CCGPS Math Standards K-5

Common Core
Georgia
Performance
Standards
CCGPS

Mathematics

Standards
Kindergarten-Fifth Grade

Georgia Department of Education

K-12 Mathematics Introduction

The Georgia Mathematics Curriculum focuses on actively engaging the students in the development of
mathematical understanding by using manipulatives and a variety of representations, working
independently and cooperatively to solve problems, estimating and computing efficiently, and
conducting investigations and recording findings. There is a shift towards applying mathematical
concepts and skills in the context of authentic problems and for the student to understand concepts
rather than merely follow a sequence of procedures. In mathematics classrooms, students will learn to
think critically in a mathematical way with an understanding that there are many different ways to a
solution and sometimes more than one right answer in applied mathematics. Mathematics is the
economy of information. The central idea of all mathematics is to discover how knowing some things
well, via reasoning, permit students to know much else—without having to commit the information to
memory as a separate fact. It is the connections, the reasoned, logical connections that make
mathematics manageable. As a result, implementation of Common Core Georgia Performance Standards
places a greater emphasis on problem solving, reasoning, representation, connections, and
communication.

Toward greater focus and coherence
Mathematics experiences in early childhood settings should concentrate on (1) number (which includes
whole number, operations, and relations) and (2) geometry, spatial relations, and measurement, with
more mathematics learning time devoted to number than to other topics. Mathematical process goals
should be integrated in these content areas.
—Mathematics Learning in Early Childhood, National Research Council, 2009

The composite standards [of Hong Kong, Korea and Singapore] have a number of features that can
inform an international benchmarking process for the development of K–6 mathematics standards in the
U.S. First, the composite standards concentrate the early learning of mathematics on the number,
measurement, and geometry strands with less emphasis on data analysis and little exposure to algebra.
The Hong Kong standards for grades 1–3 devote approximately half the targeted time to numbers and
almost all the time remaining to geometry and measurement.
— Ginsburg, Leinwand and Decker, 2009

Because the mathematics concepts in [U.S.] textbooks are often weak, the presentation becomes more
mechanical than is ideal. We looked at both traditional and non-traditional textbooks used in the US and
found this conceptual weakness in both.
— Ginsburg et al., 2005

There are many ways to organize curricula. The challenge, now rarely met, is to avoid those that distort
mathematics and turn off students.
— Steen, 2007

For over a decade, research studies of mathematics education in high-performing countries have
pointed to the conclusion that the mathematics curriculum in the United States must become
substantially more focused and coherent in order to improve mathematics achievement in this country.
To deliver on the promise of common standards, the standards must address the problem of a
curriculum that is “a mile wide and an inch deep.” These Standards are a substantial answer to that
challenge.

Dr. John D. Barge, State School Superintendent
August 19, 2011 Page 2 of 40
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It is important to recognize that “fewer standards” are no substitute for focused standards. Achieving
“fewer standards” would be easy to do by resorting to broad, general statements. Instead, these
Standards aim for clarity and specificity. Assessing the coherence of a set of standards is more difficult
than assessing their focus.
William Schmidt and Richard Houang (2002) have said that content standards and curricula are coherent
if they are: articulated over time as a sequence of topics and performances that are logical and reflect,
where appropriate, the sequential or hierarchical nature of the disciplinary content from which the
subject matter derives. That is, what and how students are taught should reflect not only the topics that
fall within a certain academic discipline, but also the key ideas that determine how knowledge is
organized and generated within that discipline. This implies that to be coherent, a set of content
standards must evolve from particulars (e.g., the meaning and operations of whole numbers, including
simple math facts and routine computational procedures associated with whole numbers and fractions)
to deeper structures inherent in the discipline. These deeper structures then serve as a means for
connecting the particulars (such as an understanding of the rational number system and its properties).
(emphasis added)
These Standards endeavor to follow such a design, not only by stressing conceptual understanding of
key ideas, but also by continually returning to organizing principles such as place value or the properties
of operations to structure those ideas. In addition, the “sequence of topics and performances” that is
outlined in a body of mathematics standards must also respect what is known about how students learn.
As Confrey (2007) points out, developing “sequenced obstacles and challenges for students…absent the
insights about meaning that derive from careful study of learning, would be unfortunate and unwise.” In
recognition of this, the development of these Standards began with research-based learning
progressions detailing what is known today about how students’ mathematical knowledge, skill, and
understanding develop over time.

Understanding mathematics
These Standards define what students should understand and be able to do in their study of
mathematics. Asking a student to understand something means asking a teacher to assess whether the
student has understood it. But what does mathematical understanding look like? One hallmark of
mathematical understanding is the ability to justify, in a way appropriate to the student’s mathematical
maturity, why a particular mathematical statement is true or where a mathematical rule comes from.
There is a world of difference between a student who can summon a mnemonic device to expand a
product such as (a + b)(x + y) and a student who can explain where the mnemonic comes from. The
student who can explain the rule understands the mathematics, and may have a better chance to
succeed at a less familiar task such as expanding (a + b + c)(x + y). Mathematical understanding and
procedural skill are equally important, and both are assessable using mathematical tasks of sufficient
richness.
The Standards set grade-specific standards but do not define the intervention methods or materials
necessary to support students who are well below or well above grade-level expectations. It is also
beyond the scope of the Standards to define the full range of supports appropriate for English language
learners and for students with special needs. At the same time, all students must have the opportunity
to learn and meet the same high standards if they are to access the knowledge and skills necessary in
their post-school lives. The Standards should be read as allowing for the widest possible range of
students to participate fully from the outset, along with appropriate accommodations to ensure
maximum participation of students with special education needs. For example, for students with
disabilities, reading should allow for use of Braille, screen reader technology, or other assistive devices,
while writing should include the use of a scribe, computer, or speech-to-text technology. In a similar
vein, speaking and listening should be interpreted broadly to include sign language. No set of grade-

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specific standards can fully reflect the great variety in abilities, needs, learning rates, and achievement
levels of students in any given classroom. However, the Standards do provide clear signposts along the
way to the goal of college and career readiness for all students.

How to read the grade level standards
Standards define what students should understand and be able to do. Clusters are groups of related
standards. Note that standards from different clusters may sometimes be closely related, because
mathematics is a connected subject. Domains are larger groups of related standards. Standards from
different domains may sometimes be closely related.

Number and Operations in Base Ten 3.NBT (DOMAIN)
Use place value understanding and properties of operations to
perform multi-digit arithmetic.

1. Use place value understanding to round whole numbers to the nearest 10 or 100. (STANDARD)
2. Fluently add and subtract within 1000 using strategies and algorithms based on place value, (Cluster)
properties of operations, and/or the relationship between addition and subtraction) (STANDARD)
3. 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 operations. . (STANDARD)

These Standards do not dictate curriculum or teaching methods. For example, just because topic A
appears before topic B in the standards for a given grade, it does not necessarily mean that topic A must
be taught before topic B. A teacher might prefer to teach topic B before topic A, or might choose to
highlight connections by teaching topic A and topic B at the same time. Or, a teacher might prefer to
teach a topic of his or her own choosing that leads, as a byproduct, to students reaching the standards
for topics A and B. What students can learn at any particular grade level depends upon what they have
learned before. Ideally then, each standard in this document might have been phrased in the form,
“Students who already know ... should next come to learn ....” But at present this approach is
unrealistic—not least because existing education research cannot specify all such learning pathways. Of
necessity therefore, grade placements for specific topics have been made on the basis of state and
international comparisons and the collective experience and collective professional judgment of
educators, researchers and mathematicians. One promise of common state standards is that over time
they will allow research on learning progressions to inform and improve the design of standards to a
much greater extent than is possible today. Learning opportunities will continue to vary across schools
and school systems, and educators should make every effort to meet the needs of individual students
based on their current understanding. These Standards are not intended to be new names for old ways
of doing business. They are a call to take the next step. It is time for states to work together to build on
lessons learned from two decades of standards based reforms. It is time to recognize that standards are
not just promises to our children, but promises we intend to keep.

