basic student is able to partition a rectangle into rows or columns of same-size squares and count to find the total number of them.
The Below Basic student does not meet the Basic performance level.
*2.G.J.3 Partition circles and rectangles into two, three, or four equal shares, describe the shares using the words halves, thirds, half of, a third of, etc.,
and describe the whole as two halves, three thirds, four fourths. Recognize that equal shares of identical wholes need not have the same shape.
2.G.J.3A Describing the shares using the words halves, thirds, half of, a third of, etc.
2.G.J.3B Describing the whole as two halves, three thirds, four fourths.
2.G.J.3C Recognizing that equal shares of identical wholes need not have the same shape.
In addition to Proficient, the Advanced student is able to partition symmetrical shapes into two, three, or four equal shares and describe shares using
correct fractional words or partition multiple circles and rectangles into two, three, and four equal shares and:
A. Describing the shares using the words halves, thirds, half of, a third of, etc.
B. Describing the whole as two halves, three thirds, four fourths.
C. Recognizing that equal shares of identical wholes need not have the same shape.
The Proficient student is able to partition circles and rectangles into two, three, or four equal shares by:
A. Describing the shares using the words halves, thirds, half of, a third of, etc.
B. Describing the whole as two halves, three thirds, four fourths.
C. Recognizing that equal shares of identical wholes need not have the same shape.
The Basic student is able to partition rectangles into two, three, or four equal shares by:
A. Describing the shares using the words halves, thirds, half of, a third of, etc.
B. Describing the whole as two halves, three thirds, four fourths.
The Below Basic student does not meet the Basic performance level.
Page | 35 Wyoming Department of Education e d u . w y o m i n g . g o v / s t a n d a rd s
2021 Math Wyoming Content & Performance Standards & PLDs
Companion document to the 2018 Mathematics Content Standards
Grade 3 Math Content & Performance Standards & PLDs
GRADE 3 MATH PRACTICES
MP1 Make sense of problems and persevere in solving
them.
3.MP.1 In third grade, students know that doing mathematics involves solving
problems and discussing how they solved them. Students explain to
themselves the meaning of a problem and look for ways to solve it. Third
grade students may use concrete objects or pictures to help them
conceptualize and solve problems. They may check their thinking by asking
themselves, “Does this make sense? “They listen to the strategies of others
and will try different approaches. They often will use another method to
check their answers.
MP2 Reason abstractly and quantitatively.
3.MP.2 Students recognize that a number represents a specific quantity.
They connect the quantity to written symbols and create a logical
representation of the problem at hand, considering both the appropriate units
involved and the meaning of quantities.
MP3 Construct viable arguments and critique the
reasoning of others.
3.MP.3 Students may construct arguments using concrete referents, such as
objects, pictures, and drawings. They refine their mathematical
communication skills as they participate in mathematical discussions
involving questions such as, “How did you get that? and “Why is that true?”
They explain their thinking to others and respond to others’ thinking.
MP4 Model with mathematics.
3.MP.4 Students experiment with representing problem situations in multiple
ways including numbers, words (mathematical language), drawing pictures,
using objects, acting out, making a chart, list, or graph, creating equations,
etc. Students need opportunities to connect the different representations and
explain the connections. They should be able to use all of these
representations as needed. Students should evaluate their results in the
context of the situation and reflect on whether the results make sense.
MP5 Use appropriate tools strategically.
3.MP.5 Students consider the available tools (including estimation) when
solving a mathematical problem and decide when certain tools might be
helpful. For instance, they may use graph paper to find all the possible
rectangles that have a given perimeter. They compile the possibilities into an
organized list or a table, and determine whether they have all the possible
rectangles.
MP6 Attend to precision.
3.MP.6 As students develop their mathematical communication skills, they try
to use clear and precise language in their discussions with others and in their
own reasoning. They are careful about specifying units of measure and state
the meaning of the symbols they choose. For instance, when figuring out the
area of a rectangle they record their answers in square units.
MP7 Look for and make use of structure.
3.MP.7 Students look closely to discover a pattern or structure. For example,
students use properties of operations as strategies to multiply and divide
(Commutative and Distributive Properties).
MP8 Look for and express regularity in repeated
reasoning.
3.MP.8 Students notice repetitive actions in computation and look for
shortcut methods. For example, students may use the Distributive Property
as a strategy for using products they know to solve products that they don’t
know. For example, if students are asked to find the product of 7 8, they
might decompose 7 into 5 and 2 then multiply 5 8 and 2 8 to arrive at
40 + 16 or 56. In addition, third graders continually evaluate their work by
asking themselves, “Does this make sense?”
Page | 36 Wyoming Department of Education e d u . w y o m i n g . g o v / s t a n d a rd s
2021 Math Wyoming Content & Performance Standards & PLDs
OPERATIONS AND ALGEBRAIC THINKING
Represent and solve problems involving multiplication and division.
3.OA.A.1 Represent the concept of multiplication of whole numbers using models including, but not limited to, equal-sized groups ("groups of"),
arrays, area models, repeated addition, and equal "jumps" on a number line.
In addition to Proficient, the Advanced student is able to:
• Represent the concept of multiplication of whole numbers using models and strategies in multiple ways.
• Create or write a scenario or model that represents the concept of multiplication of whole numbers.
The Proficient student is able to represent the concept of multiplication of whole numbers using models and strategies.
The Basic student is able to represent the concept of multiplication of whole numbers using models and strategies with partial success.
The Below Basic student may be able to recognize that a model represents the concept of multiplication of a whole number but may require counting
individual items or other methods to solve the problem.
3.OA.A.2 Represent the concept of division of whole numbers (resulting in whole number quotients) using models including, but not limited to,
partitioning, repeated subtraction, sharing, and inverse of multiplication.
In addition to Proficient, the Advanced student is able to:
• Represent the concept of division of whole numbers (resulting in whole number quotients) using models and strategies in multiple ways.
• Create or write a scenario or model that represents the concept of division of whole numbers (resulting in whole number quotients).
The Proficient student is able to represent the concept of division of whole numbers (resulting in whole number quotients) using models and strategies.
The Basic student is able to represent the concept of division of whole numbers (resulting in whole number quotients) using models and strategies with
partial success.
The Below Basic student may be able to recognize that a model represents the concept of division of a whole number.
3.OA.A.3 Solve multiplication and division word problems within 100 using appropriate modeling strategies and equations.
In addition to Proficient, the Advanced student is able to:
• Solve two-step multiplication and division word problems (with products and dividends) within 100 using appropriate modeling strategies and
equations.
• Solve multiplication and division word problems (with products and dividends) beyond 100 using appropriate modeling strategies and
equations.
• Write and solve a multiplication or division word problem (with products and dividends) within or beyond 100 using appropriate modeling
strategies and equations.
The Proficient student is able to solve multiplication and division word problems (with products and dividends) within 100 using appropriate modeling
strategies and equations.
The Basic student is able to, when provided a pictorial representation, solve multiplication and division word problems (with products and dividends) within
100 using appropriate modeling strategies and equations.
The Below Basic student may be able to select the appropriate operation necessary to solve a multiplication or division word problem (with products or
dividends) within 100.
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3.OA.A.4 Determine the unknown whole number in a multiplication or division equation relating three whole numbers when the unknown is a missing
factor, product, dividend, divisor, or quotient. (Students need not know formal terms.)
In addition to Proficient, the Advanced student is able to determine the unknown whole number in a multiplication or division equation relating at least one
three digit whole number when the unknown is a missing factor, product, dividend, divisor, or quotient.
The Proficient student is able to determine the unknown whole number in a multiplication or division equation relating three whole numbers when the
unknown is a missing factor, product, dividend, divisor, or quotient. (Students need not know formal terms.)
The Basic student is able to determine the unknown whole number in a multiplication or division equation relating three whole numbers when the unknown
is a missing factor, product, dividend, divisor, or quotient with partial success.
The Below Basic student may be able to, when provided pictorial support, determine the unknown whole number in a multiplication or division equation
relating three whole numbers when the unknown is a missing factor, product, dividend, divisor, or quotient.
Understand properties of multiplication and the relationship between multiplication and division.
3.OA.B.5 Apply properties of multiplication as strategies to multiply and divide. (Students need not use formal terms for these properties.)
In addition to Proficient, the Advanced student is able to apply properties of multiplication and division to:
• Multiply two-digit numbers.
• Divide dividends greater than 100 by a one-digit divisor.
The Proficient student is able to apply properties of multiplication including Commutative, Associative, and Distributive as strategies to multiply and divide.
(Students need not use formal terms for these properties.)
The Basic student is able to apply at least one of the properties of multiplication including Commutative, Associative, and Distributive as strategies to
multiply and divide.
The Below Basic student may be able to recognize the Commutative Property of multiplication.
3.OA.B.6 Understand division as an unknown-factor problem.
In addition to Proficient, the Advanced student is able to apply the relationship between multiplication and division to find an unknown with a dividend of at
least three digits.
The Proficient student is able to apply the relationship between multiplication and division to find the unknown.
The Basic student is able to apply the relationship between multiplication and division to find an unknown factor of a division problem when provided fact
families.
The Below Basic student may be able to apply the relationship between multiplication and division to find an unknown factor of a division problem when
provided fact families with partial success.
Multiply and divide within 100.
*3.OA.C.7 Fluently multiply and divide with factors 1 - 10 using mental strategies. By end of Grade 3, know automatically all products of one-digit
factors based on strategies.
In addition to Proficient, the Advanced student is able to fluently multiply and divide two numbers with at least one factor greater than 10 using mental
strategies.
The Proficient student is able to fluently multiply and divide with factors 1 - 10 using mental strategies. By end of Grade 3, know automatically all products
of one-digit factors based on strategies.
The Basic student is able to multiply and divide with factors 1 - 10 using mental strategies and/or pictorial representation.
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The Below Basic student may be able to multiply and divide with factors 1 - 10 using mental strategies and/or pictorial representation with partial success.
Solve problems involving the four operations, and identify and explain patterns in arithmetic.
*3.OA.D.8 Solve two-step word problems (limited to the whole number system) using the four basic operations. Students should apply the Order of
Operations when there are no parentheses to specify a particular order.
3.OA.D.8A Represent these problems using equations with a symbol standing for the unknown quantity.
3.OA.D.8B Assess the reasonableness of answers using mental computation and estimation strategies including rounding.
In addition to Proficient, the Advanced student is able to given an equation with at least two operations, create and solve a word problem that matches the
equation.
The Proficient student is able to solve two-step word problems (limited to the whole number system) using the four basic operations. Students should
apply the Order of Operations when there are no parentheses to specify a particular order.
A. Represent these problems using equations with a symbol standing for the unknown quantity.
B. Assess the reasonableness of answers using mental computation and estimation strategies including rounding.
The Basic student is able to, when provided a pictorial representation, solve two-step word problems (limited to the whole number system) using the four
basic operations. Students should apply the Order of Operations when there are no parentheses to specify a particular order.
The Below Basic student may be able to, when provided a pictorial representation, solve two-step word problems (limited to the whole number system)
using the four basic operations. Students should apply the Order of Operations when there are no parentheses to specify a particular order with partial
success.
3.OA.D.9 Identify arithmetic patterns and explain the relationships using properties of operations.
In addition to Proficient, the Advanced student is able to:
• Identify a characteristic of a pattern that is not explicitly given. OR
• Write an equation to find the nth term of the arithmetic pattern. OR
• Identify arithmetic patterns and explain the relationships using properties of operations from a real-world problem.
The Proficient student is able to identify arithmetic patterns and explain the relationships using properties of operations.
The Basic student is able to predict the next term of a pattern.
The Below Basic student may be able to predict the next term of a pattern with partial success.
NUMBER AND OPERATIONS IN BASE TEN
Use place value understanding and properties of operations to perform multi-digit arithmetic (a range of algorithms
may be used).
3.NBT.E.1 Use place value understanding to round whole numbers to the nearest 10 or 100.
In addition to Proficient, the Advanced student is able to:
• Round to the nearest 1,000. OR
• Round a multi-digit number to a given place value(s).
The Proficient student is able to use place value understanding to round whole numbers to the nearest 10 or 100.
The Basic student is able to round to the nearest 10 or 100 when provided a model such as a number line.
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The Below Basic student may be able to round to the nearest 10 or 100 with partial success when provided a model such as a number line.
*3.NBT.E.2 Fluently add and subtract within 1000 using strategies and algorithms based on place value, properties of addition, and/or the relationship
between addition and subtraction.
In addition to Proficient, the Advanced student is able to fluently add and subtract beyond 1,000 using strategies and algorithms based on place value,
properties of addition, and/or the relationship between addition and subtraction.
The Proficient student is able to fluently add and subtract within 1000 using strategies and algorithms based on place value, properties of addition, and/or
the relationship between addition and subtraction.
The Basic student is able to add and subtract within 1000 using strategies and algorithms based on place value, properties of addition, and/or the
relationship between addition and subtraction.
The Below Basic student may be able to add and subtract within 1000 using strategies and algorithms based on place value, properties of addition,
and/or the relationship between addition and subtraction with partial success.
3.NBT.E.3 Multiply one-digit whole numbers by multiples of 10 in the range 10-90 (e.g., , ) using strategies based on place value and
properties of multiplication.
In addition to Proficient, the Advanced student is able to multiply 2-digit whole numbers (less than 20) by multiples of 10.
The Proficient student is able to multiply one-digit whole numbers by multiples of 10 in the range 10-90 (e.g., 9 80, 5 60) using strategies based on
place value and properties of multiplication.
The Basic student is able to multiply one-digit whole numbers by multiples of 10 in the range 10-90 (e.g., 9 80, 5 60) when given a model or strategy
based on place value.
The Below Basic student may be able to multiply one-digit whole numbers by multiples of 10 in the range 10-90 (e.g., 9 80, 5 60) with partial success
when given a model or strategy based on place value.
NUMBER AND OPERATIONS - FRACTIONS
Develop understanding of fractions as numbers. (Limited to denominators 2, 3, 4, 6, and 8) *use horizontal fractions.
3.NF.F.1 Understand a fraction
as the quantity formed by 1 part when a whole is partitioned into b equal parts; understand a fraction
as the quantity
formed by a parts of size
.
In addition to Proficient, the Advanced student is able to, when given a fraction greater than 1 whole, understand a fraction
as a quantity formed by one
part when a whole is partitioned into b equal parts; understand a fraction
as the quantity formed by a parts of size
.
Assessment Boundary: Grade 3 expectations in this domain are limited to fractions with denominators 2, 3, 4, 6, and 8.
The Proficient student is able to understand a fraction
as the quantity formed by 1 part when a whole is partitioned into b equal parts; understand a
fraction
as the quantity formed by a parts of size
.
Assessment Boundary: Grade 3 expectations in this domain are limited to fractions with denominators 2, 3, 4, 6, and 8.
The Basic student is able to identify
when given a pictorial representation of a whole partitioned into equal parts.
Assessment Boundary: Grade 3 expectations in this domain are limited to fractions with denominators 2, 3, 4, 6, and 8.
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The Below Basic student may be able to identify
with partial success when given a pictorial representation of a whole partitioned into equal parts.
Assessment Boundary: Grade 3 expectations in this domain are limited to fractions with denominators 2, 3, 4, 6, and 8.
*3.NF.F.2 Understand and represent fractions on a number line diagram.
3.NF.F.2A Represent a fraction
on a number line diagram by defining the interval from 0 to 1 as the whole and partitioning it into b equal parts. Recognize
that each part has size
and that the endpoint of the part based at 0 locates the number
on the number line.
3.NF.F.2B Represent a fraction
on a number line diagram by marking off a lengths
from 0. Recognize that the resulting interval has size
and that its
endpoint locates the number
on the number line.
In addition to Proficient, the Advanced student is able to understand and represent fractions on a number line diagram when extending to fractions
beyond one whole:
A. Represent a fraction
on a number line diagram by defining the interval from 0 to beyond 1 and partitioning it into b equal parts. Recognize that
each part has size
and that the endpoint of the part based at 0 locates the number
on the number line.
B. Represent a fraction
on a number line diagram by marking off a lengths
from 0. Recognize that the resulting interval has size
and that its
endpoint locates the number
on the number line.
Assessment Boundary: Grade 3 expectations in this domain are limited to fractions with denominators 2, 3, 4, 6, and 8.
The Proficient student is able to understand and represent fractions on a number line diagram.
A. Represent a fraction
on a number line diagram by defining the interval from 0 to 1 as the whole and partitioning it into b equal parts.
Recognize that each part has size
and that the endpoint of the part based at 0 locates the number
on the number line.
B. Represent a fraction
on a number line diagram by marking off a lengths
from 0. Recognize that the resulting interval has size
and that its
endpoint locates the number
on the number line.
Assessment Boundary: Grade 3 expectations in this domain are limited to fractions with denominators 2, 3, 4, 6, and 8.
The Basic student is able to represent fractions on a number line diagram when given a number line pre-partitioned into equal parts from 0 to 1.
A. Represent a fraction
on a number line diagram. Recognize that each part has size
and that the endpoint of the part based at 0 locates the
number
on the number line.
B. Represent a fraction
on a number line diagram by marking off a lengths
from 0. Recognize that the resulting interval has size
and that its
endpoint locates the number
on the number line.
Assessment Boundary: Grade 3 expectations in this domain are limited to fractions with denominators 2, 3, 4, 6, and 8.
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The Below Basic student may be able to represent fractions on a number line diagram when given a number line pre-partitioned into equal parts from 0 to
1 with partial success.
A. Represent a fraction
on a number line diagram. Recognize that each part has size
and that the endpoint of the part based at 0 locates the
number
on the number line.
B. Represent a fraction
on a number line diagram by marking off a lengths
from 0. Recognize that the resulting interval has size
and that its
endpoint locates the number
on the number line.
Assessment Boundary: Grade 3 expectations in this domain are limited to fractions with denominators 2, 3, 4, 6, and 8.
*3.NF.F.3 Explain equivalence of fractions in special cases, and compare fractions by reasoning about their size.
3.NF.F.3A Understand two fractions as equivalent if they are the same size, or the same point on a number line.
3.NF.F.3B Recognize and generate simple equivalent fractions. Explain why the fractions are equivalent.
3.NF.F.3C Express whole numbers as fractions, and recognize fractions that are equivalent to whole numbers.
3.NF.F.3D Compare two fractions with the same numerator or the same denominator, by reasoning about their size, Recognize that valid comparisons rely on
the two fractions referring to the same whole. Record the results of comparisons with the symbols >, =, or <, and justify the conclusions.
In addition to Proficient, the Advanced student is able to compare fractions with different numerators or different denominators, by reasoning about their
size, recognize that valid comparisons rely on the two fractions referring to the same whole. Record the results of comparisons with the symbols >, =, or
<, and justify the conclusions.
Assessment Boundary: Grade 3 expectations in this domain are limited to fractions with denominators 2, 3, 4, 6, and 8.
The Proficient student is able to explain equivalence of fractions in special cases, and compare fractions by reasoning about their size.
A. Understand two fractions as equivalent if they are the same size, or the same point on a number line.
B. Recognize and generate simple equivalent fractions. Explain why the fractions are equivalent.
C. Express whole numbers as fractions, and recognize fractions that are equivalent to whole numbers.
D. Compare two fractions with the same numerator or the same denominator, by reasoning about their size. Recognize that valid comparisons rely
on the two fractions referring to the same whole. Record the results of comparisons with the symbols >, =, or <, and justify the conclusions.
Assessment Boundary: Grade 3 expectations in this domain are limited to fractions with denominators 2, 3, 4, 6, and 8.
The Basic student is able to explain equivalence of fractions in special cases, and compare fractions by reasoning about their size when given a pictorial
representation.
A. Understand two fractions as equivalent if they are the same size, or the same point on a number line.
B. Recognize and generate simple equivalent fractions. Explain why the fractions are equivalent.
C. Express whole numbers as fractions, and recognize fractions that are equivalent to whole numbers.
D. Compare two fractions with the same numerator or the same denominator, by reasoning about their size. Recognize that valid comparisons rely
on the two fractions referring to the same whole. Record the results of comparisons with the symbols >, =, or <, and justify the conclusions.
Assessment Boundary: Grade 3 expectations in this domain are limited to fractions with denominators 2, 3, 4, 6, and 8.
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The Below Basic student may be able to explain equivalence of fractions in special cases, and compare fractions by reasoning about their size with
partial success when given a pictorial representation.
A. Understand two fractions as equivalent if they are the same size, or the same point on a number line.
B. Recognize and generate simple equivalent fractions. Explain why the fractions are equivalent.
C. Express whole numbers as fractions, and recognize fractions that are equivalent to whole numbers.
D. Compare two fractions with the same numerator or the same denominator, by reasoning about their size. Recognize that valid comparisons rely
on the two fractions referring to the same whole. Record the results of comparisons with the symbols >, =, or <, and justify the conclusions.
Assessment Boundary: Grade 3 expectations in this domain are limited to fractions with denominators 2, 3, 4, 6, and 8.
MEASUREMENT AND DATA
Solve problems involving measurement and estimation of intervals of time, liquid volumes, and masses of objects.
3.MD.G.1 Use analog clocks to tell and write time to the nearest minute and measure time intervals in minutes. Solve word problems involving addition
and subtraction of time intervals in minutes.
In addition to Proficient, the Advanced student is able to:
• Use analog clocks to tell and write time to the nearest minute and measure time intervals in minutes. Solve multi-step word problems involving
addition and subtraction of time intervals in minutes. OR
• Solve multi-step word problems involving multiplication of time intervals in minutes. OR
• Solve multi-step word problems involving elapsed time.
The Proficient student is able to use analog clocks to tell and write time to the nearest minute and measure time intervals in minutes. Solve word
problems involving addition and subtraction of time intervals in minutes.
The Basic student is able to use analog clocks to tell and write time to the nearest 5 minute intervals. Solve word problems involving addition and
subtraction of time intervals of 5 minutes.
The Below Basic student may be able to use analog clocks to tell and write time to the nearest 5 minute intervals with partial success. Solve word
problems involving addition and subtraction of time intervals of 5 minutes with partial success.
3.MD.G.2 Measure and estimate liquid volumes and masses of objects using grams (g), kilograms (kg), and liters (L). (Excludes compound units such
as cm^3 and finding the geometric volume of a container.) Add, subtract, multiply, or divide to solve one-step word problems involving masses or
volumes that are given in the same units. (Excludes multiplicative comparison problems involving notions of “times as much.”)
In addition to Proficient, the Advanced student is able to:
• Solve one-step problems involving liquid measures and masses using the four operations requiring reading a measurement off of a scaled
measurement tool. OR
• Estimate the combined mass or volume of real-world objects or amounts with relative accuracy and calculate the actual mass or volume to
determine accuracy.
The Proficient student is able to measure and estimate liquid volumes and masses of objects using grams (g), kilograms (kg), and liters (L). (Excludes
compound units such as cm^3 and finding the geometric volume of a container.) Add, subtract, multiply, or divide to solve one-step word problems
involving masses or volumes that are given in the same units. (Excludes multiplicative comparison problems involving notions of “times as much.”)
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The Basic student is able to measure and estimate liquid volumes and masses of objects using grams (g), kilograms (kg), and liters (L). (Excludes
compound units such as cm^3 and finding the geometric volume of a container and excludes multiplicative comparison problems involving notions of
“times as much.”) Assessment Boundary: Whole number measurements only.
The Below Basic student may be able to measure and estimate liquid volumes and masses of objects using grams (g), kilograms (kg), and liters (L) with
partial success. (Excludes compound units such as cm^3 and finding the geometric volume of a container and excludes multiplicative comparison
problems involving notions of “times as much.”) Assessment Boundary: Whole number measurements only.
Represent and interpret data.
3.MD.H.3 Draw a scaled picture graph and a scaled bar graph to represent a data set with several categories. Solve one- and two-step “how many
more” and “how many less” problems using information presented in scaled graphs.
In addition to Proficient, the Advanced student is able to:
• Interpret how changes in the design of a graph can alter impressions of the data it represents OR
• Use a scaled picture graph and a scaled bar graph to solve multi-step problems.
The Proficient student is able to draw a scaled picture graph and a scaled bar graph to represent a data set with several categories. Solve one- and two-
step “how many more” and “how many less” problems using information presented in scaled graphs.
The Basic student is able to solve one- and two-step “how many more” and “how many less” problems using information presented in scaled picture
graphs and scaled bar graphs.
The Below Basic student may be able to solve one- and two-step “how many more” and “how many less” problems using information presented in scaled
picture graphs and scaled bar graphs with partial success.
3.MD.H.4 Generate measurement data by measuring lengths using rulers marked with halves and fourths of an inch. Use the data to create a line plot,
where the horizontal scale is marked off in appropriate units—whole numbers, halves, or quarters.
In addition to Proficient, the Advanced student is able to generate measurement data by measuring lengths using rulers marked with halves, fourths, and
eighths of an inch. Use the data to create a line plot marking it in appropriate units—whole numbers, halves, quarters, or eighths.
The Proficient student is able to generate measurement data by measuring lengths using rulers marked with halves and fourths of an inch. Use the data
to create a line plot, where the horizontal scale is marked off in appropriate units—whole numbers, halves, or quarters.
The Basic student is able to use given data to create a line plot, where the horizontal scale is marked off in appropriate units—whole numbers, halves, or
quarters.
The Below Basic student may be able to use given data to create a line plot, where the horizontal scale is marked off in appropriate units—whole
numbers, halves, or quarters with partial success.
Geometric measurement: understand concepts of area and relate area to multiplication and to addition.
3.MD.I.5 Understand area as an attribute of plane figures and understand concepts of area measurement, such as square units without gaps or
overlaps.
In addition to Proficient, the Advanced student is able to create a real-world scenario using area concepts.
The Proficient student is able to identify area as an attribute of plane figures and apply concepts of area measurement, such as square units without gaps
or overlaps.
The Basic student is able to identify concepts of area measurement to determine appropriate representations, when given appropriate and inappropriate
representations of area of a plane figure.
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The Below Basic student may be able to identify concepts of area measurement to determine appropriate representations, with partial success, when
given appropriate and inappropriate representations of area of a plane figure.
3.MD.I.6 Measure areas by counting unit squares (square cm, square m, square in., square ft., and improvised units).
In addition to Proficient, the Advanced student is able to:
• Estimate areas of non-rectangular polygons or find areas of combined rectangular figures by counting unit squares and labeling in appropriate
units (square cm, square m, square in, square ft, and improvised units).
• Find the area of a rectangle using a non-counting strategy.
The Proficient student is able to measure area by counting unit squares and labeling in appropriate units (square cm, square m, square in, square ft, and
improvised units).
The Basic student is able to measure area by tiling with unit squares and labeling in square units.
The Below Basic student may be able to measure area by tiling with unit squares and labeling in square units with partial success.
*3.MD.I.7 Relate area to the operations of multiplication and addition.
3.MD.I.7A Find the area of a rectangle with whole-number side lengths (dimensions) by multiplying them. Show that this area is the same as when counting
unit squares.
3.MD.I.7B Multiply side lengths to find areas of rectangles with whole-number side lengths in the context of solving real-world and mathematical problems, and
represent whole-number products as rectangular areas in mathematical reasoning.
3.MD.I.7C Use area models to represent the Distributive Property in mathematical reasoning. Use tiling to show in a concrete case that the area of a rectangle
with whole-number side lengths and + is the sum of and .
In addition to Proficient, the Advanced student is able to solve for a missing dimension or partial dimension when given area and one dimension and write
an equation to support their thinking.
The Proficient student is able to relate area to the operations of multiplication and addition.
A. Find the area of a rectangle with whole-number side lengths (dimensions) by multiplying them. Show that this area is the same as when
counting unit squares.
B. Multiply side lengths to find areas of rectangles with whole-number side lengths in the context of solving real world and mathematical problems,
and represent whole-number products as rectangular areas in mathematical reasoning.
C. Use area models to represent the Distributive Property in mathematical reasoning. Use tiling to show in a concrete case that the area of a
rectangle with whole-number side lengths and + is the sum of and .
The Basic student is able to Measure area by tiling with unit squares.
A. Find the area of a rectangle with whole-number side lengths (dimensions) by tiling them or using a pre-partitioned rectangle.
B. Multiply side lengths to find areas of rectangles with whole-number side lengths in the context of solving real world and mathematical problems,
and represent whole-number products as rectangular areas in mathematical reasoning when given tiling or pre-partitioned rectangles.
The Below Basic student may be able to Measure area by tiling with unit squares.
A. Find the area of a rectangle with whole-number side lengths (dimensions) by tiling them or using a pre-partitioned rectangle with partial
success.
B. Multiply side lengths to find areas of rectangles with whole-number side lengths in the context of solving real world and mathematical problems,
and represent whole-number products as rectangular areas in mathematical reasoning with partial success when given tiling or pre-partitioned
rectangles.
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Geometric measurement: recognize perimeter as an attribute of plane figures and distinguish between linear and area
measures.
3.MD.J.8 Solve real-world and mathematical problems involving perimeters of polygons, including finding the perimeter given the side lengths, finding
an unknown side length, and exhibiting rectangles with the same perimeter and different area or with the same area and different perimeter.
In addition to Proficient, the Advanced student is able to:
• Build a rectangle or polygon with given perimeter. OR
• Find multiple possible perimeters of a rectangle with a given area. OR
• Find multiple possible areas of a rectangle with a given perimeter.
The Proficient student is able to solve real-world and mathematical problems involving perimeters of polygons, including finding the perimeter given the
side lengths, finding an unknown side length, and exhibiting rectangles with the same perimeter and different area or with the same area and different
perimeter.
The Basic student is able to solve real-world and mathematical problems involving perimeters of polygons, including finding the perimeter given the side
lengths, and finding an unknown side length when given a pictorial representation.
The Below Basic student may be able to solve real-world and mathematical problems involving perimeters of polygons, including:
• Finding the perimeter, when given the side lengths, AND
• Finding an unknown side length, when given a pictorial representation.
GEOMETRY
Reason with shapes and their attributes.
3.G.K.1 Use attributes of quadrilaterals to classify rhombuses, rectangles, and squares. Understand that the shared attributes can define a larger
category (e.g., quadrilaterals). Recognize rhombuses, rectangles, and squares as examples of quadrilaterals, and draw examples of quadrilaterals that
do not belong to any of these subcategories.
In addition to Proficient, the Advanced student is able to generate possible categories and subcategories to classify and/or group polygons.
The Proficient student is able to use attributes of quadrilaterals to classify rhombuses, rectangles, and squares. Understand that the shared attributes can
define a larger category (e.g., quadrilaterals). Recognize rhombuses, rectangles, and squares as examples of quadrilaterals, and draw examples of
quadrilaterals that do not belong to any of these subcategories.
The Basic student is able to select attributes that correspond to a given quadrilateral or select quadrilaterals that correspond to given attributes.
The Below Basic student may be able to select attributes that correspond to a given quadrilateral or select quadrilaterals that correspond to given
attributes with partial success.
3.G.K.2 Partition rectangles, regular polygons, and circles into parts with equal areas. Express the area of each part as a unit fraction of the whole.
In addition to Proficient, the Advanced student is able to use a unit fraction to represent a fraction larger than one by partitioning geometric figure(s).
The Proficient student is able to partition rectangles, regular polygons, and circles into parts with equal areas. Express the area of each part as a unit
fraction of the whole.
The Basic student is able to identify the unit fraction of the whole when given a rectangle, regular polygon, or circle that has been pre-partitioned into
equal parts.
The Below Basic student may be able to identify the unit fraction of the whole with partial success when given a rectangle, regular polygon, or circle that
has been pre-partitioned into equal parts.
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Companion document to the 2018 Mathematics Content Standards
Grade 4 Math Content & Performance Standards & PLDs
GRADE 4 MATH PRACTICES
MP1 Make sense of problems and persevere in solving
them.
4.MP.1 In grade four, students know that doing mathematics involves solving
problems and discussing how they solved them. Students explain to
themselves the meaning of a problem and look for ways to solve it. Third
grade students may use concrete objects or pictures to help them
conceptualize and solve problems. They may check their thinking by asking
themselves, “Does this make sense? “They listen to the strategies of others
and will try different approaches. They will often use another method to
check their answers.
MP2 Reason abstractly and quantitatively.
4.MP.2 Students recognize that a number represents a specific quantity.
They connect the quantity to written symbols and create a logical
representation of the problem at hand, considering both the appropriate units
involved and the meaning of quantities. They extend this understanding from
whole numbers to their work with fractions and decimals. Students write
simple expressions, record calculations with numbers, and represent or
round numbers using place value concepts.
MP3 Construct viable arguments and critique the
reasoning of others.
4.MP.3 Students may construct arguments using concrete referents, such as
objects, pictures, and drawings. They explain their thinking and make
connections between models and equations. They refine their mathematical
communication skills as they participate in mathematical discussions
involving questions like, “How did you get that?” and “Why is that true?” They
explain their thinking to others and respond to others’ thinking.
MP4 Model with mathematics.
4.MP.4 Students experiment with representing problem situations in multiple
ways including numbers, words (mathematical language), drawing pictures,
using objects, acting out, making a chart, list, or graph, creating equations,
etc. Students need opportunities to connect different representations and
explain the connections. They should be able to use all of these
representations as needed. Students should evaluate their results in the
context of the situation and reflect on whether the results make sense.
MP5 Use appropriate tools strategically.
4.MP.5 Students consider the available tools (including estimation) when
solving a mathematical problem and decide when certain tools might be
helpful. For instance, they may use graph paper or a number line to
represent and compare decimals, they may use protractors to measure
angles. They use other measurement tools to understand the relative size of
units within a system and express measurements given in larger units in
terms of smaller units.
MP6 Attend to precision.
4.MP.6 As students develop their mathematical communication skills, they try
to use clear and precise language in their discussions with others and in their
own reasoning. They are careful about specifying units of measure and
stating the meaning of the symbols they choose. For instance, they use
appropriate labels when creating a line plot.
MP7 Look for and make use of structure.
4.MP.7 Students look closely to discover a pattern or structure. For instance,
students use properties of operations to explain calculations (partial products
model). They relate representations of counting problems such as tree
diagrams and arrays to the multiplication principal of counting. They generate
number or shape patterns that follow a given rule.
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MP8 Look for and express regularity in repeated reasoning.
4.MP.8 Students notice repetitive actions in computation to make generalizations Students use models to explain calculations and understand how algorithms
work. They also use models to examine patterns and generate their own algorithms. For example, students use visual fraction models to write equivalent fractions.
OPERATIONS AND ALGEBRAIC THINKING
Use the four operations with whole numbers to solve problems.
4.OA.A.1 This standard was intentionally removed by the 2018 Math Standards Review Committee.
4.OA.A.2 Multiply or divide to solve word problems involving multiplicative comparison, by using strategies including, but not limited to, drawings and
equations with a symbol for the unknown number to represent the problem, distinguishing multiplicative comparison from additive comparison.
In addition to Proficient, the Advanced student is able to recognize how many times larger one quantity is than another, when comparing two quantities.
The Proficient student is able to multiply or divide to solve word problems involving multiplicative comparison, by using strategies including, but not limited
to, drawings and equations with a symbol for the unknown number to represent the problem, distinguishing multiplicative comparison from additive
comparison.
The Basic student is able to multiply or divide to solve word problems involving multiplicative comparison when given an equation or pictorial
representation for the problem.
The Below Basic student may be able to multiply or divide to solve word problems involving multiplicative comparison with partial success, when given an
equation or pictorial representation for the problem.
*4.OA.A.3 Solve multi-step word problems posed with whole numbers, including problems in which remainders must be interpreted.
4.OA.A.3A Represent these problems using equations with a letter standing for the unknown quantity.
4.OA.A.3B Assess the reasonableness of answers using mental computation and estimation strategies including rounding.
In addition to Proficient, the Advanced student is able to create a real-world problem when given a division problem that includes a remainder, then solve
and interpret the remainder.
The Proficient student is able to solve multi-step word problems posed with whole numbers, including problems in which remainders must be interpreted.
A. Represent these problems using equations with a letter standing for the unknown quantity.
B. Assess the reasonableness of answers using mental computation and estimation strategies including rounding.
The Basic student is able to solve one-step word problems posed with whole numbers, including problems in which remainders must be interpreted.
A. Represent these problems using equations with a letter standing for the unknown quantity.
B. Assess the reasonableness of answers using mental computation and estimation strategies including rounding.
The Below Basic student may be able to solve one-step word problems posed with whole numbers with partial success, including problems in which
remainders must be interpreted.
A. Represent these problems using equations with a letter standing for the unknown quantity.
B. Assess the reasonableness of answers using mental computation and estimation strategies including rounding.
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Develop understanding of factors and multiples.
4.OA.B.4 Demonstrate an understanding of factors and multiples.
4.OA.B.4A Find all factor pairs for a whole number in the range 1-100.
4.OA.B.4B Recognize that a whole number is a multiple of each of its factors.
4.OA.B.4C Determine whether a given whole number in the range 1-100 is a multiple of a given one-digit number.
4.OA.B.4D Determine whether a given whole number in the range 1-100 is prime or composite.
In addition to Proficient, the Advanced student is able to find the prime factorization of a given number within the range of 1-100.
The Proficient student is able to demonstrate an understanding of factors and multiples.
A. Find all factor pairs for a whole number in the range 1-100.
B. Recognize that a whole number is a multiple of each of its factors.
C. Determine whether a given whole number in the range 1-100 is a multiple of a given one-digit number.
D. Determine whether a given whole number in the range 1-100 is prime or composite.
The Basic student is able to demonstrate an understanding of factors and multiples.
A. Find all factor pairs for a whole number in the range 1-25.
B. Recognize that a whole number is a multiple of each of its factors.
C. Determine whether a given whole number in the range 1-25 is a multiple of a given one-digit number.
D. Determine whether a given whole number in the range 1-25 is prime or composite.
The Below Basic student may be able to demonstrate an understanding of factors and multiples with partial success.
A. Find all factor pairs for a whole number in the range 1-25.
B. Recognize that a whole number is a multiple of each of its factors.
C. Determine whether a given whole number in the range 1-25 is a multiple of a given one-digit number.
D. Determine whether a given whole number in the range 1-25 is prime or composite.
Generate and analyze patterns.
4.OA.C.5 Given a pattern, explain a rule that the pattern follows and extend the pattern. Generate a number or shape pattern that follows a given rule.
Identify apparent features of the pattern that were not explicit in the rule itself.
In addition to Proficient, the Advanced student is able to generate a pattern, write a rule, and predict a term in a number or shape pattern.
The Proficient student is able to, given a pattern, explain the rule that the pattern follows and extend the pattern. Generate a number or shape pattern that
follows a given rule. Identify apparent features of the pattern that were not explicit in the rule itself.
The Basic student is able to, given a pattern and the rule that the pattern follows, extend the pattern.
The Below Basic student may be able to, given a pattern and the rule that the pattern follows, extend the pattern with partial success.
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NUMBER AND OPERATIONS IN BASE TEN
Generalize place value understanding for multi-digit whole numbers (limited to numbers less than or equal to
1,000,000).
4.NBT.D.1 Recognize that in a multi-digit whole number, a digit in one place represents ten times what it represents in the place to its right.
In addition to Proficient, the Advanced student is able to:
• Recognize that a digit in one place represents a multiple of 10 times what another digit represents in the place to the right and apply this
relationship as an equation. OR
• Compare numbers beyond millions by reasoning about place value. OR
• Extend reasoning about place value to go beyond one place value to the right.
The Proficient student is able to recognize that in a multi-digit whole number, a digit in one place represents ten times what it represents in the place to its
right.
The Basic student is able to recognize that in a multi-digit whole number, a digit in one place represents ten times what it represents in the place to its
right when given a visual model or representation.
The Below Basic student may be able to recognize, that in a multi-digit whole number, a digit in one place represents ten times what it represents in the
place to its right with partial success, when given a visual model or representation.
4.NBT.D.2 Read and write multi-digit whole numbers using base-ten numerals, number names, and expanded form. Compare two multi-digit numbers
based on meanings of the digits in each place, using >, =, and < symbols.
In addition to Proficient, the Advanced student is able to apply understanding of numbers beyond 1,000,000 to a real-world context or situation and
compare them using >, =, and < symbols.
The Proficient student is able to read and write multi-digit whole numbers using base-ten numerals, number names, and expanded form. Compare two
multi-digit numbers based on meanings of the digits in each place, using >, =, and < symbols.
The Basic student is able to read and write multi-digit whole numbers up to 10,000 using base-ten numerals, number names, and expanded form.
Compare two multi-digit numbers based on meanings of the digits in each place, using >, =, and < symbols.
The Below Basic student may be able to read and write multi-digit whole numbers up to 10,000 using base-ten numerals, number names, and expanded
form with partial success. Compare two multi-digit numbers based on meanings of the digits in each place, using >, =, and < symbols with partial
success.
4.NBT.D.3 Use place value understanding to round multi-digit whole numbers to any place.
In addition to Proficient, the Advanced student is able to:
• Explain how to use the digits in multi-digit whole numbers to round numbers up to 1,000,000. OR
• Use an example of rounding and explain how it is helpful in computation. OR
• Justify the appropriate place value to which the student would round in a given situation.
The Proficient student is able to use place value understanding to round multi-digit whole numbers up to 1,000,000 to any place.
The Basic student is able to use place value understanding to round multi-digit whole numbers up to 1,000,000 to any place when provided a model such
as a number line with benchmark numbers.
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The Below Basic student may be able to use place value understanding to round multi-digit whole numbers up to 1,000,000 to any place, with partial
success, when provided a model such as a number line with benchmark numbers.
Use place value understanding and properties of operations to perform multi-digit arithmetic (limited to whole numbers
less than or equal to 1,000,000).
4.NBT.E.4 Add and subtract multi-digit whole numbers using place value strategies including the standard algorithm.
In addition to Proficient, the Advanced student is able to extend the standard algorithm for addition and subtraction of whole numbers to the addition and
subtraction of decimal values.
The Proficient student is able to add and subtract multi-digit whole numbers using place value strategies including the standard algorithm.
The Basic student is able to add or subtract two or more numbers, using place value strategies including the standard algorithm, whose sum or difference
is less than 10,000.
The Below Basic student may be able to add or subtract two or more numbers, using place value strategies, with partial success, including the standard
algorithm, whose sum or difference is less than 10,000.
*4.NBT.E.5 Use strategies based on place value and the properties of multiplication.
4.NBT.E.5A Multiply a whole number of up to four digits by a one-digit whole number.
4.NBT.E.5B Multiply a pair of two-digit numbers.
4.NBT.E.5C Use appropriate models to explain the calculation, such as by using equations, rectangular arrays, and/or area models.
In addition to Proficient, the Advanced student is able to multiply a whole number of up to four digits by a one-digit whole number or a pair of two-digit
numbers using more than one model or strategy and defend the efficiency of the strategy used.
The Proficient student is able to use strategies based on place value and the properties of multiplication to:
A. Multiply a whole number of up to four digits by a one-digit whole number.
B. Multiply a pair of two-digit numbers.
C. Use appropriate models to explain the calculation, such as by using equations, rectangular arrays, ratio tables, or area models.
The Basic student is able to use strategies based on place value and the properties of multiplication to:
A. Multiply a whole number of up to four digits by a one-digit whole number when given a partially completed model.
B. Multiply a pair of two-digit numbers when given a partially completed model.
The Below Basic student may be able to use strategies based on place value and the properties of multiplication with partial success to:
A. Multiply a whole number of up to four digits by a one-digit whole number when given a partially completed model.
B. Multiply a pair of two-digit numbers when given a partially completed model.
*4.NBT.E.6 Use strategies based on place value, the properties of multiplication, and/or the relationship between multiplication and division to find
quotients and remainders with up to four-digit dividends and one-digit divisors. Use appropriate models to explain the calculation, such as by using
equations, rectangular arrays, and/or area models.
In addition to Proficient, the Advanced student is able to create a real-world situation that can be modeled using a given division problem or find quotients
and remainders with up to four-digit dividends and one-digit divisors using more than one model or strategy and defend the efficiency of the strategy used.
The Proficient student is able to use strategies based on place value, the properties of multiplication, and/or the relationship between multiplication and
division to find quotients and remainders with up to four-digit dividends and one-digit divisors. Use appropriate models to explain the calculation, such as
by using equations, rectangular arrays, ratio tables, or area models.
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The Basic student is able to use partially completed models to find quotients and remainders with up to four-digit dividends and one-digit divisors.
The Below Basic student may be able to use partially completed models to find quotients and remainders with up to four-digit dividends and one-digit
divisors with partial success.
NUMBER AND OPERATIONS - FRACTIONS
Extend understanding of fraction equivalence and ordering (limited to denominators 2, 3, 4, 5, 6, 8, 10, 12, and 100).