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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 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).

1 Make sense of problems and persevere in solving them.
Mathematically proficient students start by explaining to themselves the meaning of a problem and
looking for entry points to its solution. They analyze givens, constraints, relationships, and goals. They
make conjectures about the form and meaning of the solution and plan a solution pathway rather than
simply jumping into a solution attempt. They consider analogous problems, and try special cases and
simpler forms of the original problem in order to gain insight into its solution. They monitor and
evaluate their progress and change course if necessary. Older students might, depending on the context
of the problem, transform algebraic expressions or change the viewing window on their graphing
calculator to get the information they need. Mathematically proficient students can explain
correspondences between equations, verbal descriptions, tables, and graphs or draw diagrams of
important features and relationships, graph data, and search for regularity or trends. Younger students
might rely on using concrete objects or pictures to help conceptualize and solve a problem.
Mathematically proficient students check their answers to problems using a different method, and they
continually ask themselves, “Does this make sense?” They can understand the approaches of others to
solving complex problems and identify correspondences between different approaches.

2 Reason abstractly and quantitatively.
Mathematically proficient students make sense of quantities and their relationships in problem
situations. They bring two complementary abilities to bear on problems involving quantitative
relationships: the ability to decontextualize—to abstract a given situation and represent it symbolically
and manipulate the representing symbols as if they have a life of their own, without necessarily
attending to their referents—and the ability to contextualize, to pause as needed during the
manipulation process in order to probe into the referents for the symbols involved. Quantitative
reasoning entails habits of creating a coherent representation of the problem at hand; considering the
units involved; attending to the meaning of quantities, not just how to compute them; and knowing and
flexibly using different properties of operations and objects.

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

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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
specifying units of measure, and labeling axes to clarify the correspondence with quantities in a
problem. They calculate accurately and efficiently, express 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.

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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 x2 + 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)(x2 + x + 1), and (x –
1)(x3 + 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.

Connecting the Standards for Mathematical Practice to the Standards for Mathematical Content
The Standards for Mathematical Practice describe ways in which developing student practitioners of the
discipline of mathematics increasingly ought to engage with the subject matter as they grow in
mathematical maturity and expertise throughout the elementary, middle and high school years.
Designers of curricula, assessments, and professional development should all attend to the need to
connect the mathematical practices to mathematical content in mathematics instruction. The Standards
for Mathematical Content are a balanced combination of procedure and understanding. Expectations
that begin with the word “understand” are often especially good opportunities to connect the practices
to the content. Students who lack understanding of a topic may rely on procedures too heavily. Without
a flexible base from which to work, they may be less likely to consider analogous problems, represent
problems coherently, justify conclusions, apply the mathematics to practical situations, use technology
mindfully to work with the mathematics, explain the mathematics accurately to other students, step
back for an overview, or deviate from a known procedure to find a shortcut. In short, a lack of
understanding effectively prevents a student from engaging in the mathematical practices. In this
respect, those content standards which set an expectation of understanding are potential “points of
intersection” between the Standards for Mathematical Content and the Standards for Mathematical
Practice. These points of intersection are intended to be weighted toward central and generative
concepts in the school mathematics curriculum that most merit the time, resources, innovative
energies, and focus necessary to qualitatively improve the curriculum, instruction, assessment,
professional development, and student achievement in mathematics.

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Mathematics | Kindergarten
In Kindergarten, instructional time should focus on two critical areas: (1) representing, relating, and
operating on whole numbers, initially with sets of objects; (2) describing shapes and space. More
learning time in Kindergarten should be devoted to number than to other topics.

Content standards for Kindergarten are arranged within the following domains and clusters:

Counting and Cardinality
• Know number names and the count sequence.
• Count to tell the number of objects.
• Compare numbers.

Operations and Algebraic Thinking
• Understand addition as putting together and adding to, and understand subtraction as taking apart
and taking from.

Number and Operations in Base Ten
• Work with numbers 11–19 to gain foundations for place value.

Measurement and Data
• Describe and compare measurable attributes.
• Classify objects and count the number of objects in categories.

Geometry
• Identify and describe shapes.
• Analyze, compare, create, and compose shapes.

Standards for Mathematical Practice
Mathematical Practices are listed with each grade’s mathematical content standards to reflect the need to connect
the mathematical practices to mathematical content in instruction.

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 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).

Students are expected to:
1. Make sense of problems and persevere in solving them.
In Kindergarten, students begin to build the understanding 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. Younger 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?” or they may try another strategy.

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2. Reason abstractly and quantitatively.
Younger students begin to recognize that a number represents a specific quantity. Then, they connect the quantity
to written symbols. Quantitative reasoning entails creating a representation of a problem while attending to the
meanings of the quantities.

3. Construct viable arguments and critique the reasoning of others.
Younger students construct arguments using concrete referents, such as objects, pictures, drawings, and actions.
They also begin to develop 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.

4. Model with mathematics.
In early grades, students experiment with representing problem situations in multiple ways including numbers,
words (mathematical language), drawing pictures, using objects, acting out, 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.

5. Use appropriate tools strategically.
Younger students begin to consider the available tools (including estimation) when solving a mathematical
problem and decide when certain tools might be helpful. 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.

6. Attend to precision.
As kindergarteners begin to develop their mathematical communication skills, they try to use clear and precise
language in their discussions with others and in their own reasoning.

7. Look for and make use of structure.
Younger students begin to discern a 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. They also recognize that 3 + 2 = 5 and 2 + 3 = 5.

8. Look for and express regularity in repeated reasoning.
In the early grades, 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). In addition, students continually check their work by asking
themselves, “Does this make sense?”

Counting and Cardinality K.CC

Know number names and the count sequence.

MCCK.CC.1 Count to 100 by ones and by tens.

MCCK.CC.2 Count forward beginning from a given number within the known sequence (instead of
having to begin at 1).

MCCK.CC.3 Write numbers from 0 to 20. Represent a number of objects with a written numeral 0-20
(with 0 representing a count of no objects).

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Count to tell the number of objects.

MCCK.CC.4 Understand the relationship between numbers and quantities; connect counting to
cardinality.
a. When counting objects, say the number names in the standard order, pairing each object with
one and only one number name and each number name with one and only one object.
b. Understand that the last number name said tells the number of objects counted. The number
of objects is the same regardless of their arrangement or the order in which they were
counted.
c. Understand that each successive number name refers to a quantity that is one larger.

MCCK.CC.5 Count to answer “how many?” questions about as many as 20 things arranged in a line, a
rectangular array, or a circle, or as many as 10 things in a scattered configuration; given a number
from 1–20, count out that many objects.

Compare numbers.

MCCK.CC.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.1

MCCK.CC.7 Compare two numbers between 1 and 10 presented as written numerals.

Operations and Algebraic Thinking K.OA

Understand addition as putting together and adding to, and understand subtraction as taking apart
and taking from.

MCCK.OA.1 Represent addition and subtraction with objects, fingers, mental images, drawings2,
sounds (e.g., claps), acting out situations, verbal explanations, expressions, or equations.

MCCK.OA.2 Solve addition and subtraction word problems, and add and subtract within 10, e.g., by
using objects or drawings to represent the problem.

MCCK.OA.3 Decompose numbers less than or equal to 10 into pairs in more than one way, e.g., by
using objects or drawings, and record each decomposition by a drawing or equation (e.g., 5 = 2 + 3 and
5 = 4 + 1).

MCCK.OA.4 For any number from 1 to 9, find the number that makes 10 when added to the given
number, e.g., by using objects or drawings, and record the answer with a drawing or equation.

MCCK.OA.5 Fluently add and subtract within 5.

1
Include groups with up to ten objects.
2
Drawings need not show details, but should show the mathematics in the problem.

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Number and Operations in Base Ten K.NBT

Work with numbers 11–19 to gain foundations for place value.

MCCK.NBT.1 Compose and decompose numbers from 11 to 19 into ten ones and some further ones,
e.g., by using objects or drawings, and record each composition or decomposition by a drawing or
equation (e.g., 18 = 10 + 8); understand that these numbers are composed of ten ones and one, two,
three, four, five, six, seven, eight, or nine ones.

Measurement and Data K.MD

Describe and compare measurable attributes.

MCCK.MD.1 Describe measurable attributes of objects, such as length or weight. Describe several
measurable attributes of a single object.

MCCK.MD.2 Directly compare two objects with a measurable attribute in common, to see which
object has “more of”/“less of” the attribute, and describe the difference. For example, directly
compare the heights of two children and describe one child as taller/shorter.

Classify objects and count the number of objects in each category.

MCCK.MD.3 Classify objects into given categories; count the numbers of objects in each category and
sort the categories by count.3

Geometry K.G

Identify and describe shapes (squares, circles, triangles, rectangles, hexagons, cubes, cones, cylinders,
and spheres).

MCCK.G.1 Describe objects in the environment using 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.

MCCK.G.2 Correctly name shapes regardless of their orientations or overall size.

MCCK.G.3 Identify shapes as two-dimensional (lying in a plane, “flat”) or three-dimensional (“solid”).

3
Limit category counts to be less than or equal to 10.

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Analyze, compare, create, and compose shapes.

MCCK.G. 4 Analyze and compare two- and three-dimensional shapes, in different sizes and
orientations, using informal language to describe their similarities, differences, parts (e.g., number of
sides and vertices/“corners”) and other attributes (e.g., having sides of equal length).

MCCK.G. 5 Model shapes in the world by building shapes from components (e.g., sticks and clay balls)
and drawing shapes.

MCCK.G. 6 Compose simple shapes to form larger shapes. For example, “Can you join these two
triangles with full sides touching to make a rectangle?”

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Mathematics | Grade 1
In Grade 1, instructional time should focus on four critical areas: (1) developing understanding of
addition, subtraction, and strategies for addition and subtraction within 20; (2) developing
understanding of whole number relationships and place value, including grouping in tens and ones; (3)
developing understanding of linear measurement and measuring lengths as iterating length units; and
(4) reasoning about attributes of, and composing and decomposing geometric shapes.

Content standards for Grade 1 are arranged within the following domains and clusters:

• Represent and solve problems involving addition and subtraction.
• Understand and apply properties of operations and the relationship between addition and
subtraction.
• Add and subtract within 20.
• Work with addition and subtraction equations.

• Extend the counting sequence.
• Understand place value.
• Use place value understanding and properties of operations to add and subtract.

• Measure lengths indirectly and by iterating length units.
• Tell and write time.
• Represent and interpret data.

Geometry
• Reason with shapes and their attributes.



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In first grade, students realize 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. Younger 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 are willing to try other approaches.

Younger students recognize that a number represents a specific quantity. They connect the quantity to written
symbols. Quantitative reasoning entails creating a representation of a problem while attending to the meanings of
the quantities.

First graders construct arguments using concrete 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,” and “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.


In first grade, students begin to consider the available tools (including estimation) when solving a mathematical
problem and decide when certain tools might be helpful. For instance, first graders decide it might be best to use
colored chips to model an addition problem.

As young children begin to develop their mathematical communication skills, they try to use clear and precise
language in their discussions with others and when they explain their own reasoning.

First graders begin to discern a pattern or structure. For instance, if students recognize 12 + 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.

In the early grades, students notice repetitive actions in counting and computation, etc. When children have
multiple opportunities to add and subtract “ten” and multiples of “ten” they notice the pattern and gain a better
understanding of place value. Students continually check their work by asking themselves, “Does this make
sense?”

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Operations and Algebraic Thinking 1.OA

Represent and solve problems involving addition and subtraction.

MCC1.OA.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,
e.g., by using objects, drawings, and equations with a symbol for the unknown number to represent
the problem.4

MCC1.OA.2 Solve word problems that call for addition of three whole numbers whose sum is less than
or equal to 20, e.g., by using objects, drawings, and equations with a symbol for the unknown number
to represent the problem.

Understand and apply properties of operations and the relationship between addition and
subtraction.

MCC1.OA.3 Apply properties of operations as strategies to add and subtract.5
Examples: If 8 + 3 = 11 is known, then 3 + 8 = 11 is also known. (Commutative property of addition.) To
add 2 + 6 + 4, the second two numbers can be added to make a ten, so 2 + 6 + 4 = 2 + 10 = 12.
(Associative property of addition.)

MCC1.OA.4 Understand subtraction as an unknown-addend problem. For example, subtract 10 – 8 by
finding the number that makes 10 when added to 8.

Add and subtract within 20

MCC1.OA.5 Relate counting to addition and subtraction (e.g., by counting on 2 to add 2).

MCC1.OA.6 Add and subtract within 20, demonstrating fluency for addition and subtraction within 10.
Use strategies such as counting on; making ten (e.g., 8 + 6 = 8 + 2 + 4 = 10 + 4 = 14); decomposing a
number leading to a ten (e.g., 13 – 4 = 13 – 3 – 1 = 10 – 1 = 9); using the relationship between addition
and subtraction (e.g., knowing that 8 + 4 = 12, one knows 12 – 8 = 4); and creating equivalent but
easier or known sums (e.g., adding 6 + 7 by creating the known equivalent 6 + 6 + 1 = 12 + 1 = 13).

Work with addition and subtraction equations

MCC1.OA.7 Understand the meaning of the equal sign, and determine if equations involving addition
and subtraction are true or false. For example, which of the following equations are true and which
are false? 6 = 6, 7 = 8 – 1, 5 + 2 = 2 + 5, 4 + 1 = 5 + 2.

MCC1.OA.8 Determine the unknown whole number in an addition or subtraction equation relating to
three whole numbers. For example, determine the unknown number that makes the equation true in
each of the equations 8 + ? = 11, 5 = □ – 3, 6 + 6 = ∆.

4
See Glossary, Table 1
5
Students need not use formal terms for these properties. Problems should be within 20.

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Number and Operations in Base Ten 1.NBT

Extend the counting sequence

MCC1.NBT.1 Count to 120, starting at any number less than 120. In this range, read and write
numerals and represent a number of objects with a written numeral.
Understand place value

MCC1.NBT.2 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).