4.NF.F.1 Explain why a fraction
is equivalent to a fraction
by using visual fraction models, with attention to how the number and size of the parts
differ even though the two fractions themselves are the same size. Use this principle to recognize and generate equivalent fractions.
In addition to Proficient, the Advanced student is able to use knowledge of equivalent fractions to solve real-world problems.
The Proficient student is able to explain why a fraction
is equivalent to a fraction
by using visual fraction models, with attention to how the number
and size of the parts differ even though the two fractions themselves are the same size. Use this principle to recognize and generate equivalent fractions.
The Basic student is able to identify equivalent fractions when given visual fraction models, with attention to how the number and size of the parts differ
even though the two fractions themselves are the same size.
The Below Basic student may be able to identify equivalent fractions, with partial success, when given visual fraction models, with attention to how the
number and size of the parts differ even though the two fractions themselves are the same size.
*4.NF.F.2 Compare two fractions with different numerators and different denominators by creating common denominators or numerators, or by
comparing to a benchmark fraction such as
.
4.NF.F.2A Recognize that comparisons are valid only when the two fractions refer to the same whole.
4.NF.F.2B Record the results of comparisons with symbols >, =, or <.
4.NF.F.2C Justify the conclusions by using a visual fraction model.
In addition to Proficient, the Advanced student is able to compare more than two fractions with different numerators and different denominators by
creating common denominators or numerators, or by comparing to a benchmark fraction such as .
A. Recognize that comparisons are valid only when the fractions refer to the same whole.
B. Record the results of comparisons on a number line.
C. Justify the conclusions by using a visual fraction model.
The Proficient student is able to compare two fractions with different numerators and different denominators by creating common denominators or
Assessment Boundary: Grade 4 expectations in this domain are limited to fractions with denominators 2, 3, 4, 6, 8, 10, 12, and 100.
numerators, or by comparing to a benchmark fraction such as
.
A. Recognize that comparisons are valid only when the two fractions refer to the same whole.
B. Record the results of comparisons with symbols >, =, or <.
C. Justify the conclusions by using a visual fraction model.
Assessment Boundary: Grade 4 expectations in this domain are limited to fractions with denominators 2, 3, 4, 6, 8, 10, 12, and 100.
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The Basic student is able to compare two fractions with different numerators and different denominators by creating common denominators or
numerators, or by comparing to a benchmark fraction such as
when provided pre-partitioned visual models.
A. Recognize that comparisons are valid only when the two fractions refer to the same whole.
B. Record the results of comparisons with symbols >, =, or <.
Assessment Boundary: limit fractions to have denominators 2, 3, 4, 6, 8, 10, and 100.
The Below Basic student may be able to compare two fractions with different numerators and different denominators by creating common denominators
or numerators, or by comparing to a benchmark fraction such as
, with partial success, when provided pre-partitioned visual models.
A. Recognize that comparisons are valid only when the two fractions refer to the same whole.
B. Record the results of comparisons with symbols >, =, or <.
Assessment Boundary: limit fractions to have denominators 2, 3, 4, 6, 8, 10, and 100.
Build fractions from unit fractions by applying and extending previous understandings of operations on whole numbers
(limited to denominators 2, 3, 4, 5, 6, 8, 10, 12, and 100).
*4.NF.G.3 Understand a fraction a/b with > as a sum of unit fractions (1/b).
4.NF.G.3A Understand addition and subtraction of fractions as joining and separating parts referring to the same whole.
4.NF.G.3B Decompose a fraction into a sum of fractions with the same denominator in more than one way, recording each decomposition by an equation.
Justify decompositions by using a visual fraction model.
4.NF.G.3C Add and subtract mixed numbers with like denominators by replacing each mixed number with an equivalent fraction, and/or by using properties of
addition and the relationship between addition and subtraction.
4.NF.G.3D Solve word problems involving addition and subtraction of fractions referring to the same whole and having like denominators.
In addition to Proficient, the Advanced student is able to:
A-B. Use properties of operations and inverse operations to add or subtract two fractions with like denominators including mixed numbers.
C. Identify and represent addition and subtraction of fractions with like denominators in multiple ways.
D. Solve two-step problems involving addition or subtraction of fractions with like denominators in mathematical or real-world contexts.
The Proficient student is able to understand a fraction a/b with > 1 as a sum of unit fractions (
).
A. Understand addition and subtraction of fractions as joining and separating parts referring to the same whole.
B. Decompose a fraction into a sum of fractions with the same denominator in more than one way, recording each decomposition by an equation.
Justify decompositions by using a visual fraction model.
C. Add and subtract mixed numbers with like denominators by replacing each mixed number with an equivalent fraction, and/or by using
properties of addition and the relationship between addition and subtraction.
D. Solve word problems involving addition and subtraction of fractions referring to the same whole and having like denominators.
The Basic student is able to with a pictorial representation:
A-B. Interpret a fraction as a sum or a difference of unit fractions.
C. Convert mixed numbers into equivalent fractions.
D. Solve one-step problems involving addition or subtraction of fractions with like denominators in mathematical contexts.
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The Below Basic student may be able to, with a pictorial representation:
A-B. Interpret a fraction as a sum or a difference of unit fractions with partial success.
C. Convert mixed numbers into equivalent fractions with partial success.
D. Solve one-step problems involving addition or subtraction of fractions with like denominators in mathematical contexts with partial success.
4.NF.G.4 Apply and extend an understanding of multiplication by multiplying a whole number and a fraction.
4.NF.G.4A Understand a fraction
as a multiple of
.
4.NF.G.4B Understand a multiple of
as a multiple of
, and use this understanding to multiply a fraction by a whole number.
4.NF.G.4C Solve real-world problems involving multiplication of a fraction by a whole number, using visual fraction models and equations to represent the
problem.
In addition to Proficient, the Advanced student is able to generalize and explain the multiplication of a whole number and a fraction as (/) = (
)/ by creating a visual fraction model in the context of a real-world problem.
The Proficient student is able to apply and extend an understanding of multiplication by multiplying a whole number and a fraction.
A. Understand a fraction
as a multiple of
.
B. Understand a multiple of
as a multiple of
, and use this understanding to multiply a fraction by a whole number.
C. Solve real-world problems involving multiplication of a fraction by a whole number, using visual fraction models and equations to represent the
problem.
The Basic student is able to apply and extend an understanding of multiplication by multiplying a whole number and a fraction when given a pictorial
representation.
A. Understand a fraction
as a multiple of
.
B. Understand a multiple of
as a multiple of
, and use this understanding to multiply a fraction by a whole number.
C. Solve real-world problems involving multiplication of a fraction by a whole number, given visual fraction models and equations to represent the
problem.
The Below Basic student may be able to apply and extend an understanding of multiplication by multiplying a whole number and a fraction when given a
pictorial representation.
A. Understand a fraction
as a multiple of
with partial success.
B. Understand a multiple of
as a multiple of
, and use this understanding to multiply a fraction by a whole number with partial success.
C. Solve real-world problems involving multiplication of a fraction by a whole number, with partial success, when given visual fraction models and
equations to represent the problem.
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Understand decimal notation for fractions, and compare decimal fractions.
4.NF.H.5 Express a fraction with denominator 10 as an equivalent fraction with denominator 100, and use this technique to add two fractions with
respective denominators 10 and 100.
In addition to Proficient, the Advanced student is able to express a fraction with denominator 10 as an equivalent fraction with denominator 100 with
fractions greater than one whole, and use this technique to add two fractions with respective denominators 10 and 100.
The Proficient student is able to express a fraction with denominator 10 as an equivalent fraction with denominator 100, and use this technique to add two
fractions with respective denominators 10 and 100.
The Basic student is able to express a fraction with denominator 10 as an equivalent fraction with denominator 100, and use this technique to add two
fractions with respective denominators 10 and 100 when given a visual model.
The Below Basic student may be able to express a fraction with denominator 10 as an equivalent fraction with denominator 100, and use this technique to
add two fractions with respective denominators 10 and 100 with partial success when given a visual model.
4.NF.H.6 Use decimal notation for fractions with denominators 10 or 100.
In addition to Proficient, the Advanced student is able to:
• Apply decimal notation for fractions with denominators 10 or 100 with fractions greater than one whole. OR
• Apply decimal notation for fractions with a denominator of 1000.
The Proficient student is able to apply decimal notation for fractions with denominators 10 or 100.
The Basic student is able to apply decimal notation for fractions with denominators 10 or 100 when given a visual model.
The Below Basic student may be able to apply decimal notation for fractions with denominators 10 or 100 with partial success when given a visual model.
4.NF.H.7 Compare and order decimal numbers to hundredths and justify by using concrete and visual models. Record the results of comparisons with
the words "is greater than," "is equal to," "is less than," and with the symbols >, =, and <.
In addition to Proficient, the Advanced student is able to compare and order decimal numbers greater than one and/or to thousandths and justify by using
concrete and visual models. Record the results of comparisons with the words "is greater than," "is equal to," "is less than," and with the symbols >, =,
and <.
The Proficient student is able to compare and order decimal numbers to hundredths and justify by using concrete and visual models. Record the results
of comparisons with the words "is greater than," "is equal to," "is less than," and with the symbols >, =, and <.
The Basic student is able to compare and order decimal numbers to hundredths when given visual models. Record the results of comparisons with the
words "is greater than," "is equal to," "is less than," and with the symbols >, =, and <.
The Below Basic student may be able to compare and order decimal numbers to hundredths with partial success when given visual models. Record the
results of comparisons with the words "is greater than," "is equal to," "is less than," and with the symbols >, =, and < with partial success.
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MEASUREMENT AND DATA
Solve problems involving measurement and conversion of measurements from a larger unit to a smaller unit.
4.MD.I.1 Know relative sizes of measurement units within one system of units including, but not limited to, km, m, cm; kg, g; lb., oz.; l L, ml; hr., min,
sec; ft., in., gal., qt. pt., c. Within a single system of measurement, express measurements in a larger unit in terms of a smaller unit. Record
measurement equivalents in a two-column table.
In addition to Proficient, the Advanced student is able to estimate a given unit of measurement from one system to another (i.e., km vs. miles, kg vs lbs,
etc.).
The Proficient student is able to know relative sizes of measurement units within one system of units including, but not limited to, km, m, cm; kg, g; lb, oz.;
l L, ml; hr, min, sec; ft., in., gal., qt. pt., c. Within a single system of measurement, express measurements in a larger unit in terms of a smaller unit. Record
measurement equivalents in a two-column table.
The Basic student is able to, within a single system of measurement, express measurements in a larger unit in terms of a smaller unit when given a
labeled model or two-column table.
The Below Basic student may be able to, within a single system of measurement, express measurements in a larger unit in terms of a smaller unit with
partial success when given a labeled model or two-column table.
4.MD.I.2 Use the four operations to solve word problems involving distances, intervals of time, liquid volumes, masses of objects, and money,
including problems involving simple fractions or decimals, and problems that require expressing measurements given in a larger unit in terms of a
smaller unit. Represent measurement quantities using diagrams such as number line diagrams that feature a measurement scale.
Assessment Boundary: Use denominators of 2, 4, 8 and decimals up to hundredths.
In addition to Proficient, the Advanced student is able to create and solve real-world scenarios using the four operations involving distances, intervals of
time, liquid volumes, masses of objects, and money, including problems involving simple fractions or decimals, and problems that require expressing
measurements given in a larger unit in terms of a smaller unit or smaller units in terms of larger units. Represent measurement quantities using diagrams
such as number line diagrams that feature a measurement scale.
Assessment Boundary: Use denominators of 2, 4, 8 and decimals up to hundredths.
The Proficient student is able to use the four operations to solve word problems involving distances, intervals of time, liquid volumes, masses of objects,
and money, including problems involving simple fractions or decimals, and problems that require expressing measurements given in a larger unit in terms
of a smaller unit. Represent measurement quantities using diagrams such as number line diagrams that feature a measurement scale.
Assessment Boundary: Use denominators of 2, 4, 8 and decimals up to hundredths.
The Basic student is able to use the four operations to solve one-step word problems (involving distances, intervals of time, liquid volumes, masses of
objects, and money) which require expressing measurements given in a larger unit in terms of a smaller unit when given diagrams that feature a
measurement scale.
Assessment Boundary: Use denominators of 2, 4, 8 and decimals up to hundredths.
The Below Basic student may be able to use the four operations to solve one-step word problems (involving distances, intervals of time, liquid volumes,
masses of objects, and money) which require expressing measurements given in a larger unit in terms of a smaller unit with partial success, when given
diagrams that feature a measurement scale.
Assessment Boundary: Use denominators of 2, 4, 8 and decimals up to hundredths.
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*4.MD.I.3 Apply the area and perimeter formulas for rectangles in real-world and mathematical problems.
In addition to Proficient, the Advanced student is able to:
• Apply the formula for area of a rectangle to find the area of a right triangle. OR
• Find the largest possible area of a rectangle when given a specific perimeter. OR
• Find the largest possible perimeter of a rectangle when given a specific area.
The Proficient student is able to apply the area and perimeter formulas for rectangles in real-world and mathematical problems.
The Basic student is able to determine the area and perimeter of rectangles when given a labeled (length and width) pictorial representation.
The Below Basic student may be able to determine the area and perimeter of rectangles with partial success when given a labeled (length and width)
pictorial representation.
Represent and interpret data.
4.MD.J.4 Make a line plot to display a data set of measurements in fractions of a unit (
, , ). Solve problems involving addition and subtraction of
fractions by using information presented in line plots.
In addition to Proficient, the Advanced student is able to
• Solve problems involving multiplication of fractions by using information presented in line plots. AND
• Use multiplicative thinking to generalize data from a line plot.
The Proficient student is able to make a line plot to display a data set of measurements in fractions of a unit (
, , ). Solve problems involving addition
and subtraction of fractions by using information presented in line plots.
The Basic student is able to make a line plot to:
• Display a data set of measurements in fractions of a unit (
, , ). OR
• Solve problems involving addition and subtraction of fractions by using information presented in given line plots.
The Below Basic student may be able to:
• Make a line plot to display a data set of measurements in fractions of a unit (
, , ) with partial success. OR
• Solve problems involving addition and subtraction of fractions by using information presented in given line plots with partial success.
Geometric measurement: understand concepts of angle and measure angles.
4.MD.K.5
4.MD.K.5A Regarding angles, recognize angles as geometric shapes that are formed wherever two rays share a common endpoint.
4.MD.K.5B Regarding angles, understand concepts of angle measurement. An angle is measured with reference to a circle with its center at the
common endpoint of the rays.
In addition to Proficient, the Advanced student is able to apply understanding of angles as geometric shapes and find examples of angles in the real world
for different angle measurements.
The Proficient student is able to:
A. Recognize angles as geometric shapes that are formed wherever two rays share a common endpoint.
B. Understand concepts of angle measurement. An angle is measured with reference to a circle with its center at the common endpoint of the rays.
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The Basic student is able to:
A. Recognize angles as geometric shapes that are formed wherever two rays share a common endpoint. OR
B. Understand concepts of angle measurement. An angle is measured with reference to a circle with its center at the common endpoint of the rays.
The Below Basic student may be able to:
A. Recognize angles as geometric shapes that are formed wherever two rays share a common endpoint with partial success. OR
B. Understand concepts of angle measurement with partial success. An angle is measured with reference to a circle with its center at the common
endpoint of the rays.
4.MD.K.6 Measure angles in whole-number degrees using a protractor. Sketch angles of specified measure.
In addition to Proficient, the Advanced student is able to measure angles in whole-number degrees greater than 180 degrees using a protractor. Sketch
angles of specified measure in whole-number degrees greater than 180 degrees.
The Proficient student is able to measure angles in whole-number degrees using a protractor. Sketch angles of specified measure.
The Basic student is able to:
• Measure angles in whole-number degrees using a protractor. OR
• Sketch angles of specified measure.
The Below Basic student may be able to:
• Measure angles in whole-number degrees with partial success using a protractor. OR
• Sketch angles of specified measure with partial success.
4.MD.K.7 Solve addition and subtraction problems to find unknown angles on a diagram in real-world and mathematical problems.
In addition to Proficient, the Advanced student is able to:
• Solve addition and subtraction problems to find more than one unknown angle (may include angles greater than 180 degrees) on a diagram.
OR
• Create a diagram from a real-world problem to find unknown angles.
The Proficient student is able to solve addition and subtraction problems to find unknown angles on a diagram in real-world and mathematical problems.
The Basic student is able to solve addition and subtraction problems to find unknown angles on a diagrams where the sum of the angles is 90 or 180
degrees in real-world and mathematical problems.
The Below Basic student may be able to solve addition and subtraction problems to find unknown angles on a diagrams where the sum of the angles is
90 or 180 degrees in real-world and mathematical problems with partial success.
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GEOMETRY
Draw and identify lines and angles, and classify shapes by properties of their lines and angles.
4.G.L.1 Draw points, lines, line segments, rays, angles (right, acute, obtuse), and perpendicular and parallel lines. Identify these in two-dimensional
figures.
In addition to Proficient, the Advanced student is able to create a two-dimensional shape when given specific attributes.
The Proficient student is able to draw points, lines, line segments, rays, angles (right, acute, obtuse), and perpendicular and parallel lines. Identify these
in two-dimensional figures.
The Basic student is able to identify points, lines, line segments, rays, angles (right, acute, obtuse), and perpendicular and parallel lines in two-
dimensional figures.
The Below Basic student may be able to identify points, lines, line segments, rays, angles (right, acute, obtuse), and perpendicular and parallel lines in
two-dimensional figures with partial success.
4.G.L.2 Classify two-dimensional figures based on the presence or absence of parallel or perpendicular lines, or the presence or absence of angles of a
specified size. Recognize right triangles as a category, and identify right triangles.
In addition to Proficient, the Advanced student is able to create two-dimensional figures based on the presence or absence of parallel or perpendicular
lines, or the presence or absence of angles of specified size; classify triangles and justify their reasoning.
The Proficient student is able to classify two-dimensional figures based on the presence or absence of parallel or perpendicular lines, or the presence or
absence of angles of a specified size. Recognize right triangles as a category and identify right triangles.
The Basic student is able to:
• Identify two-dimensional figures based on the presence or absence of parallel or perpendicular lines, or the presence or absence of angles of a
specified size. OR
• Recognize right triangles as a category and identify right triangles.
The Below Basic student may be able to:
• Identify two-dimensional figures based on the presence or absence of parallel or perpendicular lines, or the presence or absence of angles of a
specified size with partial success. OR
• Recognize right triangles as a category and identify right triangles with partial success.
4.G.L.3 Identify line-symmetric figures. Recognize and draw lines of symmetry for two-dimensional figures.
In addition to Proficient, the Advanced student is able to create a figure with a given number of lines of symmetry.
The Proficient student is able to identify line-symmetric figures. Recognize and draw lines of symmetry for two-dimensional figures.
The Basic student is able to:
• Identify line-symmetric figures. OR
• Recognize and draw lines of symmetry for two-dimensional figures.
The Below Basic student may be able to:
• Identify line-symmetric figures with partial success OR
• Recognize and draw lines of symmetry for two-dimensional figures with partial success.
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Companion document to the 2018 Mathematics Content Standards
Grade 5 Math Content & Performance Standards & PLDs
GRADE 5 MATH PRACTICES
MP1 Make sense of problems and persevere in solving
them.
5.MP.1 In grade five, students solve problems by applying their
understanding of operations with whole numbers, decimals, and fractions
including mixed numbers. They solve problems related to volume and
measurement conversions. Students seek the meaning of a problem and
look for efficient ways to represent and solve it. They may check their
thinking by asking themselves, “What is the most efficient way to solve the
problem?”, “Does this make sense?”, and “Can I solve the problem in a
different way?”
MP2 Reason abstractly and quantitatively.
5.MP.2 Students recognize that a number represents a specific quantity.
They connect quantities to written symbols and create logical representation
of the problem at hand, while considering both the appropriate units involved
and the meaning of quantities. They extend this understanding from whole
numbers to their work with fractions and decimals. Students write simple
expressions that record calculations with numbers and represent or round
numbers using place value concepts.
MP3 Construct viable arguments and critique the
reasoning of others.
5.MP.3 Students may construct arguments using concrete referents, such as
objects, pictures, and drawings. They explain calculations based upon
models and properties of operations and rules that generate patterns. They
demonstrate and explain the relationship between volume and multiplication.
They refine their mathematical communication skills as they participate in
mathematical discussions involving questions like, “How did you get that?”
and “Why is that true?” They explain their thinking to others and respond to
others’ thinking.
MP4 Model with mathematics.
5.MP.4 Students experiment with representing problem situations in multiple
ways including numbers, words (mathematical language), drawing pictures,
using objects, making a chart, list, or graph, to create equations, etc.
Students need opportunities to connect the different representations and
explain the connections. They should be able to use all of these
representations as needed. Fifth graders should evaluate their results in the
context of the situation and whether the results make sense. They also
evaluate the utility of models to determine which models are most useful and
efficient to solve problems.
MP5 Use appropriate tools strategically.
5.MP.5 Fifth graders consider the available tools, including estimation, when
solving a mathematical problem and decide when certain tools might be
helpful. For instance, they may use unit cubes to fill a rectangular prism and
then use a ruler to measure the dimensions. They use graph paper to
accurately create graphs and solve problems, or to make predictions from
real-world data.
MP6 Attend to precision.
5.MP.6 Students continue to refine their mathematical communication skills
by using clear and precise language in their discussions with others and in
their own reasoning. Students use appropriate terminology when referring to
expressions, fractions, geometric figures, and coordinate grids. They are
careful about specifying units of measure and state the meaning of the
symbols they choose. For instance, when figuring out the volume of a
rectangular prism, they record their answers in cubic units.
MP7 Look for and make use of structure.
5.MP.7 Students look closely to discover a pattern or structure. For instance,
students use properties of operations as strategies to add, subtract, multiply
and divide with whole numbers, fractions, and decimals. They examine
numerical patterns and relate them to a rule or a graphical representation.
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MP8 Look for and express regularity in repeated reasoning.
5.MP.8 Students use repeated reasoning to understand algorithms and make generalizations about patterns. Students connect place value and their prior work
with operations to understand algorithms to fluently multiply multi-digit numbers and to perform all operations with decimals to hundredths. Students explore
operations with fractions with visual models and begin to formulate generalizations.
OPERATIONS AND ALGEBRAIC THINKING
Write, interpret, and/or evaluate numerical expressions.
*5.OA.A.1 Use parentheses, brackets, or braces in numerical expressions, and evaluate expressions with these symbols.
In addition to Proficient, the Advanced student is able to create and evaluate numerical expressions that use two or more types of grouping symbols to
complete the simplification of numerical expressions.
The Proficient student is able to use parentheses, brackets, or braces in numerical expressions, and evaluate expressions with these symbols.
The Basic student is able to evaluate two-step numerical expressions with no grouping symbols.
The Below Basic student may be able to evaluate two-step numerical expressions with no grouping symbols with partial success.
5.OA.A.2 Write simple expressions requiring parentheses that record calculations with numbers, and interpret numerical expressions without
evaluating them.
In addition to Proficient, the Advanced student is able to, using the order and properties of operations, determine whether expressions are equivalent and
justify their thinking.
The Proficient student is able to write simple expressions requiring parentheses that record calculations with numbers, and interpret numerical
expressions without evaluating them.
The Basic student is able to:
• Write simple expressions requiring parentheses that record calculations with numbers. OR
• Interpret numerical expressions without evaluating them.
The Below Basic student may be able to:
• Write simple expressions requiring parentheses that record calculations with numbers with partial success. OR
• Interpret numerical expressions without evaluating them with partial success.
Analyze patterns and relationships.
5.OA.B.3 Generate two numerical patterns with each pattern having its own rule. Explain informally the relationship(s) between corresponding terms in
the two patterns.
5.OA.B.3A Form ordered pairs consisting of corresponding terms from the two patterns.
5.OA.B.3B Graph the ordered pairs on a coordinate plane.
In addition to Proficient, the Advanced student is able to identify and explain features between the corresponding terms of two numerical patterns not
explicitly given in the rule.
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The Proficient student is able to generate two numerical patterns with each pattern having its own rule. Explain informally the relationship(s) between
corresponding terms in the two patterns.
A. Form ordered pairs consisting of corresponding terms from the two patterns.
B. Graph the ordered pairs on a coordinate plane.
The Basic student is able to extend two numerical patterns with each pattern having its own rule. Form ordered pairs consisting of corresponding terms
from the two patterns.
The Below Basic student may be able to extend two numerical patterns with each pattern having its own rule with partial success. Form ordered pairs
consisting of corresponding terms from the two patterns with partial success.
NUMBER AND OPERATIONS IN BASE TEN
Understand the place value system.
5.NBT.C.1 Recognize that in a multi-digit number, a digit in one place represents 10 times as much as it represents in the place to its right and
of
what it represents in the place to its left.
In addition to Proficient, the Advanced student is able to:
• Recognize that given two different digits in a multi-digit number, one digit can represent a multiple of 100 times the digit two places to its right,
and a multiple of times the digit two places to its left. OR
• Extend reasoning about place value to go two or more places to the left and right.
The Proficient student is able to recognize that in a multi-digit number, a digit in one place represents 10 times as much as it represents in the place to its
right and
of what it represents in the place to its left.
The Basic student is able to recognize that in a multi-digit number, a digit in one place represents 10 times as much as it represents in the place to its right
and
of what it represents in the place to its left when given a visual model or representation.
The Below Basic student may be able to recognize that in a multi-digit number, a digit in one place represents 10 times as much as it represents in the
place to its right and
of what it represents in the place to its left with partial success, when given a visual model or representation.
5.NBT.C.2 Explain patterns in the number of zeros of the product when multiplying a number by powers of 10, and explain patterns in the placement of
the decimal point when a decimal is multiplied or divided by a power of 10. Use whole number exponents to denote powers of 10.
In addition to Proficient, the Advanced student is able to:
• Explain patterns in the placement of the decimal point when a decimal is multiplied or divided by a power of 10 and use whole number
exponents to denote powers of 10. OR
• Compare two powers of 10 expressed exponentially. OR
• Apply understanding of exponents to real-world examples of scientific notation.
The Proficient student is able to explain patterns in the number of zeros of the product when multiplying a number by powers of 10, and explain patterns
in the placement of the decimal point when a decimal is multiplied or divided by a power of 10. Use whole number exponents to denote powers of 10.
The Basic student is able to continue a pattern of a number multiplied and divided by a power of 10.
The Below Basic student may be able to continue a pattern of a number multiplied and divided by a power of 10 with partial success.
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5.NBT.C.3 Read, write, and compare decimals to thousandths.
5.NBT.C.3A Read and write decimals to thousandths using base-ten numerals, number names, and expanded form.
5.NBT.C.3B Compare two decimals to thousandths based on meanings of the digits in each place, using >, =, and < symbols.
In addition to Proficient, the Advanced student is able to:
• Relate decimals to real-world scenarios with scientific notation. OR
• Create numbers in different forms, order them, and justify reasoning for order. OR
• Apply what is known about comparing decimals to be able to analyze and justify errors in comparisons.
The Proficient student is able to:
A. Read and write decimals to thousandths using base-ten numerals, number names, and expanded form.
B. Compare two decimals to thousandths based on meanings of the digits in each place, using >, =, and < symbols.
The Basic student is able to:
A. Read and write decimals to thousandths using base-ten numerals, number names, and expanded form. OR
B. Compare two decimals to thousandths based on meanings of the digits in each place, using >, =, and < symbols.
The Below Basic student may be able to:
A. Read and write decimals to thousandths using base-ten numerals, number names, and expanded form with partial success. OR
B. Compare two decimals to thousandths based on meanings of the digits in each place, using >, =, and < symbols with partial success.
5.NBT.C.4 Use place value understanding to round decimals to any place to a given place.
Assessment Boundary: Limit place value to the thousandths.
In addition to Proficient, the Advanced student is able to:
• Explain how to use the digits in multi-digit decimal numbers to round numbers to any place. OR
• Use an example of rounding decimals and explain how it is helpful in computation. OR
• Justify the appropriate place value to which the student would round in a given situation.
Assessment Boundary: Limit place value to the thousandths.
The Proficient student is able to use place value understanding to round decimals to any place to a given place.
Assessment Boundary: Limit place value to the thousandths.
The Basic student is able to use place value understanding to round decimals to any place when provided a model such as a number line with benchmark
numbers.
Assessment Boundary: Limit place value to the thousandths.
The Below Basic student may be able to use place value understanding to round decimals to any place with partial success, when provided a model such
as a number line with benchmark numbers.
Assessment Boundary: Limit place value to the thousandths.
Perform operations with multi-digit whole numbers and with decimals to hundredths.
5.NBT.D.5 Multiply multi-digit whole numbers using place value strategies including the standard algorithm.
In addition to Proficient, the Advanced student is able to justify how the various place value strategies for multiplication relate to the standard algorithm.
The Proficient student is able to multiply multi-digit whole numbers using place value strategies including the standard algorithm.
The Basic student is able to multiply multi-digit whole numbers using place value strategies.
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The Below Basic student may be able to multiply multi-digit whole numbers using place value strategies with partial success.
5.NBT.D.6 Find whole-number quotients with up to four-digit dividends and two-digit divisors, using strategies based on place value, the properties of
multiplication, and/or the relationship between multiplication and division, including the standard algorithm. Use appropriate models to Illustrate and
explain the calculation, such as equations, rectangular arrays, and/or area models.
Assessment Boundary: The standard algorithm for division will not be assessed.
In addition to Proficient, the Advanced student is able to:
• Create a real-world situation that can be modeled using a given division problem. OR
• Find quotients and remainders with up to four-digit dividends and two-digit divisors using more than one model or strategy and defend the
efficiency of the strategy used. OR
• Justify how the various place value strategies for division relate to the standard algorithm.
Assessment Boundary: The standard algorithm for division will not be assessed.
The Proficient student is able to find whole-number quotients with up to four-digit dividends and two-digit divisors, using strategies based on place value,
the properties of multiplication, and/or the relationship between multiplication and division, including the standard algorithm. Use appropriate models to
Illustrate and explain the calculation, such as equations, rectangular arrays, and/or area models. Assessment Boundary: The standard algorithm for
division will not be assessed.
The Basic student is able to:
• Find whole-number quotients with up to four-digit dividends and two-digit divisors when given a partially completed model. OR
• Find quotients with up to four-digit dividends and one-digit divisors, using strategies based on place value, the properties of multiplication,
and/or the relationship between multiplication and division. Use appropriate models to Illustrate and explain the calculation, such as equations,
rectangular arrays, and/or area models.
Assessment Boundary: The standard algorithm for division will not be assessed.
The Below Basic student may be able to:
• Find whole-number quotients with up to four-digit dividends and two-digit divisors with partial success when given a partially completed model.
OR
• Find quotients with up to four-digit dividends and one-digit divisors, using strategies based on place value, the properties of multiplication,
and/or the relationship between multiplication and division with partial success. Use appropriate models to Illustrate and explain the calculation,
such as equations, rectangular arrays, and/or area models with partial success.
Assessment Boundary: The standard algorithm for division will not be assessed.
*5.NBT.D.7 Add, subtract, multiply, and divide decimals to hundredths using concrete models or drawings, and strategies based on place value,
properties of operations, and/or the relationship between addition and subtraction; Relate the strategy to a written method and explain the reasoning
used.
In addition to Proficient, the Advanced student is able to:
• Explain patterns in the base-ten system when finding sums and differences with decimals and how it is consistent with the standard algorithm.
OR
• Investigate and draw conclusions about decimal placement when finding products and quotients with decimals.
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The Proficient student is able to add, subtract, multiply, and divide decimals to hundredths using concrete models or drawings, and strategies based on
place value, properties of operations, and/or the relationship between addition and subtraction; Relate the strategy to a written method and explain the
reasoning used.
The Basic student is able to add, subtract, multiply, and divide decimals to hundredths when given concrete models or drawings, and strategies based on
place value, properties of operations, and/or the relationship between addition and subtraction.
The Below Basic student may be able to add, subtract, multiply, and divide decimals to hundredths with partial success when given concrete models or
drawings, and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction.
NUMBER AND OPERATIONS - FRACTIONS
Use equivalent fractions as a strategy to add and subtract fractions.
5.NF.E.1 Add and subtract fractions with unlike denominators (including mixed numbers) by replacing given fractions with equivalent fractions in such
a way as to produce an equivalent sum or difference of fractions with like denominators.
In addition to Proficient, the Advanced student is able to add or subtract three or more fractions with unlike denominators (including mixed numbers) that
require regrouping by replacing the given fractions with equivalent fractions and justify the denominators chosen.
The Proficient student is able to add and subtract fractions with unlike denominators (including mixed numbers) by replacing given fractions with
equivalent fractions in such a way as to produce an equivalent sum or difference of fractions with like denominators.
The Basic student is able to add and subtract fractions with unlike denominators by replacing given fractions with equivalent fractions in such a way as to
produce an equivalent sum or difference of fractions with like denominators when given a visual model.
The Below Basic student may be able to add and subtract fractions with unlike denominators by replacing given fractions with equivalent fractions in such
a way as to produce an equivalent sum or difference of fractions with like denominators, with partial success, when given a visual model.
*5.NF.E.2 Solve word problems involving addition and subtraction of fractions referring to the same whole, including cases of unlike denominators,
e.g., by using visual fraction models or equations to represent the problem. Use benchmark fractions and number sense of fractions to estimate
mentally and assess the reasonableness of answers.
In addition to Proficient, the Advanced student is able to solve real-world problems involving addition or subtraction with at least 3 or more fractions with
unlike denominators (including mixed numbers) by replacing given fractions with equivalent fractions in such a way as to produce an equivalent sum or
difference of fractions with like denominators. Use benchmark fractions and number sense of fractions to estimate mentally and assess the
reasonableness of answers.
The Proficient student is able to solve word problems involving addition and subtraction of fractions referring to the same whole, including cases of unlike
denominators, e.g., by using visual fraction models or equations to represent the problem. Use benchmark fractions and number sense of fractions to
estimate mentally and assess the reasonableness of answers.
The Basic student is able to solve one-step mathematical problems involving addition and subtraction of fractions referring to the same whole, including
cases of unlike denominators, when given visual fraction models or equations to represent the problem. Use benchmark fractions and number sense of
fractions to estimate mentally and assess the reasonableness of answers.
The Below Basic student may be able to solve one-step mathematical problems involving addition and subtraction of fractions referring to the same
whole, including cases of unlike denominators with partial success, when given visual fraction models or equations to represent the problem. Use
benchmark fractions and number sense of fractions to estimate mentally and assess the reasonableness of answers.
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Apply and extend previous understandings of multiplication and division to multiply and divide fractions.
5.NF.F.3 Interpret a fraction as division of the numerator by the denominator (a/b = a ÷ b). Solve word problems involving division of whole numbers
leading to answers in the form of fractions or mixed numbers by using visual fraction models or equations to represent the problem.
In addition to Proficient, the Advanced student is able to create a real-world situation and visual model to demonstrate understanding between division of
whole numbers and fractions.
The Proficient student is able to interpret a fraction as division of the numerator by the denominator (
= ÷ ). Solve word problems involving division
of whole numbers leading to answers in the form of fractions or mixed numbers by using visual fraction models or equations to represent the problem.
The Basic student is able to interpret a fraction as division of the numerator by the denominator (
= ÷ ) when given a visual model. Solve word
problems involving division of whole numbers leading to answers in the form of fractions or mixed numbers when given visual fraction models or equations
to represent the problem.
The Below Basic student may be able to interpret a fraction as division of the numerator by the denominator (
= ÷ ) with partial success, when given
a visual model. Solve word problems involving division of whole numbers leading to answers in the form of fractions or mixed numbers, with partial
success, when given visual fraction models or equations to represent the problem.
5.NF.F.4 Extend the concept of multiplication to multiply a fraction or whole number by a fraction.
5.NF.F.4A Recognize the relationship between multiplying fractions and finding the areas of rectangles with fractional side lengths.
5.NF.F.4B Interpret multiplication of a fraction by a whole number and a whole number by a fraction and compute the product.
5.NF.F.4C Interpret multiplication in which both factors are fractions less than one and compute the product.
In addition to Proficient, the Advanced student is able to:
A. Draw a visual representation to show understanding of how to multiply a fraction by a fraction OR find a missing fractional side length when
given an area and one side length of a rectangle.
B. Estimate the result of multiplying a whole number by a fraction less than one, by a fraction equal to one, or by a fraction greater than one and
justify the estimation with a visual model or description. Predict the sizes of the factors based on the product without performing the indicated
multiplication.
The Proficient student is able to extend the concept of multiplication to multiply a fraction or whole number by a fraction.
A. Recognize the relationship between multiplying fractions and finding the areas of rectangles with fractional side lengths.
B. Interpret multiplication of a fraction by a whole number and a whole number by a fraction and compute the product.
C. Interpret multiplication in which both factors are fractions less than one and compute the product.
The Basic student is able to:
A. Recognize the relationship between multiplying fractions and finding the areas of rectangles with fractional side lengths when given a visual
model.
B. Interpret multiplication of a fraction by a whole number and a whole number by a fraction and compute the product when given a visual model.
C. Interpret multiplication in which both factors are fractions less than one and compute the product when given a visual model.
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The Below Basic student may be able to:
A. Recognize the relationship between multiplying fractions and finding the areas of rectangles with fractional side lengths, with partial success,
when given a visual model.
B. Interpret multiplication of a fraction by a whole number and a whole number by a fraction and compute the product, with partial success, when
given a visual model.
C. Interpret multiplication in which both factors are fractions less than one and compute the product, with partial success, when given a visual
model.
5.NF.F.5 Justify the reasonableness of a product when multiplying with fractions.
5.NF.F.5A Estimate the size of the product based on the size of the two factors.
5.NF.F.5B Explain why multiplying a given number by a number greater than 1 (improper fractions, mixed numbers, whole numbers) results in a product larger
than the given number.
5.NF.F.5C Explain why multiplying a given number by a fraction less than 1 results in a product smaller than the given number.
5.NF.F.5D Explain why multiplying the numerator and denominator by the same number has the same effect as multiplying the fraction by 1.
In addition to Proficient, the Advanced student is able to estimate the size of a factor based on the product and the other factor without performing the
indicated multiplication and justify the reasonableness of a product when multiplying with fractions by creating a visual model or explanation.
The Proficient student is able to justify the reasonableness of a product when multiplying with fractions.
A. Estimate the size of the product based on the size of the two factors.
B. Explain why multiplying a given number by a number greater than 1 (improper fractions, mixed numbers, whole numbers) results in a product
larger than the given number.
C. Explain why multiplying a given number by a fraction less than 1 results in a product smaller than the given number.
D. Explain why multiplying the numerator and denominator by the same number has the same effect as multiplying the fraction by 1.
The Basic student is able to estimate the size of the product based on the size of the two factors.
The Below Basic student may be able to estimate the size of the product based on the size of the two factors with partial success.
*5.NF.F.6 Solve real-world problems involving multiplication of fractions and mixed numbers by using visual fraction models or equations to represent
the problem.
In addition to Proficient, the Advanced student is able to:
• Solve multi-step real-world problems involving multiplication of fractions including mixed numbers with multiple strategies or representations.
OR
• Create and solve real-world problems involving multiplication of fractions and mixed numbers by creating visual fraction models or equations to
represent the problem.
The Proficient student is able to solve real-world problems involving multiplication of fractions and mixed numbers by using visual fraction models or
equations to represent the problem.
The Basic student is able to solve real-world problems involving multiplication of fractions and whole numbers when given visual fraction models or
equations to represent the problem.
The Below Basic student may be able to solve real-world problems involving multiplication of fractions and whole numbers, with partial success, when
given visual fraction models or equations to represent the problem.
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5.NF.F.7 Extend the concept of division to divide unit fractions and whole numbers by using visual fraction models and equations.
5.NF.F.7A Interpret division of a unit fraction by a non-zero whole number and compute the quotient.
5.NF.F.7B Interpret division of a whole number by a unit fraction and compute the quotient.
5.NF.F.7C Solve real-world problems involving division of unit fractions by non-zero whole numbers and division of whole numbers by unit fractions by using
visual fraction models and equations to represent the problem.
In addition to Proficient, the Advanced student is able to:
• Create and solve a real-world problem and justify how the model/equation relates to the problem. OR
• Solve real-world problems involving division of unit fractions by non-zero whole numbers and division of whole numbers using multiple
representations.
The Proficient student is able to extend the concept of division to divide unit fractions and whole numbers by using visual fraction models and equations.
A. Interpret division of a unit fraction by a non-zero whole number and compute the quotient.
B. Interpret division of a whole number by a unit fraction and compute the quotient.
C. Solve real-world problems involving division of unit fractions by non-zero whole numbers and division of whole numbers by unit fractions by
using visual fraction models and equations to represent the problem.
The Basic student is able to:
A. Interpret division of a unit fraction by a non-zero whole number and compute the quotient when given a visual model.
B. Interpret division of a whole number by a unit fraction and compute the quotient when given a visual model.
The Below Basic student may be able to:
A. Interpret division of a unit fraction by a non-zero whole number and compute the quotient with partial success when given a visual model.
B. Interpret division of a whole number by a unit fraction and compute the quotient with partial success when given a visual model.
MEASUREMENT AND DATA
Convert like measurement units within a given measurement system.
5.MD.G.1 Understand a coordinate system.
In addition to Proficient, the Advanced student is able to:
• Convert different-sized standard measurement units, within a given measurement system, requiring multiple conversions. Solve real-world
problems with multiple steps involving these conversions. OR
• Create and solve multi-step real-world problems by converting different-sized standard measurement units within a given measurement system.
OR
• Convert different-sized standard measurement units within multiple measurement systems to solve real-world problems.
The Proficient student is able to solve multi-step real-world problems by converting among different-sized standard measurement units within a given
measurement system.
The Basic student is able to solve multi-step real-world problems by converting among different-sized standard measurement units, within a given
measurement system, when given the conversion equivalence.
The Below Basic student may be able to solve multi-step real-world problems by converting among different-sized standard measurement units, within a
given measurement system, with partial success, when given the conversion equivalence.
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Represent and interpret data.
5.MD.H.2 Make a line plot to display a data set of measurements in fractions of a unit (
, , ). Use operations on fractions to solve problems involving
information presented in line plots.
In addition to Proficient, the Advanced student is able to:
• Predict ways in which operations with fractions, and fractions on a line plot, would change if the data set were changed. AND
• Use three or more operations with fractions to solve problems involving information presented in line plots.
The Proficient student is able to make a line plot to display a data set of measurements in fractions of a unit (
, , ). Use operations on fractions to solve
problems involving information presented in line plots.
The Basic student is able to:
• Make a line plot to display a data set of measurements in fractions of a unit (
, , ). OR
• Use operations on fractions to solve problems involving information presented in line plots.
The Below Basic student may be able to:
• Make a line plot to display a data set of measurements in fractions of a unit (
, , ) with partial success. OR
• Use operations on fractions to solve problems involving information presented in line plots with partial success.
Geometric measurement: understand concepts of volume and relate volume to multiplication and to addition.
5.MD.I.3 Recognize volume as an attribute of three-dimensional figures and understand concepts of volume measurement such as "unit cube" and a
volume of n cubic units.
In addition to Proficient, the Advanced student is able to explain how to find missing dimension(s) when given volume and justify.
The Proficient student is able to recognize volume as an attribute of three-dimensional figures and understand concepts of volume measurement such as
"unit cube" and a volume of n cubic units.
The Basic student is able to identify the use of volume and the appropriate measurement for a given situation or model.
The Below Basic student may be able to identify the use of volume and the appropriate measurement for a given situation or model with partial success.
5.MD.I.4 Measure volumes by counting unit cubes, using cubic cm, cubic in, cubic ft., and improvised units.
In addition to Proficient, the Advanced student is able to:
• Use an efficient counting strategy (not counting one unit at a time) and determine appropriate measurement units. OR
• Explain how you would find volume in a compound figure.
The Proficient student is able to measure volumes by counting unit cubes, using cubic cm, cubic in, cubic ft., and improvised units.
The Basic student is able to measure volumes by counting unit cubes, using cubic cm, cubic in, cubic ft, and improvised units when given a partially
completed visual model.
The Below Basic student may be able to measure volumes by counting unit cubes, using cubic cm, cubic in, cubic ft, and improvised units, with partial
success, when given a partially completed visual model.
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*5.MD.I.5 Relate volume to the operations of multiplication and solve real-world and mathematical problems involving volume.
5.MD.I.5A Find the volume of a right rectangular prism with whole number dimensions by multiplying them. Show that this volume is the same as when
counting unit cubes.
5.MD.I.5B Find volumes of right rectangular prisms with whole-number edge lengths in the context of solving real-world and mathematical problems given the
formulas = ()()() and = ()() for rectangular prisms.