MCC1.NBT.3 Compare two two-digit numbers based on meanings of the tens and ones digits,
recording the results of comparisons with the symbols >, =, and <.

Use place value understanding and properties of operations to add and subtract.

MCC1.NBT.4 Add within 100, including adding a two-digit number and a one-digit number, and adding
a two-digit number and a multiple of 10, 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. Understand that in adding
two-digit numbers, one adds tens and tens, ones and ones; and sometimes it is necessary to compose
a ten.

MCC1.NBT.5 Given a two-digit number, mentally find 10 more or 10 less than the number, without
having to count; explain the reasoning used.

MCC1.NBT.6 Subtract multiples of 10 in the range 10-90 from multiples of 10 in the range 10-90
(positive or zero differences), 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.

Measurement and Data 1.MD

Measure lengths indirectly and by iterating length units

MCC1.MD.1 Order three objects by length; compare the lengths of two objects indirectly by using a
third object.

MCC1.MD.2 Express the length of an object as a whole number of length units, by laying multiple
copies of a shorter object (the length unit) end to end; understand that the length measurement of an
object is the number of same-size length units that span it with no gaps or overlaps. Limit to contexts
where the object being measured is spanned by a whole number of length units with no gaps or
overlaps.

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Tell and write time.

MCC1.MD.3 Tell and write time in hours and half-hours using analog and digital clocks.

Represent and interpret data.

MCC1.MD.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.

Geometry 1.G

Reason with shapes and their attributes.

MCC1.G.1 Distinguish between defining attributes (e.g., triangles are closed and three-sided) versus
non-defining attributes (e.g., color, orientation, overall size); build and draw shapes to possess
defining attributes.

MCC1.G.2 Compose two-dimensional shapes (rectangles, squares, trapezoids, triangles, half-circles,
and quarter-circles) or three-dimensional shapes (cubes, right rectangular prisms, right circular cones,
and right circular cylinders) to create a composite shape, and compose new shapes from the
composite shape.6

MCC1.G.3 Partition circles and rectangles into two and four equal shares, describe the shares using
the words halves, fourths, and quarters, and use the phrases half of, fourth of, and quarter of.
Describe the whole as two of, or four of the shares. Understand for these examples that decomposing
into more equal shares creates smaller shares.

6
Students do not need to learn formal names such as “right rectangular prism.”

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In Grade 2, instructional time should focus on four critical areas: (1) extending understanding of base-
ten notation; (2) building fluency with addition and subtraction; (3) using standard units of measure; and
(4) describing and analyzing shapes.


• Represent and solve problems involving addition and subtraction.
• Add and subtract within 20.
• Work with equal groups of objects to gain foundations for multiplication.

• Understand place value.
• Use place value understanding and properties of operations to add and subtract.

• Measure and estimate lengths in standard units.
• Relate addition and subtraction to length.
• Work with time and money.

Geometry




In second grade, students realize 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. 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.

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Younger students recognize that a number represents a specific quantity. They connect the quantity to written
symbols. Quantitative reasoning entails creating a representation of a problem while attending to the meanings of
the quantities. Second graders begin to know and use different properties of operations and objects.

Second graders may construct arguments using concrete referents, 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,” and “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
appropriate questions.


In second grade, students consider the available tools (including estimation) when solving a mathematical problem
and decide when certain tools might be better suited. For instance, second graders may decide to solve a problem
by drawing a picture rather than writing an equation.

As children begin to develop their mathematical communication skills, they try to use clear and precise language in
their discussions with others and when they explain their own reasoning.

Second graders look for patterns. For instance, they adopt mental math strategies based on patterns (making ten,
fact families, doubles).

Second graders look for patterns. For instance, they adopt mental math strategies based on patterns (making ten,
fact families, doubles).


Represent and solve problems involving addition and subtraction.

MCC2.OA.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, e.g., by using drawings and equations with a symbol for the unknown
number to represent the problem.7

Add and subtract within 20.

MCC2.OA.2 Fluently add and subtract within 20 using mental strategies.8 By end of Grade 2, know
from memory all sums of two one-digit numbers.

7
See Glossary, Table 1.
8
See standard 1.OA.6 for a list of mental strategies.

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Work with equal groups of objects to gain foundations for multiplication.

MCC2.OA.3 Determine whether a group of objects (up to 20) has an odd or even number of members,
e.g., by pairing objects or counting them by 2s; write an equation to express an even number as a sum
of two equal addends.

MCC2.OA.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.


Understand place value.

MCC2.NBT.1 Understand that the three digits of a three-digit number represent amounts of hundreds,
tens, and ones; e.g., 706 equals 7 hundreds, 0 tens, and 6 ones. Understand the following as special
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).

MCC2.NBT.2 Count within 1000; skip-count by 5s, 10s, and 100s.

MCC2.NBT.3 Read and write numbers to 1000 using base-ten numerals, number names, and expanded
form.

MCC2.NBT.4 Compare two three-digit numbers based on meanings of the hundreds, tens, and ones
digits, using >, =, and < symbols to record the results of comparisons.

Use place value understanding and properties of operations to add and subtract.

MCC2.NBT.5 Fluently add and subtract within 100 using strategies based on place value, properties of
operations, and/or the relationship between addition and subtraction.

MCC2.NBT.6 Add up to four two-digit numbers using strategies based on place value and properties of
operations.

MCC2.NBT.7 Add and subtract within 1000, 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. Understand that in adding or subtracting three-digit numbers,
one adds or subtracts hundreds and hundreds, tens and tens, ones and ones; and sometimes it is
necessary to compose or decompose tens or hundreds.

MCC2.NBT.8 Mentally add 10 or 100 to a given number 100–900, and mentally subtract 10 or 100
from a given number 100–900.

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MCC2.NBT.9 Explain why addition and subtraction strategies work, using place value and the
properties of operations.9


Measure and estimate lengths in standard units.

MCC2.MD.1 Measure the length of an object by selecting and using appropriate tools such as rulers,
yardsticks, meter sticks, and measuring tapes.

MCC2.MD.2 Measure the length of an object twice, using length units of different lengths for the two
measurements; describe how the two measurements relate to the size of the unit chosen.

MCC2.MD.3 Estimate lengths using units of inches, feet, centimeters, and meters.

MCC2.MD.4 Measure to determine how much longer one object is than another, expressing the
length difference in terms of a standard length unit.

Relate addition and subtraction to length.

MCC2.MD.5 Use addition and subtraction within 100 to solve word problems involving lengths that
are given in the same units, e.g., by using drawings (such as drawings of rulers) and equations with a
symbol for the unknown number to represent the problem.

MCC2.MD.6 Represent whole numbers as lengths from 0 on a number line diagram with equally
spaced points corresponding to the numbers 0, 1, 2, ..., and represent whole-number sums and
differences within 100 on a number line diagram.

MCC2.MD.7 Tell and write time from analog and digital clocks to the nearest five minutes, using a.m.
and p.m.

MCC2.MD.8 Solve word problems involving dollar bills, quarters, dimes, nickels, and pennies, using $
and ¢ symbols appropriately. Example: If you have 2 dimes and 3 pennies, how many cents do you
have?

Represent and interpret data

MCC2.MD.9 Generate measurement data by measuring lengths of several objects to the nearest
whole unit, or by making repeated measurements of the same object. Show the measurements by
making a line plot, where the horizontal scale is marked off in whole-number units.

9
Explanations may be supported by drawings or objects.

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MCC2.MD.10 Draw a picture graph and a bar graph (with single-unit scale) to represent a data set
with up to four categories. Solve simple put-together, take-apart, and compare problems10 using
information presented in a bar graph.
Geometry 2.G


MCC2.G.1 Recognize and draw shapes having specified attributes, such as a given number of angles or
a given number of equal faces.11 Identify triangles, quadrilaterals, pentagons, hexagons, and cubes.

MCC2.G.2 Partition a rectangle into rows and columns of same-size squares and count to find the total
number of them.

MCC2.G.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

10
11
Sizes are compared directly or visually, not compared by measuring.

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In Grade 3, instructional time should focus on four critical areas: (1) developing understanding of
multiplication and division and strategies for multiplication and division within 100; (2) developing
understanding of fractions, especially unit fractions (fractions with numerator 1); (3) developing
understanding of the structure of rectangular arrays and of area; and (4) describing and analyzing two-
dimensional shapes.


• Represent and solve problems involving multiplication and division.
• Understand properties of multiplication and the relationship between multiplication and division.
• Multiply and divide within 100.
• Solve problems involving the four operations, and identify and explain patterns in arithmetic.

• Use place value understanding and properties of operations to perform multi-digit arithmetic.

Number and Operations—Fractions
• Develop understanding of fractions as numbers.

• Solve problems involving measurement and estimation of intervals of time, liquid volumes, and
masses of objects.
• Geometric measurement: understand concepts of area and relate area to multiplication and to
addition.
• Geometric measurement: recognize perimeter as an attribute of plane figures and distinguish
between linear and area measures.

Geometry



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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 graders 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.

Third graders should 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.

In third grade, 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 like “How did you get that?” and “Why is that true?” They explain their thinking to others and respond
to others’ thinking.

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
They should be able to use all of these representations as needed. Third graders should evaluate their results in
the context of the situation and reflect on whether the results make sense.

Third 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 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

As third graders 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.

In third grade, students look closely to discover a pattern or structure. For instance, students use properties of
operations as strategies to multiply and divide (commutative and distributive properties).

Students in third grade should notice repetitive actions in computation and look for more 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 x 8, they might decompose 7 into
5 and 2 and then multiply 5 x 8 and 2 x 8 to arrive at 40 + 16 or 56. In addition, third graders continually evaluate
their work by asking themselves, “Does this make sense?”

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Represent and solve problems involving multiplication and division.

MCC3.OA.1 Interpret products of whole numbers, e.g., interpret 5 × 7 as the total number of objects
in 5 groups of 7 objects each. For example, describe a context in which a total number of objects can
be expressed as 5 × 7.

MCC3.OA.2 Interpret whole-number quotients of whole numbers, e.g., interpret 56 ÷ 8 as the number
of objects in each share when 56 objects are partitioned equally into 8 shares, or as a number of
shares when 56 objects are partitioned into equal shares of 8 objects each. For example, describe a
context in which a number of shares or a number of groups can be expressed as 56 ÷ 8.

MCC3.OA.3 Use multiplication and division within 100 to solve word problems in situations involving
equal groups, arrays, and measurement quantities, e.g., by using drawings and equations with a
symbol for the unknown number to represent the problem.12

MCC3.OA.4 Determine the unknown whole number in a multiplication or division equation relating
three whole numbers. For example, determine the unknown number that makes the equation true in
each of the equations 8 × ? = 48, 5 = □ ÷ 3, 6 × 6 = ?. × ? = 48, 5 = □ ÷ 3, 6 × 6 = ?.

Understand properties of multiplication and the relationship between multiplication and division.

MCC3.OA.5 Apply properties of operations as strategies to multiply and divide.13 Examples: If 6 × 4 =
24 is known, then 4 × 6 = 24 is also known. (Commutative property of multiplication.) 3 × 5 × 2 can be
found by 3 × 5 = 15, then 15 × 2 = 30, or by 5 × 2 = 10, then 3 × 10 = 30. (Associative property of
multiplication.) Knowing that 8 × 5 = 40 and 8 × 2 = 16, one can find 8 × 7 as 8 × (5 + 2) = (8 × 5) + (8 ×
2) = 40 + 16 = 56. (Distributive property.)

MCC3.OA.6 Understand division as an unknown-factor problem. For example, find 32 ÷ 8 by finding
the number that makes 32 when multiplied by 8.

Multiply and divide within 100

MCC3.OA.7 Fluently multiply and divide within 100, using strategies such as the relationship between
multiplication and division (e.g., knowing that 8 × 5 = 40, one knows 40 ÷ 5 = 8) or properties of
operations. By the end of Grade 3, know from memory all products of two one-digit numbers.

Solve problems involving the four operations, and identify and explain patterns in arithmetic.
MCC3.OA.8 Solve two-step word problems using the four operations. Represent these problems using
equations with a letter standing for the unknown quantity. Assess the reasonableness of answers
using mental computation and estimation strategies including rounding.14

12
13
Students need not use formal terms for these properties.
14
This standard is limited to problems posed with whole numbers and having whole-number answers; students should know
how to perform operations in the conventional order where there are no parentheses to specify a particular order (Order of
Operations).

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MCC3.OA.9 Identify arithmetic patterns (including patterns in the addition table or multiplication
table), and explain them using properties of operations. For example, observe that 4 times a number
is always even, and explain why 4 times a number can be decomposed into two equal addends.


Use place value understanding and properties of operations to perform multi-digit arithmetic.15

MCC3.NBT.1 Use place value understanding to round whole numbers to the nearest 10 or 100.

MCC3.NBT.2 Fluently add and subtract within 1000 using strategies and algorithms based on place
value, properties of operations, and/or the relationship between addition and subtraction.

MCC3.NBT.3 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 operations.

Number and Operations – Fractions16 3.NF

Develop understanding of fractions as numbers.

MCC3.NF.1 Understand a fraction 1/b as the quantity formed by 1 part when a whole is partitioned
into b equal parts; understand a fraction a/b as the quantity formed by a parts of size 1/b.

MCC3.NF.2 Understand a fraction as a number on the number line; represent fractions on a number
line diagram.
a. Represent a fraction 1/b 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 1/b and that the
endpoint of the part based at 0 locates the number 1/b on the number line.
b. Represent a fraction a/b on a number line diagram by marking off a lengths 1/b from 0.
Recognize that the resulting interval has size a/b and that its endpoint locates the number a/b
on the number line.

MCC3.NF.3 Explain equivalence of fractions in special cases, and compare fractions by reasoning
about their size.
a. Understand two fractions as equivalent (equal) if they are the same size, or the same point on
a number line.
b. Recognize and generate simple equivalent fractions, e.g., 1/2 = 2/4, 4/6 = 2/3). Explain why
the fractions are equivalent, e.g., by using a visual fraction model.
c. Express whole numbers as fractions, and recognize fractions that are equivalent to whole
numbers. Examples: Express 3 in the form 3 = 3/1; recognize that 6/1 = 6; locate 4/4 and 1 at
the same point of a number line diagram.
d. Compare two fractions with the same numerator or the same denominator by reasoning
about their size. Recognize that comparisons are valid only when the two fractions refer to the

15
A range of algorithms will be used.
16
Grade 3 expectations in this domain are limited to fractions with denominators of 2, 3, 4, 6, and 8.

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same whole. Record the results of comparisons with the symbols >, =, or <, and justify the
conclusions, e.g., by using a visual fraction model.