In addition to Proficient, the Advanced student is able to:
• Solve real-world problems by finding volumes of solid figures composed of two non-overlapping right rectangular prisms by adding the volumes
of the non-overlapping parts. OR
• Find the base or height when given volume and some of the dimensions. OR
• Find multiple possible dimensions of a right rectangular prism with a given volume.
The Proficient student is able to relate volume to the operations of multiplication and solve real-world and mathematical problems involving volume.
A. Find the volume of a right rectangular prism with whole number dimensions by multiplying them. Show that this volume is the same as when
counting unit cubes.
B. Find volumes of right rectangular prisms with whole-number edge lengths in the context of solving real-world and mathematical problems given
the formulas = ()()() and = ()() for rectangular prisms.
The Basic student is able to:
A. Find the volume of a right rectangular prism with whole number dimensions by multiplying them. Show that this volume is the same as when
counting unit cubes. OR
B. Find volumes of right rectangular prisms with whole-number edge lengths in the context of solving real-world and mathematical problems given
the formulas = ()()() and = ()() for rectangular prisms.
The Below Basic student may be able to:
A. Find the volume of a right rectangular prism with whole number dimensions by multiplying them with partial success. OR
B. Find volumes of right rectangular prisms with whole-number edge lengths in the context of solving real-world and mathematical problems given
the formulas = ()()() and = ()() for rectangular prisms with partial success.
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GEOMETRY
Graph points on the coordinate plane to solve real-world and mathematical problems.
5.G.J.1 Understand a coordinate system.
5.G.J.1A The x- and y- axes are perpendicular number lines that intersect at 0 (the origin).
5.G.J.1B Any point on the coordinate plane can be represented by its coordinates.
5.G.J.1C The first number in an ordered pair is the x-coordinate and represents the horizontal distance from the origin.
5.G.J.1D The second number in an ordered pair is the y-coordinate and represents the vertical distance from the origin.
In addition to Proficient, the Advanced student is able to show understanding of the coordinate system in real-world situations (first quadrant only).
The Proficient student is able to understand a coordinate system.
A. The x- and y- axes are perpendicular number lines that intersect at 0 (the origin).
B. Any point on the coordinate plane can be represented by its coordinates.
C. The first number in an ordered pair is the x-coordinate and represents the horizontal distance from the origin.
D. The second number in an ordered pair is the y-coordinate and represents the vertical distance from the origin.
The Basic student is able to identify the components of a coordinate system.
A. The x- and y- axes are perpendicular number lines that intersect at 0 (the origin).
B. Any point on the coordinate plane can be represented by its coordinates.
C. The first number in an ordered pair is the x-coordinate and represents the horizontal distance from the origin.
D. The second number in an ordered pair is the y-coordinate and represents the vertical distance from the origin.
The Below Basic student may be able to identify the components of a coordinate system with partial success.
A. The x- and y- axes are perpendicular number lines that intersect at 0 (the origin).
B. Any point on the coordinate plane can be represented by its coordinates.
C. The first number in an ordered pair is the x-coordinate and represents the horizontal distance from the origin.
D. The second number in an ordered pair is the y-coordinate and represents the vertical distance from the origin.
5.G.J.2 Plot and interpret points in the first quadrant of the coordinate plane to represent real-world and mathematical situations.
In addition to Proficient, the Advanced student is able to describe the x- and y-coordinate's position when mathematical operations are performed on the
coordinates in any of the four quadrants.
The Proficient student is able to plot and interpret points in the first quadrant of the coordinate plane to represent real-world and mathematical situations.
The Basic student is able to locate a point in the first quadrant using an ordered pair.
The Below Basic student may be able to locate a point in the first quadrant using an ordered pair with partial success.
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Classify two-dimensional figures into categories based on their properties.
5.G.K.3 Understand that attributes belonging to a category of two-dimensional figures also belong to all subcategories of that category.
Assessment Boundary: Use polygons only.
In addition to Proficient, the Advanced student is able to formulate logical arguments to show that attributes belonging to a category of two-dimensional
figures also belong to all subcategories of that category.
Assessment Boundary: Use polygons only.
The Proficient student is able to understand that attributes belonging to a category of two-dimensional figures also belong to all subcategories of that
category.
Assessment Boundary: Use polygons only.
The Basic student is able to classify two-dimensional figures into basic subcategories.
Assessment Boundary: Use polygons only.
The Below Basic student may be able to classify two-dimensional figures into basic subcategories with partial success.
Assessment Boundary: Use polygons only.
5.G.K.4 Classify polygons in a hierarchy based on properties.
In addition to Proficient, the Advanced student is able to construct polygons according to given attributes.
The Proficient student is able to classify polygons in a hierarchy based on properties.
The Basic student is able to:
• Classify two-dimensional figures into two basic subcategories. OR
• Identify specific properties of the subcategories.
The Below Basic student may be able to:
• Classify two-dimensional figures into two basic subcategories with partial success. OR
• Identify specific properties of the subcategories with partial success.
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Companion document to the 2018 Mathematics Content Standards
Grade 6 Math Content & Performance Standards & PLDs
GRADE 6 MATH PRACTICES
MP1 Make sense of problems and persevere in solving
them.
6.MP.1 In grade 6, students solve problems involving ratios and rates and
discuss (verbally or in writing) how they solve them. Students analyze the
problem (including what is given, not given, and what is being asked), identify
what strategies are needed, recognize multiple pathways to a solution, and
make an initial attempt to solve the problem. Students analyze the result for
validity and refine strategies if necessary.
MP2 Reason abstractly and quantitatively.
6.MP.2 Students recognize a wide variety of real-world contexts through the
use of real numbers and variables in mathematical expressions, equations,
and inequalities. Students begin to contextualize to understand the meaning
of the number or variable as it relates to the problem.
MP3 Construct viable arguments and critique the
reasoning of others.
6.MP.3 Students begin to contextualize to understand the meaning of the
number or variable as it relates to the problem. They make conjectures,
explore validity, reason mathematically, justify, evaluate their own thinking.
MP4 Model with mathematics.
6.MP.4 Students can clearly show their work by using diagrams, words,
symbols or pictures. They are able to identify important quantities in a
practical situation and map their relationships using tools such as, diagrams,
two-way tables, graphs, flowcharts or formulas. They can recognize and
analyze those relationships mathematically to draw conclusions. They can
interpret their mathematical results of problems involving non-negative
rational numbers in the context of the situation and reflect on whether the
results make sense.
MP5 Use appropriate tools strategically.
6.MP.5 Students consider available tools (including estimation, concrete
models, and technology), and decide when certain tools might be helpful.
They choose the representation (table, graph, equation, words) that best
suits the problem. Students use concrete models to develop insight into
ratios and other concepts. Students extend this insight to more abstract
representations, including pictures and symbols. Students understand the
limitations of each tool. Tools might include: unifix cubes, fraction bars, base-
ten blocks, number lines, graph paper, calculator, paper and pencil, and
others.
MP6 Attend to precision.
6.MP.6 Students continue to refine their mathematical communication and
reasoning skills by using clear language in their discussions with others.
Students define variables, including their relationship, specify units of
measure, and label each axis accurately. Students use appropriate
terminology when referring to rates, ratios, geometric figures, data displays,
and components of expressions, equations or inequalities. Students use
appropriate symbols, labels, and units of measure when solving problems
with calculations that are accurate and efficient. The answer to the problem
matches what was asked in the problem.
MP7 Look for and make use of structure.
6.MP.7 Students routinely seek patterns or structure to model and solve
problems. They recognize that patterns exist in ratio tables. Students notice
patterns and identify strategies for creating equivalent expressions. Students
identify complicated expressions or figures as compositions of simple parts.
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MP8 Look for and express regularity in repeated reasoning.
6.MP.8 Students use repeated reasoning to understand algorithms and make generalizations about patterns. They construct examples and models that confirm
their generalization. They develop short cuts and check for reasonableness of answers. Students ask questions such as, "How would we verify that?" and "How is
this similar to patterns with whole numbers?"
RATIOS AND PROPORTIONAL RELATIONSHIPS
Understand ratio concepts and use ratio reasoning to solve problems.
6.RP.A.1 Understand the concept of a ratio and use ratio language to describe a ratio relationship between two quantities.
In addition to Proficient, the Advanced student is able to manipulate and make connections between different representations for ratio relationships.
The Proficient student is able to interpret a ratio relationship between two quantities, including part-to-part and part-to-whole.
The Basic student is able to write a ratio relationship between two quantities.
The Below Basic student may be able to identify a ratio using a mathematical or verbal representation.
6.RP.A.2 Understand the concept of a unit rate
associated with a ratio : with , and use rate language in the context of a ratio relationship.
In addition to Proficient, the Advanced student is able to interpret a unit rate from a visual representation using unit rate language.
Assessment Boundary: Do not use complex fractions or negatives.
The Proficient student is able to write a unit rate to compare two quantities using rational numbers and use unit rate language to describe two quantities in
the context of a ratio relationship.
Assessment Boundary: Do not use complex fractions or negatives.
The Basic student is able to write a unit rate to compare two quantities using whole numbers.
Assessment Boundary: Do not use complex fractions or negatives.
The Below Basic student may be able to identify unit rates.
Assessment Boundary: Do not use complex fractions or negatives.
*6.RP.A.3 Use ratio and rate reasoning to solve real-world and mathematical problems.
6.RP.A.3A Make tables of equivalent ratios relating quantities with whole-number measurements, find missing values in the tables, and plot the pairs of values
on the coordinate plane. Use tables to compare ratios.
6.RP.A.3B Solve unit rate problems including those involving unit pricing and constant speed.
6.RP.A.3C Understand that a percentage is a rate per 100 and use this to solve problems involving wholes, parts, and percentages.
6.RP.A.3D Use ratio reasoning to convert measurement units; convert units appropriately when multiplying or dividing quantities.
In addition to Proficient, the Advanced student is able to:
A. Extend ratio and rate reasoning beyond what is displayed in a table or graph.
B. Solve unit rate problems that require determining a unit rate with a positive rational numerator and whole number denominator.
C. In mathematical and real-world contexts solve two-step problems involving wholes, parts, and percentages.
D. Use ratio reasoning to convert measurement units and transform units appropriately when multiplying and dividing in two-step problems.
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The Proficient student is able to:
A. Make tables of equivalent ratios relating quantities with whole number measurements and plot the pairs of values on the coordinate plane. Use
tables to compare ratios.
B. Solve unit rate problems with whole number measurements including those involving unit pricing and constant speed.
C. In mathematical and real-world contexts solve one-step problems involving wholes, parts, and percentages.
D. Use ratio reasoning to convert measurement units and to transform units appropriately when multiplying or dividing quantities in one-step
problems.
The Basic student is able to:
A. Make a table of equivalent ratios relating quantities with whole number measurements.
B. Solve unit rate problems given the unit rate with whole number measurements.
C. Solve one-step problems involving wholes, parts, or percentages.
D. Use ratio reasoning in a one-step problem to convert measurement units within the same system.
The Below Basic student may be able to:
A. Find missing values in a table.
B. Identify unit rate with whole number measurements.
C. Identify the percent of a quantity as a rate per hundred.
D. Identify ratio relationships using measurement units within the same system.
THE NUMBER SYSTEM
Apply and extend previous understandings of multiplication and division to divide fractions by fractions.
*6.NS.B.1 Interpret and compute quotients of fractions, and solve word problems involving division of fractions by fractions by using visual fraction
models and equations to represent the problem.
In addition to Proficient, the Advanced student is able to interpret and compute quotients of fractions, and solve word problems involving division of
fractions by fractions by using visual fraction models and equations to represent the problem in a real-world context.
The Proficient student is able to interpret and compute quotients of fractions, and solve word problems involving division of fractions by fractions by using
visual fraction models and equations to represent the problem.
The Basic student is able to compute quotients of a fraction by a unit fraction. Students solve problems in mathematical contexts involving division of a
fraction by a unit fraction.
The Below Basic student may be able to solve problems in mathematical contexts involving division of a whole number by a unit fraction.
6.NS.B.2 Divide multi-digit numbers using efficient and generalizable procedures including, but not limited to the standard algorithm.
Assessment Boundary: Use up to 5-digit dividends, 2-digit divisors.
In addition to Proficient, the Advanced student is able to divide multi-digit numbers with fractional remainders using efficient and generalizable procedures
including, but not limited to the standard algorithm and explain the reasonableness of the result.
Assessment Boundary: Use up to 5-digit dividends, 2-digit divisors.
The Proficient student is able to divide multi-digit numbers using efficient and generalizable procedures including, but not limited to the standard
algorithm.
Assessment Boundary: Use up to 5-digit dividends, 2-digit divisors.
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The Basic student is able to divide three-digit or four-digit dividends by two-digit divisors resulting in no remainder using efficient and generalizable
procedures including, but not limited to the standard algorithm.
Assessment Boundary: Use up to 5-digit dividends, 2-digit divisors.
The Below Basic student may be able to divide two-digit dividends by one-digit divisors resulting in no remainder using efficient and generalizable
procedures, including, but not limited to the standard algorithm.
Assessment Boundary: Use up to 5-digit dividends, 2-digit divisors.
Compute fluently with multi-digit numbers and find common factors and multiples.
*6.NS.C.3 Add, subtract, multiply, and divide manageable multi-digit decimals using efficient and generalizable procedures including, but not limited to
the standard algorithm for each operation.
In addition to Proficient, the Advanced student is able to add, subtract, multiply, and divide multi-digit decimals using efficient and generalizable
procedures including, but not limited to the standard algorithm for each operation and explain the reasonableness of the answer.
Assessment Boundary: Limit decimals in the given values to the hundredths place.
The Proficient student is able to add, subtract, multiply, and divide multi-digit decimals using efficient and generalizable procedures including, but not
limited to the standard algorithm for each operation.
Assessment Boundary: Limit decimals in the given values to the hundredths place.
The Basic student is able to add, subtract, multiply, and divide decimals to tenths using efficient and generalizable procedures including, but not limited to
the standard algorithm for each operation.
The Below Basic student may be able to add, subtract, multiply, or divide decimals to tenths using efficient and generalizable procedures including, but
not limited to the standard algorithm for each operation.
6.NS.C.4 Find common factors and multiples using two whole numbers.
6.NS.C.4A Find the greatest common factor of two whole numbers less than or equal to 100.
6.NS.C.4B Find the least common multiple of two whole numbers less than or equal to 12.
6.NS.C.4C Use the Distributive Property to express a sum of two whole numbers 1–100 with a common factor as a multiple of a sum of two whole numbers
with no common factor.
In addition to Proficient, the Advanced student is able to find common factors and multiples using two whole numbers.
A. Find the greatest common factor of multiple whole numbers less than or equal to 100.
B. Find the least common multiple of three whole numbers less than or equal to 12.
C. Explain why two given expressions written in factored and distributed form are equivalent using appropriate mathematical language.
The Proficient student is able to find common factors and multiples using two whole numbers.
A. Find the greatest common factor of two whole numbers less than or equal to 100.
B. Find the least common multiple of two whole numbers less than or equal to 12.
C. Use the Distributive Property to express a sum of two whole numbers 1–100 with a common factor as a multiple of a sum of two whole numbers
with no common factor.
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The Basic student is able to find common factors and multiples using two whole numbers.
A. Find the greatest common factor of two whole numbers less than or equal to 24.
B. Find the least common multiple of two prime numbers less than or equal to 12.
C. Use the Distributive Property to express a sum of two whole numbers 1–24 with a common factor as a multiple of a sum of two whole numbers
with no common factor.
The Below Basic student may be able to find common factors and multiples using two whole numbers.
A. Given a visual model, identify the greatest common factor of two whole numbers less than or equal to 24.
B. Given multiples, identify the least common multiple of two whole numbers less than or equal to 12.
C. Given the common factors, use the Distributive Property to express a sum of two whole numbers 1–24 with a common factor as a multiple of a
sum of two whole numbers with no common factor.
Apply and extend previous understandings of numbers to the system of rational numbers.
6.NS.D.5 Understand that positive and negative numbers are used together to describe quantities having opposite directions or values and use them to
represent quantities in real-world contexts, explaining the meaning of 0 in each situation.
In addition to Proficient, the Advanced student is able to recognize patterns and make generalizations about characteristics of positive and negative
numbers in real-world contexts (may use any rational number, including fractions and decimals).
The Proficient student is able to understand that positive and negative integers are used together to describe quantities having opposite directions or
values and use them to represent quantities in real-world contexts, explaining the meaning of 0 in each situation.
The Basic student is able to place integers on the number line. In a given situation (e.g., elevation, sea level), student is able to determine the meaning of
0.
The Below Basic student may be able to place integers on the number line (with whole-number increments), extending the counting pattern to integers.
6.NS.D.6 Extend the understanding of the number line to include all rational numbers and apply this concept to the coordinate plane.
6.NS.D.6A Understand the concept of opposite numbers, including 0, and their relative locations on the number line.
6.NS.D.6B Understand that signs of numbers in ordered pairs indicate locations in quadrants of the coordinate plane; recognize that when two ordered pairs
differ only by signs, the locations of the points are related by reflections across one or both axes.
6.NS.D.6C Find and position rational numbers on a horizontal or vertical number line diagram; find and position pairs of rational numbers on a coordinate
plane.
In addition to Proficient, the Advanced student is able to:
A. Create a real-world situation to demonstrate opposite numbers.
B. Given one ordered pair, create a reflection across the x- and y-axis.
C. Find and position rational numbers on a horizontal or vertical number line diagram when the scale is not given; find and position pairs of rational
numbers on a coordinate plane when the scale is not given.
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The Proficient student is able to extend the understanding of the number line to include all rational numbers and apply this concept to the coordinate
plane.
A.
Represent the concept of opposite numbers, including 0, and their relative locations on the number line in real-world contexts.
B. Given ordered pairs that differ by a sign with x-coordinate and/or y-coordinate, recognize that the location of the points are related by reflections
across one or both axes.
C. Find and position rational numbers on a horizontal or vertical number line diagram when the scale is given; find and position pairs of rational
numbers on a coordinate plane when the scale is given.
The Basic student is able to:
A. Place two opposite numbers on a number line in a mathematical context.
B. Identify/determine the quadrant from a given coordinate.
C. Graph integer values on a horizontal or vertical number line and graph ordered pairs of integers in all four quadrants of a coordinate plane.
The Below Basic student may be able to:
A. Place the opposite number on a number line given one.
B-C. Plot whole-number ordered pairs in the first quadrant of a coordinate plane (with one-unit increments on both axes).
6.NS.D.7 Understand ordering and absolute value of rational numbers.
6.NS.D.7A Interpret statements of inequality as statements about the relative position of two numbers on a number line diagram.
6.NS.D.7B Write, interpret, and explain statements of order for rational numbers in real-world contexts.
6.NS.D.7C Understand the absolute value of a rational number as its distance from 0 on the number line; interpret absolute value as magnitude for a positive
or negative quantity in a real-world situation.
6.NS.D.7D Distinguish comparisons of absolute value from statements about order.
In addition to Proficient, the Advanced student is able to apply ordering and absolute value of rational numbers (including percent form).
A. Create statements of inequality about the relative position of two numbers on a (vertical or horizontal) number line diagram.
B. Write, interpret, and explain statements of order for rational numbers in real-world contexts.
C. Describe the absolute value of a rational number as its distance from 0 on the number line; interpret absolute value as magnitude for a positive
or negative quantity in a real-world situation.
D. Distinguish comparisons of absolute value from statements about order.
The Proficient student is able to understand ordering and absolute value of rational numbers.
A. Interpret statements of inequality as statements about the relative position of two numbers on a (vertical or horizontal) number line diagram.
B. Write, interpret, and explain statements of order for rational numbers in real-world contexts.
C. Describe the absolute value of a rational number as its distance from 0 on the number line; interpret absolute value as magnitude for a positive
or negative quantity in a real-world situation.
D. Distinguish comparisons of absolute value from statements about order.
The Basic student is able to understand ordering and absolute value of integers.
A. Interpret statements of inequality as statements about the relative position of two numbers on a (vertical or horizontal) number line diagram.
B. Write, interpret, and explain statements of order for integers in real-world contexts.
C. Describe the absolute value of an integer as its distance from 0 on the number line; interpret absolute value as magnitude for a positive or
negative quantity in a real-world situation.
D. Distinguish comparisons of absolute value from statements about order.
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The Below Basic student may be able to understand ordering and absolute value of whole numbers.
A. Interpret statements of inequality as statements about the relative position of two numbers on a (vertical or horizontal) number line diagram.
B. Write, interpret, and explain statements of order for whole numbers in real-world contexts.
C. Describe the absolute value of a whole number as its distance from 0 on the number line; interpret absolute value as magnitude for a positive
quantity in a real-world situation.
D. Distinguish comparisons of absolute value from statements about order.
*6.NS.D.8 Solve real-world and mathematical problems by graphing points in all four quadrants of the coordinate plane. Find distances between points
with the same first coordinate or the same second coordinate; relate absolute value and distance.
In addition to Proficient, the Advanced student is able to solve real-world and mathematical problems by graphing non-integer points in all four quadrants
of the coordinate plane. Find distances between non-integer points with the same first coordinate or the same second coordinate; relate absolute value
and distance.
The Proficient student is able to solve real-world and mathematical problems by graphing points in all four quadrants of the coordinate plane. Find
distances between points with the same first coordinate or the same second coordinate; relate absolute value and distance.
The Basic student is able to solve real-world and mathematical problems by graphing points in all four quadrants of the coordinate plane. Find distances
between points with the same first coordinate or the same second coordinate in the same quadrant.
The Below Basic student may be able to solve real-world and mathematical problems by graphing points in quadrant one of the coordinate plane. Find
distances between points with the same first coordinate or the same second coordinate.
EXPRESSIONS AND EQUATIONS
Apply and extend previous understandings of arithmetic to algebraic expressions.
6.EE.E.1 Write and evaluate numerical expressions involving whole-number exponents.
In addition to Proficient, the Advanced student is able to write and evaluate numerical multi-step expressions involving the Distributive Property and whole
number exponents.
The Proficient student is able to write and evaluate multi-step numerical expressions involving whole number exponents.
The Basic student is able to write and evaluate two-step numerical expressions involving one whole number exponent.
The Below Basic student may be able to write and evaluate one-step numerical expressions involving one whole number exponent.
*6.EE.E.2 Write, read, and evaluate expressions in which letters stand for numbers.
6.EE.E.2A Write expressions that record operations with numbers and with letters standing for numbers.
6.EE.E.2B Identify parts of an expression using mathematical terms (sum, difference, term, product, factor, quotient, coefficient, constant).
6.EE.E.2C Use Order of Operations to evaluate algebraic expressions using positive rational numbers and whole-number exponents. Include expressions that
arise from formulas in real-world problems.
In addition to Proficient, the Advanced student is able to write, read, and evaluate expressions in which letters stand for numbers.
A. Write algebraic expressions using grouping symbols.
B. Create an expression given mathematical terms (sum, difference, term, product, factor, quotient, coefficient, constant).
C. Use Order of Operations to justify the evaluation of algebraic expressions that contain positive rational numbers and whole-number exponents.
Include expressions that arise from formulas relative to sixth grade standards in real-world problems.
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The Proficient student is able to write, read, and evaluate expressions in which letters stand for numbers.
A. Write two-step algebraic expressions.
B. Identify parts of an expression using mathematical terms (sum, difference, term, product, factor, quotient, coefficient, constant).
C. Use Order of Operations to evaluate algebraic expressions using positive rational numbers and whole-number exponents. Include expressions
that arise from formulas relative to sixth grade standards in real-world problems.
The Basic student is able to write, read, and evaluate expressions in which letters stand for numbers.
A. Write one-step algebraic expressions.
B. Identify a part of an expression using mathematical terms (sum, difference, term, product, factor, quotient, coefficient, constant).
C. Use Order of Operations to evaluate algebraic expressions using whole numbers and whole-number exponents. Include expressions that arise
from formulas relative to sixth grade geometry standards in real-world problems.
The Below Basic student may be able to write, read, and evaluate expressions in which letters stand for numbers.
A. Identify an algebraic expression.
B. Identify the number of terms in a mathematical expression.
C. Use Order of Operations to evaluate algebraic expressions using whole numbers.
6.EE.E.3 Apply the properties of operations to generate equivalent expressions.
In addition to Proficient, the Advanced student is able to justify why two expressions are equivalent using the properties of operations.
The Proficient student is able to apply the properties of operations to generate equivalent expressions (Commutative Property, Associative Property,
Distributive Property, Additive Identity Property, Multiplicative Identity Property, and Zero Product Property).
The Basic student is able to identify the properties of operations in equivalent expressions (Commutative Property, Associative Property, Distributive
Property, Additive Identity Property, Multiplicative Identity Property, and Zero Product Property).
The Below Basic student may be able to identify the properties of operations in equivalent expressions (Commutative Property, Associative Property,
Distributive Property).
6.EE.E.4 Identify when two expressions are equivalent.
In addition to Proficient, the Advanced student is able to generate equivalent expressions.
The Proficient student is able to identify when two expressions are equivalent with positive rational numbers.
The Basic student is able to identify a model that matches an expression.
The Below Basic student may be able to identify when two numerical expressions are equivalent.
Reason about and solve one-variable equations and inequalities.
6.EE.F.5 Understand a solution to an equation or an inequality makes the equation or inequality true. Use substitution to determine whether a given
number in a specified set makes an equation or inequality true.
In addition to Proficient, the Advanced student is able to use substitution to determine whether a given rational number in a specified set makes a two-
step equation or a two-step inequality true.
The Proficient student is able to use substitution to determine whether a given non-negative rational number in a specified set makes a one-step equation
or a one-step inequality true.
The Basic student is able to use substitution to determine whether a given whole number in a specified set makes a one-step equation or a one-step
inequality true.
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The Below Basic student may be able to use substitution to determine whether a given whole number in a specified set makes a one-step equation true.
6.EE.F.6 Use variables to represent unknown numbers and write expressions when solving a real-world or mathematical problem.
In addition to Proficient, the Advanced student is able to use variables to represent unknown numbers and write multiple expressions to represent the
same real-world or mathematical problem and justify that they are equivalent.
The Proficient student is able to use variables to represent unknown numbers and write expressions to represent real-world or mathematical problems.
The Basic student is able to use variables to represent unknown numbers and write one-step expressions to represent real-world or mathematical
problems.
The Below Basic student may be able to identify a variable representing an unknown number.
*6.EE.F.7 Write and solve real-world and mathematical problems in the form of one-step, linear equations involving nonnegative rational numbers.
In addition to Proficient, the Advanced student is able to solve problems in both real-world and mathematical contexts by writing and solving equations in
the form of two-step, linear equations involving nonnegative rational numbers.
The Proficient student is able to solve problems in both real-world and mathematical contexts by writing and solving equations in the form of one-step,
linear equations involving nonnegative rational numbers.
The Basic student is able to solve problems in both real-world and mathematical contexts by writing and solving equations in the form of one-step, linear
equations involving whole numbers.
The Below Basic student may be able to solve equations in the form of one-step, linear equations involving whole numbers.
6.EE.F.8 Write an inequality of the form > or < to represent a constraint or condition in a real-world or mathematical problem. Recognize that
inequalities of the form > or < have infinitely many solutions; represent solutions of such inequalities on number line diagrams.
In addition to Proficient, the Advanced student is able to write an inequality of the form or (with the variable on either side) to represent a
constraint or condition in a real-world or mathematical problem. Recognize that inequalities have infinitely many solutions and represent solutions by
graphing on a number line.
The Proficient student is able to write an inequality of the form > or < (with the variable on either side) to represent a constraint or condition in a
real-world or mathematical problem. Recognize that inequalities have infinitely many solutions and represent solutions by graphing on a number line.
The Basic student is able to write an inequality of the form > or < (with the variable on the left) to represent a constraint or condition in a real-
world or mathematical problem. Represent solutions by graphing on a number line.
The Below Basic student may be able to recognize the value on an inequality of the form > or < (with the variable on the left) on number line
diagrams.
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Represent and analyze quantitative relationships between dependent and independent variables.
6.EE.G.9 Use variables to represent two quantities in a real-world problem that change in relationship to one another; write an equation to express one
quantity (dependent variable), in terms of the other quantity (independent variable). Analyze their relationship using graphs and tables, and relate these
to the equation.
In addition to Proficient, the Advanced student is able to create a real-world problem from a given table, graph, or equation and justify the relationship
between the dependent and independent variables.
The Proficient student is able to use variables to represent two quantities in a real-world problem that change in relationship to one another; write an
equation to express one quantity (dependent variable) in terms of the other quantity (independent variable). Analyze their relationship using graphs and
tables, and relate these to the equation.
The Basic student is able to describe the relationship between dependent and independent variables from a table or graph.
The Below Basic student may be able to complete a table from a given equation.
GEOMETRY
Solve real-world and mathematical problems involving area, surface area, and volume.
*6.G.H.1 Find area of right triangles, other triangles, special quadrilaterals, and polygons by composing into rectangles or decomposing into triangles
and other shapes; apply these techniques in the context of solving real-world and mathematical problems.
In addition to Proficient, the Advanced student is able to find a missing dimension given the area of a triangle or a special quadrilateral and all but one
dimension.
The Proficient student is able to find area of right triangles, other triangles, special quadrilaterals, and polygons by composing into rectangles or
decomposing into triangles and other shapes; apply these techniques in the context of solving real-world and mathematical problems.
The Basic student is able to find area of right triangles, other triangles, squares, and rectangles using their respective formulas or strategies.
The Below Basic student may be able to decompose a polygon into right triangles and rectangles.
6.G.H.2 Find the volume of a right rectangular prism with fractional edge lengths in the context of solving real-world and mathematical problems by
applying the formulas V = (l)(w)(h) and V = (B)(h), and label with appropriate units.
In addition to Proficient, the Advanced student is able to find a missing dimension when given the volume of a right rectangular prism with at least one
fractional edge length.
The Proficient student is able to find the volume of a right rectangular prism with fractional edge lengths in the context of solving real-world and
mathematical problems by applying the formulas = ()()() and = ()(), and label with appropriate units.
The Basic student is able to find the volume of a right rectangular prism with one fractional edge length in the context of solving real-world and
mathematical problems by applying the formulas = ()()() and = ()(), and label with appropriate units.
The Below Basic student may be able to find the volume of a right rectangular prism with whole number edge lengths.
6.G.H.3 Draw polygons in the coordinate plane given coordinates for the vertices; use coordinates to find the length of a side joining points with the
same first coordinate or the same second coordinate. Apply these techniques in the context of solving real-world and mathematical problems.
In addition to Proficient, the Advanced student is able to create and graph a polygon with a given perimeter, area, or dimensions on a coordinate plane;
justify the solution by listing ordered pairs and dimensions of the polygon.
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The Proficient student is able to draw polygons in the coordinate plane given coordinates for the vertices; use coordinates to find the length of a side
joining points with the same first coordinate or the same second coordinate. Apply these techniques in the context of solving real-world and mathematical
problems including perimeter and area.
The Basic student is able to draw a right triangle or a rectangle in the coordinate plane given coordinates for the vertices; use coordinates to find the
length of a side joining points with the same first coordinate or the same second coordinate. Apply these techniques in the context of mathematical
problems involving area.
The Below Basic student may be able to use coordinates to find the length of a side joining points with the same first coordinate or the same second
coordinate when given a rectangle in the coordinate plane with coordinates for the vertices. Apply these techniques in the context of mathematical
problems involving area.
6.G.H.4 Represent three-dimensional figures using nets made up of rectangles and triangles, and use the nets to find the surface area of these figures
in the context of solving real-world and mathematical problems.
In addition to Proficient, the Advanced student is able to:
• Create nets with different dimensions that represent right rectangular prisms with the same surface area. OR
• Compare nets with a fixed volume to determine maximum or minimum surface area.
The Proficient student is able to represent three-dimensional figures using nets made up of rectangles and triangles, and use the nets to find the surface
area of right triangular prisms, right rectangular prisms, and right rectangular pyramids (given lateral height) in the context of solving real-world and
mathematical problems.
The Basic student is able to represent three-dimensional figures using nets made up of rectangles and triangles, and use the nets to find the surface area
of right rectangular prisms.
The Below Basic student may be able to identify the net of a right prism.
STATISTICS AND PROBABILITY
Develop understanding of statistical variability.
6.SP.I.1 Recognize a statistical question as one that anticipates variability in the data related to the question and accounts for it in the answers.
In addition to Proficient, the Advanced student is able to create a statistical question and explain the variability in the answer.
The Proficient student is able to recognize a statistical question as one that anticipates variability in the data related to the question and accounts for it in
the answers.
The Basic student is able to change a question from a nonstatistical question to a statistical question.
The Below Basic student may be able to recognize a statistical question from a list of questions.
6.SP.I.2 Understand that a set of data collected to answer a statistical question has a distribution which can be described by its center, spread, and
overall shape.
In addition to Proficient, the Advanced student is able to make generalizations about a numerical data set collected to answer a statistical question that
has a distribution which can be described by its center (median, mode, or mean), spread (range or interquartile range), and overall shape (symmetry).
The Proficient student is able to show that a visual representation of a set of data collected to answer a statistical question has a distribution which can be
described by its center (median, mode, or mean), spread (range or interquartile range), and overall shape (symmetry).
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The Basic student is able to show that a visual representation of a set of data collected to answer a statistical question has a distribution which can be
described by its spread (range or interquartile range) and overall shape (symmetry).
The Below Basic student may be able to show that a visual representation of a set of data collected to answer a statistical question has a distribution
which can be described by its overall shape (symmetry).
6.SP.I.3 Recognize that a measure of center for a numerical data set summarizes all of its values with a single number, while a measure of variation
describes how its values vary with a single number.
In addition to Proficient, the Advanced student is able to:
• Create questions that could be answered from the measures of center or the measures of variation. OR
• Determine the kinds of questions that can be answered using the measures of center and the measures of variation.
The Proficient student is able to compare and contrast the measures of center for a numerical data set that summarizes all of its values with a single
number and the measures of variation that describe how its values vary with a single number.
The Basic student is able to compare and contrast the measures of center (mean or median) for a numerical data set that summarizes all of its values with
a single number and the measure of variation (range) that describes how its values vary with a single number.
The Below Basic student may be able to identify a measure of center (mean or median) for a numerical data set that summarizes all of its values with a
single number.
Summarize and describe distributions.
6.SP.J.4 Display numerical data in plots on a number line, including dot plots, stem-and-leaf plots, histograms, and box plots.
In addition to Proficient, the Advanced student is able to determine the appropriate display to represent numerical data (e.g., plot on a number line,
including dot plots or histograms) and justify the choice.
The Proficient student is able to display numerical data in plots on a number line, including dot plots, stem-and-leaf plots, histograms, and box plots.
The Basic student is able to identify from a given data set a corresponding representation including dot plots, stem-and-leaf plots, histograms, and box
plots.
The Below Basic student may be able to match a visual representation of data to the name of the graph (dot plot, stem-and-leaf plot, histogram, or box
plot).
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*6.SP.J.5 Summarize numerical data sets in relation to their real-world context.
6.SP.J.5A Report the sample size.
6.SP.J.5B Describe the context of the data under investigation, including how it was measured and its units of measurement.
6.SP.J.5C Find quantitative measures of center (median, mode and mean) and variability (range and interquartile range). Describe any overall pattern
(including outliers, clusters, and distribution), with reference to the context in which the data was gathered.
6.SP.J.5D Justify the choice of measures of center (median, mode, or mean) based on the shape of the data distribution and the context in which the data was
gathered.
In addition to Proficient, the Advanced student is able to compare and make generalizations about two different sets of numerical data in relation to their
real-world context including:
A. Report sample size.
B. Describe the context of the data under investigation, including how it was measured and its units of measurement.
C. Find quantitative measures of center (median, mode, and mean) and variability (range and interquartile range). Describe any overall pattern
(including outliers, clusters, and distribution), with reference to the context in which the data was gathered.
D. Justify the choice of measures of center (median, mode, or mean) based on the shape of the data distribution and the context in which the data
was gathered.
The Proficient student is able to summarize numerical data sets in relation to their real-world context.
A. Report the sample size.
B. Describe the context of the data under investigation, including how it was measured and its units of measurement.
C. Find quantitative measures of center (median, mode, and mean) and variability (range and interquartile range). Describe any overall pattern
(including outliers, clusters, and distribution), with reference to the context in which the data was gathered.
D. Justify the choice of measures of center (median, mode, or mean) based on the shape of the data distribution and the context in which the data
was gathered.
The Basic student is able to summarize numerical data sets in relation to their real-world context.
A. Report sample size.
B. Describe the context of the data under investigation, including how it was measured and its units of measurement.
C. Find quantitative measures of center (median, mode, and mean) and variability (range and interquartile range).
The Below Basic student may be able to summarize numerical data sets in relation to their real-world context by finding quantitative measures of center
(median, mode, and mean) and variability (range).
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Companion document to the 2018 Mathematics Content Standards
Grade 7 Math Content & Performance Standards & PLDs
GRADE 7 MATH PRACTICES
MP1 Make sense of problems and persevere in solving
them.
7.MP.1 In grade 7, students solve real-world problems involving ratios, rates,
proportions, rational numbers and geometric concepts and discuss (verbally
or in writing) how they solve them. Students analyze the problem (including
what is given, not given, and what is being asked), identify what strategies
are needed, choose an appropriate pathway, then make an initial attempt to
solve the problem. Students analyze the result for validity and refine
strategies if necessary.
MP2 Reason abstractly and quantitatively.
7.MP.2 Students represent a wide variety of real-world contexts through the
use of real numbers and variables in mathematical expressions, equations,
and inequalities. Students contextualize to understand the meaning of the
number or variable as related to the problem and decontextualize to
manipulate symbolic representations by applying properties of operations.
MP3 Construct viable arguments and critique the
reasoning of others.
7.MP.3 Students construct arguments using verbal or written explanations
that involve solving problems with rational numbers. They make conjectures,
explore validity, reason mathematically, justify, evaluate their own thinking
and the thinking of other students.
MP4 Model with mathematics.
7.MP.4 Students can clearly show their work by using diagrams, words,
symbols or pictures. They are able to identify important quantities in a
practical situation and map their relationships using such tools as diagrams,
two-way tables, graphs, flowcharts and/or formulas. They can analyze those
relationships mathematically to draw conclusions. They interpret their
mathematical results of problems involving rational numbers in the context of
the situation and reflect on whether the results make sense.
MP5 Use appropriate tools strategically.
7.MP.5 Students consider available tools (including estimation, concrete
models, and technology as appropriate), and decide when certain tools might
be helpful. Students develop more efficacy with technology. They choose the
representation (table, graph, equation, words) that best suits the problem.
Students use concrete models to develop insight into proportions and other
concepts. Students then extend this insight to more abstract representations,
including pictures and symbols. Students understand the limitations of each
tool. Tools might include: integer tiles, algebra tiles, geometric nets, number
lines, graphing technology, scientific calculator, paper and pencil, and others.
MP6 Attend to precision.
7.MP.6 Students continue to refine their mathematical communication skills
by using clear and precise language in their discussions with others and in
their own reasoning. Students define variables, including their relationship,
specify units of measure, and label each axis accurately. Student use
appropriate terminology when referring to rates, ratios, proportions,
probability models, geometric figures, data displays, and components of
expressions, equations or inequalities. Students use appropriate symbols,
labels, and units of measure when solving problems with calculations that are
accurate and efficient. Answer to the problem matches what was asked in
the problem.
MP7 Look for and make use of structure.
7.MP.7 Students routinely seek patterns or structure to model and solve
problems. They recognize that patterns exist in ratio tables and make
connections with the constant of proportionality in a table and the slope of a
graph. Students recognize patterns and identify and develop strategies for
creating equivalent expressions. Students identify complicated expressions
or figures as compositions of simple parts.
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MP8 Look for and express regularity in repeated reasoning.
7.MP.8 Students routinely seek patterns or structure to model and solve problems. They apply properties to solve problems based upon patterns they have
identified. Students examine patterns to generate equations and describe relationships. Students simplify complicated expressions into simple terms. Students
recognize the effects of transformations and describe them in terms of congruence and similarity.
RATIOS AND PROPORTIONAL RELATIONSHIPS
Analyze proportional relationships and use them to solve real-world and mathematical problems.
7.RP.A.1 Compute unit rates, including those involving complex fractions, with like or different units.
In addition to Proficient, the Advanced student is able to compute multi-step unit rates, including those involving complex fractions, with like or different
units.
The Proficient student is able to compute unit rates, including those involving complex fractions, with like or different units.
The Basic student is able to compute unit rates, including those involving complex fractions, with like units.
The Below Basic student may be able to compute unit rates, including those involving integers, with like units.
*7.RP.A.2 Recognize and represent proportional relationships between quantities.
7.RP.A.2A Decide whether two quantities in a table or graph are in a proportional relationship.
7.RP.A.2B Identify the constant of proportionality (unit rate) in tables, graphs, equations, diagrams, and verbal descriptions of proportional relationships.
7.RP.A.2C Represent proportional relationships with equations.
7.RP.A.2D Explain what a point (, ) on the graph of a proportional relationship means in terms of the situation, with special attention to the points (0, 0) and
(1, ) where is the unit rate.
In addition to Proficient, the Advanced student is able to recognize and represent proportional relationships between quantities.
A. Justify proportionality in a table or graph by extrapolating and/or interpolating ratios that fit the proportional relationship.
B. Compare the constant of proportionality (unit rate) when given two different representations (tables, graphs, equations, diagrams, and verbal
descriptions).
C. Create a scenario with a real-world context that represents a given proportional equation.
D. Extrapolate or interpolate coordinates of another point which follows the proportional relationship and explain the reasoning.
The Proficient student is able to recognize and represent proportional relationships between quantities.
A. Decide whether two quantities in a table or graph are in a proportional relationship.
B. Identify the constant of proportionality (unit rate) in tables, graphs, equations, diagrams, and verbal descriptions of proportional relationships.
C. Represent proportional relationships with equations.
D. Explain what a point (, ) on the graph of a proportional relationship means in terms of the situation, with special attention to the points (0, 0)
and (1, ) where is the unit rate.
The Basic student is able to recognize and represent proportional relationships between quantities.
A. Identify characteristics of proportionality from a proportional table (consistent unit rates and passes through the origin).
B. Calculate the constant of proportionality (unit rate) from a verbal descriptions of a proportional relationship.
C. Represent proportional relationships with equations when the unit rate is given.
D. Identify the unit rate on the graph of a proportional relationship when given the coordinate (1, ) where is the unit rate.
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The Below Basic student may be able to recognize and represent proportional relationships between quantities.
A. Identify characteristics of proportionality from a proportional graph (linear and passing through the origin).
B. Identify that the unit rate and the constant of proportionality are the same value.
C. Identify an equation that represents a proportional relationship.
D. Identify coordinates on a graph in the proportional relationship.
*7.RP.A.3 Solve multi-step real-world and mathematical problems involving ratios and percentages.
In addition to Proficient, the Advanced student is able to represent and solve multi-step real-world and mathematical problems involving ratios and
percentages in multiple ways.
The Proficient student is able to solve multi-step real-world and mathematical problems involving ratios and percentages (e.g., simple interest, tax,
markups and markdowns, gratuities and commissions, fees, percent increase and decrease, percent error).
The Basic student is able to solve one-step real-world and mathematical problems involving ratios and percentages.
The Below Basic student may be able to identify a proportion that can be used to solve real-world or mathematical problems involving ratios and
percentages.
THE NUMBER SYSTEM
Apply and extend previous understandings of operations with fractions to add, subtract, multiply, and divide rational
numbers.
7.NS.B.1 Apply and extend previous understandings of addition and subtraction to add and subtract rational numbers.
7.NS.B.1A Describe situations in which opposite quantities combine to make 0 (the additive identity).
7.NS.B.1B Understand that + represents the distance || from whose placement is determined by the sign of . Interpret sums of rational numbers by
describing real-world contexts.
7.NS.B.1C Show that a number and its opposite have a sum of 0 (are additive inverses).
7.NS.B.1D Understand subtraction of rational numbers as adding the additive inverse, = + (). Apply this principal in real-world contexts.
7.NS.B.1E Apply properties of addition as strategies to add and subtract rational numbers.
In addition to Proficient, the Advanced student is able to add and subtract rational numbers showing more than one strategy (e.g., number line diagrams,
subtraction as adding the additive inverse, properties of addition).
The Proficient student is able to add and subtract two rational numbers and interpret sums of rational numbers by describing them in real-world contexts.
The Basic student is able to add and subtract two integers.
The Below Basic student may be able to add and subtract two integers given a number line or manipulatives.
7.NS.B.2 Apply and extend previous understandings of multiplication and division and of fractions to multiply and divide rational numbers.
7.NS.B.2A 1. Understand that the multiplicative inverse of a number is its reciprocal and their product is equal to one (the multiplicative identity). 2. Understand
positive and negative sign rules for multiplying rational numbers. Interpret products of rational numbers by describing real-world contexts.
7.NS.B.2B Understand that integers can be divided, provided that the divisor is not 0, and every quotient of integers is a rational number. Recognize that if p
and q are integers then – (/) = (– )/ = /(– ). Interpret quotients of rational numbers by describing real-world contexts.
7.NS.B.2C Apply properties of multiplication (Commutative, Associative, Distributive, or Properties of Identity and Inverse elements) to multiply and divide
rational numbers.
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7.NS.B.2D Convert a rational number to a decimal. Recognize that rational numbers can be written as fractions or decimal numbers that terminate or repeat.
In addition to Proficient, the Advanced student is able to multiply and divide rational numbers showing more than one strategy (e.g., as repeated addition,
as repeated subtraction, on a number line, properties of multiplication).
The Proficient student is able to multiply and divide two rational numbers and interpret products and quotients of rational numbers by describing real-
world contexts.
• Understand that integers can be divided, provided that the divisor is not 0. Recognize that if p and q are integers then – (/) = (– )/ =
/(– ).
• Convert a rational number to a decimal. Recognize that rational numbers can be written as fractions or decimal numbers that terminate or
repeat.
The Basic student is able to multiply and divide two integers. Convert a rational number to a decimal and recognize terminating or repeating decimals.
The Below Basic student may be able to multiply and divide two integers given a number line or manipulatives.
*7.NS.B.3 Solve real-world and mathematical problems involving the four arithmetic operations with rational numbers. (Computations with rational
numbers extend the rules for manipulating fractions to complex fractions.)
In addition to Proficient, the Advanced student is able to solve multi-step real-world and mathematical problems involving the four arithmetic operations
with different representations of rational numbers (fractions, decimals, percentages, or integers).
The Proficient student is able to solve real-world and mathematical problems involving the four arithmetic operations with rational numbers.
(Computations with rational numbers extend the rules for manipulating fractions to complex fractions.)
The Basic student is able to solve real-world and mathematical problems involving the four arithmetic operations with integers.
The Below Basic student may be able to determine if an answer will be positive or negative in a one- or two- step real-world or mathematical problem
involving the four arithmetic operations with integers.
EXPRESSIONS AND EQUATIONS
Use properties of operations to generate equivalent expressions.
7.EE.C.1 Apply properties of operations as strategies to add, subtract, factor, and expand linear expressions with rational coefficients.
In addition to Proficient, the Advanced student is able to create more than one equivalent expression when adding, subtracting, factoring, and expanding
rational linear expressions and justify equivalence based on properties or a visual representation.
The Proficient student is able to add, subtract, factor, and expand linear expressions with rational coefficients.
The Basic student is able to combine like terms or use the Distributive Property to simplify integer expressions.
The Below Basic student may be able to identify like terms in an expression.
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7.EE.C.2 Recognize that algebraic expressions may have a variety of equivalent forms that reveal different information, and determine an appropriate
form for a given real-world situation.
In addition to Proficient, the Advanced student is able to write different equivalent forms from real-world situations and explain why they are equivalent,
referring to what the value represents for each term based on the context of the problem.
The Proficient student is able to identify equivalent forms of expressions from real-world situations, and interpret what the value represents for each term
based on the context of the problem.
The Basic student is able to interpret what the value represents for each term based on the context of a real-world problem.
The Below Basic student may be able to tell what the variable represents in the context of a real-world problem in an algebraic expression.
Solve real-life and mathematical problems using numerical and algebraic expressions and equations.
7.EE.D.3 Solve multi-step real-world and mathematical problems involving rational numbers. Include fraction bars as a grouping symbol.
In addition to Proficient, the Advanced student is able to recognize a numerical expression that could be used to solve a multi-step real-world or
mathematical problem involving rational numbers. Include fraction bars as a grouping symbol.
The Proficient student is able to solve multi-step real-world and mathematical problems involving rational numbers. Include fraction bars as a grouping
symbol.
The Basic student is able to solve two-step real-world and mathematical problems involving rational numbers. Include fraction bars as a grouping symbol.
The Below Basic student may be able to solve two-step real-world and mathematical problems involving integer values.
*7.EE.D.4 Apply the concepts of linear equations and inequalities in one variable to real-world and mathematical situations.
7.EE.D.4A Write and fluently solve linear equations of the form + = ( + ) = where , , and are rational numbers.
7.EE.D.4B Write and solve multi-step linear equations that include the use of the Distributive Property and combining like terms. Exclude equations that
contain variables on both sides.
7.EE.D.4C Write and solve two-step linear inequalities. Graph the solution set on a number line and interpret its meaning.
7.EE.D.4D Identify and justify the steps for solving multi-step linear equations and two-step linear inequalities.
In addition to Proficient, the Advanced student is able to apply the concepts of linear equations and inequalities in one variable to real-world and
mathematical situations.
A. Write and fluently solve linear equations of the form + = and ( + ) = where , , and are rational numbers. Recognize and
verify other equivalent forms of the equation.
B. Create a real-world scenario when given a multi-step linear equation that includes the use of the Distributive Property and combining like terms.
C. Write and solve a multi-step linear inequality that includes the use of the Distributive Property or combining like terms. Graph the solution set on
a number line and interpret its meaning.
D. Identify and justify multiple ways for solving multi-step linear equations and inequalities that include the use of the Distributive Property and
combining like terms.
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The Proficient student is able to apply the concepts of linear equations and inequalities in one variable to real-world and mathematical situations.
A. Write and fluently solve linear equations of the form + = and ( + ) = where , , and are rational numbers.
B. Write and solve multi-step linear equations that include the use of the Distributive Property and combining like terms. Exclude equations that
contain variables on both sides.
C. Write and solve two-step linear inequalities. Graph the solution set on a number line and interpret its meaning.
D. Identify and justify the steps for solving multi-step linear equations and two-step linear inequalities.
The Basic student is able to apply the concepts of linear equations and inequalities in one variable to real-world and mathematical situations.
A. Write and fluently solve linear equations of the form + = and ( + ) = where , , and are integers.
B. Solve multi-step linear equations that include the use of the Distributive Property or combining like terms. Exclude equations that contain
variables on both sides.
C. Solve two-step linear inequalities. Graph the solution set on a number line.
D. Identify and justify the steps for solving two-step linear equations and two-step linear inequalities with whole numbers.
The Below Basic student may be able to apply the concepts of linear equations and inequalities in one variable to real-world and mathematical situations.
A. Write and solve one-step linear equations with whole numbers.
B. Use the Distributive Property or combine like terms when given a multi-step linear equation to simplify the equation. Exclude equations that
contain variables on both sides.
C. Solve one-step linear inequalities with whole numbers. Graph the solution set on a number line.
D. Identify and justify the steps for solving one-step linear equations and one-step linear inequalities with whole numbers.
GEOMETRY
Draw, construct, and describe geometrical figures and describe the relationships between them.
7.G.E.1 Solve problems involving scale drawings of geometric figures, including computing actual lengths and areas from a scale drawing.
In addition to Proficient, the Advanced student is able to identify and reproduce scale drawing(s) at different scales with respect to the dimensions of the
actual figure.
The Proficient student is able to solve problems involving scale drawings of geometric figures, including computing actual lengths and areas from a scale
drawing.
The Basic student is able to compute a single dimension from a scale drawing of a geometric figure.
The Below Basic student may be able to compute a single dimension from a scale drawing of a simple geometric figure with a scale factor of 2, 3, 5, or 10
when given a visual representation.
7.G.E.2 Draw geometric shapes with given conditions using a variety of tools (e.g., ruler and protractor, or technology). Focus on constructing
triangles from three measures of angles or sides, noticing when the conditions determine a unique triangle, more than one triangle, or no triangle.
In addition to Proficient, the Advanced student is able to explain the conditions for a unique triangle, more than one triangle, or no triangle.
The Proficient student is able to draw geometric shapes with given conditions using a variety of tools (e.g., ruler and protractor, or technology). Focus on
constructing triangles from three measures of angles or sides, noticing when the conditions determine a unique triangle, more than one triangle, or no
triangle.
The Basic student is able to draw a triangle using a variety of tools when given one angle measure or one side length.
The Below Basic student may be able to identify the type of triangle with respect to angle measures and side measures.
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7.G.E.3 Describe the two-dimensional figures that result from slicing three-dimensional figures parallel to the base, as in plane sections of right
rectangular prisms and right rectangular pyramids.
In addition to Proficient, the Advanced student is able to describe the two-dimensional figure that results from slicing a three-dimensional figure parallel,
perpendicular, or oblique to the base as in plane sections of right rectangular prisms, right rectangular pyramids, or cylinders.
The Proficient student is able to describe the two-dimensional figure (shape and size in relation to the base) that results from slicing a three-dimensional
figure parallel to the base, as in plane sections of right rectangular prisms and right rectangular pyramids.
The Basic student is able to identify the two-dimensional figure that result from slicing a three-dimensional figure parallel to the base, as in plane sections
of right rectangular prisms and right rectangular pyramids when given a visual representation.
The Below Basic student may be able to identify the two-dimensional figure that result from slicing a three-dimensional figure parallel to the base, as in
plane sections of right rectangular prisms when given a visual representation where the cross-section is drawn.
Solve real-life and mathematical problems involving angle measure, area, surface area, and volume.
*7.G.F.4 Investigate the concept of circles.
7.G.F.4A Demonstrate an understanding of the proportional relationships between diameter, radius, and circumference of a circle.
7.G.F.4B Understand that is defined by the constant of proportionality between the circumference and diameter.
7.G.F.4C Given the formulas for circumference and area of circles, solve real-world and mathematical problems.
In addition to Proficient, the Advanced student is able to determine how much area or circumference changes based on a change in radius or diameter.
Assessment Boundary: Specify if calculations should be performed with 3.14 or the button. Specify the place value to which answers must be rounded.
Do not include solutions in terms of .
The Proficient student is able to investigate the concept of circles.
A. Demonstrate an understanding of the proportional relationships between diameter, radius, and circumference of a circle.
B. Understand that is defined by the constant of proportionality between the circumference and diameter.
C. Given the formulas for circumference and area of circles, solve real-world and mathematical problems.
Assessment Boundary: Specify if calculations should be performed with 3.14 or the button. Specify the place value to which answers must be rounded.
Do not include solutions in terms of .
The Basic student is able to investigate the concept of circles.
A. Find radius when given the diameter and find diameter when given radius.
C. Given the formulas for circumference and area of circles, solve mathematical problems.
Assessment Boundary: Specify if calculations should be performed with 3.14 or the button. Specify the place value to which answers must be rounded.
Do not include solutions in terms of .
The Below Basic student may be able to investigate the concept of circles.
A. Identify parts of a circle (radius, diameter, center, and circumference).
C. Given a word problem, identify which formula would be used to solve the problem (area or circumference).
Assessment Boundary: Specify if calculations should be performed with 3.14 or the button. Specify the place value to which answers must be rounded.
Do not include solutions in terms of .
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7.G.F.5 Use facts about supplementary, complementary, vertical, and adjacent angles in a multi-step problem to write and solve simple equations for
an unknown angle in a figure.
In addition to Proficient, the Advanced student is able to write and solve equations for unknown angles in a complex diagram using facts about
supplementary, complementary, vertical, and adjacent angles.
The Proficient student is able to use facts about supplementary, complementary, vertical, and adjacent angles in a multi-step problem to write and solve
simple equations for an unknown angle in a figure.
The Basic student is able to use facts about supplementary, complementary, and vertical angles to find missing angle measurements given a verbal
description or visual representation.
The Below Basic student may be able to identify supplementary, complementary, vertical, and adjacent angles when given a visual representation.
*7.G.F.6 Solve real-world and mathematical problems involving.
7.G.F.6A Area and surface area of objects composed of triangles and quadrilaterals;
7.G.F.6B Volume of objects composed only of right prisms having triangular or quadrilateral bases.
In addition to Proficient, the Advanced student is able to solve real-world and mathematical problems involving:
A. i. Find a missing dimension when given the area of objects composed of triangles, quadrilaterals, circles, and semi-circles.
ii. Find a missing dimension when given the surface area of objects composed of triangles and quadrilaterals.
B. Find a missing dimension when given the volume of objects composed only of right prisms having triangular or quadrilateral bases.
The Proficient student is able to solve real-world and mathematical problems involving:
A. Area and surface area of objects composed of triangles and quadrilaterals;
B. Volume of objects composed only of right prisms having triangular or quadrilateral bases.
The Basic student is able to solve real-world and mathematical problems involving:
A. i. Area of objects composed of triangles and/or parallelograms.
ii. Surface area of objects where the base is an equilateral triangle or rectangle.
B. Volume of right prisms having triangular and rectangular bases.
The Below Basic student may be able to solve real-world and mathematical problems involving:
A. i. Area of triangles and/or parallelograms.
ii. Surface area of right prisms and/or right pyramids when given the nets with the areas labeled.
B. Volume of right rectangular prisms.
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STATISTICS AND PROBABILITY
Use random sampling to draw inferences about a population.
*7.SP.G.1 Solve real-world and mathematical problems involving:
7.SP.G.1A Understand that a sample is a subset of a population.
7.SP.G.1B Differentiate between random and non-random sampling.
7.SP.G.1C Understand that generalizations from a sample are valid only if the sample is representative of the population.
7.SP.G.1D Understand that random sampling is used to gather a representative sample and tends to support valid inferences about the population.
In addition to Proficient, the Advanced student is able to solve real-world and mathematical problems involving:
A. Generating a sample that is a subset of a population.
B. Creating the parameters for a random sample.
C. Writing a generalization and justifying its validity, given a population and a sample.
D. Analyzing multiple random samples to explain why there might be differences in inferences from the data or analyzing errors in data collection
performed by others.
The Proficient student is able to solve real-world and mathematical problems involving:
A. Describing a sample that is a subset of a population.
B. Differentiating between random and non-random sampling.
C. Determining if a generalization is valid by justifying whether or not the sample is representative of the population.
D. Determining if inferences about the population are valid based on how the given sample was collected.
The Basic student is able to solve real-world and mathematical problems involving:
A. Identifying a sample that is a subset of a population.
B. Identifying a random sample.
C. Identifying a generalization based on a sample that is representative of the population.
D. Identifying a random sampling that supports valid inferences about the given population.
The Below Basic student may be able to solve real-world and mathematical problems involving:
A. Defining sample or population.
B. Defining random sample.
C. Identifying a representative sample for a given population.
D. Defining valid inference.
7.SP.G.2 Draw inferences about a population by collecting multiple random samples of the same size to investigate variability in estimates of the
characteristic of interest.
In addition to Proficient, the Advanced student is able to draw inferences about a population by collecting multiple random samples of the same size to
investigate variability in estimates of the characteristic of interest and justify variability in context of the situation.
The Proficient student is able to draw inferences about a population by collecting multiple random samples of the same size to investigate variability in
estimates of the characteristic of interest.
The Basic student is able to draw inferences about a population given data from a random sample and recognize that random samples produce variability.
The Below Basic student may be able to define variability.
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Draw informal comparative inferences about two populations.
7.SP.H.3 Visually compare the centers, spreads, and overlap of two displays of data (e.g., back-to-back stem and leaf plots, dot plots, histograms, box
plots) that are graphed on the same scale and draw inferences about this data.
In addition to Proficient, the Advanced student is able to visually compare the centers, spreads, and overlap of two displays of data (e.g., back-to-back
stem and leaf plots, dot plots, histograms, box plots) that are graphed on different scales or in different representations and draw inferences about this
data.
The Proficient student is able to visually compare the centers, spreads, and overlap of two displays of data (e.g., back-to-back stem and leaf plots, dot
plots, histograms, box plots) that are graphed on the same scale and draw inferences about this data.
The Basic student is able to visually compare the centers and spreads of two displays of data (e.g., dot plots, box plots) that are graphed on the same
scale and draw inferences about this data.
The Below Basic student may be able to visually compare the centers and spreads of two box plots that are graphed on the same scale.
*7.SP.H.4 Given measures of center and variability (mean, median and/or mode; range, interquartile range, and/or standard deviation), for numerical
data from random samples, draw appropriate informal comparative inferences about two populations.
In addition to Proficient, the Advanced student is able to given measures of center and variability (mean, median, and/or mode; range, interquartile range,
and/or standard deviation), for numerical data from random samples, draw appropriate informal comparative inferences about multiple populations and
determine to which population(s) given values are most likely to correspond.
The Proficient student is able to given measures of center and variability (mean, median, and/or mode; range, interquartile range, and/or standard
deviation), for numerical data from random samples, draw appropriate informal comparative inferences about two populations.
The Basic student is able to given measures of center and variability (mean and/or median; range), for numerical data from random samples, identify valid
comparisons about two populations.
The Below Basic student may be able to identify measures of center and variability.
Investigate chance processes and develop, use, and evaluate probability models.
7.SP.I.5 Find and interpret the probability of a random event. Understand that the probability of a random event is a number between, and including, 0
and 1 that expresses the likelihood of the event occurring.
In addition to Proficient, the Advanced student is able to justify why probability cannot be greater than 1 or less than 0.
The Proficient student is able to find and interpret the probability of a random event. Understand that the probability of a random event is a number
between, and including, 0 and 1 that expresses the likelihood of the event occurring.
The Basic student is able to interpret the meaning of a given probability.
The Below Basic student may be able to categorize events using certain, likely, equal chance, unlikely, or impossible.
7.SP.I.6 Collect multiple samples to compare the relationship between theoretical and experimental probabilities for simple events.
In addition to Proficient, the Advanced student is able to recognize and justify why the experimental probability approaches the theoretical probability as
the relative frequency of an event increases.
The Proficient student is able to collect multiple samples to compare the relationship between theoretical and experimental probabilities for simple events.
The Basic student is able to collect one sample to compare the relationship between theoretical and experimental probabilities for a simple event.
The Below Basic student may be able to identify theoretical and experimental probabilities for a simple event.
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7.SP.I.7 Apply the concepts of theoretical and experimental probabilities for simple events.
7.SP.I.7A Develop a uniform probability model by assigning equal probability to all outcomes, and use the model to determine probabilities of events.
7.SP.I.7B Develop a probability model (which may not be uniform) by observing frequencies in data generated from a chance process.
7.SP.I.7C Compare probabilities from a model to observed frequencies; if the agreement is not good, explain possible sources of the discrepancies.
In addition to Proficient, the Advanced student is able to given probabilities or outcomes, create the corresponding probability experiment.
The Proficient student is able to apply the concepts of theoretical and experimental probabilities for simple events.
A. Develop a uniform probability model by assigning equal probability to all outcomes, and use the model to determine probabilities of events.
B. Develop a probability model (which may not be uniform) by observing frequencies in data generated from a chance process.
C. Compare probabilities from a model to observed frequencies; if the agreement is not good, explain possible sources of the discrepancies.
The Basic student is able to given the theoretical and experimental probabilities from a model, compare probabilities; if the agreement is not good, explain
possible sources of the discrepancies.
The Below Basic student may be able to explain the difference between experimental and theoretical probability.
7.SP.I.8 Find probabilities of compound events using organized lists, tables, and tree diagrams.
7.SP.I.8A Understand that, just as with simple events, the probability of a compound event is the fraction of outcomes in the sample space for which the
compound event occurs.
7.SP.I.8B Represent sample spaces for compound events using methods such as organized lists, tables, and tree diagrams. For an event described in
everyday language (e.g., “rolling double sixes”), identify the outcomes in the sample space which compose the event.
In addition to Proficient, the Advanced student is able to find the probabilities of compound dependent events.
The Proficient student is able to find probabilities of compound events using organized lists, tables, and tree diagrams.
A. Understand that, just as with simple events, the probability of a compound event is the fraction of outcomes in the sample space for which the
compound event occurs.
B. Represent sample spaces for compound events using methods such as organized lists, tables, and tree diagrams. For an event described in
everyday language (e.g., “rolling double sixes”), identify the outcomes in the sample space which compose the event.
Assessment Boundary: Only include independent events.
The Basic student is able to complete an organized list, a table, or a tree diagram and find the probability of a compound event.
Assessment Boundary: Only include independent events.
The Below Basic student may be able to find the probability of a compound event when given the sample space.
Assessment Boundary: Only include independent events.
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Companion document to the 2018 Mathematics Content Standards
Grade 8 Math Content & Performance Standards & PLDs
GRADE 8 MATH PRACTICES
MP1 Make sense of problems and persevere in solving
them.
8.MP.1 In grade 8, students solve real-world problems through the
application of algebraic and geometric concepts and discuss (verbally or in
writing) how they solve them. Students analyze the problem (including what
is given, not given, and what is being asked), identify what strategies are
needed, choose the most efficient pathway, then make an initial attempt to
solve the problem. Students analyze the result for validity and refine
strategies if necessary.
MP2 Reason abstractly and quantitatively.
8.MP.2 Students represent a wide variety of real-world contexts through the
use of real numbers and variables in mathematical expressions, equations,
and inequalities. Students examine patterns in data and assess the degree of
linearity of functions. Students contextualize to understand the meaning of
the number or variable as related to the problem and decontextualize to
manipulate symbolic representations by applying properties of operations.
MP3 Construct viable arguments and critique the
reasoning of others.
8.MP.3 Students construct arguments using verbal or written explanations
that involve solving problems with real numbers. They make conjectures,
explore validity, reason mathematically, justify, evaluate their own thinking
and analytically critique the reasoning of other students.
MP4 Model with mathematics.
8.MP.4 Students can clearly show their work by using diagrams, words,
symbols or pictures. They are able to identify important quantities in a
practical situation and map their relationships using such tools as diagrams,
two-way tables, graphs, flowcharts and formulas. They can analyze those
relationships mathematically to draw conclusions. They routinely interpret
their mathematical results of problems involving real numbers in the context
of the situation and reflect on whether the results make sense, possibly
improving the model if it has not served its purpose.
MP5 Use appropriate tools strategically.
8.MP.5 Students consider available tools (including estimation, concrete
models, and technology as appropriate), and decide when certain tools might
be helpful. Students can interpret results provided by technology. They
choose the representation (table, graph, equation, words) that best suits the
problem. Students use concrete models to develop insight into linear
equations and other concepts. Students then extend this insight to more
abstract representations, including pictures and symbols. Students
understand the limitations of each tool. Tools might include: integer tiles,
algebra tiles, geometric nets, number lines, graphing technology, scientific
calculator, paper and pencil, and others.
MP6 Attend to precision.
8.MP.6 Students continue to refine their mathematical communication skills
by using clear and precise mathematical language in their discussions with
others and in their own reasoning. Students define variables, including their
relationship, specify units of measure, and label each axis accurately.
Students use appropriate terminology when referring to the number system,
functions, geometric figures, and data displays. Students use appropriate
symbols, labels, and units of measure when solving problems with
calculations that are accurate and efficient. Answer to the problem matches
what was asked in the problem.
MP7 Look for and make use of structure.
8.MP.7 Students routinely seek patterns or structure to model and solve
problems. They apply properties to solve problems based upon patterns they
have identified. Students examine patterns to generate equations and
describe relationships. Students simplify complicated expressions into simple
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terms. Students recognize the effects of transformations and describe them
in terms of congruence and similarity.
MP8 Look for and express regularity in repeated reasoning.
8.MP.8 Students use repeated reasoning to understand algorithms and make generalizations about patterns. They develop efficient strategies for solving problems
and check for reasonableness of answers. Students ask questions such as, "What evidence supports that conclusion?"
THE NUMBER SYSTEM
Know that there are numbers that are not rational, and approximate them by rational numbers.
*8.NS.A.1 Know that numbers that are not rational are called irrational. Understand informally that every number has a decimal expansion; for rational
numbers show that the decimal expansion repeats eventually, and convert a decimal expansion which repeats eventually into a rational number.
Explore the real number system and its appropriate usage in real-world situations.
8.NS.A.1A Make comparisons between rational and irrational numbers.
8.NS.A.1B Understand that all real numbers have a decimal expansion.
8.NS.A.1C Model the hierarchy of the real number system, including natural, whole, integer, rational, and irrational numbers.
8.NS.A.1D Convert repeating decimals to fractions.
In addition to Proficient, the Advanced student is able to compare and contrast properties of rational and irrational numbers.
The Proficient student is able to know that numbers that are not rational are called irrational. Show that every number has a decimal expansion; for
rational numbers show that the decimal expansion repeats eventually, and convert a decimal expansion which repeats eventually into a rational number.
Explore the real number system and its appropriate usage in real-world situations.
A. Make comparisons between rational and irrational numbers.
B. Show that real numbers (excluding irrational numbers) have a decimal expansion.
C. Model the hierarchy of the real number system, including natural, whole, integer, rational, and irrational numbers.
D. Convert repeating decimals to fractions.
The Basic student is able to know that numbers that are not rational are called irrational. Show that every number has a decimal expansion; for rational
numbers show that the decimal expansion repeats eventually, and convert a decimal expansion which repeats eventually into a rational number. Explore
the real number system and its appropriate usage in real-world situations.
A. Make comparisons between rational and irrational numbers.
B. Show that real numbers (excluding irrational numbers) have a decimal expansion.
C. Model the hierarchy of the real number system, including natural, whole, integer, rational, and irrational numbers.
The Below Basic student may be able to know that numbers that are not rational are called irrational. Show that every number has a decimal expansion;
for rational numbers show that the decimal expansion repeats eventually, and convert a decimal expansion which repeats eventually into a rational
number. Explore the real number system and its appropriate usage in real-world situations. Make comparisons between rational and irrational numbers.
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8.NS.A.2 Use rational approximations of irrational numbers to compare the size of irrational numbers, locate them approximately on a number line
diagram, and estimate the value of expressions.
In addition to Proficient, the Advanced student is able to using estimation strategies, order a set of irrational numbers and explain your reasoning.
The Proficient student is able to use rational approximations of irrational numbers to compare the size of irrational numbers, locate them approximately on
a number line diagram, and estimate the value of expressions.
The Basic student is able to use rational approximations to locate irrational numbers on a number line and estimate the value of expressions.
The Below Basic student may be able to use rational approximations to locate irrational numbers on a number line.
EXPRESSIONS AND EQUATIONS
Work with radicals and integer exponents.
8.EE.B.1 Understand and apply the Laws of Exponents (i.e. Product Rule, Quotient Rule, Power to a Power, Product to a Power, Quotient to a Power,
Zero Power Property, negative exponents) to generate equivalent numerical expressions limited to integer exponents.
In addition to Proficient, the Advanced student is able to apply all of the following Laws of exponents to generate equivalent algebraic expressions limited
to integer exponents.
• Product Rule.
• Quotient Rule.
• Power to a Power.
• Product to a Power.
• Quotient to a Power
• Zero Power Property.
• Negative Exponents.
Assessment Boundary: Limit to no more than two rules per problem.
The Proficient student is able to apply all of the following Laws of Exponents to generate equivalent numerical expressions limited to integer exponents.
• Product Rule.
• Quotient Rule.
• Power to a Power.
• Product to a Power.
• Quotient to a Power.
• Zero Power Property.
• Negative Exponents.
The Basic student is able to apply all of the following Laws of Exponents to generate equivalent numerical expressions limited to integer exponents.
• Product Rule.
• Quotient Rule.
• Power to a Power.
• Zero Power Property.
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The Below Basic student may be able to apply all of the following Laws of Exponents to generate equivalent numerical expressions limited to integer
exponents.
• Product Rule.
• Quotient Rule.
8.EE.B.2 Investigate concepts of square and cube roots.
Assessment Boundary: Include perfect squares up to 144 and perfect cubes up to 125.
8.EE.B.2A Use radical notation, if applicable, to represent the exact solutions to equations of the form
= and
= where p is a positive rational
number and q is any rational number.
8.EE.B.2B Evaluate square roots of small perfect squares and cube roots of small perfect cubes.
8.EE.B.2C Recognize that square roots of non-perfect squares and the cube roots of non-perfect cubes are irrational.
In addition to Proficient, the Advanced student is able to investigate concepts of square and cube roots. Use radical notation, if applicable, to represent
the exact solutions to equations of the form ² + = and ³ + = where a, p, and q are positive rational numbers such that and are
greater than or equal to zero.
Assessment Boundary: Include perfect squares up to 144 and perfect cubes up to 125.
The Proficient student is able to investigate concepts of square and cube roots.
A. Use radical notation, if applicable, to represent the exact solutions to equations of the form
= and
= where is a positive rational
number and is any rational number.
B. Evaluate square roots of small perfect squares and cube roots of small perfect cubes.
C. Recognize that square roots of non-perfect squares and the cube roots of non-perfect cubes are irrational.
Assessment Boundary: Include perfect squares up to 144 and perfect cubes up to 125.
The Basic student is able to investigate concepts of square and cube roots.
A. Use radical notation, if applicable, to represent the exact solutions to equations of the form
= and
= where p is a positive rational
number and q is any rational number.
B. Evaluate square roots of small perfect squares and cube roots of small perfect cubes.
Assessment Boundary: Include perfect squares up to 144 and perfect cubes up to 125.
The Below Basic student may be able to evaluate square roots of small perfect squares and cube roots of small perfect cubes.
Assessment Boundary: Include perfect squares up to 144 and perfect cubes up to 125.
8.EE.B.3 Explore the relationship between quantities in decimal and scientific notation.
8.EE.B.3A Express very large and very small quantities, , in scientific notation in the form 10
= where 1 < 10 and b is an integer.
8.EE.B.3B Translate between decimal notation and scientific notation.
8.EE.B.3C Estimate and compare the relative size of two quantities in scientific notation.
In addition to Proficient, the Advanced student is able to compare the relative size of two quantities written in decimal and scientific notation in a real-world
context.
The Proficient student is able to explore the relationship between quantities in decimal and scientific notation.
A. Express very large and very small quantities, p, in scientific notation in the form 10
= where 1 < 10 and b is an integer.
B. Translate between decimal notation and scientific notation.
C. Estimate and compare the relative size of two quantities in scientific notation.
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The Basic student is able to explore the relationship between quantities in decimal and scientific notation.
A. Express very large and very small quantities, p, in scientific notation in the form 10
= where 1 < 10 and b is an integer.
B. Translate between decimal notation and scientific notation.
The Below Basic student may be able to express very large and very small quantities, p, in scientific notation in the form 10
= where 1 < 10
and b is an integer.
8.EE.B.4 Apply the concepts of decimal and scientific notation to real-world and mathematical problems.
8.EE.B.4A Select appropriate units of measure when representing answers in scientific notation.
8.EE.B.4B Interpret scientific notation that has been generated by a variety of technologies.
In addition to Proficient, the Advanced student is able to select appropriate units of measure when multiplying and dividing numbers written in scientific
notation in real-world and mathematical problems.
The Proficient student is able to apply the concepts of decimal and scientific notation to real-world and mathematical problems.
A. Select appropriate units of measure when representing answers in scientific notation.
B. Interpret scientific notation that has been generated by a variety of technologies.
The Basic student is able to select appropriate units of measure when representing answers in decimal and scientific notation in real-world and
mathematical problems.
The Below Basic student may be able to select appropriate units of measure when representing answers in scientific notation in real-world and
mathematical problems.
Understand the connections between proportional relationships, lines, and linear equations.
*8.EE.C.5 Graph proportional relationships, interpreting the unit rate as the slope of the graph. Compare two different proportional relationships
represented in different ways.
In addition to Proficient, the Advanced student is able to create a table, graph, and equation of a proportional relationship given a written description.
The Proficient student is able to graph proportional relationships, interpreting the unit rate as the slope of the graph. Compare two different proportional
relationships represented in different ways.
The Basic student is able to graph proportional relationships, interpreting the unit rate as the slope of the graph.
The Below Basic student may be able to graph proportional relationships from a table of values.
8.EE.C.6 Explain why the slope m is the same between any two distinct points on a non-vertical line in the coordinate plane; derive the equation y =mx
for a line through the origin and the equation = + for a line intercepting the vertical axis at (, ).
In addition to Proficient, the Advanced student is able to derive the equation = + for a line intercepting the vertical axis at (0, ) from a written
description.
The Proficient student is able to explain why the slope m is the same between any two distinct points on a non-vertical line in the coordinate plane; derive
the equation = for a line through the origin and the equation = + for a line intercepting the vertical axis at (0, ).
The Basic student is able to explain why the slope m is the same between any two distinct points on a non-vertical line in the coordinate plane; derive the
equation = for a line through the origin.
The Below Basic student may be able to explain why the slope m is the same between any two distinct points on a non-vertical line in the coordinate
plane.
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Analyze and solve linear equations and pairs of simultaneous linear equations.
*8.EE.D.7 Extend concepts of linear equations and inequalities in one variable to more complex multi-step equations and inequalities in real-world and
mathematical situations.
8.EE.D.7A Solve linear equations and inequalities with rational number coefficients that include the use of the Distributive Property, combining like terms, and
variable terms on both sides.
8.EE.D.7B Recognize the three types of solutions to linear equations: one solution, infinitely many solutions, or no solutions.
8.EE.D.7C Generate linear equations with the three types of solutions.
8.EE.D.7D Justify why linear equations have a specific type of solution.
In addition to Proficient, the Advanced student is able to create and solve a multi-step equation or inequality in a real-world context given a written
description.
The Proficient student is able to extend concepts of linear equations and inequalities in one variable to more complex multi-step equations and
inequalities in real-world and mathematical situations.
A. Solve linear equations and inequalities with rational number coefficients that include the use of the Distributive Property, combining like terms,
and variable terms on both sides.
B. Recognize the three types of solutions to linear equations: one solution, infinitely many solutions, or no solutions.
C. Generate linear equations with the three types of solutions.
D. Justify why linear equations have a specific type of solution.
The Basic student is able to extend concepts of linear equations and inequalities in one variable to more complex multi-step equations and inequalities in
real-world and mathematical situations.
A. Solve linear equations and inequalities with rational number coefficients that include the use of the Distributive Property, combining like terms,
and variable terms on both sides.
B. Recognize the three types of solutions to linear equations: one solution, infinitely many solutions, or no solutions.
The Below Basic student may be able to solve linear equations and inequalities with rational number coefficients that include the use of the Distributive
Property, combining like terms, and variable terms on one side.
*8.EE.D.8 Analyze and solve pairs of simultaneous linear equations.
8.EE.D.8A Understand that solutions to a system of two linear equations in two variables correspond to points of intersection of their graphs, because points of
intersection satisfy both equations simultaneously.
8.EE.D.8B Solve systems of two linear equations in two variables with integer solutions by graphing the equations.
8.EE.D.8C Solve simple real-world and mathematical problems leading to two linear equations in two variables given = + form with integer solutions.
In addition to Proficient, the Advanced student is able to create and solve systems of two linear equations from a real-world context that requires
simplification to write in the form = + .
The Proficient student is able to analyze and solve a system of linear equations.
A. Show that solutions to a system of two linear equations in two variables correspond to points of intersection of their graphs, because points of
intersection satisfy both equations simultaneously, including systems with one, infinitely many, and no solutions.
B. Solve systems of two linear equations in two variables with integer solutions by graphing the equations.
C. Solve simple real-world and mathematical problems leading to two linear equations in two variables given = + form with integer
solutions.
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The Basic student is able to analyze and solve a system of linear equations.
A. Show that solutions to a system of two linear equations in two variables correspond to points of intersection of their graphs, because points of
intersection satisfy both equations simultaneously, including systems with one, infinitely many, and no solutions.
B. Solve systems of two linear equations in two variables with integer solutions by graphing the equations.
The Below Basic student may be able to given a graph, identify the solution to a system of two linear equations, including systems with one, infinitely
many, and no solutions.
FUNCTIONS
Define, evaluate, and compare functions.
8.F.E.1 Understand that a function is a rule that assigns to each input exactly one output. The graph of a function is the set of ordered pairs consisting
of an input and the corresponding output.
In addition to Proficient, the Advanced student is able to compare relations in different forms to determine if they represent a function. (Function notation
is not required in Grade 8.)
The Proficient student is able to determine if a relation represented by a graph, a table, a mapping diagram, and a set of ordered pairs is a function.
(Function notation is not required in Grade 8.)
The Basic student is able to determine that a relation represented by a table or a set of ordered pairs is a function by demonstrating each input has
exactly one output. (Function notation is not required in Grade 8.)
The Below Basic student may be able to recognize the input and output values of a relation in a table or a set of ordered pairs. (Function notation is not
required in Grade 8.)
*8.F.E.2 Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal
descriptions).
In addition to Proficient, the Advanced student is able to compare properties (intercepts, domain, and range) of one linear function and one non-linear
function each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions).
The Proficient student is able to compare properties (intercepts, domain, and range) of two linear functions each represented in a different way
(algebraically, graphically, numerically in tables, or by verbal descriptions).
The Basic student is able to compare properties (intercepts, domain, and range) of two linear functions each represented graphically or numerically in
tables.
The Below Basic student may be able to compare the domains of two linear functions each represented graphically or numerically in tables.
8.F.E.3 Interpret the equation = + as defining a linear function, whose graph is a straight line; give examples of functions that are not linear.
In addition to Proficient, the Advanced student is able to identify linear and non-linear functions represented by verbal or written descriptions.
The Proficient student is able to recognize the equation = + as defining a linear function, whose graph is a straight line; write examples of
equations and sketch graphs of functions that are not linear.
The Basic student is able to recognize the equation = + as defining a linear function, whose graph is a straight line and sketch graphs of
functions that are not linear.
The Below Basic student may be able to recognize the equation = + as defining a linear function, whose graph is a straight line.
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Use functions to model relationships between quantities.
*8.F.F.4 Apply the concepts of linear functions to real-world and mathematical situations.
8.F.F.4A Understand that the slope is the constant rate of change and the y-intercept is the point where = 0.
8.F.F.4B Determine the slope and the y-intercept of a linear function given multiple representations, including two points, tables, graphs, equations, and verbal
descriptions.
8.F.F.4C Construct a function in slope-intercept form that models a linear relationship between two quantities.
8.F.F.4D Interpret the meaning of the slope and the y-intercept of a linear function in the context of the situation.
In addition to Proficient, the Advanced student is able to apply the concepts of linear functions to real-world and mathematical situations. Analyze the
meaning of the slopes and the y-intercepts of two linear functions given as a written description and justify conclusions in the context of the situation.
The Proficient student is able to apply the concepts of linear functions to real-world and mathematical situations.
A. Recognize that the slope is the constant rate of change and the y-intercept is the point where = 0 from an equation, graph, table, and verbal
description.
B. Determine the slope and the y-intercept of a linear function given multiple representations, including two points, tables, graphs, equations, and
verbal descriptions.
C. Construct a function in slope-intercept form that models a linear relationship between two quantities.
D. Interpret the meaning of the slope and the y-intercept of a linear function in the context of the situation.
The Basic student is able to apply the concepts of linear functions to real-world and mathematical situations.
A. Recognize that the slope is the constant rate of change and the y-intercept is the point where = 0 from an equation and a graph.
B. Determine the slope and the y-intercept of a linear function given multiple representations, including graphs and equations in slope-intercept
form.
C. Identify a function in slope-intercept form that models a linear relationship between two quantities.
D. Interpret the meaning of the slope of a linear function in the context of the situation.
The Below Basic student may be able to apply the concepts of linear functions to real-world and mathematical situations.
A. Recognize that the slope is the constant rate of change and the y-intercept is the point where = 0 from a graph.
B. Determine the slope and the y-intercept of a linear function given an equation in slope-intercept form.
C. Match a function in slope-intercept form to the model of a linear relationship between two quantities.
8.F.F.5 Describe qualitatively the functional relationship between two quantities by analyzing a graph where the function is increasing, decreasing,
constant, linear, or nonlinear. Sketch a graph that exhibits the qualitative features of a function that has been described verbally.
In addition to Proficient, the Advanced student is able to describe qualitatively the functional relationship between two quantities by analyzing a written
real-world scenario where the function is increasing, decreasing, constant, linear, or nonlinear. Sketch a graph that exhibits the qualitative features of a
function that has been described in a real-world scenario.
The Proficient student is able to describe qualitatively the functional relationship between two quantities by analyzing a graph where the function is
increasing, decreasing, constant, linear, or nonlinear. Sketch a graph that exhibits the qualitative features of a function that has been described verbally.
The Basic student is able to describe qualitatively the functional relationship between two quantities by analyzing a graph where the function is increasing,
decreasing, constant, linear, or nonlinear.
The Below Basic student may be able to describe qualitatively the functional relationship between two quantities by labeling a graph where the function is
increasing, decreasing, constant, linear, or nonlinear when given a word bank.
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GEOMETRY
Understand congruence and similarity using physical models, transparencies, or geometry software.
8.G.G.1 Verify experimentally the properties of rotations, reflections, and translations.
8.G.G.1A Lines are taken to lines, and line segments to line segments of the same length.
8.G.G.1B Angles are taken to angles of the same measure.
8.G.G.1C Parallel lines are taken to parallel lines.
In addition to Proficient, the Advanced student is able to verify experimentally the properties of rotations, reflections, and translations.
A. Write a sequence of transformations that takes a line to a line, and line segment to a line segment of the same length.
B. Write a sequence of transformations that takes an angle to an angle of the same measure.
C. Write a sequence of transformations that takes parallel lines to parallel lines.
The Proficient student is able to verify experimentally the properties of rotations, reflections, and translations.
A. Demonstrate that lines are taken to lines, and line segments to line segments of the same length.
B. Demonstrate that angles are taken to angles of the same measure.
C. Demonstrate that parallel lines are taken to parallel lines.
The Basic student is able to verify experimentally the properties of rotations, reflections, and translations.
A. Select the transformation that shows lines are taken to lines, and line segments to line segments of the same length.
B. Select the transformation that shows angles are taken to angles of the same measure.
C. Select the transformation that shows parallel lines are taken to parallel lines.
The Below Basic student may be able to verify experimentally the properties of rotations, reflections, and translations.
A. Select the translation that shows lines are taken to lines, and line segments to line segments of the same length.
B. Select the translation that shows angles are taken to angles of the same measure.
C. Select the translation that shows parallel lines are taken to parallel lines.
8.G.G.2 Recognize through visual comparison that a two-dimensional figure is congruent to another if the second can be obtained from the first by a
sequence of rotations, reflections, and translations; given two congruent figures, describe a sequence that exhibits the congruence between them.
In addition to Proficient, the Advanced student is able to recognize through visual comparison that a two-dimensional figure is congruent to another if the
second can be obtained from the first by a sequence of three or more transformations (rotations, reflections, and translations); given two congruent figures,
describe a sequence of three or more transformations that exhibits the congruence between them.
The Proficient student is able to recognize through visual comparison that a two-dimensional figure is congruent to another if the second can be obtained
from the first by a sequence of at most two transformations (rotations, reflections, and translations); given two congruent figures, describe a sequence of at
most two transformations that exhibits the congruence between them.
The Basic student is able to recognize through visual comparison that a two-dimensional figure is congruent to another if the second can be obtained from
the first by a sequence of at most two transformations (reflections and translations); given two congruent figures, describe a sequence of at most two
transformations that exhibits the congruence between them.
The Below Basic student may be able to recognize through visual comparison that a two-dimensional figure is congruent to another if the second can be
obtained from the first by a transformation (reflection or translation); given two congruent figures, describe a transformation that exhibits the congruence
between them.
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8.G.G.3 Describe the effect of dilations, translations, rotations and reflections on two-dimensional figures using coordinates.
In addition to Proficient, the Advanced student is able to describe and justify the sequence of dilations, translations, rotations, and reflections performed
on the pre-image to determine the image on a coordinate plane.
The Proficient student is able to describe the effect of dilations, translations, rotations, and reflections on two-dimensional figures using coordinates.
The Basic student is able to describe the effect of dilations, translations, and reflections on two-dimensional figures using coordinates.
The Below Basic student may be able to describe the effect of translations and reflections on two-dimensional figures using coordinates.
8.G.G.4 Recognize through visual comparison that a two-dimensional figure is similar to another if the second can be obtained from the first by a
sequence of rotations, reflections, translations, and dilations; given two similar two-dimensional figures, describe a sequence that exhibits the
similarity between them.
In addition to Proficient, the Advanced student is able to describe and justify a sequence of rotations, reflections, translations, and dilations that maintains
similarity between the pre-image and determined image.
The Proficient student is able to recognize through visual comparison that a two-dimensional figure is similar to another if the second can be obtained
from the first by a sequence of rotations, reflections, translations, and dilations; given two similar two-dimensional figures, describe a sequence that
exhibits the similarity between them.