Solve problems involving measurement and estimation of intervals of time, liquid volumes, and
masses of objects.

MCC3.MD.1 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, e.g., by representing
the problem on a number line diagram.

MCC3.MD.2 Measure and estimate liquid volumes and masses of objects using standard units of
grams (g), kilograms (kg), and liters (l).17 Add, subtract, multiply, or divide to solve one-step word
problems involving masses or volumes that are given in the same units, e.g., by using drawings (such
as a beaker with a measurement scale) to represent the problem.18


MCC3.MD.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 bar graphs. For example, draw a bar graph in which each square in
the bar graph might represent 5 pets.

MCC3.MD.4 Generate measurement data by measuring lengths using rulers marked with halves and
fourths of an inch. Show the data by making a line plot, where the horizontal scale is marked off in
appropriate units— whole numbers, halves, or quarters.

Geometric Measurement: understand concepts of area and relate area to multiplication and to
addition.

MCC3.MD.5 Recognize area as an attribute of plane figures and understand concepts of area
measurement.
a. A square with side length 1 unit, called “a unit square,” is said to have “one square unit” of
area, and can be used to measure area.
b. A plane figure which can be covered without gaps or overlaps by n unit squares is said to have
an area of n square units.

MCC3.MD.6 Measure areas by counting unit squares (square cm, square m, square in, square ft, and
improvised units).

MCC3.MD.7 Relate area to the operations of multiplication and addition.
a. Find the area of a rectangle with whole-number side lengths by tiling it, and show that the
area is the same as would be found by multiplying the side lengths.

17 3
Excludes compound units such as cm and finding the geometric volume of a container.
18
Excludes multiplicative comparison problems (problems involving notions of “times as much”; see Glossary, Table 2).

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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 tiling to show in a concrete case that the area of a rectangle with whole-number side
lengths a and b + c is the sum of a × b and a × c. Use area models to represent the distributive
property in mathematical reasoning.
d. Recognize area as additive. Find areas of rectilinear figures by decomposing them into non-
overlapping rectangles and adding the areas of the non-overlapping parts, applying this
technique to solve real world problems.

Geometric measurement: recognize perimeter as an attribute of plane figures and distinguish
between linear and area measures.

MCC3.MD.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 areas or with the same area and different
perimeters.

Geometry 3.G


MCC3.G.1 Understand that shapes in different categories (e.g., rhombuses, rectangles, and others)
may share attributes (e.g., having four sides), and 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.

MCC3.G.2 Partition shapes into parts with equal areas. Express the area of each part as a unit fraction
of the whole. For example, partition a shape into 4 parts with equal area, and describe the area of
each part as 1/4 of the area of the shape.

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In Grade 4, instructional time should focus on three critical areas: (1) developing understanding and
fluency with multi-digit multiplication, and developing understanding of dividing to find quotients
involving multi-digit dividends; (2) developing an understanding of fraction equivalence, addition and
subtraction of fractions with like denominators, and multiplication of fractions by whole numbers; (3)
understanding that geometric figures can be analyzed and classified based on their properties, such as
having parallel sides, perpendicular sides, particular angle measures, and symmetry


• Use the four operations with whole numbers to solve problems.
• Gain familiarity with factors and multiples.
• Generate and analyze patterns.

• Generalize place value understanding for multi-digit whole numbers.
• Use place value understanding and properties of operations to perform multi-digit arithmetic.

• Extend understanding of fraction equivalence and ordering.
• Build fractions from unit fractions by applying and extending previous understandings of operations
on whole numbers.
• Understand decimal notation for fractions, and compare decimal fractions.

• Solve problems involving measurement and conversion of measurements from a larger unit to a
smaller unit.
• Geometric measurement: understand concepts of angle and measure angles.

Geometry
• Draw and identify lines and angles, and classify shapes by properties of their lines and angles.



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In fourth 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. Fourth graders 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.

Fourth graders should recognize that a number represents a specific quantity. They connect the quantity to written
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.

In fourth grade, 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.

(mathematical language), drawing pictures, using objects, 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. Fourth graders should evaluate their results in the context of
the situation and reflect on whether the results make sense.

Fourth 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 graph paper or a number line to represent
and compare decimals and 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.

As fourth graders 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, they use appropriate labels when creating a line plot.

In fourth grade, 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.

Students in fourth grade should 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.

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Use the four operations with whole numbers to solve problems.

MCC4.OA.1 Interpret a multiplication equation as a comparison, e.g., interpret 35 = 5 × 7 as a
statement that 35 is 5 times as many as 7 and 7 times as many as 5. Represent verbal statements of
multiplicative comparisons as multiplication equations.

MCC4.OA.2 Multiply or divide to solve word problems involving multiplicative comparison, e.g., by
using drawings and equations with a symbol for the unknown number to represent the problem,
distinguishing multiplicative comparison from additive comparison.19

MCC4.OA.3 Solve multistep word problems posed with whole numbers and having whole-number
answers using the four operations, including problems in which remainders must be interpreted.
Represent these problems using equations with a letter standing for the unknown quantity. Assess
the reasonableness of answers using mental computation and estimation strategies including
rounding.

Gain familiarity with factors and multiples.

MCC4.OA.4 Find all factor pairs for a whole number in the range 1–100. Recognize that a whole
number is a multiple of each of its factors. Determine whether a given whole number in the range 1–
100 is a multiple of a given one-digit number. Determine whether a given whole number in the range
1–100 is prime or composite.

Generate and analyze patterns.

MCC4.OA.5 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. For example, given the rule “Add 3” and the
starting number 1, generate terms in the resulting sequence and observe that the terms appear to
alternate between odd and even numbers. Explain informally why the numbers will continue to
alternate in this way.

Number and Operations in Base Ten20 4.NBT

Generalize place value understanding for multi-digit whole numbers.

MCC4.NBT.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. For example, recognize that 700 ÷ 70 = 10 by applying
concepts of place value and division.

MCC4.NBT.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 to record the results of comparisons.

19
20
Grade 4 expectations in this domain are limited to whole numbers less than or equal to 1,000,000.

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MCC4.NBT.3 Use place value understanding to round multi-digit whole numbers to any place.

Use place value understanding and properties of operations to perform multi-digit arithmetic.

MCC4.NBT.4 Fluently add and subtract multi-digit whole numbers using the standard algorithm.

MCC4.NBT.5 Multiply a whole number of up to four digits by a one-digit whole number, and multiply
two two-digit numbers, using strategies based on place value and the properties of operations.
Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models.

MCC4.NBT.6 Find whole-number quotients and remainders with up to four-digit dividends and one-
digit divisors, using strategies based on place value, the properties of operations, and/or the
relationship between multiplication and division. Illustrate and explain the calculation by using
equations, rectangular arrays, and/or area models.

Number and Operations – Fractions21 4.NF

Extend understanding of fraction equivalence and ordering.

MCC4.NF.1 Explain why a fraction a/b is equivalent to a fraction (n × a)/(n × b) 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.

MCC4.NF.2 Compare two fractions with different numerators and different denominators, e.g., by
creating common denominators or numerators, or by comparing to a benchmark fraction such as 1/2.
Recognize that comparisons are valid only when the two fractions refer to the same whole. Record
the results of comparisons with symbols >, =, or <, and justify the conclusions, e.g., by using a visual
fraction model.

Build fractions from unit fractions by applying and extending previous understandings of operations
on whole numbers.