The Basic student is able to recognize through visual comparison that a two-dimensional figure is similar to another if the second can be obtained from the
first by a sequence of reflections, translations, and dilations; given two similar two-dimensional figures, describe a sequence that exhibits the similarity
between them.
The Below Basic student may be able to recognize through visual comparison that a two-dimensional figure is similar to another if the second can be
obtained from a dilation.
8.G.G.5 Use informal arguments to establish facts about the angle sum and exterior angle of triangles, about the angles created when parallel lines are
cut by a transversal, and the angle-angle criterion for similarity of triangles.
In addition to Proficient, the Advanced student is able to use informal arguments to establish that two triangles are similar using the angle-angle criterion
for similarity of triangles.
The Proficient student is able to use informal arguments to establish facts about the angle sum and exterior angle of triangles, about the angles created
when parallel lines are cut by a transversal, and the angle-angle criterion for similarity of triangles.
The Basic student is able to use informal arguments to establish facts about the angles created when parallel lines are cut by a transversal and the angle-
angle criterion for similarity of triangles.
The Below Basic student may be able to use informal arguments to establish facts about the angles created when parallel lines are cut by a transversal.
Understand and apply the Pythagorean Theorem.
8.G.H.6 Use models or diagrams to explain the Pythagorean Theorem and its converse.
In addition to Proficient, the Advanced student is able to model a proof of the Pythagorean Theorem and its converse using a pictorial representation.
The Proficient student is able to use models or diagrams to explain the Pythagorean Theorem and its converse.
The Basic student is able to identify the Pythagorean Theorem and its converse.
The Below Basic student may be able to identify the Pythagorean Theorem.
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*8.G.H.7 Apply the Pythagorean Theorem to determine unknown side lengths in right triangles in real-world and mathematical problems.
In addition to Proficient, the Advanced student is able to apply the Pythagorean Theorem to determine unknown side lengths in right triangles in multi-
step real-world and mathematical problems.
The Proficient student is able to apply the Pythagorean Theorem to determine unknown side lengths in right triangles in real-world and mathematical
problems.
The Basic student is able to apply the Pythagorean Theorem in mathematical problems by setting up the equation ² + ² = ² and solving for either leg.
The Below Basic student may be able to apply the Pythagorean Theorem in mathematical problems by setting up the equation ² + ² = ² and only
solving for the hypotenuse.
8.G.H.8 Apply the Pythagorean Theorem to find the distance between two points in a coordinate system.
In addition to Proficient, the Advanced student is able to apply the distance formula to calculate the distance between two points in a coordinate system.
The Proficient student is able to apply the Pythagorean Theorem to find the distance between two points in a coordinate system.
The Basic student is able to apply the Pythagorean Theorem to find the distance between two points plotted on a coordinate plane.
The Below Basic student may be able to apply the Pythagorean Theorem to find the distance between two points given the right triangle drawn on a
coordinate plane.
Solve real-world and mathematical problems involving volume of cylinders, cones and spheres.
8.G.I.9 Given the formulas, solve real-world and mathematical problems involving volume and surface area of cylinders.
In addition to Proficient, the Advanced student is able to, given the formulas:
• Solve for a component part (radius or height) given the volume of a cylinder. OR
• Determine the volume of a cone or sphere. AND
• Determine the volume of a composite figure containing two or more cones, cylinders, or spheres.
Assessment Boundary: Specify calculations should be performed with the pi button. Limit the place value to up to the thousandths place or written in
terms of pi.
The Proficient student is able to, given the formulas, solve real-world and mathematical problems involving volume and surface area of cylinders.
Assessment Boundary: Specify calculations should be performed with the pi button. Limit the place value to up to the thousandths place or written in
terms of pi.
The Basic student is able to, given the formulas, solve mathematical problems involving volume and surface area of cylinders.
Assessment Boundary: Specify calculations should be performed with the pi button. Limit the place value to up to the thousandths place.
The Below Basic student may be able to, given the formulas, solve mathematical problems involving volume of cylinders.
Assessment Boundary: Specify calculations should be performed with the pi button. Limit the place value to up to the thousandths place.
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STATISTICS AND PROBABILITY
Investigate patterns of association in bivariate data.
8.SP.J.1 Construct and interpret scatter plots for bivariate measurement data to investigate patterns of association between two quantities. Describe
the association by form (linear / nonlinear), direction (positive / negative), strength (correlation), and unusual features.
In addition to Proficient, the Advanced student is able to interpret and compare scatter plots for bivariate measurement data by comparing their
association by form (linear / nonlinear), direction (positive / negative), strength (correlation), and unusual features.
The Proficient student is able to construct and interpret scatter plots for bivariate measurement data to investigate patterns of association between two
quantities. Describe the association by form (linear / nonlinear), direction (positive / negative), strength (correlation), and unusual features.
The Basic student is able to construct and interpret scatter plots for bivariate measurement data to investigate patterns of association between two
quantities. Describe the linear association by direction (positive / negative) and strength (correlation).
The Below Basic student may be able to interpret scatter plots for bivariate measurement data to investigate patterns of association between two
quantities. Describe the linear association by direction (positive / negative) and strength (correlation).
8.SP.J.2 Know that straight lines are widely used to model relationships between two quantitative variables. For scatter plots that suggest a linear
association, informally fit a straight line, and informally assess the model fit by judging the closeness of the data points to the line.
In addition to Proficient, the Advanced student is able to know that straight lines are widely used to model relationships between two quantitative
variables. For scatter plots that suggest a linear association, informally fit a straight line, and informally assess the model by plotting the residuals.
The Proficient student is able to know that straight lines are widely used to model relationships between two quantitative variables. For scatter plots that
suggest a linear association, informally fit a straight line, and informally assess the model fit by judging the closeness of the data points to the line.
The Basic student is able to know that straight lines are widely used to model relationships between two quantitative variables. For scatter plots that
suggest a linear association, informally fit a straight line.
The Below Basic student may be able to know that straight lines are widely used to model relationships between two quantitative variables. For scatter
plots that suggest a linear association, identify a line of best fit.
8.SP.J.3 Use an equation of a linear model to solve problems in the context of bivariate measurement data, interpreting the slope and intercept.
In addition to Proficient, the Advanced student is able to draw a line of best fit and create its equation in the context of the bivariate measurement data
when given a scatter plot.
The Proficient student is able to interpret the slope and y-intercept in the context of the bivariate measurement data when given a scatter plot with a line
of best fit and an equation.
The Basic student is able to interpret the slope in the context of the bivariate measurement data when given a scatter plot with a line of best fit and an
equation.
The Below Basic student may be able to match an equation of a line of best fit to bivariate measurement data when given scatter plot with a line of best
fit.
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8.SP.J.4 Understand that patterns of association can also be seen in bivariate categorical data by displaying frequencies and relative frequencies in a
two-way table.
8.SP.J.4A Construct and interpret a two-way table summarizing data on two categorical variables collected from the same subjects.
8.SP.J.4B Use relative frequencies calculated for rows or columns to describe possible association between the two variables.
In addition to Proficient, the Advanced student is able to understand that patterns of association can also be seen in bivariate categorical data by
displaying frequencies and relative frequencies in a two-way table and justifying conclusions about the frequencies and relative frequencies of data in a
two-way table.
The Proficient student is able to understand that patterns of association can also be seen in bivariate categorical data by displaying frequencies and
relative frequencies in a two-way table.
A. Construct and interpret a two-way table summarizing data on two categorical variables collected from the same subjects.
B. Use relative frequencies calculated for rows or columns to describe possible association between the two variables.
The Basic student is able to understand that patterns of association can also be seen in bivariate categorical data by constructing and interpreting a two-
way table summarizing data on two categorical variables collected from the same subjects.
The Below Basic student may be able to understand that patterns of association can also be seen in bivariate categorical data by completing a two-way
table summarizing data on two categorical variables collected from the same subjects.
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Companion document to the 2018 Mathematics Content Standards
High School Math Content & Performance Standards &
PLDs
GRADE HS MATH PRACTICES
MP1 Make sense of problems and persevere in solving
them.
HS.MP.1 Students start to examine problems by explaining to themselves
the meaning of a problem and restating the problem in their own words.
These students analyze the given information in the problem, including
constraints, relationships, and goals. Students make conjectures about the
form and meaning of the solution, devise a plan, and solve. They will
consider both similar problems, and simpler forms of the original problem, in
order to gain insight and efficiency in problem solving. Students monitor and
evaluate their progress and change course if necessary. Students may utilize
algebraic methods or technology. Students explain relationships between
equations and the following: descriptions/situations, tables, and graphs.
Students produce diagrams of important features and relationships, graph
data, and search for patterns or trends. They check answers to problems and
continually ask if the solution makes sense in context. They understand
different approaches to solving complex problems and identify
correspondences between different approaches.
MP2 Reason abstractly and quantitatively.
HS.MP.2 Students seek to make sense of quantities and explore
relationships in problem situations. Students represent a given situation by
defining and manipulating variables. Students consider the units involved and
attend to the meaning of quantities in addition to computational reasoning --
knowing and using the different properties of operations.
MP3 Construct viable arguments and critique the
reasoning of others.
HS.MP.3 Students understand and use stated assumptions, definitions, and
previously established results in constructing arguments. Students make
conjectures and build logical progressions of statements to explore the truth
of their conjectures. They are able to analyze situations through
decomposition and produce counterexample(s) if necessary. Students justify
their conclusions, communicate these conclusions, and respond to
arguments of others. Students make plausible arguments by reasoning
inductively about the data and take into account the context from which the
data arose. Students are able to compare the effectiveness of two plausible
arguments, and distinguish correct logic from flawed logic. If there is a flaw in
an argument, then they explain why the logic is flawed. Students determine a
general process and/or domain to which an argument applies. The students
listen or read the arguments of others, decide whether the argument makes
sense, and ask useful questions to clarify or improve the arguments.
MP4 Model with mathematics.
HS.MP.4 Students apply their mathematical knowledge to solve problems
arising in everyday life, society, and the workplace. Students may use
geometry to solve a design problem or they may use a function to describe
how one quantity of interest depends on another. Students may use
assumptions and approximations to simplify a complicated situation and
realize these may need revision later. Students identify important
relationships between quantities in a practical situation and map these
relationships using tools such as: diagrams, two-way tables, graphs,
flowcharts, and formulas. Students analyze those relationships
mathematically to draw conclusions and interpret the results in the context of
the situation. Students are reflective of the results and may improve the
model if it has not served the purpose.
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MP5 Use appropriate tools strategically.
HS.MP.5 Students consider appropriate tools when solving a mathematical
problem, including but not limited to: a) pencil and paper, b) concrete models,
c) ruler, d) protractor, e) calculator, f) spreadsheet, and g) analytical software
applications. Students familiar with mathematical tools make sound decisions
about when each of these tools may be helpful and recognize both the
insight to be gained and the limitations of the tool. Students may use a
graphing calculator to analyze graphs of functions knowing that technology
can enable them to visualize the results of varying assumptions, explore
consequences, and compare predictions with data. Students may identify
relevant external mathematical resources, such as digital content located on
a website, and use those resources to pose or solve problems. They are able
to use technological tools to explore and deepen their understanding of
concepts.
MP6 Attend to precision.
HS.MP.6 Students communicate using mathematically correct definitions in
their own reasoning and in discussions with others. They state the meaning
of symbols they choose, specify units of measure, and label axes in order to
clarify the correspondence with quantities in a problem. Students accurately
NUMBER AND QUANTITY
THE REAL NUMBER SYSTEM
Extend the properties of exponents to rational exponents.
and efficiently calculate. They express numerical answers with the degree of
precision appropriate for the problem context.
MP7 Look for and make use of structure.
HS.MP.7 Students look closely to discern a pattern or structure and
holistically consider the overview. Students may shift perspectives if needed
to gain understanding of the pattern or structure. Students in algebra may
use patterns to create equivalent expressions, factor and solve equations,
compose functions, and transform figures. They may consider certain
algebraic expressions as single objects or as being composed of several
objects. Students in geometry recognize the significance of an existing line in
a geometric figure and may use the strategy of drawing an auxiliary line for
solving problems.
MP8 Look for and express regularity in repeated
reasoning.
HS.MP.8 Students notice repeated calculations, look for general expressions
to annotate the calculation, and consider potential shortcuts. Students
maintain oversight of a process as they work to solve problems, derive
formulas, or make generalizations, while attending to details. They assess
the reasonableness of their intermediate results.
N.RN.A.1 Explain how the meaning of the definition of rational exponents follows from extending the properties of integer exponents to those values,
allowing for a notation for radicals in terms of rational exponents.
In addition to Proficient, the Advanced student is able to prove, use, and explain the properties of rational exponents (which are an extension of the
properties of integer exponents) and extend to real-world context.
The Proficient student is able to explain and use the meaning of rational exponents in terms of properties of integer exponents and use proper notation for
radicals in terms of rational exponents.
The Basic student is able to use proper notation for radicals in terms of rational exponents, but is unable to explain the meaning.
The Below Basic student may be able to use proper notation and use structure for integer exponents only.
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*N.RN.A.2 Rewrite expressions involving radicals and rational exponents using the properties of exponents.
In addition to Proficient, the Advanced student is able to compare contexts where radical form is preferable to rational exponents, and vice versa.
The Proficient student is able to rewrite expressions involving radicals and rational exponents, using the properties of exponents.
The Basic student is able to identify equivalent forms of expressions involving rational exponents (but is not able to rewrite or find the product of multiple
radical expressions).
The Below Basic student may be able to convert radical notation to rational exponent notation.
Use properties of rational and irrational numbers.
N.RN.B.3 Explain why the sum or product of rational numbers is rational; that the sum of a rational number and an irrational number is irrational; and
that the product of a nonzero rational number and an irrational number is irrational.
In addition to Proficient, the Advanced student is able to generalize the rules for sum and product properties of rational and irrational numbers.
The Proficient student is able to explain why the sum or product of rational numbers is rational; that the sum of a rational number and an irrational number
is irrational; and that the product of a nonzero rational number and an irrational number is irrational.
The Basic student is able to:
• Explain why the sum or product of rational numbers is rational. OR
• That the sum of a rational number and an irrational number is irrational. OR
• That the product of a nonzero rational number and an irrational number is irrational.
The Below Basic student may be able to explain why adding and multiplying two rational numbers results in a rational number.
QUANTITIES
Reason quantitatively and use units to solve problems.
*N.Q.C.1 Use units as a way to understand problems and to guide the solution of multi-step problems; choose and interpret units consistently in
formulas; and choose and interpret the scale and the origin in graphs and data displays.
In addition to Proficient, the Advanced student is able to explain or defend their use of units as a way to understand problems and to guide the solution of
multi-step problems; to explain and/or defend their choice of units consistently in formulas; and/or to explain and/or defend their choice of the scale and
the origin in graphs and data displays.
The Proficient student is able to use units as a way to understand problems and to guide the solution of multi-step problems; choose and interpret units
consistently in formulas; and choose and interpret the scale and the origin in graphs and data displays.
The Basic student inconsistently uses units as a way to understand problems and/or to guide the solution of problems; chooses and/or interprets units
inconsistently in formulas; and/or inconsistently chooses and/or interprets the scale and the origin in graphs and data displays.
The Below Basic student may need guidance to use units as a way to understand problems and to guide the solution of problems; to choose and interpret
units in formulas; and/or to choose and interpret the scale and the origin in graphs and data displays.
*N.Q.C.2 Define appropriate quantities for the purpose of descriptive modeling.
In addition to Proficient, the Advanced student is able to explain or defend their choice of quantities for the purpose of descriptive modeling.
The Proficient student is able to define appropriate quantities for the purpose of descriptive modeling.
The Basic student inconsistently defines quantities for the purpose of descriptive modeling.
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The Below Basic student may need guidance to define quantities for the purpose of descriptive modeling.
N.Q.C.3 Choose a level of accuracy appropriate to limitations on measurement when reporting quantities.
In addition to Proficient, the Advanced student is able to explain or defend their choice of level of accuracy appropriate to limitations on measurement
when reporting quantities.
The Proficient student is able to choose a level of accuracy appropriate to limitations on measurement when reporting quantities.
The Basic student inconsistently chooses a level of accuracy appropriate to limitations on measurement when reporting quantities.
The Below Basic student needs guidance to choose a level of accuracy appropriate to limitations on measurement when reporting quantities.
THE COMPLEX NUMBER SYSTEM
Perform arithmetic operations with complex numbers.
N.CN.D.1 Know there is a complex number such that
= , and every complex number has the form + with and real.
In addition to Proficient, the Advanced student is able to make generalizations about the powers of to write complex numbers in the form + with
and being real numbers.
The Proficient student is able to know there is a complex number such that
= 1, and every complex number has the form + with and real.
The Basic student incorrectly and/or inconsistently applies 1 for and/or
.
The Below Basic student has a limited understanding of
= 1 and needs guidance to correctly use and/or
.
N.CN.D.2 Use the relation
= – and the Commutative, Associative, and Distributive Properties to add, subtract, and multiply complex numbers.
In addition to Proficient, the Advanced student is able to perform arithmetic operations (add, subtract, multiply, divide) with complex numbers to include
other powers of and + .
The Proficient student uses the relation
= 1 and the Commutative, Associative, and Distributive Properties to add, subtract, and multiply complex
numbers.
The Basic student inconsistently uses the relation
= 1 and the Commutative, Associative, and Distributive Properties to add, subtract, and multiply
complex numbers.
The Below Basic student inconsistently uses the relation
= 1 and the Commutative and Associative Properties to add and subtract complex
numbers.
N.CN.D.3 (+) Find the conjugate of a complex number; use conjugates to find moduli and quotients of complex numbers.
(+) In addition to Proficient, the Advanced student is able to simplify complex number expressions that involve a quotient and at least one other operation.
(+) The Proficient student is able to find the conjugate of a complex number; use conjugates to find moduli and quotients of complex numbers.
(+) The Basic student is able to, given the conjugate of a complex number, find the quotients of complex numbers.
(+) The Below Basic student does not meet the basic performance level.
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Represent complex numbers and their operations on the complex plane.
N.CN.E.4 (+) Represent complex numbers on the complex plane in rectangular and polar form (including real and imaginary numbers), and explain why
the rectangular and polar forms of a given complex number represent the same number.
(+) In addition to Proficient, the Advanced student is able to:
• Given a complex number in rectangular form, convert it to polar form. AND
• Given a complex number in polar form, convert it to rectangular form.
(+) The Proficient student is able to represent complex numbers on the complex plane in rectangular and polar form (including real and imaginary
numbers), and explain why the rectangular and polar forms of a given complex number represent the same number.
(+) The Basic student is able to represent complex numbers on the complex plane in rectangular form.
(+) The Below Basic student does not meet the basic performance level.
N.CN.E.5 (+) Represent addition, subtraction, multiplication, and conjugation of complex numbers geometrically on the complex plane; use properties
of this representation for computation. For example, ( +
)
= because ( +
) has modulus 2 and argument 120°.
(+) In addition to Proficient, the Advanced student is able to compare and contrast the algebraic and geometric approaches to finding the nth root of all
real numbers.
(+) The Proficient student is able to represent addition, subtraction, multiplication, and conjugation of complex numbers geometrically on the complex
plane; use properties of this representation for computation. For example, (1 +
3)
= 8 because (1 +
3) has modulus 2 and argument 120°.
(+) The Basic student is able to represent addition and subtraction of complex numbers geometrically on the complex plane; use properties of this
representation for computation.
(+) The Below Basic student does not meet the basic performance level.
N.CN.E.6 (+) Calculate the distance between numbers in the complex plane as the modulus of the difference, and the midpoint of a segment as the
average of the numbers at its endpoints.
(+) In addition to Proficient, the Advanced student is able to compare and contrast how distance and midpoint between two numbers are represented on
the Cartesian plane versus the complex plane.
(+) The Proficient student is able to calculate the distance between numbers in the complex plane as the modulus of the difference, and the midpoint of a
segment as the average of the numbers at its endpoints.
(+) The Basic student is able to calculate the distance between numbers in the complex plane as the modulus of the difference.
(+) The Below Basic student does not meet the basic performance level.
Use complex numbers in polynomial identities and equations.
*N.CN.F.7 Solve quadratic equations with real coefficients that have complex solutions.
In addition to Proficient, the Advanced student is able to:
• Write a quadratic equation with real coefficients in standard form, when given a complex solution (recognizing that complex solutions to
quadratic equations come in conjugate pairs). OR
• Determine the relationship between the solutions, the discriminant, and the graph of a quadratic equation with real coefficients.
The Proficient student is able to solve quadratic equations with real coefficients that have complex solutions.
The Basic student is able to solve quadratic equations with real coefficients that have pure imaginary solutions.
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The Below Basic student may be able to determine which graphs have real solutions and which graphs have complex solutions when given a series of
graphical representations of quadratic equations with real coefficients.
N.CN.F.8 (+) Extend polynomial identities to the complex numbers. For example, rewrite
+ as ( + )( – ).
(+) In addition to Proficient, the Advanced student is able to:
• Explore patterns in polynomial identities that are expressed with complex numbers. OR
• Compare and contrast how polynomial identities are expressed in the real and the complex number system.
(+) The Proficient student is able to extend polynomial identities to the complex numbers. For example, rewrite
+ 4 as ( + 2)( – 2).
(+) The Basic student is able to match equivalent forms of polynomial identities expressed with complex numbers. For example,
+ 4 as ( +
2)( – 2).
(+) The Below Basic student does not meet the basic performance level.
N.CN.F.9 (+) Know the Fundamental Theorem of Algebra; show that it is true for quadratic polynomials.
(+) In addition to Proficient, the Advanced student is able to explore the solutions to quadratic polynomials with complex coefficients to identify patterns
supporting the Fundamental Theorem of Algebra.
(+) The Proficient student is able to know the Fundamental Theorem of Algebra; show that it is true for quadratic polynomials.
(+) The Basic student is able to solve a given quadratic polynomial with real coefficients and discuss how the Fundamental Theorem of Algebra is
validated.
(+) The Below Basic student does not meet the basic performance level.
VECTOR AND MATRIX QUANTITIES
Represent and model with vector quantities.
N.VM.G.1 (+) Recognize vector quantities as having both magnitude and direction. Represent vector quantities by directed line segments, and use
appropriate symbols for vectors and their magnitudes (e.g., , ||, ||||, ).
(+) In addition to Proficient, the Advanced student is able to create real-world examples in different units that are represented as vectors. [This is identified
as having a cross-curricular connection to physics.]
(+) The Proficient student is able to recognize vector quantities as having both magnitude and direction. Represent vector quantities by directed line
segments, and use appropriate symbols for vectors and their magnitudes (e.g. , , ||, ||||, ).
(+) The Basic student is able to identify quantities that could be represented with a vector when given a real-world example (e.g., I drove 10 mph versus I
drove west at 10 mph).
(+) The Below Basic student does not meet the basic performance level.
N.VM.G.2 (+) Find the components of a vector by subtracting the coordinates of an initial point from the coordinates of a terminal point.
(+) In addition to Proficient, the Advanced student is able to model a real-world situation where subtracting the initial and terminal points of a vector would
be applied.
(+) The Proficient student is able to find the components of a vector by subtracting the coordinates of an initial point from the coordinates of a terminal
point.
(+) The Basic student is able to identify the initial and terminal points when a vector is represented graphically.
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(+) The Below Basic student does not meet the basic performance level.
*N.VM.G.3 (+) Solve problems involving velocity and other quantities that can be represented by vectors.
(+) In addition to Proficient, the Advanced student is able to create and solve real-world problems involving velocity and other quantities that can be
represented by vectors.
(+) The Proficient student is able to solve problems involving velocity and other quantities that can be represented by vectors.
(+) The Basic student is able to solve problems involving velocity that can be represented by vectors.
(+) The Below Basic student does not meet the basic performance level.
Perform operations on vectors.
*N.VM.H.4 (+) Add and subtract vectors.
N.VM.H.4A Add vectors end-to-end, component-wise, and by the parallelogram rule. Understand that the magnitude of a sum of two vectors is typically not the
sum of the magnitudes.
N.VM.H.4B Given two vectors in magnitude and direction form, determine the magnitude and direction of their sum.
N.VM.H.4C Understand vector subtraction – as + (– ), where (– ) is the additive inverse of w, with the same magnitude as w and pointing in the
opposite direction. Represent vector subtraction graphically by connecting the tips in the appropriate order, and perform vector subtraction component-wise.
(+) In addition to Proficient, the Advanced student is able to create a real-world problem that requires addition or subtraction of two or more vectors, find
the resultant vector, and interpret the magnitude and direction of the resultant vector.
(+) The Proficient student is able to add and subtract vectors.
A. Add vectors end-to-end, component-wise, and by the parallelogram rule. Understand that the magnitude of a sum of two vectors is typically not
the sum of the magnitudes.
B. Given two vectors in magnitude and direction form, determine the magnitude and direction of their sum.
C. Understand vector subtraction – as + (– ), where (– ) is the additive inverse of w, with the same magnitude as w and pointing in the
opposite direction. Represent vector subtraction graphically by connecting the tips in the appropriate order, and perform vector subtraction
component-wise.
(+) The Basic student is able to add vectors end-to-end, component-wise, and by the parallelogram rule. Understand that the magnitude of a sum of two
vectors is typically not the sum of the magnitudes.
(+) The Below Basic student does not meet the basic performance level.
N.VM.H.5 (+) Multiply a vector by a scalar.
N.VM.H.5A Represent scalar multiplication graphically by scaling vectors and possibly reversing their direction; perform scalar multiplication component-wise.
N.VM.H.5B Compute the magnitude of a scalar multiple using |||| = ||. Compute the direction of knowing that when || 0, the direction of is
either along (for > 0) or against (for < 0).
(+) In addition to Proficient, the Advanced student is able to create a real-world problem that requires scalar multiplication, find the resultant vector, and
interpret the magnitude and direction of the resultant vector.
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(+) The Proficient student is able to multiply a vector by a scalar.
A. Represent scalar multiplication graphically by scaling vectors and possibly reversing their direction; perform scalar multiplication component-
wise.
B. Compute the magnitude of a scalar multiple using |||| = ||. Compute the direction of knowing that when || 0, the direction of
is either along (for > 0) or against (for < 0).
(+) The Basic student is able to represent scalar multiplication graphically by scaling vectors and possibly reversing their direction; perform scalar
multiplication component-wise.
(+) The Below Basic student does not meet the basic performance level.
Perform operations on matrices and use matrices in applications.
*N.VM.I.6 (+) Use matrices to represent and manipulate data.
(+) In addition to Proficient, the Advanced student is able to using data from a real-world situation, create a matrix, analyze the situation, and make
decisions.
(+) The Proficient student is able to use matrices to represent and manipulate data.
(+) The Basic student is able to state what the data represents for elements in a given matrix.
(+) The Below Basic student does not meet the basic performance level.
N.VM.I.7 (+) Multiply matrices by scalars to produce new matrices.
(+) In addition to Proficient, the Advanced student is able to, in a real-world situation involving scalar multiplication, place data in a matrix, identify and
interpret the scalar, and interpret the results.
(+) The Proficient student is able to multiply matrices by scalars to produce new matrices.
(+) The Basic student is able to multiply matrices by integer scalars to produce new matrices.
(+) The Below Basic student does not meet the basic performance level.
N.VM.I.8 (+) Add, subtract, and multiply matrices of appropriate dimensions.
(+) In addition to Proficient, the Advanced student is able to apply addition, subtraction, and/or multiplication of matrices to real-world situations.
Recommendation: Allow use of technology for computations of matrices with m and n greater than 3.
(+) The Proficient student is able to add, subtract, and multiply matrices of appropriate dimensions.
Recommendation: Allow use of technology for computations of matrices with m and n greater than 3.
(+) The Basic student is able to add and subtract matrices of dimensions limited to m and n less than or equal to 3.
(+) The Below Basic student does not meet the basic performance level.
N.VM.I.9 (+) Understand that, unlike multiplication of numbers, matrix multiplication for square matrices is not a commutative operation, but still
satisfies the Associative and Distributive Properties.
(+) In addition to Proficient, the Advanced student is able to compare and contrast the Commutative, Associative, and Distributive Properties for the
addition, subtraction, and multiplication of non-square matrices versus the addition, subtraction, and multiplication of real-numbers.
(+) The Proficient student is able to understand that, unlike multiplication of numbers, matrix multiplication for square matrices is not a commutative
operation, but still satisfies the Associative and Distributive Properties.
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(+) The Basic student is able to calculate , , (), (), ( + ), + and determine which expressions are equivalent when given 2 × 2
matrices A, B, and C.
(+) The Below Basic student does not meet the basic performance level.
N.VM.I.10 (+) Understand that the zero and identity matrices play a role in matrix addition and multiplication similar to the role of 0 and 1 in the real
numbers. The determinant of a square matrix is nonzero if and only if the matrix has a multiplicative inverse.
(+) In addition to Proficient, the Advanced student is able to calculate the determinant, ||, of a square matrix. Find the inverse,
, using the
determinant, ||. Show that (
) is equal to the identity matrix (I).
Recommendation: Allow use of technology for computations of matrices with m greater than or equal to 3.
(+) The Proficient student is able to understand that the zero and identity matrices play a role in matrix addition and multiplication similar to the role of 0
and 1 in the real numbers. The determinant of a square matrix is nonzero if and only if the matrix has a multiplicative inverse.
(+) The Basic student is able to calculate , , + 0, 0 + , and determine which expressions are equivalent to when given 2 × 2 matrices , the zero
matrix (0), and the identity matrix ().
(+) The Below Basic student does not meet the basic performance level.
N.VM.I.11 (+) Multiply a vector (regarded as a matrix with one column) by a matrix of suitable dimensions to produce another vector. Work with
matrices as transformations of vectors.
(+) In addition to Proficient, the Advanced student is able to create a polygon on a coordinate grid, develop the appropriate matrix to represent the
polygon, use vectors to translate the shape, and graph the translated polygon on the same coordinate grid.
Recommendation: Use technology for polygons with more than 3 sides.
(+) The Proficient student is able to multiply a vector (regarded as a matrix with one column) by a matrix of suitable dimensions to produce another vector.
Work with matrices as transformations of vectors.
(+) The Basic student is able to multiply a vector (regarded as a matrix with one column) by a matrix of suitable dimensions to produce another vector.
(+) The Below Basic student does not meet the basic performance level.
N.VM.I.12 (+) Work with 2 X 2 matrices as transformations of the plane, and interpret the absolute value of the determinant in terms of area.
(+) In addition to Proficient, the Advanced student is able to create a polygon on a coordinate grid, develop the appropriate matrix to represent the
polygon, use vectors to transform the shape, graph the transformed polygon on the same coordinate grid, find the area for each polygon, and compare the
areas. Recommendation: Use technology for polygons with more than 3 sides.
(+) The Proficient student is able to work with 2 × 2 matrices as transformations of the plane, and interpret the absolute value of the determinant in terms
of area.
(+) The Basic student is able to identify the transformation that is created by a given 2 × 2 transformation matrix.
(+) The Below Basic student does not meet the basic performance level.
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Companion document to the 2018 Mathematics Content Standards
ALGEBRA
SEEING STRUCTURE IN EXPRESSIONS
Interpret the structure of expressions.
A.SSE.A.1 Interpret expressions that represent a quantity in terms of its context.
A.SSE.A.1A Interpret parts of an expression, such as terms, factors, and coefficients.
A.SSE.A.1B Interpret complicated expressions by viewing one or more of their parts as a single entity.
In addition to Proficient, the Advanced student is able to:
• Interpret the effect of changes made to a term, a factor, or a coefficient in an expression. OR
• Compose complicated expressions from simpler ones and decompose complicated expressions into simpler ones.
The Proficient student is able to interpret expressions that represent a quantity in terms of its context.
A. Interpret parts of an expression, such as terms, factors, and coefficients.
B. Interpret complicated expressions by viewing one or more of their parts as a single entity.
The Basic student is able to interpret expressions that represent a quantity in terms of its context by interpreting parts of an expression, such as terms,
factors, and coefficients.
The Below Basic student may be able to interpret expressions that represent a quantity in terms of its context by interpreting parts of an expression, such
as terms, factors, or coefficients.
A.SSE.A.2 Use the structure of an expression to identify ways to rewrite it.
In addition to Proficient, the Advanced student is able to explain why various forms of equivalent expressions are more advantageous in a given situation.
The Proficient student is able to use the structure of an expression to identify ways to rewrite it.
The Basic student is able to rewrite an expression into another equivalent form.
The Below Basic student may be able to determine if two given expressions are equivalent.
Write expressions in equivalent forms to solve problems.
*A.SSE.B.3 Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression.
A.SSE.B.3A Factor a quadratic expression to reveal the zeros of the function it defines.
A.SSE.B.3B Complete the square in a quadratic expression to reveal the maximum or minimum value of the function it defines.
A.SSE.B.3C Use the properties of exponents to transform expressions for exponential functions. Apply the concepts of decimal and scientific notation to solve
real-world and mathematical problems.
i. Multiply and divide numbers expressed in both decimal and scientific notation.
ii. Add and subtract numbers in scientific notation with the same integer exponent.
In addition to Proficient, the Advanced student is able to choose and produce an equivalent form of an expression to reveal and explain properties of the
quantity represented by the expression.
A. Factor a quadratic expression to reveal the zeros of the function it defines and interpret the results in a real-world context.
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B. Complete the square in a quadratic expression to reveal the maximum or minimum value of the function it defines and interpret the results in a
real-world context.
The Proficient student is able to choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by
the expression.
A. Factor a quadratic expression to reveal the zeros of the function it defines.
B. Complete the square in a quadratic expression to reveal the maximum or minimum value of the function it defines.
C. Use the properties of exponents to transform expressions for exponential functions. Apply the concepts of decimal and scientific notation to
solve real-world and mathematical problems.
I. Multiply and divide numbers expressed in both decimal and scientific notation.
II. Add and subtract numbers in scientific notation with the same integer exponent.
The Basic student is able to choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the
expression.
• Factor a quadratic expression to reveal the zeros of the function it defines. AND
• Use the properties of exponents to transform expressions for exponential functions.
The Below Basic student may be able to choose and produce an equivalent form of an expression to reveal and explain properties of the quantity
represented by the expression.
• Factor a quadratic expression to reveal the zeros of the function it defines. OR
• Use the properties of exponents to transform expressions for exponential functions.
A.SSE.B.4 Derive the formula for the sum of a finite geometric series (when the common ratio is not 1), and use the formula to solve problems.
In addition to Proficient, the Advanced student is able to:
• Use the formula for the sum of a finite geometric series (when the common ratio is not 1) to solve real-world problems. OR
• Write the series in proper summation notation.
The Proficient student is able to derive the formula for the sum of a finite geometric series (when the common ratio is not 1), and use the formula to solve
problems.
The Basic student is able to use the given formula for the sum of a finite geometric series (when the common ratio is not 1) to solve problems.
The Below Basic student may be able to:
• Write a geometric sequence as a finite geometric series and calculate its sum. AND
• Identify the common ratio and initial term of a finite geometric series (when the common ratio is not 1).
Assessment Boundary: n is less than 10 in the geometric sequence.
ARITHMETIC WITH POLYNOMIALS AND RATIONAL EXPRESSIONS
Perform arithmetic operations on polynomials.
*A.APR.C.1 Understand that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition,
subtraction, and multiplication; add, subtract, and multiply polynomials.
In addition to Proficient, the Advanced student is able to:
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• Rewrite a polynomial expression involving multiplication, and addition or subtraction, into an equivalent polynomial expression in standard form.
OR
• Generalize a pattern when adding, subtracting, and multiplying polynomials of a varying number of terms.
The Proficient student is able to understand that polynomials form a system analogous to the integers, namely, they are closed under the operations of
addition, subtraction, and multiplication; add, subtract, and multiply polynomials.
The Basic student is able to add, subtract, and multiply polynomials.
The Below Basic student may be able to add, subtract, and multiply binomials.
Understand the relationship between zeros and factors of polynomial.
A.APR.D.2 Know and apply the Remainder Theorem: For a polynomial p(x) and a number a, the remainder on division by ( – ) is (), so () = if
and only if ( – ) is a factor of ().
In addition to Proficient, the Advanced student is able to find a missing coefficient , where is a real number, of a polynomial (), when given ( – )
is a factor of () or when given ().
The Proficient student is able to know and apply the Remainder Theorem: For a polynomial () and a number a, the remainder on division by ( – ) is
() = 0 if and only if ( – ) is a factor of ().
Assessment Boundary: degree of () = 3
The Basic student is able to evaluate () and compare it to the remainder of () / ( – ). Explain the significance of the remainder.
Assessment Boundary: degree of () = 3
The Below Basic student may be able to evaluate () and compare it to the remainder of () / ( – ). Explain the significance of the remainder.
Assessment Boundary: degree of () = 2
*A.APR.D.3 Identify zeros of polynomials when suitable factorizations are available, and use the zeros to construct a rough graph of the function
defined by the polynomial.
In addition to Proficient, the Advanced student is able to given a graph of a polynomial function with integer x-intercepts, write the general form of the
polynomial in standard form, understanding that the polynomial could have a stretch or compression.
The Proficient student is able to identify zeros of polynomials when suitable factorizations are available, and use the zeros to construct a rough graph of
the function defined by the polynomial.
The Basic student is able to identify zeros of polynomials when suitable factorizations are available, and locate the zeros on a coordinate plane.
The Below Basic student may be able to match the polynomial to its factored form and its graph.
Use polynomial identities to solve problems.
A.APR.E.4 Prove polynomial identities and use them to describe numerical relationships.
In addition to Proficient, the Advanced student is able to use the structure of a polynomial identity to give real-world contextual meaning to the identity
(e.g., completing the square of a quadratic function to highlight the maximum or minimum, factoring a polynomial to highlight the zeros, or factoring a
trinomial to highlight base times width).
The Proficient student is able to prove polynomial identities and use them to describe numerical relationships.
The Basic student is able to evaluate each polynomial expression for given values, compare the results, and make a general statement concerning the
polynomials as identities.
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The Below Basic student may be able to given two polynomial identities, evaluate the polynomials at a given value to demonstrate that the polynomials
are identities.
A.APR.E.5 (+) Know and apply the Binomial Theorem for the expansion of ( + )
in powers of and for a positive integer , where and are any
numbers, with coefficients determined for example by Pascal’s Triangle.
(+) In addition to Proficient, the Advanced student is able to find specified terms in the expansion of ( + )
by applying properties of the binomial
theorem.
(+) The Proficient student is able to know and apply the Binomial Theorem for the expansion of ( + )
in powers of and for a positive integer ,
where and are any numbers, with coefficients determined for example by Pascal’s Triangle.
(+) The Basic student is able to expand ( + )
in powers of and for a positive integer , where and are any numbers, using Pascal’s Triangle.
(+) The Below Basic student does not meet the basic performance level.
Rewrite rational expressions.
()
A.APR.F.6 Rewrite simple rational expressions in different forms; write
()
in the form () + , where (), (), (), and () are polynomials with
() ()
the degree of () less than the degree of () using inspection, long division, or, for the more complicated examples, a computer algebra system. (i.e.
rewriting a rational expression as the quotient plus the remainder over divisor).
()
In addition to Proficient, the Advanced student is able to, given
()
= () + :
() ()
• Identify the missing coefficient from a polynomial () when given (), (), and (). OR
• Generalize the patterns obtained by dividing various degrees of polynomials.
()
The Proficient student is able to rewrite simple rational expressions in different forms; write
()
in the form () + , where (), (), (), and ()
() ()
are polynomials with the degree of () less than the degree of () using inspection, long division, or, for the more complicated examples, a computer
algebra system. (i.e. rewriting a rational expression as the quotient plus the remainder over divisor).
()
The Basic student is able to rewrite simple rational expressions in different forms; write
()
in the form () + , where (), (), (), and () are
() ()
polynomials with the degree of () less than the degree of () using inspection or long division where () is a quadratic and () is linear.
The Below Basic student may be able to:
()
• Match a rational expression in the form
()
to its equivalent form () + . OR
() ()
()
• Rewrite simple rational expressions in different forms; write
()
in the form () + , where (), (), (), and () are polynomials with
() ()
the degree of () less than the degree of () using inspection or long division where () is a quadratic and () is linear with assistance.
A.APR.F.7 (+) Understand that rational expressions form a system analogous to the rational numbers, closed under addition, subtraction,
multiplication, and division by a nonzero rational expression; add, subtract, multiply, and divide rational expressions.
(+) In addition to Proficient, the Advanced student is able to justify that rational expressions form a system analogous to the rational numbers, closed
under addition, subtraction, multiplication, and division by a nonzero rational expression.
(+) The Proficient student is able to understand that rational expressions form a system analogous to the rational numbers, closed under addition,
subtraction, multiplication, and division by a nonzero rational expression; add, subtract, multiply, and divide rational expressions.
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(+) The Basic student is able to add, subtract, multiply, and divide rational expressions.
(+) The Below Basic student does not meet the basic performance level.
CREATING EQUATIONS
Create equations that describe numbers or relationships.
*A.CED.G.1 Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic
functions, and simple rational and exponential functions.
In addition to Proficient, the Advanced student is able to create equations and inequalities in one variable and use them to solve problems. Include
compound inequalities arising from problems. Use interval notation to represent inequalities.
The Proficient student is able to create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear
and quadratic functions, and simple rational and exponential functions.
The Basic student is able to create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear,
quadratic, and simple exponential functions.
The Below Basic student may be able to create equations and inequalities in one variable and use them to solve problems. Include equations arising from
linear and quadratic functions.
*A.CED.G.2 Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels
and scales.
In addition to Proficient, the Advanced student is able to:
• Interpret the relationships between the graph and its corresponding equation in real-world contexts. OR
• Graph equations of the form + + = or ordered triples on the -plane. OR
• Graph equations from a real-world context of the form =
where is a real number and is greater than 0.
The Proficient student is able to create equations in two or more variables to represent relationships between quantities; graph equations on coordinate
axes with labels and scales.
The Basic student is able to:
• Create equations in two or more variables to represent relationships between quantities. OR
• Graph equations on coordinate axes with labels and scales.
The Below Basic student may be able to match a graph to its equation in two or more variables.
*A.CED.G.3 Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or
non-viable options in a modeling context.
In addition to Proficient, the Advanced student is able to examine and explain constraints and solutions to systems of equations and/or inequalities, and
interpret solutions as viable or non-viable options in a modeling context.
The Proficient student is able to represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret
solutions as viable or non-viable options in a modeling context.
The Basic student is able to:
• Interpret solutions as viable or non-viable options in a modeling context. AND
• Represent constraints by equations or inequalities, or by systems of equations and/or inequalities.
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The Below Basic student may be able to:
• Interpret solutions as viable or non-viable options in a modeling context. AND
• Represent constraints by equations or inequalities.
A.CED.G.4 Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations.
In addition to Proficient, the Advanced student is able to choose and explain reasoning for highlighting the quantity of interest, rearrange the formula, use
the rearranged formula to evaluate, and interpret the answer in a real-world context.
The Proficient student is able to rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations.
The Basic student is able to rearrange formulas to highlight a quantity of interest in two steps, using the same reasoning as in solving equations.
The Below Basic student may be able to rearrange formulas to highlight a quantity of interest in one step, using the same reasoning as in solving
equations.
REASONING WITH EQUATIONS AND INEQUALITIES
Understand solving equations as a process of reasoning and explain the reasoning.
A.REI.H.1 Explain each step in solving a simple equation as following from the equality of numbers asserted at the previous step, starting from the
assumption that the original equation has a solution. Construct a viable argument to justify a solution method.
In addition to Proficient, the Advanced student is able to critique the solution and justification of self and others in the steps in solving linear equations.
The Proficient student is able to explain each step in solving a simple equation as following from the equality of numbers asserted at the previous step,
starting from the assumption that the original equation has a solution. Construct a viable argument to justify a solution method.
The Basic student is able to informally describe the steps in solving multi-step linear equations.
The Below Basic student may be able to match steps to justifications when provided with the steps for solving multi-step linear equations.
*A.REI.H.2 Solve simple rational and radical equations in one variable, and give examples showing how extraneous solutions may arise.
In addition to Proficient, the Advanced student is able to critique the solutions of self and others for simple rational and radical equations in one variable.
The Proficient student is able to solve simple rational and radical equations in one variable, and give examples showing how extraneous solutions may
arise.
The Basic student is able to solve simple rational and radical equations in one variable with no extraneous solutions, and when given equations with
extraneous solution(s) demonstrate why the given solution(s) is/are not viable.
The Below Basic student may be able to solve simple rational and radical equations in one variable with no extraneous solutions.
Solve equations and inequalities in one variable.
*A.REI.I.3 Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters.
In addition to Proficient, the Advanced student is able to:
• Critique the solutions of self and others for linear inequalities in one variable, including equations with coefficients represented by letters. OR
• Demonstrate the solution of a linear equation or inequality in multiple ways (e.g., graphically, set notation, interval notation). OR
• Create and solve a real-world linear equation or inequality in one variable, determine its viable domain and solution set.
The Proficient student is able to solve linear equations and inequalities in one variable, including equations with coefficients represented by letters.
The Basic student is able to solve linear equations and inequalities in one variable, limited to numerical coefficients.
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The Below Basic student may be able to solve linear equations and inequalities in one variable, limited to numerical coefficients, where the variable is on
only one side of the equal or inequality sign.
*A.REI.I.4 Solve quadratic equations in one variable.
A.REI.I.4A Use the method of completing the square to transform any quadratic equation in x into an equation of the form ( – )
= that has the same
solutions.
A.REI.I.4B Solve quadratic equations by inspection (e.g., for
= 49), taking square roots, completing the square, the quadratic formula and factoring, as
appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as ± for real numbers and
.
A.REI.I.4C (+) Derive the quadratic formula from the general form of a quadratic equation.
In addition to Proficient, the Advanced student is able to:
• Critique different methods used by self and others for solving quadratic equations in one variable. OR
• Create and solve a real-world quadratic equation in one variable, determine its viable domain and solution. OR
• Explain the purpose for the method chosen to solve the quadratic equation in a real-world situation.
The Proficient student is able to solve quadratic equations in one variable.
A. Use the method of completing the square to transform any quadratic equation in into an equation of the form ( – )
= that has the same
solutions.
B. Solve quadratic equations by inspection (e.g., for
= 49), taking square roots, completing the square, the quadratic formula and factoring, as
appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as ± for real
numbers and .
C. (+) Derive the quadratic formula from the general form of a quadratic equation.
The Basic student is able to solve quadratic equations in one variable by inspection (e.g., for
= 49), taking square roots, the quadratic formula and
factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as ± for
real numbers and .
The Below Basic student may be able to solve quadratic equations in one variable, using the quadratic formula. Recognize when the quadratic formula
gives complex solutions and write them as ± for real numbers and .
Solve systems of equations.
A.REI.J.5 Prove that, given a system of two equations in two variables, replacing one equation by the sum of that equation and a multiple of the other
produces a system with the same solutions.
In addition to Proficient, the Advanced student is able to:
• Write a different system of two equations in two variables with the same solution as a given system of two equations in two variables. OR
• Create and solve, if possible, a system of two equations in two variables for a real-world context (include systems with one solution, infinitely
many solutions, or no solution).
The Proficient student is able to prove that given a system of two equations in two variables, replacing one equation by the sum of that equation and a
multiple of the other, produces a system with the same solutions.
The Basic student is able to solve a system of two equations in two variables with different coefficients resulting in one solution using the elimination
method.
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The Below Basic student may be able to solve a system of two equations in two variables with equal or opposite coefficients resulting in one solution
using the elimination method.
*A.REI.J.6 Estimate solutions graphically and determine algebraic solutions to linear systems, focusing on pairs of linear equations in two variables.
In addition to Proficient, the Advanced student is able to approximate solutions to a system of linear equations for a real-world situation using a table and
graph, then verify the solution algebraically and discuss the viability of the solution in the context of the problem.
The Proficient student is able to estimate solutions graphically and determine algebraic solutions to linear systems, focusing on pairs of linear equations
in two variables.
The Basic student is able to estimate solutions to linear systems graphically or determine algebraic solutions to linear systems, focusing on pairs of linear
equations in two variables.
The Below Basic student may be able to test a solution to the system in both original equations (graphically or algebraically).
*A.REI.J.7 Solve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically.
In addition to Proficient, the Advanced student is able to explore algebraically, graphically, and tabularly the number and type of the solutions of one linear
and one quadratic or two quadratics and describe findings.
The Proficient student is able to solve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically and
graphically.
The Basic student is able to solve a simple system consisting of a linear equation written in slope-intercept form and a quadratic equation written in
standard form in two variables algebraically.
The Below Basic student may be able to solve a simple system consisting of a linear equation written in slope-intercept form and a quadratic equation
written in standard form in two variables graphically.
A.REI.J.8 (+) Represent a system of linear equations as a single matrix equation in a vector variable.
(+) In addition to Proficient, the Advanced student is able to represent a real-world situation using a system of linear equations as a single matrix equation
in a vector variable.
(+) The Proficient student is able to represent a system of linear equations as a single matrix equation in a vector variable.
(+) The Basic student is able to identify a system of linear equations given a matrix equation in a vector variable.
(+) The Below Basic student does not meet the basic performance level.
A.REI.J.9 (+) Find the inverse of a matrix if it exists and use it to solve systems of linear equations (using technology for matrices of dimension ×
or greater).
(+) In addition to Proficient, the Advanced student is able to find the inverse of a matrix if it exists and use it to solve a real-world problem involving
systems of linear equations (using technology for matrices of dimension 3 × 3 or greater).
(+) The Proficient student is able to find the inverse of a matrix if it exists and use it to solve systems of linear equations (using technology for matrices of
dimension 3 × 3 or greater).
(+) The Basic student is able to solve systems of linear equations written as a single matrix equation in a vector variable when given the inverse of a
matrix (using technology for matrices of dimension 3 × 3 or greater).
(+) The Below Basic student does not meet the basic performance level.
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Represent and solve equations and inequalities graphically.
A.REI.K.10 Understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane.
In addition to Proficient, the Advanced student is able to create and graph two equations of different degrees that pass through the same two specific
points. Describe the relationship between the solution sets for each equation and the solution for the system.
The Proficient student is able to understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane (i.e.,
connect algebraic and graphical representations of an equation in two variables).
The Basic student is able to create and verify a set of ordered pairs that lie on the graph when given an equation.
The Below Basic student may be able to determine which ordered pairs do or do not lie on the graph of the equation when given an equation and a set of
ordered pairs.
A.REI.K.11 Explain why the x-coordinates of the points where the graphs of the equations = () and = () intersect are the solutions of the
equation () = (); find the solutions approximately, e.g., using technology to graph the functions, make tables of values, or find successive
approximations. Include cases where () and/or () are linear, polynomial, rational, absolute value, exponential, and logarithmic functions.
In addition to Proficient, the Advanced student is able to, given a table of ordered pairs and graph for both a linear = () and quadratic = () where
the intersection is a non-integer coordinate pair, describe a method to find the equations and solutions matching the depicted graphs and tables. Find the
solutions using the generated equations. Describe the accuracy of the solution algebraically and graphically. Describe how to improve the method.
The Proficient student is able to explain why the x-coordinates of the points where the graphs of the equations = () and = () intersect are the
solutions of the equation () = (); find the solutions approximately, e.g., using technology to graph the functions, make tables of values, or find
successive approximations. Include cases where () and/or () are linear, polynomial, rational, absolute value, exponential, and logarithmic functions.
The Basic student is able to explain why the x-coordinates of the points where the graphs of the equations = () and = () intersect are the
solutions of the equation f(x) = g(x); find the solutions approximately, e.g., using technology to graph the functions or make tables of values. Include cases
where () and/or () are linear, quadratic, absolute value, and exponential.
The Below Basic student may be able to explain why the x-coordinates of the points where the graphs of the equations = () and = () intersect
are the solutions of the equation () = (); find the solutions approximately, e.g., using technology to graph the functions or make tables of values.
Include cases where () and/or () are linear, quadratic, and exponential.
A.REI.K.12 Graph the solutions to a linear inequality in two variables as a half-plane (excluding the boundary in the case of a strict inequality), and
graph the solution set to a system of linear inequalities in two variables as the intersection of the corresponding half-planes.
In addition to Proficient, the Advanced student is able to:
• Write a system of linear inequalities that includes a set of specified points in a region and explain the choice of strict inequality or non-strict
inequality notation. OR
• Create a system of linear inequalities that will optimize the solution when given a real-life scenario. OR
• Write the inequalities that describe the boundaries of that scenario and discuss the optimal solution when given a graphical representation of a
real-life scenario.
The Proficient student is able to graph the solutions to a linear inequality in two variables as a half-plane (excluding the boundary in the case of a strict
inequality), and graph the solution set to a system of linear inequalities in two variables as the intersection of the corresponding half-planes.
The Basic student is able to graph the solution to a linear inequality in two variables as a half-plane (excluding the boundary in the case of strict
inequality).
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The Below Basic student may be able to match the solution to a linear inequality in two variables to its graph half-plane (excluding the boundary in the
case of strict inequality).
Companion document to the 2018 Mathematics Content Standards
FUNCTIONS
INTERPRETING FUNCTIONS
Understand the concept of a function and use function notation.
*F.IF.A.1 Understand that a function from one set (called the domain) to another set (called the range) assigns to each element of the domain exactly
one element of the range. If is a function and is an element of its domain, then () denotes the output of corresponding to the input . The graph
of is the graph of the equation = ().
In addition to Proficient, the Advanced student is able to use multiple representations to generalize ways to define functions and non-functions.
The Proficient student is able to demonstrate that a function's domain is assigned to exactly one element of the range in equations, tables, graphs, and
context.
The Basic student is able to demonstrate that a function's domain is assigned to exactly one element of the range in equations, tables, and graphs.
The Below Basic student may be able to demonstrate that a function's domain is assigned to exactly one element of the range in tables and graphs.
F.IF.A.2 Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context.
In addition to Proficient, the Advanced student is able to create context from a given domain and range and use function notation to write an equation,
graph the function, and draw a picture that models the context.
The Proficient student is able to use function notation, evaluate functions for inputs in their domain, and interpret statements that use function notation in
terms of a context.
The Basic student is able to use function notation and evaluate functions for inputs in their domain.
The Below Basic student may be able to evaluate equations for specific values and match the equation to function notation.
F.IF.A.3 Recognize that sequences are functions, sometimes defined recursively, whose domain is a subset of the integers.
In addition to Proficient, the Advanced student is able to write the explicit and recursive forms of a sequence describing linear and exponential situations.
Express the sequence graphically and in a table.
The Proficient student is able to recognize that sequences are functions, sometimes defined recursively, whose domain is a subset of the integers.
The Basic student is able to write an informal recursive description of a sequence.
The Below Basic student may be able to write the first n terms when given an informal recursive description of a sequence.
Interpret functions that arise in applications in terms of the context.
F.IF.B.4 For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and
sketch graphs showing key features given a verbal description of the relationship.
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In addition to Proficient, the Advanced student is able to graph and label the key features of the function and write the function in function notation when
given key features from a linear, exponential, or quadratic context.
The Proficient student is able to, for a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of
the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the
function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity.
The Basic student is able to, for linear, quadratic, and exponential functions that model a relationship between two quantities, interpret key features of
graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include:
intercepts; intervals where the function is increasing, decreasing, maximums and minimums; symmetries; end behavior.
The Below Basic student may be able to, for linear, quadratic, and exponential functions that model a relationship between two quantities, interpret key
features of graphs in terms of the quantities, and sketch graphs showing key features given the function. Key features include: intercepts; intervals where
the function is increasing, decreasing, maximums and minimums; symmetries; end behavior.
F.IF.B.5 Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes.
In addition to Proficient, the Advanced student is able to:
• Write and graph a function for a given context where the domain meets given parameters. Express the domain of the function using interval
and/or set notation as appropriate. OR
• Given the graph of a function, write its domain and range using interval and/or set notation as appropriate.
The Proficient student is able to relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes.
The Basic student is able to match the domain of a function to its graph and explain why the selected domain is applicable to the situation.
The Below Basic student may be able to match the domain of a function to its graph.
F.IF.B.6 Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the
rate of change from a graph.
In addition to Proficient, the Advanced student is able to:
• Analyze the difference between the rates of change of different types of functions. OR
• Compare average rates of change over different intervals of the same function. OR
• Generalize how the average rate of change differs between different function types.
The Proficient student is able to calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified
interval. Estimate the rate of change from a graph.
The Basic student is able to calculate and interpret the average rate of change of a linear, exponential, or quadratic function over a specified interval
presented as a graph, an equation, or a table.
The Below Basic student may be able to calculate and interpret the average rate of change of a linear, exponential, or quadratic function over a specified
interval presented as a graph.
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Analyze functions using different representations.
*F.IF.C.7 Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more
complicated cases.
F.IF.C.7A Graph linear and quadratic functions and show intercepts, maxima, and minima.
F.IF.C.7B Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions.
F.IF.C.7C Graph polynomial functions, identifying zeros when suitable factorizations are available, and showing end behavior.
F.IF.C.7D (+) Graph rational functions, identifying zeros and asymptotes when suitable factorizations are available, and showing end behavior.
F.IF.C.7E Graph exponential and logarithmic functions, showing intercepts and end behavior.
F.IF.C.7F (+) Graph trigonometric functions, showing period, midline, and amplitude.
In addition to Proficient, the Advanced student is able to create different representations of linear, quadratic, and exponential functions when given one of
the following representations: graphical, tabular, or algebraic. Compare and contrast all three function types identifying key features while referencing the
representations.
The Proficient student is able to graph linear, quadratic, and exponential functions expressed symbolically and show appropriate key features of the graph
showing intercepts, maxima, and minima, and end behavior.
A. Graph linear and quadratic functions and show intercepts, maxima, and minima.
B. Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions.
C. Graph polynomial functions, identifying zeros when suitable factorizations are available, and showing end behavior.
D. (+) Graph rational functions, identifying zeros and asymptotes when suitable factorizations are available, and showing end behavior.
E. Graph exponential and logarithmic functions, showing intercepts and end behavior
F. (+) Graph trigonometric functions, showing period, midline, and amplitude.
The Basic student is able to identify appropriate key features of linear, quadratic, and exponential functions from a graph showing intercepts (linear,
quadratic, and exponential), maximum or minimum (quadratic), and end behavior (linear, quadratic, and end behavior).
A. Graph linear and quadratic functions and show intercepts (linear and quadratic) and maxima or minima (quadratic).
B. Graph square root, cube root, and absolute value functions.
C. Graph polynomial functions, identifying zeros and showing end behavior.
E. Graph exponential and logarithmic functions, showing intercepts and end behavior.
D. (+) Graph rational functions, identifying zeros and asymptotes, and showing end behavior.
F. (+) Graph trigonometric functions, showing period, midline, and amplitude.
The Below Basic student may be able to match descriptions of key features of linear, quadratic, and exponential functions to the appropriate parts of the
graph including intercepts (linear, quadratic, and exponential), maximum or minimum (quadratic), and end behavior (linear, quadratic, and end behavior).
A. For linear and quadratic functions, identify intercepts (linear and quadratic) and maxima or minima (quadratic).
B. For square root, cube root, and absolute value functions identify intercepts, symmetry, and end behavior.
C. For polynomial functions, identify intercepts and end behavior.
E. For exponential and logarithmic functions, identify intercepts and end behavior.
D. (+) For rational functions, identify zeros, asymptotes, and end behavior.
F. (+) For trigonometric functions with no vertical shift, identify amplitude and period using a set of x-intercepts.
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F.IF.C.8 Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function.
F.IF.C.8A Use the process of factoring and completing the square in a quadratic function to show zeros, extreme values, and symmetry of the graph, and
interpret these in terms of a context.
F.IF.C.8B Use the properties of exponents to interpret expressions for exponential functions.
In addition to Proficient, the Advanced student is able to write a function defined by an expression in different but equivalent forms to reveal and explain
different properties of the function.
A. Compare and contrast factored, vertex, and standard form to discuss the appropriate form of various properties (zeros, extreme values, and
symmetry). Write a quadratic function in different forms to reveal the appropriate properties and compare those properties to the function's graph
or table of values in a real-world context.
B. Write an exponential function from a real-world context and use the properties of exponents to interpret the expression in the context of the
situation.
The Proficient student is able to write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the
function.
A. Use the process of factoring and completing the square in a quadratic function to show zeros, extreme values, and symmetry of the graph, and
interpret these in terms of a context.
B. Use the properties of exponents to interpret expressions for exponential functions.
The Basic student is able to write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the
function.
A. Use the process of factoring to show zeros. Compare standard and vertex forms of a quadratic function to show extreme values and symmetry
of the graph. Interpret these in terms of a context.
B. Use the properties of exponents to evaluate expressions for exponential functions.
The Below Basic student may be able to write a function defined by an expression in different but equivalent forms to reveal and explain different
properties of the function.
A. Match the factored form to its graph to identify zeros. Match vertex form of a quadratic function to its graph to show extreme values and
symmetry of the graph.
B. Match properties of exponential functions to the appropriate part of the exponential expression.
F.IF.C.9 Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal
descriptions).
In addition to Proficient, the Advanced student is able to create different representations of two functions (algebraically, graphically, numerically in tables,
or by verbal description) from a given representation. Compare and contrast properties of those functions, specifically highlighting what each
representation reveals.
The Proficient student is able to compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables,
or by verbal descriptions).
The Basic student is able to compare properties of two functions when given only two different representations at a time (algebraically, graphically,
numerically in tables, or by verbal descriptions).
The Below Basic student may be able to match properties of two functions when given only two different representations at a time (algebraically,
graphically, numerically in tables, or by verbal descriptions).
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BUILDING FUNCTIONS
Build a function that models a relationship between two quantities.
*F.BF.D.1 Write a function that describes a relationship between two quantities.
F.BF.D.1A Determine an explicit expression, a recursive process, or steps for calculation from a context.
F.BF.D.1B Recognize situations in which one quantity changes at a constant rate per unit interval relative to another.
F.BF.D.1C (+) Compose functions. For example, if () is the temperature in the atmosphere as a function of height, and () is the height of a weather
balloon as a function of time, then (()) is the temperature at the location of the weather balloon as a function of time.
In addition to Proficient, the Advanced student is able to write a function that describes a relationship between two quantities.
A. Write two or more explicit expressions to express a single sequence shown pictorially and compare their features.
B. Graphically and tabularly combine standard function types using arithmetic operations.
C. (+) Determine which field properties hold under compositions. Determine under what conditions the Commutative Property holds for
composition.
The Proficient student is able to write a function that describes a relationship between two quantities.
A. Determine an explicit expression, a recursive process, or steps for calculation from a context.
B. Combine standard function types using arithmetic operations.
C. (+) Compose functions. For example, if () is the temperature in the atmosphere as a function of height, and () is the height of a weather
balloon as a function of time, then (()) is the temperature at the location of the weather balloon as a function of time.
The Basic student is able to write a function that describes a relationship between two quantities.
A. Determine an explicit expression, a recursive process, or steps for calculation from a context for linear and exponential relationships.
B. Combine standard function types using arithmetic operations for addition, subtraction, and multiplication of binomials.
C. (+) Compose functions numerically and graphically, and interpret the solution in context.
The Below Basic student may be able to write a function that describes a relationship between two quantities.
A. Determine an explicit expression, a recursive process, or steps for calculation from a context for linear relationships.
B. Combine standard function types using arithmetic operations for addition, subtraction, and multiplication of linear binomials.
C. (+) Compose functions numerically, and interpret the solution in context.
F.BF.D.2 (+) Write arithmetic and geometric sequences both recursively and with an explicit formula, use them to model situations, and translate
between the two forms.
(+) In addition to Proficient, the Advanced student is able to:
• Compare the graphical representation, tabular representation, explicit and recursive formulas, and contextual representation for arithmetic and
geometric sequences. Draw parallels to linear and exponential functions, respectively. AND/OR
• Describe a real-world situation that can be modeled linearly or exponentially and develop the explicit or recursive formulas to model the
situation.
(+) The Proficient student is able to write arithmetic and geometric sequences both recursively and with an explicit formula, use them to model situations,
and translate between the two forms.
(+) The Basic student is able to write arithmetic and geometric sequences both recursively and with an explicit formula given a modeling situation.
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(+) The Below Basic student may be able to write an explicit or recursive formula for the model of an arithmetic and geometric sequence given the other
formula.
Build new functions from existing functions.
*F.BF.E.3 Identify the effect on the graph of replacing () by () + , (), (), and ( + ) for specific values of (both positive and negative);
find the value of given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include
recognizing even and odd functions from their graphs and algebraic expressions for them.
In addition to Proficient, the Advanced student is able to:
• Write the equation for a transformed parent function given the graph or verbal description of the transformations. OR
• Write a description of the transformations using function notation given a verbal description of the transformations of () or the original ()
graph and its transformed graph. OR
• Generalize effects of a transformation on domains and ranges, including effects on ordered pairs, when exploring all of the different types of
transformations in multiple representations of ().
The Proficient student is able to identify the effect on the graph of replacing () by () + , (), (), and ( + ) for specific values of (both
positive and negative); find the value of given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using
technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them.
The Basic student is able to identify the effect on the graph of replacing () by () + , (), and ( + ) for specific values of k (both positive and
negative); find the value of given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology.
The Below Basic student may be able to match the equation to the graph showing the effect on the graph of replacing () by () + , (), and
( + ) for specific values of (both positive and negative); write a description of the transformation. Experiment with cases and illustrate an explanation
of the effects on the graph using technology.
F.BF.E.4 Find inverse functions.
F.BF.E.4A Write an expression for the inverse of a simple, invertible function (). Understand that an inverse function can be obtained by expressing the
dependent variable of one function as the independent variable of another, as and are inverse functions, if and only if, () = and () = , for all
values of x in the domain of and all values of in the domain of .
F.BF.E.4B (+) Verify by composition that one function is the inverse of another.
F.BF.E.4C (+) Read values of an inverse function from a graph or a table, given that the function has an inverse.
F.BF.E.4D (+) Produce an invertible function from a non-invertible function by restricting the domain.
In addition to Proficient, the Advanced student is able to find inverse functions.
A. Create a model of a function and its inverse from a real-world context.
B. (+) Compare the equation, table, and graph when composing two functions (()) and (()). Determine what happens to the equations,
table values, and graphs of (()) and (()) when and are inverses.
C. (+) Create a table for two functions in such a way that one is invertible and the other is not. Create graphs for the functions and connect
features of the graph to properties of invertible functions.
D. (+) Show the relationship between the properties of a function and its inverse (domain, range, increasing, decreasing, asymptotes, intercepts).
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The Proficient student is able to find inverse functions.
A. Write an expression for the inverse of a simple, invertible function (). Understand that an inverse function can be obtained by expressing the
dependent variable of one function as the independent variable of another, as f and g are inverse functions, if and only if, () = and () = ,
for all values of in the domain of and all values of in the domain of .
B. (+) Verify by composition that one function is the inverse of another.
C. (+) Read values of an inverse function from a graph or a table, given that the function has an inverse.
D. (+) Produce an invertible function from a non-invertible function by restricting the domain.
The Basic student is able to understand inverse functions.
A. Match expressions for a function and its inverse.
B. (+) Identify the compositions that shows and are inverses.
C. (+) Identify values of an inverse function from a table, given that the function has an inverse.
D. (+) Restrict the domain of a non-invertible function to make it invertible given a graph.
The Below Basic student may be able to understand inverse functions.
A. Match the graph and/or table of a function to the graph and/or table of its inverse.
B. (+) Show that one function may be the inverse of another by using numerical composition.
C. (+) Create a table for an inverse of a function, given a function table.
D. (+) Determine if a function will have an inverse from a given graph.
F.BF.E.5 (+) Build new functions from existing functions. Understand the inverse relationship between exponents and logarithms and use this
relationship to solve problems involving logarithms and exponents.
(+) In addition to Proficient, the Advanced student is able to solve real-world problems involving logarithms and exponents.
(+) The Proficient student is able to understand the inverse relationship between exponents and logarithms and use this relationship to solve problems
involving logarithms and exponents.
(+) The Basic student is able to:
• Given an expression written in logarithmic form, write the equivalent expression in exponential form. AND
• Given an expression in exponential form, write the equivalent expression in logarithmic form.
(+) The Below Basic student does not meet the basic performance level.
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LINEAR, QUADRATIC, AND EXPONENTIAL MODELS
Construct and compare linear, quadratic, and exponential models and solve problems.
*F.LE.F.1 Distinguish between situations that can be modeled with linear functions and with exponential functions.
F.LE.F.1A Verify that linear functions grow by equal differences over equal intervals and that exponential functions grow by equal factors over equal intervals.
F.LE.F.1B Recognize situations in which one quantity changes at a constant rate per unit interval relative to another.
F.LE.F.1C Recognize situations in which a quantity grows or decays by a constant percent rate per unit interval relative to another.
In addition to Proficient, the Advanced student is able to distinguish between situations that can be modeled with linear functions and with exponential
functions.
A. Write an equation, sketch a graph, and create a table of values given linear and exponential real-world contexts and explain how the growth
changes.
B. Predict and compare values along a continuum of linear and exponential functions in real-world contexts.
C. Compare and contrast the rates of change of linear and exponential functions. Write a generalization about rates of change.
The Proficient student is able to distinguish between situations that can be modeled with linear functions and with exponential functions.
A. Verify that linear functions grow by equal differences over equal intervals and that exponential functions grow by equal factors over equal
intervals.
B. Recognize situations in which one quantity changes at a constant rate per unit interval relative to another.
C. Recognize situations in which a quantity grows or decays by a constant percent rate per unit interval relative to another.
The Basic student is able to distinguish between situations that can be modeled with linear functions and with exponential functions.
A. Demonstrate that linear functions grow by equal differences over equal intervals and that exponential functions grow by equal factors over
equal intervals from a table of values.
B. Match situations in which one quantity changes at a constant rate per unit interval relative to another.
C. Match situations in which a quantity grows or decays by a constant percent rate per unit interval relative to another.
The Below Basic student may be able to distinguish between situations that can be modeled with linear functions and with exponential functions.
A. Informally show that linear functions grow by equal differences over equal intervals by calculating rate of change for two sets of order pairs.
Informally show that exponential functions grow by equal factors over equal intervals by showing multiplicative growth for two sets of ordered
pairs.
B. Identify graphs in which one quantity changes at a constant rate per unit interval relative to another.
C. Identify graphs in which a quantity grows or decays by a constant percent rate per unit interval relative to another.
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*F.LE.F.2 Construct linear and exponential functions using a graph, a description of a relationship, or two input-output pairs (include reading these
from a table).
In addition to Proficient, the Advanced student is able to:
• Explain how exponential and linear function values differ as approaches positive or negative infinity. AND
• Relate algebraic representations of linear and exponential functions to the explicit and recursive forms of arithmetic and geometric sequences,
respectively. AND/OR
• Create a real-world scenario and develop linear or exponential tables or graphs representing the scenario.
The Proficient student is able to construct linear and exponential functions using a graph, a description of a relationship, or two input-output pairs (include
reading these from a table).
The Basic student is able to construct linear and exponential functions using two of the following representations: a graph, a description of a relationship,
or two input-output pairs (include reading these from a table).
The Below Basic student may be able to match linear and exponential functions to their graph, description of a relationship, and two input-output pairs
(include reading these from a table).
F.LE.F.3 Observe using graphs and tables that a quantity increasing exponentially eventually exceeds a quantity increasing linearly, quadratically, or
(more generally) as a polynomial function.
In addition to Proficient, the Advanced student is able to use an algebraic argument or informal proof to show that an increasing exponential function
eventually exceeds an increasing linear function.
The Proficient student is able to observe using graphs and tables that a quantity increasing exponentially eventually exceeds a quantity increasing
linearly, quadratically, or (more generally) as a polynomial function.
The Basic student is able to identify which function eventually exceeds the others when given the graphs and tables of increasing linear, quadratic, and
exponential functions.
The Below Basic student may be able to compare the output values for increasing -values of increasing linear, quadratic, and exponential functions to
determine which function has the greatest output value.
F.LE.F.4 For exponential models, express as a logarithm the solution to
= where , , and are numbers and the base is 2, 10, or ; evaluate
the logarithm using technology.
In addition to Proficient, the Advanced student is able to construct an exponential model using given contextual data. Predict the input from a graph or
table then use logarithms to find the exact solution of an independent value from the context, given a dependent value. Compare and discuss the exact
solution to a graphical or tabular solution.
The Proficient student is able to, for exponential models, express as a logarithm the solution to
= where , , and are numbers and the base is
2, 10, or ; evaluate the logarithm using technology.
The Basic student is able to, for exponential models, express as a logarithm the solution to
= where and are numbers and the base is 2, 10,
or ; evaluate the logarithm using technology.
The Below Basic student may be able to, for exponential models, express as a logarithm the solution to
= where is a real number and the base
is 2, 10, or ; evaluate the logarithm using technology.
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Interpret expressions for functions in terms of the situation they model.
F.LE.G.5 Interpret the parameters in a linear or exponential function in terms of a context.
In addition to Proficient, the Advanced student is able to model real-world scenarios with linear and exponential functions with appropriate parameters.
The Proficient student is able to interpret the parameters in a linear or exponential function in terms of a context.
The Basic student is able to identify appropriate parameters for linear or exponential functions in terms of a context.
The Below Basic student may be able to match parameters for linear or exponential functions to the appropriate parts of the graph.
TRIGONOMETRIC FUNCTIONS
Extend the domain of trigonometric functions using the unit circle.
*F.TF.H.1 (+) Understand radian measure of an angle as the length of the arc on the unit circle subtended by the angle.
(+) In addition to Proficient, the Advanced student is able to solve problems using radian measure.
(+) The Proficient student is able to demonstrate that radian measure of an angle is the length of the arc on the unit circle subtended by the angle.
(+) The Basic student is able to convert between degree and radian measures.
(+) The Below Basic student does not meet the basic performance level.
F.TF.H.2 (+) Explain how the unit circle in the coordinate plane enables the extension of trigonometric functions to all real numbers, interpreted as
radian measures of angles traversed counterclockwise around the unit circle.
(+) In addition to Proficient, the Advanced student is able to explain how the unit circle in the coordinate plane enables the extension of trigonometric
functions to all real numbers, interpreted as radian measures of angles traversed clockwise and counterclockwise around the unit circle.
(+) The Proficient student is able to explain how the unit circle in the coordinate plane enables the extension of trigonometric functions to all real numbers,
interpreted as radian measures of angles traversed counterclockwise around the unit circle.
(+) The Basic student is able to explain how the unit circle in the coordinate plane enables the extension of the sine and cosine functions to the special
angles (e.g.,
, , , ), interpreted as radian measures of angles traversed counterclockwise once around the unit circle.
(+) The Below Basic student does not meet the basic performance level.
*F.TF.H.3 (+) Use special triangles to determine geometrically the values of sine, cosine, and tangent for
, , and
, and use the unit circle to express
the values of sine, cosine, and tangent for , + , and in terms of their values for , where is any real number.
(+) In addition to Proficient, the Advanced student is able to use special triangles to determine geometrically the values of the six trigonometric functions
for
, , and
, and use the unit circle to express the values of the six trigonometric functions for , + , and 2 in terms of their values for ,
where is any real number.
(+) The Proficient student is able to use special triangles to determine geometrically the values of sine, cosine, and tangent for
, , and
, and use the
unit circle to express the values of sine, cosine, and tangent for , + , and 2 in terms of their values for , where is any real number.
(+) The Basic student is able to use special triangles to determine geometrically the values of sine and cosine for
, , and
, and use the unit circle to
express the values of sine and cosine for the reference angles of
, , and
.
(+) The Below Basic student does not meet the basic performance level.
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F.TF.H.4 (+) Use the unit circle to explain symmetry (odd and even) and periodicity of trigonometric functions.
(+) In addition to Proficient, the Advanced student is able to verify the relationships for symmetry (odd and even) of trigonometric functions holds for
values of theta outside of the first quadrant of the unit circle.
(+) The Proficient student is able to use the unit circle to explain symmetry (odd and even) and periodicity of trigonometric functions.
(+) The Basic student is able to use the unit circle to explain periodicity of trigonometric functions.
(+) The Below Basic student does not meet the basic performance level.
Model periodic phenomena with trigonometric functions.
*F.TF.I.5 (+) Choose trigonometric functions to model periodic phenomena with specified amplitude, frequency, and midline.
(+) In addition to Proficient, the Advanced student is able to model trigonometric functions of periodic phenomena by identifying amplitude, frequency, and
midline, and writing the equation when given a data set.
(+) The Proficient student is able to choose trigonometric functions to model periodic phenomena with specified amplitude, frequency, and midline.
(+) The Basic student is able to choose trigonometric functions to model periodic phenomena with specified amplitude and midline.
(+) The Below Basic student does not meet the basic performance level.
F.TF.I.6 (+) Understand that restricting a trigonometric function to a domain on which it is always increasing or always decreasing allows its inverse to
be constructed.
(+) In addition to Proficient, the Advanced student is able to identify key features of an inverse trigonometric function when given a trigonometric function
whose domain is always increasing or always decreasing.
(+) The Proficient student is able to understand that restricting a trigonometric function to a domain on which it is always increasing or always decreasing
allows its inverse to be constructed.
(+) The Basic student is able to understand that restricting the sine and cosine functions to a domain on which it is always increasing or always decreasing
allows its inverse to be constructed.
(+) The Below Basic student does not meet the basic performance level.
F.TF.I.7 (+) Use inverse functions to solve trigonometric equations that arise in modeling contexts; evaluate the solutions using technology, and
interpret them in terms of the context.
(+) In addition to Proficient, the Advanced student is able to use inverse functions to solve trigonometric equations that arise in modeling contexts;
extrapolate solutions to the model based on periodicity, and interpret them in terms of the context.
(+) The Proficient student is able to use inverse functions to solve trigonometric equations that arise in modeling contexts; evaluate the solutions using
technology, and interpret them in terms of the context.
(+) The Basic student is able to use inverse functions to solve trigonometric equations that arise in modeling contexts; evaluate the solutions using
technology.
(+) The Below Basic student does not meet the basic performance level.
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Prove and apply trigonometric identities.
F.TF.J.8 (+) Prove the Pythagorean identity
+
= and use it to find , , or , given , , or , and the quadrant of
the angle.
(+) In addition to Proficient, the Advanced student is able to prove the other two Pythagorean identities using
+
= 1.
(+) The Proficient student is able to prove the Pythagorean identity
+
= 1 and use it to find , , or , given , , or
, and the quadrant of the angle.
(+) The Basic student is able to use the Pythagorean identity
+
= 1 to find or given or and the quadrant of the angle.
(+) The Below Basic student does not meet the basic performance level.
F.TF.J.9 (+) Prove the addition and subtraction formulas for sine, cosine, and tangent and use them to solve problems.
(+) In addition to Proficient, the Advanced student is able to use the addition formula to prove multiple-angle identities and use them to solve problems.
(+) The Proficient student is able to prove the addition and subtraction formulas for sine, cosine, and tangent and use them to solve problems.
(+) The Basic student is able to use the addition and subtraction formulas for sine and cosine to solve problems.
(+) The Below Basic student does not meet the basic performance level.
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Companion document to the 2018 Mathematics Content Standards
GEOMETRY
CONGRUENCE
Experiment with transformations in the plane.
*G.CO.A.1 Apply precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line,
distance along a line, and distance around a circular arc.
In addition to Proficient, the Advanced student is able to apply and justify the use of precise definitions while synthetically and/or analytically solving
problems.
The Proficient student is able to apply precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined
notions of point, line, distance along a line, and distance around a circular arc.
The Basic student is able to define an angle, circle, perpendicular line, parallel line, and line segment in simple terms.
The Below Basic student may be able to identify an angle, circle, perpendicular line, parallel line, and line segment in simple terms given contextual
and/or illustrative choices.
G.CO.A.2 Represent transformations in the plane using, e.g., transparencies and geometry software; describe transformations as functions that take
points in the plane as inputs and give other points as outputs. Compare transformations that preserve distance and angle to those that do not (e.g.,
translation versus horizontal stretch).
In addition to Proficient, the Advanced student is able to generalize the representations of rigid transformations by using and recognizing transformations
in other areas of mathematics.
The Proficient student is able to represent transformations in the plane using, e.g., transparencies and geometry software; describe transformations as
functions that take points in the plane as inputs and give other points as outputs. Compare transformations that preserve distance and angle to those that
do not (e.g., translation versus horizontal stretch).
The Basic student is able to perform two of the following:
• Represent transformations in the plane using, e.g., transparencies and geometry software.
• Describe transformations as functions that take points in the plane as inputs and give other points as outputs.
• Compare transformations that preserve distance and angle to those that do not (e.g., translation versus horizontal stretch).
The Below Basic student may be able to perform one of the following:
• Represent transformations in the plane using, e.g., transparencies and geometry software.
• Describe transformations as functions that take points in the plane as inputs and give other points as outputs.
• Compare transformations that preserve distance and angle to those that do not (e.g., translation versus horizontal stretch).
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G.CO.A.3 Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections that carry it onto itself.
In addition to Proficient, the Advanced student is able to utilize reflective and rotational symmetry to describe irregular polygons or algebraic functions.
The Proficient student is able to, given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections that carry it onto
itself.
The Basic student is able to determine if some, but not all, shapes (rectangle, parallelogram, trapezoid, or regular polygon) have rotational and/or
reflective symmetry.
The Below Basic student may be able to choose a rectangle, parallelogram, trapezoid, or regular polygon and determine if it has rotational or reflective
symmetry and/or identify if a figure has been rotated or reflected when given a graph or picture.
G.CO.A.4 Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line
segments.
In addition to Proficient, the Advanced student is able to apply the definitions of rotations, reflections, and translations in terms of angles, circles,
perpendicular lines, parallel lines, and line segments to real-world situations.
The Proficient student is able to develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines,
and line segments.
The Basic student is able to develop definitions of rotations, reflections, and/or translations in terms of angles, circles, perpendicular lines, parallel lines,
and/or line segments.
The Below Basic student may be able to identify rotations, reflections, and/or translations in terms of angles, circles, perpendicular lines, parallel lines,
and/or line segments.
*G.CO.A.5 Given a geometric figure and a rotation, reflection, or translation, draw the transformed figure using, e.g., graph paper, tracing paper, or
geometry software. Specify a sequence of transformations that will carry a given figure onto another.
In addition to Proficient, the Advanced student is able to generate a geometric figure identifying the rotations, reflections, or translations used in its
creation. Investigate to determine an efficient sequence of transformations that will carry a given figure onto another.
The Proficient student is able to, given a geometric figure and a rotation, reflection, or translation, draw the transformed figure using, e.g., graph paper,
tracing paper, or geometry software. Specify a sequence of transformations that will carry a given figure onto another.
The Basic student is able to sketch the image of the figure when given a geometric figure and a transformation described in words, or given a sketch,
describe the transformation in writing or verbally.
The Below Basic student may be able to, given a figure, identify and describe the transformation performed on a given shape in simple terms.
Understand congruence in terms of rigid motions.
G.CO.B.6 Use geometric descriptions of rigid motions to transform figures and to predict the effect of a given rigid motion on a given figure; given two
figures, use the definition of congruence in terms of rigid motions to decide if they are congruent.
In addition to Proficient, the Advanced student is able to use the definition of congruence, in terms of rigid motions, to construct a viable argument that two
figures are congruent and/or predict the effect on a given rigid motion on an algebraic function.
The Proficient student is able to use geometric descriptions of rigid motions to transform figures and to predict the effect of a given rigid motion a given
figure; given two figures, use the definition of congruence in terms of rigid motions to decide if they are congruent.
The Basic student is able to recognize congruency and/or be able to predict the effect on a rigid motion on a given figure.
The Below Basic student may be able to determine if two figures are congruent and explain their reasoning.
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G.CO.B.7 Use the definition of congruence in terms of rigid motions to show that two triangles are congruent if and only if corresponding pairs of sides
and corresponding pairs of angles are congruent.
In addition to Proficient, the Advanced student is able to construct a viable argument using the definition of congruence, in terms of rigid motions, to show
that two triangles are congruent if and only if corresponding pairs of sides and corresponding pairs of angles are congruent and/or use counter-examples.
The Proficient student is able to use the definition of congruence in terms of rigid motions to show that two triangles are congruent if and only if
corresponding pairs of sides and corresponding pairs of angles are congruent.
The Basic student is able to recognize and identify that two triangles are congruent using rigid transformation.
The Below Basic student may be able to distinguish between congruent and non-congruent triangles.
G.CO.B.8 Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of congruence in terms of rigid motions.
In addition to Proficient, the Advanced student is able to explain and provide examples showing that the criteria (SSA, AAA, and SAA) do not always
prove triangles congruent.
The Proficient student is able to explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of congruence in terms of
rigid motions.
The Basic student is able to identify the criteria for triangle congruence (ASA, SAS, and/or SSS) follow from definition of congruence using rigid motions,
using tools such as rulers, protractors, distance formula, etc.
The Below Basic student may be able to identify corresponding parts in two triangles.
Prove geometric theorems.
G.CO.C.9 Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate
interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those
equidistant from the segment’s endpoints.
In addition to Proficient, the Advanced student is able to prove theorems about lines and angles using multiple representations (constructions, analytic
geometry, theorems, etc.) and/or analyze and critique proofs written by others by verifying the logic.
The Proficient student is able to prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses
parallel lines, alternate interior angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment's
endpoints.
The Basic student is able to:
• Give an informal explanation of the relationship of pairs of lines and/or angles. AND/OR
• Complete a partial proof by filling in the blanks when given either a statement or a reason.
The Below Basic student may be able to identify:
• Vertical angles are congruent;
• When a transversal crosses parallel lines, alternate interior angles are congruent; AND
• Points on a perpendicular bisector of a line segment are exactly those equidistant from the segment's endpoints.
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*G.CO.C.10 Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180 degrees; base angles of isosceles
triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a
triangle meet at a point.
In addition to Proficient, the Advanced student is able to:
• Prove theorems about triangles using multiple representations (constructions, analytic geometry, theorems, etc.) AND/OR
• Analyze and critique proofs written by others by verifying the logic.
The Proficient student is able to prove theorems about triangles. Theorems include: measure of interior angles of a triangle sum to 180 degrees; base
angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the
medians of a triangle meet at a point.
The Basic student is able to:
• Give informal explanations of triangle proofs. AND/OR
• Complete a partial proof by filling in the blanks when given either a statement or a reason.
Theorems include: measure of interior angles of a triangle sum to 180 degrees; base angles of isosceles triangles are congruent; the segment joining
midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point.
The Below Basic student may be able to identify triangle theorems. Theorems include: measure of interior angles of a triangle sum to 180 degrees; base
angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the
medians of a triangle meet at a point.
G.CO.C.11 Prove theorems about parallelograms. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a
parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals.
In addition to Proficient, the Advanced student is able to, using theorems about parallelograms and triangles, prove other conjectures about figures that
are compositions of parallelograms and/or triangles and/or circles.
The Proficient student is able to prove theorems about parallelograms. Theorems include: opposite sides are congruent, opposite angles are congruent,
the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals.
The Basic student is able to informally prove, (including by measurement or inspection) theorems about parallelograms, and:
• Complete proofs of theorems about parallelograms, AND/OR
• Order the steps of a proof of theorems about parallelograms.
The Below Basic student may be able to, given a proof and figure:
• Identify opposite and consecutive sides or angles in a figure. OR
• Identify and define congruent figures using symbols or given notation. OR
• Identify and define parallel lines using symbols or given notation. OR
• Identify and define perpendicular lines symbols or given notation. OR
• Define supplementary angles. OR
• Draw or identify diagonals of a polygon.
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Make geometric constructions.
G.CO.D.12 Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper
folding, dynamic geometric software, etc.). Copying a segment; copying an angle; bisecting a segment; bisecting an angle; constructing perpendicular
lines, including the perpendicular bisector of a line segment; and constructing a line parallel to a given line through a point not on the line.
In addition to Proficient, the Advanced student is able to use formal geometric constructions to prove theorems about parallelograms, circles, and
triangles or prove that the formal geometric construction copies a line segment; copies an angle; bisects a segment; bisects an angle; constructs
perpendicular lines, including perpendicular bisectors of a line segment; constructing a line parallel to a given line through a point not on the line.
The Proficient student is able to make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective
devices, paper folding, dynamic geometric software, etc.). Copying a segment; copying an angle; bisecting a segment; bisecting an angle; constructing
perpendicular lines, including the perpendicular bisector of a line segment; and constructing a line parallel to a given line through a point not on the line.
The Basic student is able to make some, but not all, formal geometric constructions using at least one tool or method:
• Copying a segment.
• Copying an angle.
• Bisecting a segment.
• Bisecting an angle.
• Constructing perpendicular lines, including the perpendicular bisector of a line segment.
• Constructing a line parallel to a given line through a point not on the line.
The Below Basic student may be able to make some, but not all, formal geometric constructions using at least one tool or method with assistance:
• Copying a segment.
• Copying an angle.
• Bisecting a segment.
• Bisecting an angle.
• Constructing perpendicular lines, including the perpendicular bisector of a line segment.
• Constructing a line parallel to a given line through a point not on the line.
G.CO.D.13 Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle.