MCC4.NF.3 Understand a fraction a/b with a > 1 as a sum of fractions 1/b.
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, e.g., by using a
visual fraction model. Examples: 3/8 = 1/8 + 1/8 + 1/8 ; 3/8 = 1/8 + 2/8 ; 2 1/8 = 1 + 1 + 1/8 =
8/8 + 8/8 + 1/8.
c. Add and subtract mixed numbers with like denominators, e.g., by replacing each mixed
number with an equivalent fraction, and/or by using properties of operations 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, e.g., by using visual fraction models and equations to
represent the problem.

21
Grade 4 expectations in this domain are limited to fractions with denominators of 2, 3, 4, 5, 6, 8, 10, 12, and 100.

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MCC4.NF.4 Apply and extend previous understandings of multiplication to multiply a fraction by a
whole number.
a. Understand a fraction a/b as a multiple of 1/b. For example, use a visual fraction model to
represent 5/4 as the product 5 × (1/4), recording the conclusion by the equation 5/4 = 5 ×
(1/4).
b. Understand a multiple of a/b as a multiple of 1/b, and use this understanding to multiply a
fraction by a whole number. For example, use a visual fraction model to express 3 × (2/5) as 6
× (1/5), recognizing this product as 6/5. (In general, n × (a/b) = (n × a)/b.)
c. Solve word problems involving multiplication of a fraction by a whole number, e.g., by using
visual fraction models and equations to represent the problem. For example, if each person at
a party will eat 3/8 of a pound of roast beef, and there will be 5 people at the party, how
many pounds of roast beef will be needed? Between what two whole numbers does your
answer lie?

Understand decimal notation for fractions, and compare decimal fractions.

MCC4.NF.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 100.22 For example,
express 3/10 as 30/100, and add 3/10 + 4/100 = 34/100.

MCC4.NF.6 Use decimal notation for fractions with denominators 10 or 100. For example, rewrite 0.62
as 62/100; describe a length as 0.62 meters; locate 0.62 on a number line diagram.

MCC4.NF.7 Compare two decimals to hundredths by reasoning about their size. Recognize that
comparisons are valid only when the two decimals refer to the same whole. Record the results of
comparisons with the symbols >, =, or <, and justify the conclusions, e.g., by using a visual model.


Solve problems involving measurement and conversion of measurements from a larger unit to a
smaller unit.

MCC4.MD.1 Know relative sizes of measurement units within one system of units including km, m,
cm; kg, g; lb, oz.; l, ml; hr, min, sec. 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. For
example, know that 1 ft is 12 times as long as 1 in. Express the length of a 4 ft snake as 48 in. Generate
a conversion table for feet and inches listing the number pairs (1, 12), (2, 24), (3, 36), ...

MCC4.MD.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.

22
Students who can generate equivalent fractions can develop strategies for adding fractions with unlike denominators in
general. But, addition and subtraction with unlike denominators in general is not a requirement at this grade.

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MCC4.MD.3 Apply the area and perimeter formulas for rectangles in real world and mathematical
problems. For example, find the width of a rectangular room given the area of the flooring and the
length, by viewing the area formula as a multiplication equation with an unknown factor.


MCC4.MD.4 Make a line plot to display a data set of measurements in fractions of a unit (1/2, 1/4,
1/8). Solve problems involving addition and subtraction of fractions by using information presented in
line plots. For example, from a line plot find and interpret the difference in length between the longest
and shortest specimens in an insect collection.

Geometric Measurement: understand concepts of angle and measure angles.

MCC4.MD.5 Recognize angles as geometric shapes that are formed wherever two rays share a
common endpoint, and understand concepts of angle measurement:
a. An angle is measured with reference to a circle with its center at the common endpoint of the
rays, by considering the fraction of the circular arc between the points where the two rays
intersect the circle. An angle that turns through 1/360 of a circle is called a “one-degree
angle,” and can be used to measure angles.
b. An angle that turns through n one-degree angles is said to have an angle measure of n
degrees.

MCC4.MD.6 Measure angles in whole-number degrees using a protractor. Sketch angles of specified
measure.

MCC4.MD.7 Recognize angle measure as additive. When an angle is decomposed into non-overlapping
parts, the angle measure of the whole is the sum of the angle measures of the parts. Solve addition
and subtraction problems to find unknown angles on a diagram in real world and mathematical
problems, e.g., by using an equation with a symbol for the unknown angle measure.

Geometry 4.G

Draw and identify lines and angles, and classify shapes by properties of their lines and angles.

MCC4.G.1 Draw points, lines, line segments, rays, angles (right, acute, obtuse), and perpendicular and
parallel lines. Identify these in two-dimensional figures.

MCC4.G.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.

MCC4.G.3 Recognize a line of symmetry for a two-dimensional figure as a line across the figure such
that the figure can be folded along the line into matching parts. Identify line-symmetric figures and
draw lines of symmetry.

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In Grade 5, instructional time should focus on three critical areas: (1) developing fluency with addition
and subtraction of fractions, and developing understanding of the multiplication of fractions and of
division of fractions in limited cases (unit fractions divided by whole numbers and whole numbers
divided by unit fractions); (2) extending division to 2-digit divisors, integrating decimal fractions into the
place value system and developing understanding of operations with decimals to hundredths, and
developing fluency with whole number and decimal operations; and (3) developing understanding of
volume.
• Write and interpret numerical expressions.
• Analyze patterns and relationships.

• Understand the place value system.
• Perform operations with multi-digit whole numbers and with decimals to hundredths.

• Use equivalent fractions as a strategy to add and subtract fractions.
• Apply and extend previous understandings of multiplication and division to multiply and divide
fractions.

• Convert like measurement units within a given measurement system.
• Geometric measurement: understand concepts of volume and relate volume to multiplication and
to addition.

Geometry
• Graph points on the coordinate plane to solve real-world and mathematical problems.
• Classify two-dimensional figures into categories based on their properties.




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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?”.

Fifth graders should recognize that a number represents a specific quantity. They connect quantities to written
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.

In fifth grade, 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.

(mathematical language), drawing pictures, using objects, 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. 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.

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 make
predictions from real world data.

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.

In fifth grade, 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.

Fifth graders 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 perform all operations with decimals to hundredths. Students explore operations with fractions with
visual models and begin to formulate generalizations.

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Write and interpret numerical expressions.

MCC5.OA.1 Use parentheses, brackets, or braces in numerical expressions, and evaluate expressions
with these symbols.

MCC5.OA.2 Write simple expressions that record calculations with numbers, and interpret numerical
expressions without evaluating them. For example, express the calculation “add 8 and 7, then
multiply by 2” as 2 × (8 + 7). Recognize that 3 × (18932 + 921) is three times as large as 18932 + 921,
without having to calculate the indicated sum or product.

Analyze patterns and relationships.

MCC5.OA.3 Generate two numerical patterns using two given rules. Identify apparent relationships
between corresponding terms. Form ordered pairs consisting of corresponding terms from the two
patterns, and graph the ordered pairs on a coordinate plane. For example, given the rule “Add 3” and
the starting number 0, and given the rule “Add 6” and the starting number 0, generate terms in the
resulting sequences, and observe that the terms in one sequence are twice the corresponding terms in
the other sequence. Explain informally why this is so.


Understand the place value system.

MCC5.NBT.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 1/10 of what it represents in the place to its left.

MCC5.NBT.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.