In addition to Proficient, the Advanced student is able to construct an equilateral triangle, a square, and other regular polygons (e.g., pentagon, hexagon,
octagon) inscribed in a circle and justify the tools and techniques used.
The Proficient student is able to construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle.
The Basic student is able to construct an equilateral triangle, a square, or a regular hexagon inscribed in a circle.
The Below Basic student may be able to construct an equilateral triangle, a square, or a regular hexagon inscribed in a circle with assistance (e.g., video,
ordering completed steps of a construction, etc.).
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SIMILARITY, RIGHT TRIANGLES, AND TRIGONOMETRY
Understand similarity in terms of similarity transformations.
G.SRT.E.1 Verify heuristically the properties of dilations given by a center and a scale factor.
G.SRT.E.1A A dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line passing through the center unchanged.
G.SRT.E.1B The dilation of a line segment is longer or shorter in the ratio given by the scale factor.
In addition to Proficient, the Advanced student is able to prove the properties of dilations given by a center and a scale factor.
A. A dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line passing through the center unchanged.
B. The dilation of a line segment is longer or shorter in the ratio given by the scale factor.
The Proficient student is able to understand similarity in terms of similarity transformations. Verify heuristically the properties of dilations given by a center
and a scale factor. (A heuristic approach is an approach to problem solving or discovery that employs practical method that is not guaranteed to be optimal
or perfect, but is sufficient for the immediate goals.)
A. A dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line passing through the center unchanged.
B. The dilation of a line segment is longer or shorter in the ratio given by the scale factor.
The Basic student is able to demonstrate that two figures are similar using the given center of dilation and the scale factor.
The Below Basic student may be able to recognize that two figures resulting from a dilation are similar.
G.SRT.E.2 Given two figures, use the definition of similarity in terms of similarity transformations to decide if they are similar; explain using similarity
transformations the meaning of similarity for triangles as the equality of all corresponding pairs of angles and the proportionality of all corresponding
pairs of sides.
In addition to Proficient, the Advanced student is able to, given two figures, use the definition of similarity in terms of similarity transformations and other
theorems to prove that the figures are similar; paying particular attention to the equality of corresponding angle pairs and the proportionality of
corresponding side pairs.
The Proficient student is able to, given two figures, use the definition of similarity in terms of similarity transformations to decide if they are similar; explain
using similarity transformations the meaning of similarity for triangles as the equality of all corresponding pairs of angles and the proportionality of all
corresponding pairs of sides.
The Basic student is able to, given two similar figures, demonstrate that corresponding angle pairs are congruent and that corresponding side pairs are in
proportion using transformations and/or measurement.
The Below Basic student may be able to, given two similar figures, identify congruent corresponding angle pairs and corresponding side pairs.
G.SRT.E.3 Use the properties of similarity transformations to establish the AA criterion for two triangles to be similar.
In addition to Proficient, the Advanced student is able to prove that two triangles are similar or dissimilar using the AA criterion for two similar triangles or
prove the AA criterion.
The Proficient student is able to use the properties of similarity transformations to establish the AA criterion for two triangles to be similar.
The Basic student is able to, given two triangles, determine similarity and dissimilarity using the AA criterion.
The Below Basic student may be able to, given two similar triangles identify the two pairs of angles that are congruent.
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Prove theorems involving similarity.
G.SRT.F.4 Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and
conversely; the Pythagorean Theorem proved using triangle similarity.
In addition to Proficient, the Advanced student is able to prove theorems about other regular figures by making generalizations of triangle proofs and
theorems.
The Proficient student is able to prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two
proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity.
The Basic student is able to informally prove, (including by measurement or inspection) theorems about triangles, and:
• Complete proofs of theorems about triangles. AND/OR
• Order the steps of a proof of theorems about triangles.
Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using
triangle similarity.
The Below Basic student may be able to, with assistance:
• Informally prove, (including by measurement or inspection) theorems about triangles. OR
• Complete proofs of theorems about triangles. OR
• Order the steps of a proof of theorems about triangles.
Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using
triangle similarity.
*G.SRT.F.5 Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures.
In addition to Proficient, the Advanced student is able to use congruence and similarity criteria for triangles to solve problems and explain why a geometric
figure has been incorrectly deemed congruent or similar.
The Proficient student is able to use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures.
The Basic student is able to use congruence or similarity criteria for triangles to solve problems or to prove relationships in geometric figures.
The Below Basic student may be able to, when given specific information about similarity or congruent triangles, solve problems or identify the given
relationship for the triangles.
Define trigonometric ratios and solve problems involving right triangles.
G.SRT.G.6 Understand that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of trigonometric
ratios for acute angles.
In addition to Proficient, the Advanced student is able to using data taken from several pairs of similar right triangles, establish generalities about the sides
of right triangles and their relationship to the acute angles of said triangles.
The Proficient student is able to understand that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions
of trigonometric ratios for acute angles.
The Basic student is able to demonstrate the ability to correctly orient two similar right triangles in order to identify the relationship between sides and
angles.
The Below Basic student may be able to identify (verbally or in writing) the relationship between sides and angles when given two similar right triangles,
each similarly oriented.
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G.SRT.G.7 Explain and use the relationship between the sine and cosine of complementary angles.
In addition to Proficient, the Advanced student is able to use data about side lengths of several right triangles and the properties of similar triangles, derive
generalities about the sine and cosine relationships found.
The Proficient student is able to explain and use the relationship between the sine and cosine of complementary angles.
The Basic student is able to use the relationship between the sine and cosine of complementary angles.
The Below Basic student may be able to use the sine and cosine relationships with assistance.
*G.SRT.G.8 Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems.
In addition to Proficient, the Advanced student is able to use trigonometric ratios and the Pythagorean Theorem to solve right triangles and verify
proposed solutions in applied problems.
The Proficient student is able to use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems.
The Basic student is able to solve right triangles in applied problems when given the trigonometric ratios and the Pythagorean Theorem.
The Below Basic student may be able to identify parts of a given right triangle figure that correspond to an applied problem for use in the formula when
given trigonometric ratios and/or the Pythagorean Theorem.
Apply trigonometry to general triangles.
G.SRT.H.9 (+) Derive the formula = () for the area of a triangle by drawing an auxiliary line from a vertex perpendicular to the opposite side.
(+) In addition to Proficient, the Advanced student is able to:
• Explain if the formula, = () would be appropriate for use in a compound figure and if possible, use the formula to determine the area
of the compound figure. OR
• Use the area formula in a novel way.
(+) The Proficient student is able to derive the formula = () for the area of a triangle by drawing an auxiliary line from a vertex perpendicular to
the opposite side.
(+) The Basic student is able to complete the derivation of the formula = () by choosing appropriate steps from a provided list or correctly
ordering the steps of a completed derivation.
(+) The Below Basic student may be able to use = () to find the area of a triangle when provided the formula and a labeled figure.
G.SRT.H.10 (+) Prove the Laws of Sines and Cosines and use them to solve problems.
(+) In addition to Proficient, the Advanced student is able to:
• Demonstrate instances when using the Law of Sines or the Law of Cosines would not be appropriate. OR
• Detect errors in the work of others.
(+) The Proficient student is able to prove the Law of Sines and the Law of Cosines, and use them to solve problems.
(+) The Basic student is able to use the Law of Sines and the Law of Cosines to solve problems when provided with the laws and complete proofs of the
Law of Sines and the Law of Cosines by:
• Choosing appropriate steps from a provided list. OR
• Correctly ordering the steps of completed proofs.
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(+) The Below Basic student may be able to use the Law of Sines and the Law of Cosines to solve problems when provided with the laws or complete
proofs of the Law of Sines and the Law of Cosines by:
• Choosing appropriate steps from a provided list. OR
• Correctly ordering the steps of completed proofs.
G.SRT.H.11 (+) Understand and apply the Law of Sines and the Law of Cosines to find unknown measurements in right and non-right triangles (e.g.,
surveying problems, resultant forces).
(+) In addition to Proficient, the Advanced student is able to:
• Demonstrate instances when using the Law of Sines or Law of Cosines would not be appropriate. OR
• Detect errors in the work of others.
(+) The Proficient student is able to understand and apply the Law of Sines and the Law of Cosines to find unknown measurements in right and non-right
triangles (e.g., surveying problems, resultant forces).
(+) The Basic student is able to use the Law of Sines and the Law of Cosines to solve problems when provided with the laws.
(+) The Below Basic student may be able to use the Law of Sines and Law of Cosines to solve problems when provided with the laws when given
assistance.
CIRCLES
Understand and apply theorems about circles.
G.C.I.1 Prove that all circles are similar.
In addition to Proficient, the Advanced student is able to:
• Extend the proof that all circles are similar to other appropriate curvilinear figures. OR
• Detect errors in the work of others
The Proficient student is able to prove that all circles are similar.
The Basic student is able to explain the similar relationship between two circles.
The Below Basic student may be able to recognize that two circles are similar and describe in simple terms the relationship verbally or in writing.
*G.C.I.2 Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and
circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects
the circle.
In addition to Proficient, the Advanced student is able to apply relationships among inscribed angles, radii, and chords to solve real-world problems.
Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is
perpendicular to the tangent where the radius intersects the circle.
The Proficient student is able to identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central,
inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius
intersects the circle.
The Basic student is able to identify and describe some relationships among inscribed angles, radii, and chords. Include the relationship between central,
inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius
intersects the circle.
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The Below Basic student may be able to identify or describe some relationships among inscribed angles, radii, and chords. Include the relationship
between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent
where the radius intersects the circle.
G.C.I.3 Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle.
In addition to Proficient, the Advanced student is able to:
• Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for regular polygons inscribed in a circle. OR
• Solve real-world problems using the properties described above.
The Proficient student is able to construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed
in a circle.
The Basic student is able to construct the inscribed and circumscribed circles of a triangle, and complete proofs of properties of angles for a quadrilateral
inscribed in a circle by:
• Choosing appropriate steps from a provided list. OR
• Correctly ordering the steps of completed proofs.
The Below Basic student may be able to construct the inscribed and circumscribed circles of a triangle, or complete proofs of properties of angles for a
quadrilateral inscribed in a circle by:
• Choosing appropriate steps from a provided list. OR
• Correctly ordering the steps of completed proofs.
G.C.I.4 (+) Construct a tangent line from a point outside a given circle to the circle.
(+) In addition to Proficient, the Advanced student is able to:
• Prove the construction of a tangent line from a point outside of a circle to a point of tangency is unique. OR
• Apply this concept to solve a real-world problem.
(+) The Proficient student is able to construct a tangent line from a point outside a given circle to the circle.
(+) The Basic student is able to construct a tangent line from a point outside a given circle to the circle with assistance.
(+) The Below Basic student may be able to distinguish between tangent lines from a point outside a given circle to the circle, and lines from the same or
a different point outside a given circle to the circle, that are not tangent to a circle (i.e., identify tangency).
Find arc lengths and areas of sectors of circles.
G.C.J.5 Derive using similarity the fact that the length of the arc intercepted by an angle is proportional to the radius, and define the radian measure of
the angle as the constant of proportionality; derive the formula for the area of a sector.
In addition to Proficient, the Advanced student is able to find the constant of proportionality by comparing arc lengths and sector areas of angles with
differing radii measurements. (This is an extension of G.SRT.1 – properties of dilations).
The Proficient student is able to derive, using similarity, the fact that the length of the arc intercepted by an angle is proportional to the radius, and define
the radian measure of the angle as the constant of proportionality; derive the formula for the area of a sector.
The Basic student is able to recognize a sector as a part of the whole measure of the area of a circle with the same radius. Find the area of that sector
using the formula for the area of a sector and find the arc length of the segment created by the sector.
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The Below Basic student may be able to:
• Recognize a sector as a part of the whole measure of the area of a circle with the same radius. OR
• Find the area of that sector using the formula for the area of a sector and find the arc length of segment created by the sector.
EXPRESSING GEOMETRIC PROPERTIES WITH EQUATIONS
Translate between the geometric description and the equation for a conic section.
G.GPE.K.1 Derive the equation of a circle of given center and radius using the Pythagorean Theorem; complete the square to find the center and radius
of a circle given by an equation.
In addition to Proficient, the Advanced student is able to use algebraic techniques to draw connections between distance formula, Pythagorean Theorem,
completing the square, and transformations of functions to write the equation of a circle and to develop logical arguments for the standard form of a circle.
The Proficient student is able to derive the equation of a circle of given center and radius using the Pythagorean Theorem; complete the square to find
the center and radius of a circle given by an equation.
The Basic student is able to derive the equation of a circle of given center and radius using the Pythagorean Theorem or complete the square to find the
center and radius of a circle given by an equation.
The Below Basic student may be able to derive the equation of a circle of given center and radius using the Pythagorean Theorem with assistance or
complete the square to find the center and radius of a circle given by an equation with assistance.
G.GPE.K.2 (+) Derive the equation of a parabola given a focus and directrix.
(+) In addition to Proficient, the Advanced student is able to draw connections between standard form and conic form of parabola equations or detect
errors in others' derivation of equations.
(+) The Proficient student is able to derive the equation of a parabola given a focus and directrix.
(+) The Basic student is able to derive the equation of a parabola given a focus and directrix by:
• Choosing appropriate steps from a provided list. OR
• Correctly ordering the steps of completed proofs.
(+) The Below Basic student may be able to, with assistance, derive the equation of a parabola given a focus and directrix by:
• Choosing appropriate steps from a provided list. OR
• Correctly ordering the steps of completed proofs.
G.GPE.K.3 (+) Derive the equations of ellipses and hyperbolas given the foci, using the fact that the sum or difference of distances from the foci is
constant.
(+) In addition to Proficient, the Advanced student is able to:
• Draw connections between the standard forms and the conic forms of ellipses and hyperbolas. OR
• Detect errors in others' derivations of said equations.
(+) The Proficient student is able to derive the equations of ellipses and hyperbolas given the foci, using the fact that the sum or difference of distances
from the foci is constant.
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(+) The Basic student is able to derive the equations of ellipses and hyperbolas given the foci, using the fact that the sum or difference of distances from
the foci is constant by:
• Choosing appropriate steps from a provided list. OR
• Correctly ordering the steps of completed proofs.
(+) The Below Basic student may be able to, with assistance, derive the equations of ellipses and hyperbolas given the foci, using the fact that the sum or
difference of distances from the foci is constant by:
• Choosing appropriate steps from a provided list. OR
• Correctly ordering the steps of completed proofs.
Use coordinates to prove simple geometric theorems algebraically.
G.GPE.L.4 Use coordinates to prove simple geometric theorems algebraically.
In addition to Proficient, the Advanced student is able to:
• Use coordinates to prove complex geometric theorems. OR
• Detect errors in the proofs of others.
The Proficient student is able to use coordinates to prove simple geometric theorems algebraically.
The Basic student is able to use coordinates to prove simple geometric theorems algebraically by:
• Choosing appropriate steps from a provided list. OR
• Correctly ordering the steps of completed proofs.
The Below Basic student may be able to, with assistance, use coordinates to prove simple geometric theorems algebraically by:
• Choosing appropriate steps from a provided list. OR
• Correctly ordering the steps of completed proofs.
*G.GPE.L.5 Prove the slope criteria for parallel and perpendicular lines and use them to solve geometric problems (e.g., find the equation of a line
parallel or perpendicular to a given line that passes through a given point).
In addition to Proficient, the Advanced student is able to:
• Make observations and develop logical arguments about the relationship of lines found in real-world contexts (parallel and perpendicular).
• Analyze and critique the work of others in using and proving slope criteria.
The Proficient student is able to prove the slope criteria for parallel and perpendicular lines and use them to solve geometric problems (e.g., find the
equation of a line parallel or perpendicular to a given line that passes through a given point).
The Basic student is able to informally prove the slope criteria for parallel or perpendicular lines and use them to solve geometric problems (e.g., find the
equation of a line parallel or perpendicular to a given line that passes through a given point).
The Below Basic student may be able to verify perpendicular and parallel lines when given different representations (e.g., graphs, tables, equations) and
use them to solve geometric problems.
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G.GPE.L.6 Find the point on a directed line segment between two given points that partitions the segment in a given ratio.
In addition to Proficient, the Advanced student is able to explore and draw conclusions about patterns found among directed line segments or geometric
figures of different dimensions and different ratios (e.g., patterns in the coordinates, patterns in actual lengths, patterns when changing units).
The Proficient student is able to find the point on a directed line segment between two given points that partitions the segment in a given ratio.
The Basic student is able to find the point on a directed line segment between two given points that partitions the segment in a given common unit ratio
(e.g.,
, , ).
The Below Basic student may be able to given a directed line segment, identify the point between two given points for common parts and wholes in a
whole ratio using unit ratios (e.g.,
, , ).
*G.GPE.L.7 Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, (e.g., using the distance formula).
In addition to Proficient, the Advanced student is able to use coordinates to compute areas of polygons and verify the technique using alternative
methods.
The Proficient student is able to use coordinates to compute perimeters of polygons and areas of triangles and rectangles (e.g., using the distance
formula).
The Basic student is able to use coordinates to compute perimeters and areas of triangles and rectangles.
The Below Basic student may be able to use coordinates to compute perimeters or areas of right triangles or rectangles.
GEOMETRIC MEASUREMENT AND DIMENSION
Explain volume formulas and use them to solve problems.
G.GMD.M.1 Give an informal argument for the formulas for the circumference of a circle, area of a circle, volume of a cylinder, pyramid, and cone. Use
dissection arguments, Cavalieri’s principle, and informal limit arguments.
In addition to Proficient, the Advanced student is able to:
• Critique the method and verify the logic used by others. OR
• Use dissection arguments, Cavalieri's principle, and informal limit arguments to find the area or volume of real-world irregular figures (e.g.,
horseshoe, hand, putting green, area under a curve).
The Proficient student is able to give an informal argument for the formulas for the circumference of a circle, area of a circle, volume of a cylinder,
pyramid, and cone. Use dissection arguments, Cavalieri's principle, and informal limit arguments.
The Basic student is able to, given a figure, identify the unit shapes that could be used to determine the area and use the sum of the parts to determine a
method that approximates the area. Given a three dimensional figure, identify the unit shapes that could be used to determine the volume and use the sum
of the parts to determine a method that approximates the volume.
The Below Basic student may be able to, given a figure, identify the unit shapes that could be used to determine the area. Given a three dimensional
figure, identify the unit shapes that could be used to determine the volume.
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G.GMD.M.2 (+) Give an informal argument using Cavalieri’s Principle for the formulas for the volume of a sphere and other solid figures.
(+) In addition to Proficient, the Advanced student is able to give an informal argument using Cavalieri’s principle for the formulas for the volume of a
sphere and other solid figures and explain their use in real-world situations.
(+) The Proficient student is able to give an informal argument using Cavalieri’s principle for the formulas for the volume of a sphere and other solid
figures.
(+) The Basic student is able to, given several sliced solid figures with dimensions labeled, give an informal argument to explain why some figures have
equal volumes.
(+) The Below Basic student may be able to, given several sliced solid figures with dimensions labeled, identify the figures that have equal volumes.
*G.GMD.M.3 Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems.
In addition to Proficient, the Advanced student is able to use volume formulas for cylinders, pyramids, cones, and spheres to solve problems with
compound shapes built from cylinders, pyramids, cones, and/or spheres in real-world contexts.
The Proficient student is able to use volume formulas for cylinders, pyramids, cones, and spheres to solve problems.
The Basic student is able to use given volume formulas and shapes with all of the dimensions labeled for cylinders, pyramids, cones, and spheres to solve
problems.
The Below Basic student may be able to use given volume formulas and shapes with all of the dimensions labeled for cylinders, pyramids, cones, or
spheres to solve problems.
Visualize relationships between two-dimensional and three-dimensional objects.
G.GMD.M.4 Identify the shapes of two-dimensional cross-sections of three-dimensional objects, and identify three-dimensional objects generated by
rotations of two-dimensional objects.
In addition to Proficient, the Advanced student is able to:
• Use an irregular two-dimensional slice and generate the three dimensional object created from rotating it about an axis. OR
• Explore the slicing of commonly shaped solids and analyze the patterns you find when making different slices. OR
• Show where to slice a cube or cylinder to get a minimum and maximum number of sides of the two dimensional cross-sections.
The Proficient student is able to identify the shapes of two dimensional cross-sections of three dimensional objects, and identify three dimensional objects
generated by rotations of two-dimensional objects.
The Basic student is able to identify the shapes of two dimensional cross-sections of three dimensional objects, or identify three dimensional objects
generated by rotations of two-dimensional objects.
The Below Basic student may be able to identify the shapes of two dimensional cross-sections of three dimensional objects when sliced horizontally or
vertically, or identifies three dimensional objects generated by rotations about a horizontal or vertical edge of two-dimensional objects.
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MODELING WITH GEOMETRY
Apply geometric concepts in modeling situations.
G.MG.O.1 Use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a tree trunk or a human torso as a cylinder).
In addition to Proficient, the Advanced student is able to combine three-dimensional shapes to create a real-world object and estimate the volume.
The Proficient student is able to use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a tree trunk or a human
torso as a cylinder).
The Basic student is able to choose the appropriate combination of geometric shapes to describe a specified object.
The Below Basic student may be able to choose the appropriate geometric shape to describe a specified object.
*G.MG.O.2 Apply concepts of density based on area and volume in modeling situations (e.g., persons per square mile, BTUs per cubic foot).
In addition to Proficient, the Advanced student is able to critique the reasoning of others/self and apply concepts of density (density = mass/volume)
based on area and volume in modeling situations and compare results to real-world data, if available, to make adjustments to estimation methods.
The Proficient student is able to apply concepts of density based on area and volume in modeling situations (e.g., persons per square mile, BTUs per
cubic foot).
The Basic student is able to, given two different unit rates of density based on area or volume, compare totals when given the overall area or volume (e.g.,
given persons per square mile estimate the total for each given area and compare them, given BTUs per cubic foot estimate the total for more than one
larger volume and compare them).
The Below Basic student may be able to, given a unit rate of density based on area or volume, find an estimated total in a modeling situation (e.g., given
persons per square mile estimate the total for a given area, given BTUs per cubic foot estimate the total for a larger volume).
*G.MG.O.3 Apply geometric methods to solve design problems (e.g., designing an object or structure to satisfy physical constraints or minimize cost;
working with typographic grid systems based on ratios).
In addition to Proficient, the Advanced student is able to design an object or structure to satisfy physical constraints and minimize cost and justify the
reasoning.
The Proficient student is able to apply geometric methods to solve design problems (e.g., designing an object or structure to satisfy physical constraints
or minimize cost; working with typographic grid systems based on ratios).
The Basic student is able to, given volume and surface area formulas, (prisms, pyramids, and spheres) determine the amount of material required to
create a structure with specific physical constraints.
The Below Basic student may be able to identify which geometric attribute(s) need(s) to be calculated and use given volume and surface area formulas,
(prisms, pyramids, and spheres) to determine the amount of material required to create a structure with specific physical constraints.
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Companion document to the 2018 Mathematics Content Standards
STATISTICS AND PROBABILITY
INTERPRETING CATEGORICAL AND QUANTITATIVE DATA
Summarize, represent, and interpret data on a single count or measurement variable.
S.ID.A.1 Represent data with plots on the real number line (dot plots, histograms, and box plots) by hand or using technology.
In addition to Proficient, the Advanced student is able to compare and contrast the different data representations (dot plots, histograms, and box plots) to
determine what information can be gleaned or lost from each and justify the most appropriate representation to use.
The Proficient student is able to represent data with plots on the real number line (dot plots, histograms, and box plots) by hand or using technology.
The Basic student is able to represent data with plots on the real number line (using dot plots, histograms, or box plots).
The Below Basic student may be able to represent data with plots on the real number line (using a specified representation: dot plots and/or histograms
and/or box plots) and given a pictorial or verbal example of each type that is to be created.
*S.ID.A.2 Use statistics appropriate to the shape of the data distribution to compare center (median, mean) and spread (interquartile range, standard
deviation) of two or more different data sets.
In addition to Proficient, the Advanced student is able to find the appropriate visual representation and justify the appropriate measure of center and
spread, given two or more sets of data.
The Proficient student is able to use statistics appropriate to the shape of the data distribution to compare center (median, mean) and spread
(interquartile range, standard deviation) of two or more different data sets.
The Basic student is inconsistently able to use appropriate terminology to describe similarities and differences when comparing center (median, mean)
and spread (interquartile range, standard deviation) of two or more different data sets.
The Below Basic student may be able to informally compare the similarities and differences when comparing center (median, mean) and spread
(interquartile range, standard deviation) of two or more different data sets.
S.ID.A.3 Interpret differences in shape, center, and spread in the context of the data sets, accounting for possible effects of extreme data points
(outliers).
In addition to Proficient, the Advanced student is able to given different shapes, context, and statistics for sets of data, discern and predict the differences
caused by omission or inclusion of extreme data points (outliers) in data sets, including those with uniform or near uniform values.
The Proficient student is able to interpret differences in shape, center, and spread in the context of the data sets, accounting for possible effects of
extreme data points (outliers).
The Basic student is able to interpret differences in any two of the following: shape, center, or spread in the context of the data sets, accounting for
possible effects of extreme data points (outliers).
The Below Basic student may be able to when comparing different data sets using the same representation, Identify the impact of extreme data points
(outliers) on the shape (skewed, symmetrical, or constant), center (mean or median), or spread (interquartile range or standard deviation).
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S.ID.A.4 (+) Use the mean and standard deviation of a data set to fit it to a normal distribution and to estimate population percentages. Recognize that
there are data sets for which such a procedure is not appropriate. Use the Empirical Rule, calculators, spreadsheets, and/or tables to estimate areas
under the normal curve.
(+) In addition to Proficient, the Advanced student is able to, using the mean, standard deviation, other statistics, and visual representations of several
data sets, compare the visual representations to determine and justify the appropriateness of fitting to a normal curve to estimate population percentages.
(+) The Proficient student is able to use the mean and standard deviation of a data set to fit it to a normal distribution and to estimate population
percentages. Recognize that there are data sets for which such a procedure is not appropriate. Use the Empirical Rule, calculators, spreadsheets, and/or
tables to estimate areas under the normal curve.
(+) The Basic student is able to use the mean and standard deviation of a data set to attempt to fit it to a normal distribution and to estimate population
percentages, not recognizing when such procedures might not be appropriate. Use the Empirical Rule, calculators, spreadsheets, and/or tables to make
population estimates.
(+) The Below Basic student may be able to, given a normal distribution, estimate population percentages using the Empirical Rule, calculators,
spreadsheets, and/or tables.
S.ID.B.5 (+) Summarize categorical data for two categories in two-way frequency tables. Interpret relative frequencies in the context of the data
(including joint, marginal, and conditional relative frequencies). Recognize possible associations in the data, and use inferential statistical techniques
to show association.
(+) In addition to Proficient, the Advanced student is able to identify real-world problems where chi-square analysis (i.e., goodness of fit, homogeneity, and
independence) would be appropriate and use chi-square analysis to solve these problems.
(+) The Proficient student is able to summarize categorical data for two categories in two-way frequency tables. Interpret relative frequencies in the
context of the data (including joint, marginal, and conditional relative frequencies). Recognize possible associations in the data, and use inferential
statistical techniques to show association.
(+) The Basic student is able to summarize categorical data for two categories in two-way frequency tables. Interpret relative frequencies in the context of
the data (including joint, marginal, and/or conditional relative frequencies).
(+) The Below Basic student does not meet the basic performance level.
*S.ID.B.6 Represent data on two quantitative variables on a scatter plot, and describe how the variables are related.
S.ID.B.6A Use a function to describe data trends to solve problems in the context of the data. Use given functions or choose a function suggested by the
context. Emphasize linear, quadratic, and exponential models.
S.ID.B.6B (+) Informally assess the fit of a function by plotting and analyzing residuals.
S.ID.B.6C Using technology, fit a least squares linear regression function for a scatter plot that suggests a linear association.
In addition to Proficient, the Advanced student is able to:
• Determine the best model and justify by describing the pros and cons of the data representation for two quantitative variables on a scatter plot
using alternative linear, quadratic, and exponential models. AND
• Determine the effect of removing extreme values (outliers) on the model. Make an argument for or against removing extreme values (outliers).
AND
• Discuss appropriate use of the model to make predictions while attending to precision (correct data entry errors).
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The Proficient student is able to represent data on two quantitative variables on a scatter plot, and describe how the variables are related.
A. Use a function to describe data trends to solve problems in the context of the data. Use given functions or choose a function suggested by the
context. Emphasize linear, quadratic, and exponential models.
B. (+) Informally assess the fit of a function by plotting and analyzing residuals.
C. Using technology, fit a least squares linear regression function for a scatter plot that suggests a linear association.
The Basic student is able to:
• Given two models (linear, quadratic, or exponential) determine which model best represents the data by informally assessing the fit of a
function by plotting and/or analyzing residuals. OR
• Given a data set, use technology to create a scatter plot and the least squares regression function.
The Below Basic student may be able to represent data on two quantitative variables on a scatter plot and informally determine if a linear, quadratic, or
exponential is a best fit.
Interpret linear models.
*S.ID.C.7 Interpret the slope (rate of change) and the intercept (constant term) of a linear model in the context of the data.
In addition to Proficient, the Advanced student is able to:
• Make predictions using the rate of change and the constant term of a linear model in the context of the data. Determine and explain when
extrapolation is appropriate or inappropriate. OR
• Identify a data source, formulate questions about the data, and explain in context how the slope and intercept would be interpreted.
The Proficient student is able to interpret the slope (rate of change) and the intercept (constant term) of a linear model in the context of the data.
The Basic student is able to determine the slope (rate of change) and the intercept (constant term) of a linear model in the context of the data.
The Below Basic student may be able to, given the slope (rate of change) and the intercept (constant term) of a linear model, locate these in a scatter plot
of the same data.
S.ID.C.8 Compute (using technology) and interpret the correlation coefficient of a linear fit.
In addition to Proficient, the Advanced student is able to compare different scatter plots with the same correlation coefficient to determine if a linear model
best fits the data by examining shape and statistics and justify reasoning.
The Proficient student is able to compute (using technology) and interpret the correlation coefficient of a linear fit.
The Basic student is able to compute (using technology) or interpret the correlation coefficient of a linear fit.
The Below Basic student may be able to compute (using technology) or interpret the correlation coefficient of a linear fit with the assistance of written or
pictorial guided steps or video.
S.ID.C.9 Distinguish between correlation and causation.
In addition to Proficient, the Advanced student is able to:
• Support or refute claims of causation from a real-world example (e.g., newspaper, website) with the understanding that a strong correlation
does not imply causation. OR
• Research or create two sets of data and their context to demonstrate how correlation and causation could be confused. Explain the reasons for
confusion.
The Proficient student is able to distinguish between correlation and causation.
The Basic student is able to identify the existence or nonexistence of causation in the context of a correlated problem.
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The Below Basic student may be able to, when provided with two disparate examples, determine which demonstrates correlation.
MAKING INFERENCES AND JUSTIFYING CONCLUSIONS
Understand and evaluate random processes underlying statistical experiments.
S.IC.D.1 (+) Understand statistics as a process for making inferences about population parameters based on a random sample from that population.
(+) In addition to Proficient, the Advanced student is able to select a random sample from a real-world population and use statistics to make appropriate
inferences.
(+) The Proficient student is able to use statistics as a process for making inferences about population parameters based on a random sample from that
population.
(+) The Basic student is able to draw logical conclusions about a population when provided with descriptive statistics from a random sample from that
population.
(+) The Below Basic student does not meet the basic performance level.
S.IC.D.2 (+) Decide if a specified model is consistent with results from a given data-generating process, e.g., using simulation.
(+) In addition to Proficient, the Advanced student is able to generate or estimate a model consistent with results from a given data-generating process,
e.g., using simulation.
(+) The Proficient student is able to decide if a specified model is consistent with results from a given data-generating process, e.g., using simulation.
(+) The Basic student is able to match a plot for data from a specified real-world situation with a given model.
(+) The Below Basic student does not meet the basic performance level.
Make inferences and justify conclusions from sample surveys, experiments, and observational studies.
S.IC.E.3 (+) Recognize the purposes of and differences among sample surveys, experiments, and observational studies; explain how randomization
relates to each.
(+) In addition to Proficient, the Advanced student is able to draw multiple random samples to complete a survey, experiment, or observational study.
Compare and discuss the results.
(+) The Proficient student is able to recognize the purposes of and differences among sample surveys, experiments, and observational studies; explain
how randomization relates to each.
(+) The Basic student is able to explain how results can be biased if the sample is not randomly selected, e.g., a convenience sample vs. a random
sample.
(+) The Below Basic student does not meet the basic performance level.
*S.IC.E.4 (+) Use data from a sample survey to estimate a population mean or proportion; develop a margin of error through the use of simulation
models for random sampling.
(+) In addition to Proficient, the Advanced student is able to develop a confidence interval for the population mean or proportion using the data from the
sample survey and relate it to the margin of error from the simulation.
(+) The Proficient student is able to use data from a sample survey to estimate a population mean or proportion; develop a margin of error through the
use of simulation models for random sampling.
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(+) The Basic student is able to use data from a sample survey to estimate a population mean or proportion and use the formula to calculate the margin of
error.
(+) The Below Basic student does not meet the basic performance level.
S.IC.E.5 (+) Use data from a randomized experiment to compare two treatments; use simulations to decide if differences between parameters are
significant.
(+) In addition to Proficient, the Advanced student is able to do a statistical analysis (i.e., t-test) of the data from a randomized experiment to compare two
treatments. Report results to determine if the differences between the parameters are significant.
(+) The Proficient student is able to use data from a randomized experiment to compare two treatments; use simulations to decide if differences between
parameters are significant.
(+) The Basic student is able to, given a plot comparing the two treatments from a randomized experiment, construct a logical argument about whether or
not the parameters (proportion or mean) would be different.
(+) The Below Basic student does not meet the basic performance level.
*S.IC.E.6 (+) Evaluate reports based on data.
(+) In addition to Proficient, the Advanced student is able to evaluate a report discussing the sampling technique, the data collection instruments, the
assumptions of the statistical analysis used, the data analysis, and the accuracy of the conclusions drawn.
(+) The Proficient student is able to (+) evaluate reports based on data.
(+) The Basic student is able to evaluate reports by identifying the type of sampling done and comment on its appropriateness. Discuss if there is data
provided to support the conclusions.
(+) The Below Basic student does not meet the basic performance level.
CONDITIONAL PROBABILITY AND THE RULES OF PROBABILITY
Understand independence and conditional probability and use them to interpret data.
*S.CP.F.1 Describe events as subsets of a sample space (the set of outcomes) using characteristics (or categories) of the outcomes, or as unions,
intersections, or complements of other events ("or," "and," "not").
In addition to Proficient, the Advanced student is able to:
• Compare at least two different representations (e.g., Venn Diagram, two-way table, set notation, verbal description) of events as subsets of a
sample space (the set of outcomes) using characteristics (or categories) of the outcomes, or as unions, intersections, or complements of other
events ("or," "and," "not"). OR
• Develop questions that can be answered using unions, intersections, or complements.
The Proficient student is able to describe events as subsets of a sample space (the set of outcomes) using characteristics (or categories) of the
outcomes, or as unions, intersections, or complements of other events ("or," "and," "not").
The Basic student is able to describe events as subsets using three of the four characteristics:
• Outcomes.
• Unions.
• Intersection.
• Complements.
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The Below Basic student may be able to describe events as subsets using two of the four characteristics:
• Outcomes.
• Unions.
• Intersection.
• Complements.
*S.CP.F.2 (+) Understand that two events and are independent if the probability of and occurring together is the product of their probabilities,
and use this characterization to determine if they are independent.
(+) In addition to Proficient, the Advanced student is able to identify real-world situations where () and () can be used to determine if the events
and are independent by deriving the probabilities (), (), ( ) and interpret results.
(+) The Proficient student is able to understand that two events and are independent if the probability of and occurring together is the product of
their probabilities, and use this characterization to determine if they are independent.
(+) The Basic student is able to determine if events and are independent when given probabilities (), (), and ( ).
(+) The Below Basic student does not meet the basic performance level.
S.CP.F.3 (+) Understand the conditional probability of given as ( )/(), and interpret independence of and as saying that the
conditional probability of given is the same as the probability of , and the conditional probability of given is the same as the probability of .
(+) In addition to Proficient, the Advanced student is able to identify real-world situations where () and () can be used to determine if the events
and are independent by deriving the probabilities (), (), (|), (|), and ( ) and interpret results.
(+) The Proficient student is able to (+) understand the conditional probability of given as ( )/(), and interpret independence of and as
saying that the conditional probability of given is the same as the probability of , and the conditional probability of given is the same as the
probability of .
(+) The Basic student is able to determine if events & are independent when given probabilities (), (), (|), (|), and ( ).
(+) The Below Basic student does not meet the basic performance level.
S.CP.F.4 (+) Construct and interpret two-way frequency tables of data when two categories are associated with each object being classified. Use the
two-way table as a sample space to decide if events are independent and to approximate conditional probabilities.
(+) In addition to Proficient, the Advanced student is able to identify a real-world situation and collect data that is appropriate for constructing a two-way
frequency table. Construct and interpret the two-way frequency table of data when two categories are associated with each object being classified.
Analyze and describe the results.
(+) The Proficient student is able to construct and interpret two-way frequency tables of data when two categories are associated with each object being
classified. Use the two-way table as a sample space to decide if events are independent and to approximate conditional probabilities.
(+) The Basic student is able to interpret a given two-way frequency table of data when two categories are associated with each object being classified.
Use the two-way table as a sample space to decide if events are independent and to approximate conditional probabilities.
(+) The Below Basic student does not meet the basic performance level.
*S.CP.F.5 Recognize and explain the concepts of conditional probability and independence in everyday language and everyday situations.
In addition to Proficient, the Advanced student is able to identify a real-world situation that uses independence and identify a real-world situation that uses
dependence.
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The Proficient student is able to recognize and explain the concepts of conditional probability and independence in everyday language and everyday
situations.
The Basic student is able to recognize the concepts of conditional probability or independence in everyday language and everyday situations.
The Below Basic student may be able to, with assistance, recognize the concepts of conditional probability or independence in everyday language and
everyday situations.
Use the rules of probability to compute probabilities of compound events in a uniform probability model.
S.CP.G.6 (+) Find the conditional probability of given as the fraction of ’s outcomes that also belong to , and interpret the answer in terms of the
model.
(+) In addition to Proficient, the Advanced student is able to explain why (|) is different than (|) in a real-world situation.
(+) The Proficient student is able to find the conditional probability of given as the fraction of ’s outcomes that also belong to , and interpret the
answer in terms of the model.
(+) The Basic student is able to calculate conditional probability of given when given probabilities (), (), (|), (|), and ( ).
(+) The Below Basic student does not meet the basic performance level.
S.CP.G.7 (+) Apply the Addition Rule, ( ) = () + () – ( ), and interpret the answer in terms of the model.
(+) In addition to Proficient, the Advanced student is able to identify real-word situations where the addition rule would apply, derive the appropriate
probabilities, solve the problem, and interpret the results.
(+) The Proficient student is able to apply the addition rule, ( ) = () + () – ( ), and interpret the answer in terms of the model.
(+) The Basic student is able to apply the addition rule and interpret results when given probabilities (), (), and ( ).
(+) The Below Basic student does not meet the basic performance level.
S.CP.G.8 (+) Apply the general Multiplication Rule in a uniform probability model, ( ) = [()][(|)] = [()][(|)], and interpret the
answer in terms of the model.
(+) In addition to Proficient, the Advanced student is able to identify real-word situations where the multiplication rule would apply, derive the appropriate
probabilities, solve the problem and interpret the results.
(+) The Proficient student is able to apply the general Multiplication Rule in a uniform probability model, ( ) = [()] [(|)] =
[()][(|)], and interpret the answer in terms of the model.
(+) The Basic student is able to apply the multiplication rule and interpret results when given probabilities (), (), (|), and (|).
(+) The Below Basic student does not meet the basic performance level.
S.CP.G.9 (+) Use permutations and combinations to compute probabilities of compound events and solve problems.
(+) In addition to Proficient, the Advanced student is able to identify a real-world problem that requires permutations and combinations to compute the
probability of a compound event, derive the required values, solve the problem, and interpret the results.
(+) The Proficient student is able to use permutations and combinations to compute probabilities of compound events and solve problems.
(+) The Basic student is able to determine the probability of a compound event when given the values of the permutations and the combinations
applicable to the problem.
(+) The Below Basic student does not meet the basic performance level.
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USE PROBABILITY TO MAKE DECISIONS
Calculate expected values and use them to solve problems.
S.MD.H.1 (+) Define a random variable for a quantity of interest by assigning a numerical value to each event in a sample space; graph the
corresponding probability distribution using the same graphical displays as for data distributions.
(+) In addition to Proficient, the Advanced student is able to construct a statistical experiment by identifying an appropriate random variable, listing the
events in the sample space, calculating the probability distribution, and constructing the appropriate graphical display. Interpret the results.
(+) The Proficient student is able to define a random variable for a quantity of interest by assigning a numerical value to each event in a sample space;
graph the corresponding probability distribution using the same graphical displays as for data distributions.
(+) The Basic student is able to, given a random variable for a quantity of interest and the numerical value for each event in the sample space; graph the
corresponding probability distribution using the same graphical displays as for data distributions.
(+) The Below Basic student does not meet the basic performance level.
S.MD.H.2 (+) Calculate the expected value of a random variable; interpret it as the mean of the probability distribution.
(+) In addition to Proficient, the Advanced student is able to gather data for a real-world situation where expected value could be used to make an
advantageous decision (e.g., lottery ticket, card game, dice game, investments). Calculate the expected value and explain or support the decision.
(+) The Proficient student is able to calculate the expected value of a random variable; interpret it as the mean of the probability distribution.
(+) The Basic student is able to calculate expected value when given the formula () = (
(
)) and a table with
and (
).
(+) The Below Basic student does not meet the basic performance level.
*S.MD.H.3 (+) Develop a probability distribution for a random variable defined for a sample space in which theoretical probabilities can be calculated;
find the expected value. For example, find the theoretical probability distribution for the number of correct answers obtained by guessing on all five
questions of a multiple-choice test where each question has four choices, and find the expected grade under various grading schemes.
(+) In addition to Proficient, the Advanced student is able to construct a statistical experiment with a random variable and develop a table of
, (
)
values, and determine the expected value. Interpret the results.
(+) The Proficient student is able to develop a probability distribution for a random variable defined for a sample space in which theoretical probabilities
can be calculated; find the expected value. For example, find the theoretical probability distribution for the number of correct answers obtained by guessing
on all five questions of a multiple-choice test where each question has four choices, and find the expected grade under various grading schemes.
(+) The Basic student is able to develop a probability distribution for a random variable defined for a sample space in which theoretical probabilities can be
calculated.
(+) The Below Basic student does not meet the basic performance level.
S.MD.H.4 (+) Develop a probability distribution for a random variable defined for a sample space in which probabilities are assigned empirically; find
the expected value.
(+) In addition to Proficient, the Advanced student is able to simulate a random process by defining the sample space and developing the probability
distribution for a random variable. Calculate the expected value and interpret the results.
(+) The Proficient student is able to develop a probability distribution for a random variable defined for a sample space in which probabilities are assigned
empirically; find the expected value.
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(+) The Basic student is able to complete a probability distribution for a random variable from a simple experiment (e.g., a coin flip, tossing a die, drawing
a card) defined for a sample space in which probabilities are assigned empirically.
(+) The Below Basic student does not meet the basic performance level.
Use probability to evaluate outcomes of decisions.
*S.MD.I.5 (+) Weigh the possible outcomes of a decision by assigning probabilities to payoff values and finding expected values.
S.MD.I.5A Find the expected payoff for a game of chance. For example, find the expected winnings from a state lottery ticket or a game at a fast-food
restaurant.
S.MD.I.5B Evaluate and compare strategies on the basis of expected values. For example, compare a high-deductible versus a low-deductible automobile
insurance policy using various, but reasonable, chances of having a minor or a major accident.
(+) In addition to Proficient, the Advanced student is able to gather data to make the advantageous decision for a real-world situation. Explain reasoning
for the decision.
(+) The Proficient student is able to weigh the possible outcomes of a decision by assigning probabilities to payoff values and finding expected values.
A. Find the expected payoff for a game of chance. For example, find the expected winnings from a state lottery ticket or a game at a fast-food
restaurant.
B. Evaluate and compare strategies on the basis of expected values. For example, compare a high-deductible versus a low-deductible automobile
insurance policy using various, but reasonable, chances of having a minor or a major accident.