MCC5.NBT.3 Read, write, and compare decimals to thousandths.
a. Read and write decimals to thousandths using base-ten numerals, number names, and
expanded form, e.g., 347.392 = 3 × 100 + 4 × 10 + 7 × 1 + 3 × (1/10) + 9 × (1/100) + 2 × (1/1000).
b. Compare two decimals to thousandths based on meanings of the digits in each place, using >,
=, and < symbols to record the results of comparisons.

MCC5.NBT.4 Use place value understanding to round decimals to any place.

Perform operations with multi-digit whole numbers and with decimals to hundredths.

MCC5.NBT.5 Fluently multiply multi-digit whole numbers using the standard algorithm.

MCC5.NBT.6 Find whole-number quotients of whole numbers with up to four-digit dividends and two-
digit divisors, using strategies based on place value, the properties of operations, and/or the
relationship between multiplication and division. Illustrate and explain the calculation by using
equations, rectangular arrays, and/or area models.

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MCC5.NBT.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.

Number and Operations – Fractions 5.NF

Use equivalent fractions as a strategy to add and subtract fractions.

MCC5.NF.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. For example, 2/3 + 5/4 = 8/12 + 15/12 = 23/12. (In
general, a/b + c/d = (ad + bc)/bd.)

MCC5.NF.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. For example, recognize an incorrect result 2/5 + ½ = 3/7, by
observing that 3/7 < ½.

Apply and extend previous understandings of multiplication and division to multiply and divide
fractions.

MCC5.NF.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, e.g., by using visual fraction models or equations to represent the problem. For
example, interpret ¾ as the result of dividing 3 by 4, noting that ¾ multiplied by 4 equals 3, and that
when 3 wholes are shared equally among 4 people each person has a share of size ¾. If 9 people want
to share a 50-pound sack of rice equally by weight, how many pounds of rice should each person get?
Between what two whole numbers does your answer lie?

MCC5.NF.4 Apply and extend previous understandings of multiplication to multiply a fraction or
whole number by a fraction.
a. Interpret the product (a/b) × q as a parts of a partition of q into b equal parts; equivalently, as
the result of a sequence of operations a × q ÷ b. For example, use a visual fraction model to
show (2/3) × 4 = 8/3, and create a story context for this equation. Do the same with (2/3) ×
(4/5) =8/15. (In general, (a/b) × (c/d) = ac/bd.)
b. Find the area of a rectangle with fractional side lengths by tiling it with unit squares of the
appropriate unit fraction side lengths, and show that the area is the same as would be found
by multiplying the side lengths. Multiply fractional side lengths to find areas of rectangles, and
represent fraction products as rectangular areas.

MCC5.NF.5 Interpret multiplication as scaling (resizing), by:
a. Comparing the size of a product to the size of one factor on the basis of the size of the other
factor, without performing the indicated multiplication.
b. Explaining why multiplying a given number by a fraction greater than 1 results in a product
greater than the given number (recognizing multiplication by whole numbers greater than 1 as
a familiar case); explaining why multiplying a given number by a fraction less than 1 results in

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a product smaller than the given number; and relating the principle of fraction equivalence
a/b = (n×a)/(n×b) to the effect of multiplying a/b by 1.

MCC5.NF.6 Solve real world problems involving multiplication of fractions and mixed numbers, e.g.,
by using visual fraction models or equations to represent the problem.

MCC5.NF.7 Apply and extend previous understandings of division to divide unit fractions by whole
numbers and whole numbers by unit fractions. 23
a. Interpret division of a unit fraction by a non-zero whole number, and compute such quotients.
For example, create a story context for (1/3) ÷ 4, and use a visual fraction model to show the
quotient. Use the relationship between multiplication and division to explain that (1/3) ÷ 4 =
1/12 because (1/12) × 4 = 1/3.
b. Interpret division of a whole number by a unit fraction, and compute such quotients. For
example, create a story context for 4 ÷ (1/5), and use a visual fraction model to show the
quotient. Use the relationship between multiplication and division to explain that 4 ÷ (1/5) =
20 because 20 × (1/5) = 4.
c. Solve real world problems involving division of unit fractions by non-zero whole numbers and
division of whole numbers by unit fractions, e.g., by using visual fraction models and
equations to represent the problem. For example, how much chocolate will each person get if
3 people share 1/2 lb of chocolate equally? How many 1/3-cup servings are in 2 cups of
raisins?


Convert like measurement units within a given measurement system.

MCC5.MD.1 Convert among different-sized standard measurement units within a given measurement
system (e.g., convert 5 cm to 0.05 m), and use these conversions in solving multi-step, real world
problems.


MCC5.MD.2 Make a line plot to display a data set of measurements in fractions of a unit (1/2, 1/4,
1/8). Use operations on fractions for this grade to solve problems involving information presented in
line plots. For example, given different measurements of liquid in identical beakers, find the amount of
liquid each beaker would contain if the total amount in all the beakers were redistributed equally.

Geometric Measurement: understand concepts of volume and relate volume to multiplication and
division.

MCC5.MD.3 Recognize volume as an attribute of solid figures and understand concepts of volume
measurement.
a. A cube with side length 1 unit, called a “unit cube,” is said to have “one cubic unit” of volume,
and can be used to measure volume.

23
Students able to multiply fractions in general can develop strategies to divide fractions in general, by reasoning about the
relationship between multiplication and division. But division of a fraction by a fraction is not a requirement at this grade.

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b. A solid figure which can be packed without gaps or overlaps using n unit cubes is said to have
a volume of n cubic units.

MCC5.MD.4 Measure volumes by counting unit cubes, using cubic cm, cubic in, cubic ft, and
improvised units.

MCC5.MD.5 Relate volume to the operations of multiplication and addition and solve real world and
mathematical problems involving volume.
a. Find the volume of a right rectangular prism with whole-number side lengths by packing it
with unit cubes, and show that the volume is the same as would be found by multiplying the
edge lengths, equivalently by multiplying the height by the area of the base. Represent
threefold whole-number products as volumes, e.g., to represent the associative property of
multiplication.
b. Apply the formulas V = l × w × h and V = b × h for rectangular prisms to find volumes of right
rectangular prisms with whole number edge lengths in the context of solving real world and
mathematical problems.
c. Recognize volume as additive. Find volumes of solid figures composed of two non-overlapping
right rectangular prisms by adding the volumes of the non-overlapping parts, applying this
technique to solve real world problems.

Geometry 5.G

Graph points on the coordinate plane to solve real-world and mathematical problems.

MCC5.G.1 Use a pair of perpendicular number lines, called axes, to define a coordinate system, with
the intersection of the lines (the origin) arranged to coincide with the 0 on each line and a given point
in the plane located by using an ordered pair of numbers, called its coordinates. Understand that the
first number indicates how far to travel from the origin in the direction of one axis, and the second
number indicates how far to travel in the direction of the second axis, with the convention that the
names of the two axes and the coordinates correspond (e.g., x-axis and x-coordinate, y-axis and y-
coordinate).

MCC5.G.2 Represent real world and mathematical problems by graphing points in the first quadrant
of the coordinate plane, and interpret coordinate values of points in the context of the situation.

Classify two-dimensional figures into categories based on their properties.

MCC5.G.3 Understand that attributes belonging to a category of two-dimensional figures also belong
to all subcategories of that category. For example, all rectangles have four right angles and squares
are rectangles, so all squares have four right angles.

MCC5.G.4 Classify two-dimensional figures in a hierarchy based on properties.

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CCGPS Math Standards K-5

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