(+) The Basic student is able to weigh the possible outcomes of a decision by assigning probabilities to payoff values and/or finding expected values. Find
the expected payoff for a game of chance. For example, find the expected winnings from a state lottery ticket or a game at a fast- food restaurant.
(+) The Below Basic student does not meet the basic performance level.
Calculate expected values and use them to solve problems.
S.MD.I.6 (+) Use probabilities to make fair decisions (e.g., drawing by lots, using a random number generator).
(+) In addition to Proficient, the Advanced student is able to identify real-world situations where probability could be used to make fair and unfair
decisions. Explain the reasoning.
(+) The Proficient student is able to use probabilities to make fair decisions (e.g., drawing by lots, using a random number generator).
(+) The Basic student is able to compare given probability distributions. State which distribution would result in a fair decision.
(+) The Below Basic student does not meet the basic performance level.
*S.MD.I.7 (+) Analyze decisions and strategies using probability concepts (e.g., product testing, medical testing, pulling a hockey goalie at the end of a
game).
(+) In addition to Proficient, the Advanced student is able to analyze decisions and strategies using probability concepts, identify the advantages and
disadvantages of the possible decisions, and justify the best choice (e.g., product testing, medical testing, pulling a hockey goalie at the end of a game).
(+) The Proficient student is able to analyze decisions and strategies using probability concepts (e.g., product testing, medical testing, pulling a hockey
goalie at the end of a game).
(+) The Basic student is able to analyze decisions and strategies using probability concepts when given the probabilities (e.g., product testing, medical
testing, pulling a hockey goalie at the end of a game).
(+) The Below Basic student does not meet the basic performance level.
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Snapshot of the set of Math Performance Standards (PS).
The PS are the subset of Proficient PLDs found throughout this document in blue highlight.
GRADE K MATH PERFORMANCE STANDARDS
*K.CC.A.1 The Proficient student is able to: A1. Count to 100 by ones,
starting at one; A2. Count to 100 by multiples of ten, starting at
ten; B. Count backwards by ones from 20.
*K.CC.B.4 The Proficient student is able to count and tell the number of
objects in a range from 10 to 39.
*K.OA.D.2 The Proficient student is able to solve word problems using
objects and drawings to find sums up to 10 and differences
within 10.
*K.OA.D.3 The Proficient student is able to decompose numbers less than
or equal to 10 in more than one way.
*K.MD.G.3 The Proficient student is able to classify objects into given
categories; count the numbers of objects in each category and
sort the categories by count. (Limit category counts to be less
than or equal to 10.)
GRADE 1 MATH PERFORMANCE STANDARDS
*1.OA.A.1 The Proficient student is able to use addition and subtraction
within 20 to solve word problems involving situations of adding
to, taking from, putting together, taking apart, and comparing,
with unknowns in all positions, by using objects, drawings, or
equations with a symbol for the unknown number to represent
the problem.
*1.OA.C.6 The Proficient student is able to add and subtract within 20,
demonstrating fluency in addition and subtraction within 10. Use
strategies such as counting on; making ten using the
relationship between addition and subtraction.
*1.NBT.E.1 The Proficient student is able to extend the number sequences
to 120. In this range: A. Count forward and backward, starting at
any number less than 120; B. Read numerals; C. Write
numerals; D. Represent a number of objects with a written
numeral.
*1.NBT.G.4 The Proficient student is able to add within 100, using concrete
models or drawings and strategies based on place value: A.
Including adding a two-digit number and a one-digit number; B.
Adding a two-digit number and a multiple of 10; C. Understand
that in adding two-digit numbers, adds tens and tens, ones and
ones; and sometimes it is necessary to compose a ten: D.
Relate the strategy to a written method and explain the
reasoning used.
*1.MD.I.3A The Proficient student is able to tell and write time in hours and
half-hours using analog and digital clocks.
*1.MD.I.3B The Proficient student is able to identify U.S. coins by value
(pennies, nickels, dimes, quarters).
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GRADE 2 MATH PERFORMANCE STANDARDS
*2.OA.A.1 The Proficient student is able to use addition and subtraction
within 100 to solve one- and two-step word problems involving
situations of adding to, taking from, putting together, and taking
apart, and comparing with unknowns in all positions, by using
drawings and equations with a symbol for the unknown number
to represent the problem.
*2.OA.B.2 The Proficient student is able to fluently add and subtract
within 20 using mental strategies. By end of Grade 2, know
automatically all sums of two one-digit numbers based on
strategies.
*2.NBT.D.1 The Proficient student is able to understand that the three
digits of a three-digit number represent amounts of hundreds,
tens, and ones; and demonstrate that: A. 100 can be thought of
as a bundle of ten tens — called a “hundred;” B. The numbers
100, 200, 300, 400, 500, 600, 700, 800, 900 refer to one, two,
three, four, five, six, seven, eight, or nine hundreds (and 0 tens
and 0 ones); C. Three-digit numbers can be decomposed in
multiple ways (e.g. 524 can be decomposed as 5 hundreds, 2
tens and 4 ones or 4 hundreds, 12 tens, and 4 ones, etc.).
*2.NBT.E.7 The Proficient student is able to add and subtract within 1000,
using concrete models or drawings and strategies based on
place value, properties of addition, and/or the relationship
between addition and subtraction: A. Relate the strategy to a
written method and explain the reasoning used; B. Understand
that in adding or subtracting three-digit numbers, add or
subtract hundreds and hundreds, tens and tens, ones and
ones; C. Understand that sometimes it is necessary to compose
or decompose tens or hundreds.
*2.MD.H.8 The Proficient student is able to solve word problems up to $10
involving dollar bills, quarters, dimes, nickels, and pennies,
using $ (dollars) and ¢ (cents) symbols appropriately.
*2.G.J.3 The Proficient student is able to partition circles and rectangles
into two, three, or four equal shares by: A. Describing the
shares using the words halves, thirds, half of, a third of, etc.; B.
Describing the whole as two halves, three thirds, four fourths; C.
Recognizing that equal shares of identical wholes need not
have the same shape.
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GRADE 3 MATH PERFORMANCE STANDARDS
*3.OA.C.7 The Proficient student is able to fluently multiply and divide
with factors 1 - 10 using mental strategies. By end of Grade 3,
know automatically all products of one-digit factors based on
strategies.
*3.OA.D.8 The Proficient student is able to solve two-step word problems
(limited to the whole number system) using the four basic
operations. Students should apply the Order of Operations
when there are no parentheses to specify a particular order. A.
Represent these problems using equations with a symbol
standing for the unknown quantity; B. Assess the
reasonableness of answers using mental computation and
estimation strategies including rounding.
*3.NBT.E.2 The Proficient student is able to fluently add and subtract
within 1000 using strategies and algorithms based on place
value, properties of addition, and/or the relationship between
addition and subtraction.
*3.NF.F.2 The Proficient student is able to understand and represent
fractions on a number line diagram. A. Represent a fraction
on
a number line diagram by defining the interval from 0 to 1 as the
whole and partitioning it into b equal parts. Recognize that each
part has size
and that the endpoint of the part based at 0
locates the number
on the number line; B. Represent a
fraction
on a number line diagram by marking off a lengths
from 0. Recognize that the resulting interval has size
and that
its endpoint locates the number
on the number line.
Assessment Boundary: Grade 3 expectations in this domain
are limited to fractions with denominators 2, 3, 4, 6, and 8.
*3.NF.F.3 The Proficient student is able to explain equivalence of
fractions in special cases, and compare fractions by reasoning
about their size. A. Understand two fractions as equivalent if
they are the same size, or the same point on a number line; B.
Recognize and generate simple equivalent fractions. Explain
why the fractions are equivalent; C. Express whole numbers as
fractions, and recognize fractions that are equivalent to whole
numbers; D. Compare two fractions with the same numerator or
the same denominator, by reasoning about their size,
Recognize that valid comparisons rely on the two fractions
referring to the same whole. Record the results of comparisons
with the symbols >, =, <, and justify the conclusions.
*3.MD.I.7 The Proficient student is able to solve real world and
mathematical problems involving perimeters of polygons,
including finding the perimeter given the side lengths, finding an
unknown side length, and exhibiting rectangles with the same
perimeter and different area or with the same area and different
perimeter.
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GRADE 4 MATH PERFORMANCE STANDARDS
*4.OA.A.3 The Proficient student is able to solve multi-step word
problems posed with whole numbers, including problems in
which remainders must be interpreted. A. Represent these
problems using equations with a letter standing for the unknown
quantity; B. Assess the reasonableness of answers using
mental computation and estimation strategies including
rounding.
*4.NBT.E.5 The Proficient student is able to use strategies based on place
value and the properties of multiplication to: A. Multiply a whole
number of up to four digits by a one-digit whole number; B.
Multiply a pair of two-digit numbers; C. Use appropriate models
to explain the calculation, such as by using equations,
rectangular arrays, ratio tables, or area models.
*4.NBT.E.6 The Proficient student is able to use strategies based on place
value, the properties of multiplication, and/or the relationship
between multiplication and division to find quotients and
remainders with up to four-digit dividends and one-digit divisors.
Use appropriate models to explain the calculation, such as by
using equations, rectangular arrays, ratio tables, or area
models.
*4.NF.F.2 The Proficient student is able to compare two fractions with
different numerators and different denominators by creating
common denominators or numerators, or by comparing to a
benchmark fraction such as
. A. Recognize that comparisons
are valid only when the two fractions refer to the same whole; B.
Record the results of comparisons with symbols >, =, <; C.
Justify the conclusions by using a visual fraction model.
Assessment Boundary: Grade 4 expectations in this domain
are limited to fractions with denominators 2, 3, 4, 6, 8, 10, 12,
and 100.
*4.NF.G.3 The Proficient student is able to understand a fraction a/b with
> 1 as a sum of unit fractions (
). A. Understand addition
and subtraction of fractions as joining and separating parts
referring to the same whole; B. Decompose a fraction into a
sum of fractions with the same denominator in more than one
way, recording each decomposition by an equation. Justify
decompositions by using a visual fraction model; C. Add and
subtract mixed numbers with like denominators by replacing
each mixed number with an equivalent fraction, and/or by using
properties of addition and the relationship between addition and
subtraction; D. Solve word problems involving addition and
subtraction of fractions referring to the same whole and having
like denominators.
*4.MD.I.3 The Proficient student is able to apply the area and perimeter
formulas for rectangles in real world and mathematical
problems.
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GRADE 5 MATH PERFORMANCE STANDARDS
*5.OA.A.1 The Proficient student is able to use parentheses, brackets, or
braces in numerical expressions, and evaluate expressions with
these symbols.
*5.NBT.D.7 The Proficient student is able to add, subtract, multiply, and
divide decimals to hundredths using concrete models or
drawings, and strategies based on place value, properties of
operations, and/or the relationship between addition and
subtraction; Relate the strategy to a written method and explain
the reasoning used.
*5.NF.E.2 The Proficient student is able to solve word problems involving
addition and subtraction of fractions referring to the same
whole, including cases of unlike denominators, e.g., by using
visual fraction models or equations to represent the problem.
Use benchmark fractions and number sense of fractions to
estimate mentally and assess the reasonableness of answers.
*5.NF.F.6 The Proficient student is able to solve real-world problems
involving multiplication of fractions and mixed numbers by using
visual fraction models or equations to represent the problem.
*5.MD.I.5 The Proficient student is able to relate volume to the
operations of multiplication and solve real world and
mathematical problems involving volume. A. Find the volume of
a right rectangular prism with whole number dimensions by
multiplying them. Show that this volume is the same as when
counting unit cubes; B. Find volumes of right rectangular prisms
with whole-number edge lengths in the context of solving real
world and mathematical problems given the formulas =
()()() and = ()() for rectangular prisms.
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GRADE 6 MATH PERFORMANCE STANDARDS
*6.RP.A.3 The Proficient student is able to: A. Make tables of equivalent
ratios relating quantities with whole number measurements and
plot the pairs of values on the coordinate plane. Use tables to
compare ratios; B. Solve unit rate problems with whole number
measurements that including those involving unit pricing and
constant speed; C. In mathematical and real world contexts
solve one-step problems involving wholes, parts, and
percentages; D. Use ratio reasoning to convert measurement
units and to transform units appropriately when multiplying or
dividing quantities in one-step problems.
*6.NS.B.1 The Proficient student is able to interpret and compute
quotients of fractions, and solve word problems involving
division of fractions by fractions by using visual fraction models
or equations to represent the problem.
*6.NS.C.3 The Proficient student is able to add, subtract, multiply, and
divide multi-digit decimals using efficient and generalizable
procedures including, but not limited to the standard algorithm
for each operation. Assessment Boundary: Limit decimals in
the given values to the hundredths place.
*6.NS.D.8 The Proficient student is able to solve real-world and
mathematical problems by graphing points in all four quadrants
of the coordinate plane. Find distances between points with the
same first coordinate or the same second coordinate; relate
absolute value and distance.
*6.EE.E.2 The Proficient student is able to write, read, and evaluate
expressions in which letters stand for numbers. A. Write two-
step algebraic expressions; B. Identify parts of an expression
using mathematical terms (sum, difference, term, product,
factor, quotient, coefficient, constant); C. Use Order of
Operations to evaluate algebraic expressions using positive
rational numbers and whole-number exponents. Include
expressions that arise from formulas relative to sixth grade
standards in real-world problems.
*6.EE.F.7 The Proficient student is able to use variables to represent
unknown numbers and write one-step expressions to represent
real-world or mathematical problems.
*6.G.H.1 The Proficient student is able to find area of right triangles,
other triangles, special quadrilaterals, and polygons by
composing into rectangles or decomposing into triangles and
other shapes; apply these techniques in the context of solving
real-world and mathematical problems.
*6.SP.J.5 The Proficient student is able to summarize numerical data
sets in relation to their real-world context. A. Report the sample
size; B. Describe the context of the data under investigation,
including how it was measured and its units of measurement; C.
Find quantitative measures of center (median, mode and mean)
and variability (range and interquartile range). Describe any
overall pattern (including outliers, clusters, and distribution),
with reference to the context in which the data was gathered; D.
Justify the choice of measures of center (median, mode, or
mean) based on the shape of the data distribution and the
context in which the data was gathered.
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GRADE 7 MATH PERFORMANCE STANDARDS
*7.RP.A.2 The Proficient student is able to recognize and represent
proportional relationships between quantities. A. Decide
whether two quantities in a table or graph are in a proportional
relationship; B. Identify the constant of proportionality (unit rate)
in tables, graphs, equations, diagrams, and verbal descriptions
of proportional relationships; C. Represent proportional
relationships with equations; D. Explain what a point (, ) on
the graph of a proportional relationship means in terms of the
situation, with special attention to the points (0, 0) and (1, )
where r is the unit rate.
*7.RP.A.3 The Proficient student is able to solve multistep real world and
mathematical problems involving ratios and percentages (e.g.,
simple interest, tax, markups and markdowns, gratuities and
commissions, fees, percent increase and decrease, percent
error).
*7.NS.B.3 The Proficient student is able to solve real-world and
mathematical problems involving the four arithmetic operations
with rational numbers. (Computations with rational numbers
extend the rules for manipulating fractions to complex fractions.)
*7.EE.D.4 The Proficient student is able to apply the concepts of linear
equations and inequalities in one variable to real-world and
mathematical situations. A. Write and fluently solve linear
equations of the form + = and ( + ) = where ,
, and are rational numbers; B. Write and solve multi-step
linear equations that include the use of the distributive property
and combining like terms. Exclude equations that contain
variables on both sides; C. Write and solve two-step linear
inequalities. Graph the solution set on a number line and
interpret its meaning; D. Identify and justify the steps for solving
multi-step linear equations and two-step linear inequalities.
*7.G.F.4 The Proficient student is able to investigate the concept of
circles. A. Demonstrate an understanding of the proportional
relationships between diameter, radius, and circumference of a
circle; B. Understand that pi is defined by the constant of
proportionality between the circumference and diameter; C.
Given the formulas for circumference and area of circles, solve
real-world and mathematical problems.
*7.G.F.6 The Proficient student is able to solve real-world and
mathematical problems involving: A. i. Find a missing dimension
when given the area of objects composed of triangles,
quadrilaterals, circles, and semi-circles; ii. Find a missing
dimension when given the surface area of objects composed of
triangles and quadrilaterals; B. Find a missing dimension when
given the volume of objects composed only of right prisms
having triangular or quadrilateral bases.
*7.SP.G.1 The Proficient student is able to solve real-world and
mathematical problems involving: A. Describing a sample that is
a subset of a population; B. Differentiating between random and
non-random sampling; C. Determining if a generalization is valid
by justifying whether or not the sample is representative of the
population; D. Determining if inferences about the population
are valid based on how the given sample was collected.
*7.SP.H.4 The Proficient student is able to given measures of center and
variability (mean, median and/or mode; range, interquartile
range, and/or standard deviation), for numerical data from
random samples, draw appropriate informal comparative
inferences about two populations.
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GRADE 8 MATH PERFORMANCE STANDARDS
*8.NS.A.1 The Proficient student is able to know that numbers that are
not rational are called irrational. Show that every number has a
decimal expansion; for rational numbers show that the decimal
expansion repeats eventually, and convert a decimal expansion
which repeats eventually into a rational number. Explore the
real number system and its appropriate usage in real-world
situations. A. Make comparisons between rational and irrational
numbers; B. Show that real numbers (excluding irrational
numbers) have a decimal expansion; C. Model the hierarchy of
the real number system, including natural, whole, integer,
rational, and irrational numbers; D. Convert repeating decimals
to fractions.
*8.EE.C.5 The Proficient student is able to graph proportional
relationships, interpreting the unit rate as the slope of the graph.
Compare two different proportional relationships represented in
different ways.
*8.EE.D.7 The Proficient student is able to extend concepts of linear
equations and inequalities in one variable to more complex
multi-step equations and inequalities in real-world and
mathematical situations. A. Solve linear equations and
inequalities with rational number coefficients that include the
use of the distributive property, combining like terms, and
variable terms on both sides; B. Recognize the three types of
solutions to linear equations: one solution, infinitely many
solutions, or no solutions; C. Generate linear equations with the
three types of solutions; D. Justify why linear equations have a
specific type of solution.
*8.EE.D.8 The Proficient student is able to analyze and solve a system of
linear equations. A. Show that solutions to a system of two
linear equations in two variables correspond to points of
intersection of their graphs, because points of intersection
satisfy both equations simultaneously, including systems with
one, infinitely many, and no solutions; B. Solve systems of two
linear equations in two variables with integer solutions by
graphing the equations; C. Solve simple real-world and
mathematical problems leading to two linear equations in two
variables given = + form with integer solutions.
*8.F.E.2 The Proficient student is able to compare properties
(intercepts, domain, and range) of two linear functions each
represented in a different way (algebraically, graphically,
numerically in tables, or by verbal descriptions).
*8.F.F.4 The Proficient student is able to apply the concepts of linear
functions to real-world and mathematical situations. A.
Recognize that the slope is the constant rate of change and the
y-intercept is the point where = 0 from an equation, graph,
table, and verbal description; B. Determine the slope and the y-
intercept of a linear function given multiple representations,
including two points, tables, graphs, equations, and verbal
descriptions; C. Construct a function in slope-intercept form that
models a linear relationship between two quantities; D. Interpret
the meaning of the slope and the y-intercept of a linear function
in the context of the situation.
*8.G.H.7 The Proficient student is able to apply the Pythagorean
Theorem to determine unknown side lengths in right triangles in
real-world and mathematical problems.
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GRADE 9-12 MATH PERFORMANCE STANDARDS – NUMBER AND QUANTITY
*N.RN.A.2 The Proficient student is able to rewrite expressions involving radicals and rational exponents, using the properties of exponents.
*N.Q.C.1 The Proficient student is able to use units as a way to understand problems and to guide the solution of multi-step problems; choose and interpret
units consistently in formulas; choose and interpret the scale and the origin in graphs and data displays.
The Proficient student is able to define appropriate quantities for the purpose of descriptive modeling.
The Proficient student is able to solve quadratic equations with real coefficients that have complex solutions.
(+) The Proficient student is able to solve problems involving velocity and other quantities that can be represented by vectors.
(+) The Proficient student is able to add and subtract vectors. A. Add vectors end-to-end, component-wise, and by the parallelogram rule.
Understand that the magnitude of a sum of two vectors is typically not the sum of the magnitudes; B. Given two vectors in magnitude and direction
form, determine the magnitude and direction of their sum; C. Understand vector subtraction – as + (– ), where (– ) is the additive inverse of
w, with the same magnitude as w and pointing in the opposite direction. Represent vector subtraction graphically by connecting the tips in the
appropriate order, and perform vector subtraction component-wise.
*N.Q.C.2
*N.CN.F.7
*N.VM.G.3
*N.VM.H.4
*N.VM.I.6 (+) The Proficient student is able to use matrices to represent and manipulate data.
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GRADE 9-12 MATH PERFORMANCE STANDARDS – ALGEBRA
*A.SSE.B.3 The Proficient student is able to choose and produce an equivalent form of an expression to reveal and explain properties of the quantity
represented by the expression. A. Factor a quadratic expression to reveal the zeros of the function it defines; B. Complete the square in a quadratic
expression to reveal the maximum or minimum value of the function it defines; C. Use the properties of exponents to transform expressions for
exponential functions. Apply the concepts of decimal and scientific notation to solve real-world and mathematical problems; I. Multiply and divide
numbers expressed in both decimal and scientific notation; II. Add and subtract numbers in scientific notation with the same integer exponent.
*A.APR.C.1 The Proficient student is able to understand that polynomials form a system analogous to the integers, namely, they are closed under the operations
of addition, subtraction, and multiplication; add, subtract, and multiply polynomials.
*A.APR.D.3 The Proficient student is able to identify zeros of polynomials when suitable factorizations are available, and use the zeros to construct a rough
graph of the function defined by the polynomial.
*A.CED.G.1 The Proficient student is able to create equations and inequalities in one variable and use them to solve problems. Include equations arising from
linear and quadratic functions, and simple rational and exponential functions.
*A.CED.G.2 The Proficient student is able to create equations in two or more variables to represent relationships between quantities; graph equations on
coordinate axes with labels and scales.
*A.CED.G.3 The Proficient student is able to represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret
solutions as viable or non-viable options in a modeling context.*
*A.REI.H.2 The Proficient student is able to solve simple rational and radical equations in one variable, and give examples showing how extraneous solutions
may arise.
The Proficient student is able to solve linear equations and inequalities in one variable, including equations with coefficients represented by letters.
The Proficient student is able to solve quadratic equations in one variable. A. Use the method of completing the square to transform any quadratic
equation in x into an equation of the form ( – )
= that has the same solutions; B. Solve quadratic equations by inspection (e.g., for
= 49),
taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the
quadratic formula gives complex solutions and write them as ± for real numbers and ; C. Derive the quadratic formula from the general form
of a quadratic equation.
*A.REI.I.3
*A.REI.I.4
*A.REI.J.6 The Proficient student is able to estimate solutions graphically and determine algebraic solutions to linear systems, focusing on pairs of linear
equations in two variables.
*A.REI.J.7 The Proficient student is able to solve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically and
graphically.
Page | 173 Wyoming Department of Education e d u . w y o m i n g . g o v / s t a n d a rd s
2021 Math Wyoming Content & Performance Standards & PLDs
GRADE 9-12 MATH PERFORMANCE STANDARDS – FUNCTIONS
*F.IF.A.1 The Proficient student is able to demonstrate that a function's domain is assigned to exactly one element of the range in equations, tables, graphs,
and context.
*F.IF.C.7 The Proficient student is able to graph linear, quadratic, and exponential functions expressed symbolically and show appropriate key features of the
graph showing intercepts, maxima, and minima, and end behavior.
*F.BF.D.1 The Proficient student is able to write a function that describes a relationship between two quantities. A. Determine an explicit expression, a
recursive process, or steps for calculation from a context; B. Combine standard function types using arithmetic operations; C. (+) Compose functions.
For example, if () is the temperature in the atmosphere as a function of height, and () is the height of a weather balloon as a function of time,
then (()) is the temperature at the location of the weather balloon as a function of time.
*F.BF.E.3 The Proficient student is able to identify the effect on the graph of replacing () by () + , (), (), and ( + ) for specific values of
(both positive and negative); find the value of k given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph
using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them.
*F.LE.F.1 The Proficient student is able to distinguish between situations that can be modeled with linear functions and with exponential functions. A. Verify
that linear functions grow by equal differences over equal intervals and that exponential functions grow by equal factors over equal intervals; B.
Recognize situations in which one quantity changes at a constant rate per unit interval relative to another; C. Recognize situations in which a quantity
grows or decays by a constant percent rate per unit interval relative to another.
*F.LE.F.2 The Proficient student is able to construct linear and exponential functions using a graph, a description of a relationship, or two input-output pairs
(include reading these from a table).
*F.TF.H.1 (+) The Proficient student is able to demonstrate that radian measure of an angle is the length of the arc on the unit circle subtended by the angle.
*F.TF.H.3
and
(+) The Proficient student is able to use special triangles to determine geometrically the values of sine, cosine, tangent for
, , and use the
unit circle to express the values of sine, cosine, and tangent for – , + , and 2– in terms of their values for , where is any real number.
*F.TF.I.5 (+) The Proficient student is able to choose trigonometric functions to model periodic phenomena with specified amplitude, frequency, and midline.
Page | 174 Wyoming Department of Education e d u . w y o m i n g . g o v / s t a n d a rd s
2021 Math Wyoming Content & Performance Standards & PLDs
GRADE 9-12 MATH PERFORMANCE STANDARDS – GEOMETRY
*G.CO.A.1 The Proficient student is able to apply precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined
notions of point, line, distance along a line, and distance around a circular arc.
*G.CO.A.5 The Proficient student is able to given a geometric figure and a rotation, reflection, or translation, draw the transformed figure using, e.g., graph
paper, tracing paper, or geometry software. Specify a sequence of transformations that will carry a given figure onto another.
*G.CO.C.10 The Proficient student is able to prove theorems about triangles. Theorems include: measure of interior angles of a triangle sum to 180 degrees;
base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the
length; the medians of a triangle meet at a point.
The Proficient student is able to use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures.
The Proficient student is able to use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems.
The Proficient student is able to identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between
central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent
where the radius intersects the circle.
*G.SRT.F.5
*G.SRT.G.8
*G.C.I.2
*G.GPE.L.5 The Proficient student is able to prove the slope criteria for parallel and perpendicular lines and use them to solve geometric problems (e.g., find the
equation of a line parallel or perpendicular to a given line that passes through a given point).
*G.GPE.L.7 The Proficient student is able to use coordinates to compute perimeters of polygons and areas of triangles and rectangles, (e.g., using the distance
formula).
*G.GMD.M.3 The Proficient student is able to use volume formulas for cylinders, pyramids, cones, and spheres to solve problems.
*G.MG.O.2 The Proficient student is able to apply concepts of density based on area and volume in modeling situations (e.g., persons per square mile, BTUs
per cubic foot).
*G.MG.O.3 The Proficient student is able to apply geometric methods to solve design problems (e.g., designing an object or structure to satisfy physical
constraints or minimize cost; working with typographic grid systems based on ratios).
Page | 175 Wyoming Department of Education e d u . w y o m i n g . g o v / s t a n d a rd s
2021 Math Wyoming Content & Performance Standards & PLDs
GRADE 9-12 MATH PERFORMANCE STANDARDS – STATISTICS AND PROBABILITY
*S.ID.A.2 The Proficient student is able to use statistics appropriate to the shape of the data distribution to compare center (median, mean) and spread
(interquartile range, standard deviation) of two or more different data sets.
*S.ID.B.6 The Proficient student is able to represent data on two quantitative variables on a scatter plot, and describe how the variables are related. A. Use a
function to describe data trends to solve problems in the context of the data. Use given functions or choose a function suggested by the context.
Emphasize linear, quadratic, and exponential models; B. Informally assess the fit of a function by plotting and analyzing residuals; C. Using
technology, fit a least squares linear regression function for a scatter plot that suggests a linear association.
*S.ID.C.7 The Proficient student is able to interpret the slope (rate of change) and the intercept (constant term) of a linear model in the context of the data.
*S.IC.E.4 (+) The Proficient student is able to use data from a sample survey to estimate a population mean or proportion; develop a margin of error through
the use of simulation models for random sampling.
*S.IC.E.6 (+) The Proficient student is able to evaluate reports based on data.
*S.CP.F.1 The Proficient student is able to describe events as subsets of a sample space (the set of outcomes) using characteristics (or categories) of the
outcomes, or as unions, intersections, or complements of other events ("or," "and," "not").
*S.CP.F.2 (+) The Proficient student is able to understand that two events A and B are independent if the probability of A and B occurring together is the
product of their probabilities, and use this characterization to determine if they are independent.
*S.CP.F.5 The Proficient student is able to recognize and explain the concepts of conditional probability and independence in everyday language and everyday
situations.
*S.MD.H.3 (+) The Proficient student is able to develop a probability distribution for a random variable defined for a sample space in which theoretical
probabilities can be calculated; find the expected value. For example, find the theoretical probability distribution for the number of correct answers
obtained by guessing on all five questions of a multiple-choice test where each question has four choices, and find the expected grade under various
grading schemes.
*S.MD.I.5 (+) The Proficient student is able to weigh the possible outcomes of a decision by assigning probabilities to payoff values and finding expected
values. A. Find the expected payoff for a game of chance. For example, find the expected winnings from a state lottery ticket or a game at a fast- food
restaurant; B. Evaluate and compare strategies on the basis of expected values. For example, compare a high-deductible versus a low-deductible
automobile insurance policy using various, but reasonable, chances of having a minor or a major accident.
*S.MD.I.7 (+) The Proficient student is able to analyze decisions and strategies using probability concepts (e.g., product testing, medical testing, pulling a
hockey goalie at the end of a game).
Page | 176 Wyoming Department of Education
e d u . w y o m i n g . g o v / s t a n d a rd s
2021 Math Wyoming Content & Performance
Standards & PLDs
CONTENT STANDARDS TIED TO PS
AND TO THE WY-TOPP BLUEPRINT
Within this document, it is important to understand
the different components related to the Math
Standards and the purpose and intent of each. Each
serves a different purpose as seen in the definitions
and tables below.
Content Standards define the content and skills
students are expected to know and be able to do by
the end of the grade band or grade level. They are
built foundationally and then in learning progressions
and therefore, ALL content standards are important
for student learning. They do not dictate what
methodology or instructional materials should be
used, nor how the material is delivered.
Performance Standards are the standards all students
are expected to learn and be assessed on through the
district assessment system by the end of the grade
band or grade level. They specify the specific degree
of understanding or demonstration of the knowledge
and/or skill for a particular content standard. As such,
they employ clear action verbs and describe “how
good is good enough.” Districts are expected to give
students multiple opportunities to demonstrate
proficiency on the Performance Standards through the
District Assessment System (DAS) and provide
appropriate supports for student success. Teachers
should provide extra focus, targeted supports, and
offer multiple opportunities to demonstrate student
understanding and proficiency. This can be done
through observation, student demonstration, and
formative, interim, benchmark, and summative
assessments.
Performance Standards are tied to the Content
Standards and have been selected for the full breadth
of standards. These may be learned and assessed over
multiple courses and years, and students will only be
held accountable for those designated in the courses
for which they receive credit. The HS Math
Performance Standards cover four school years of
content and may be met within a variety of
mathematics courses, which not every Wyoming
student will necessarily take.
WY-TOPP Math Assessment Blueprint
- The state
summative test assesses students on the depth and
breadth of the standards in each grade 3-10. For
Grade 3-8, the Blueprint covers the depth and breadth
of the grade specific standards. For High School Grade
9, the Blueprint focuses on the content students are
expected to have learned by the end of 9
th
grade and
through an Algebra I course. For High School Grade
10, the Blueprint focuses on the content students are
expected to have learned by the end of 10
th
grade and
through a Geometry course.
Page | 177 Wyoming Department of Education
e d u . w y o m i n g . g o v / s t a n d a rd s
GRADE K-2 MATH
*Identified Performance Standards (PS) for the District Assessment System (DAS).
Kindergarten
Standard *Identified PS for DAS
K.CC.A.1
YES
K.CC.A.2
K.CC.A.3
K.CC.B.4
YES
K.CC.B.5
K.CC.C.6
K.CC.C.7
K.OA.D.1
K.OA.D.2
YES
K.OA.D.3
YES
K.OA.D.4
K.OA.D.5
K.NBT.E.1
K.MD.F.1
K.MD.F.2
K.MD.G.3 YES
K.MD.G.4
K.G.H.1
K.G.H.2
K.G.H.3
K.G.I.4
K.G.I.5
1st Grade
Standard *Identified PS for DAS
1.OA.A.1
YES
1.OA.A.2
1.OA.B.3
1.OA.B.4
1.OA.C.5
1.OA.C.6
YES
1.OA.D.7
1.OA.D.8
1.NBT.E.1
YES
1.NBT.F.2
1.NBT.F.3
1.NBT.G.4
YES
1.NBT.G.5
1.NBT.G.6
1.MD.H.1
1.MD.H.2
1.MD.I.3
YES
1.MD.J.4
1.G.K.1
1.G.K.2
1.G.K.3
2nd Grade
Standard *Identified PS for DAS
2.OA.A.1
YES
2.OA.B.2
YES
2.OA.C.3
2.OA.C.4
2.NBT.D.1
YES
2.NBT.D.2
2.NBT.D.3
2.NBT.D.4
2.NBT.E.5
2.NBT.E.6
2.NBT.E.7
YES
2.NBT.E.8
2.NBT.E.9
2.MD.F.1
2.MD.F.2
2.MD.F.3
2.MD.F.4
2.MD.G.5
2.MD.G.6
2.MD.H.7
2.MD.H.8
YES
2.MD.I.9
2.MD.I.10
2.G.J.1
2.G.J.2
2.G.J.3 YES
Page | 178 Wyoming Department of Education
e d u . w y o m i n g . g o v / s t a n d a rd s
GRADE 3-5 MATH
*Identified Performance Standards (PS) for the District Assessment System (DAS).
**Standard is found on the WY-TOPP Math Blueprint.
3rd Grade
Standard
*Identified
PS for DAS
**WY-TOPP
Math
Blueprint
3.OA.A.1
YES
3.OA.A.2
YES
3.OA.A.3
YES
3.OA.A.4
YES
3.OA.B.5
YES
3.OA.B.6
YES
3.OA.C.7
YES
YES
3.OA.D.8
YES
YES
3.OA.D.9
YES
3.NBT.E.1
3.NBT.E.2
YES
3.NBT.E.3
3.NF.F.1
YES
3.NF.F.2
YES
YES
3.NF.F.3
YES
YES
3.MD.G.1
3.MD.G.2
3.MD.H.3
YES
3.MD.H.4
YES
3.MD.I.5
YES
3.MD.I.6
YES
3.MD.I.7
YES
YES
3.MD.J.8
3.G.K.1
YES
3.G.K.2
YES
4th Grade
Standard
*Identified
PS for DAS
**WY-
TOPP Math
Blueprint
4.OA.A.1
YES
4.OA.A.2
YES
4.OA.A.3
YES YES
4.OA.B.4
YES
4.OA.C.5
4.NBT.D.1
YES
4.NBT.D.2
YES
4.NBT.D.3
YES
4.NBT.E.4
YES
4.NBT.E.5
YES YES
4.NBT.E.6
YES YES
4.NF.F.1
YES
4.NF.F.2
YES YES
4.NF.G.3
YES YES
4.NF.G.4
YES
4.NF.H.5
YES
4.NF.H.6
YES
4.NF.H.7
YES
4.MD.I.1
YES
4.MD.I.2
YES
4.MD.I.3
YES YES
4.MD.J.4
YES
4.MD.K.5
YES
4.MD.K.6
YES
4.MD.K.7
YES
4.G.L.1
YES
4.G.L.2
YES
4.G.L.3
YES
5th Grade
Standard
*Identifie
d PS for
DAS
**WY-TOPP
Math
Blueprint
5.OA.A.1
YES YES
5.OA.A.2
YES
5.OA.B.3
5.NBT.C.1
YES
5.NBT.C.2
YES
5.NBT.C.3
YES
5.NBT.C.4
YES
5.NBT.D.5
YES
5.NBT.D.6
YES
5.NBT.D.7
YES YES
5.NF.E.1
YES
5.NF.E.2
YES YES
5.NF.F.3
YES
5.NF.F.4
YES
5.NF.F.5
YES
5.NF.F.6
YES YES
5.NF.F.7
YES
5.MD.G.1
YES
5.MD.H.2
YES
5.MD.I.3
YES
5.MD.I.4
YES
5.MD.I.5
YES YES
5.G.J.1
YES
5.G.J.2
YES
5.G.K.3
YES
5.G.K.4
YES
Page | 179 Wyoming Department of Education
e d u . w y o m i n g . g o v / s t a n d a rd s
MIDDLE SCHOOL MATH
*Identified Performance Standards (PS) for the District Assessment System (DAS).
**Standard is found on the WY-TOPP Math Blueprint.
6th Grade
Standard
*Identified
PS for DAS
**WY-TOPP
Math
Blueprint
6.RP.A.1
YES
6.RP.A.2
YES
6.RP.A.3
YES YES
6.NS.B.1
YES YES
6.NS.B.2
YES
6.NS.C.3
YES YES
6.NS.C.4
YES
6.NS.D.5
6.NS.D.6
6.NS.D.7
6.NS.D.8
YES YES
6.EE.E.1
YES
6.EE.E2
YES YES
6.EE.E.3
YES
6.EE.E.4
YES
6.EE.F.5
YES
6.EE.F.6
YES
6.EE.F.7
YES YES
6.EE.F.8
YES
6.EE.G.9
YES
6.G.H.1
YES YES
6.G.H.2
YES
6.G.H.3
YES
6.G.H.4
YES
6.SP.I.1
YES
6.SP.I.2
YES
6.SP.I.3
YES
6.SP.J.4
YES
6.SP.J.5
YES YES
7th Grade
Standard
*Identified
PS for DAS
**WY-TOPP
Math
Blueprint
7.RP.A.1
YES
7.RP.A.2
YES YES
7.RP.A.3
YES YES
7.NS.B.1
YES
7.NS.B.2
YES
7.NS.B.3
YES YES
7.EE.C.1
YES
7.EE.C.2
YES
7.EE.D.3
YES
7.EE.D.4
YES YES
7.G.E.1
YES
7.G.E.2
YES
7.G.E.3
YES
7.G.F.4
YES YES
7.G.F.5
YES
7.G.F.6
YES YES
7.SP.G.1
YES YES
7.SP.G.2
YES
7.SP.H.3
YES
7.SP.H.4
YES YES
7.SP.I.5
7.SP.I.6
7.SP.I.7
7.SP.I.8
8th Grade
Standard
*Identified
PS for DAS
**WY-TOPP
Math
Blueprint
8.NS.A.1
YES
8.NS.A.2
8.EE.B.1
8.EE.B.2
8.EE.B.3
8.EE.B.4
8.EE.C.5
YES YES
8.EE.C.6
YES
8.EE.D.7
YES YES
8.EE.D.8
YES YES
8.F.E.1
YES
8.F.E.2
YES YES
8.F.E.3
YES
8.F.F.4
YES YES
8.F.F.5
YES
8.G.G.1
YES
8.G.G.2
YES
8.G.G.3
YES
8.G.G.4
YES
8.G.G.5
YES
8.G.H.6
YES
8.G.H.7
YES YES
8.G.H.8
YES
8.G.I.9
YES
8.SP.J.1
YES
8.SP.J.2
YES
8.SP.J.3
YES
8.SP.J.4
YES
Page | 180 Wyoming Department of Education
e d u . w y o m i n g . g o v / s t a n d a rd s
HIGH SCHOOL MATH
*Identified Performance Standards (PS) for the District Assessment System (DAS).
**Standard is found on the WY-TOPP Math Blueprint.
Number and Quantity
Standard
*Identified
PS for DAS
**WY-TOPP
Math
Blueprint
N.RN.A.1
YES
N.RN.A.2
YES YES
N.RN.B.3
YES
N.Q.C.1
YES
N.Q.C.2
YES
N.Q.C.3
N.CN.D.1
N.CN.D.2
N.CN.D.3
N.CN.E.4
N.CN.E.5
N.CN.E.6
N.CN.F.7
N.CN.F.8
N.CN.F.9
N.VM.G.1
N.VM.G.2
N.VM.G.3
YES
N.VM.H.4
YES
N.VM.H.5
N.VM.I.6
YES
N.VM.I.7
N.VM.I.8
N.VM.I.9
N.VM.I.10
N.VM.I.11
N.VM.I.12
Algebra
Standard
*Identified
PS for DAS
**WY-TOPP
Math
Blueprint
A.SSE.A.1
YES
A.SSE.A.2
A.SSE.B.3
YES YES
A.SSE.B.4
A.APR.C.1
YES
A.APR.D.2
A.APR.D.3
YES
A.APR.E.4
A.APR.E.5
A.APR.F.6
A.APR.F.7
A.CED.G.1
YES YES
A.CED.G.2
YES
A.CED.G.3
YES
A.CED.G.4
A.REI.H.1
YES
A.REI.H.2
YES
A.REI.I.3
YES YES
A.REI.I.4
YES
A.REI.J.5
YES
A.REI.J.6
YES YES
A.REI.J.7
YES
A.REI.J.8
A.REI.J.9
A.REI.K.10
A.REI.K.11
A.REI.K.12
Functions
Standard
*Identified
PS for DAS
**WY-TOPP
Math
Blueprint
F.IF.A.1
YES YES
F.IF.A.2
YES
F.IF.A.3
YES
F.IF.B.4
YES
F.IF.B.5
YES
F.IF.B.6
YES
F.IF.C.7
YES YES
F.IF.C.8
F.IF.C.9
F.BF.D.1
YES YES
F.BF.D.2
F.BF.E.3
YES
F.BF.E.4
F.BF.E.5
F.LE.F.1
YES YES
F.LE.F.2
YES
F.LE.F.3
F.LE.F.4
F.LE.F.5
F.TF.H.1
YES
F.TF.H.2
F.TF.H.3
YES
F.TF.H.4
F.TF.I.5
YES
F.TF.I.6
F.TF.I.7
F.TF.J.8
F.TF.J.9
Page | 181 Wyoming Department of Education
e d u . w y o m i n g . g o v / s t a n d a rd s
HIGH SCHOOL MATH (CONT.)
*Identified Performance Standards (PS) for the District Assessment System (DAS).
**Standard is found on the WY-TOPP Math Blueprint.
Geometry
Standard
*Identified
PS for DAS
**WY-TOPP
Math
Blueprint
G.CO.A.1
YES YES
G.CO.A.2
YES
G.CO.A.3
YES
G.CO.A.4
YES
G.CO.A.5
YES YES
G.CO.B.6
YES
G.CO.B.8
YES
G.CO.B.7
YES
G.CO.C.9
YES
G.CO.C.10
YES YES
G.CO.C.11
YES
G.CO.D.12
YES
G.CO.D13
YES
G.SRT.E.1
YES
G.SRT.E.2
YES
G.SRT.E.3
YES
G.SRT.F.4
YES
G.SRT.F.5
YES YES
G.SRT.G.6
YES
G.SRT.G.7
G.SRT.G.8
YES YES
G.SRT.H.9
G.SRT.H.10
G.SRT.H.11
G.C.I.1
YES
G.C.I.2
YES YES
G.C.I.3
G.C.I.4
G.C.J.5
G.GPE.K.1
YES
G.GPE.K.2
G.GPE.K.3
G.GPE.L.4
YES
G.GPE.L.5
YES
G.GPE.L.6
YES
G.GPE.L.7
YES YES
G.GMD.M.1
YES
G.GMD.M.2
G.GMD.M.3
YES YES
G.GMD.N.4
YES
G.MG.O.1
YES
G.MG.O.2
YES YES
G.MG.O.3
YES YES
Statistics and Probability
Standard
*Identified
PS for DAS
**WY-TOPP
Math
Blueprint
S.ID.A.1
YES
S.ID.A.2
YES YES
S.ID.A.3
S.ID.A.4
S.ID.B.5
S.ID.B.6
YES
S.ID.D.7
YES
S.ID.C.8
S.ID.C.9
YES
S.IC.D.1
S.IC.D.2
S.IC.E.3
S.IC.E.4
YES
S.IC.E.5
S.IC.E.6
YES
S.CP.F.1
YES
S.CP.F.2
YES
S.CP.F.3
S.CP.F.4
S.CP.F.5
YES
S.CP.G.6
S.CP.G.7
S.CP.G.8
S.CP.G.9
S.MD.H.1
S.MD.H.2
S.MD.H.3
YES
S.MD.H.4
S.MD.H.5
YES
S.MD.H.6
S.MD.H.7
YES
