## Standards for Mathematical Practice

As a Self-Directed Learner know that:

##### The Standards for Mathematical Practice (California Common Core)

describe varieties of expertise that mathematics educators (INCLUDING YOURSELF!) at all levels should seek to ____ in their students.  ????????

develop

These practices rest on important “____ ____ ____ ” with longstanding importance in mathematics education.  ????????

processes and proficiencies

The first of these are the National Council of Mathematics Teachers (NCTM) ____ ____ of

• problem solving,
• reasoning and proof,
• communication,
• representation, and
• connections.    ????????

process standards

The second are the strands of ____ ____ specified in the National Research Council’s report “Adding It Up”.  ????????

mathematical proficiency

The strands of mathematical proficiency are:

• strategic ____,
• conceptual ____ (comprehension of mathematical concepts, operations and relations),
• procedural ____ (skill in carrying out procedures flexibly, accurately, efficiently and appropriately), and
• productive ____
• (habitual inclination to see mathematics as sensible, useful, and worthwhile,
• coupled with a belief in diligence and one’s own efficacy).   ????????

reasoning    competence     understanding     fluency     disposition

##### 1)  Make sense of problems and persevere in solving them.

Mathematically proficient students start by ____ to themselves the meaning of a ____ and looking for entry points to its solution. ????????

explaining    problem

They ____  givens, constraints, relationships, and goals. ????????

analyze

They make ____ about the form and meaning of the solution and plan a ____  pathway rather than simply jumping into a solution attempt. ????????

conjectures   solution

They consider ____ problems, and try special cases and simpler forms of the original problem in order to gain ____  into its solution. ????????

analogous   insight

They monitor and evaluate their____ and change ____ if necessary. ????????

progress   course

Older students might, depending on the context of the problem, ____ algebraic expressions or change the viewing window on their graphing calculator to get the information they need. ????????

transform

Mathematically proficient students can explain ____ between equations, verbal descriptions, tables, and graphs or draw ____  of important features and relationships, graph data, and search for regularity or trends. ????????

correspondences   diagrams

Younger students might rely on using ____  objects or pictures to help conceptualize and solve a problem. ????????

concrete

Mathematically proficient students ____ their answers to problems using a different method, and they continually ask themselves, “Does this make ____ ?” ????????

check   sense

They can understand the ____  of others to solving complex problems and ____ correspondences between different approaches. ????????

approaches   identify

##### 2)  Reason abstractly and quantitatively.

Mathematically proficient students make sense of quantities and their ____ in problem situations. ????????

relationships

They bring two complementary abilities to bear on problems involving ____ ____. ????????

quantitative relationships

The first is the ability to ____ — ????????

de-contextualize

to ____ a given situation and represent it symbolically and manipulate the representing ____ as if they have a life of their own, without necessarily attending to their referents— ????????

abstract   symbols

and the second is the ability to ____, ????????

contextualize

to pause as needed during the manipulation process in order to probe into the ____ for the symbols involved. ????????

referents

Quantitative reasoning entails habits of creating a ____ ____ of the problem at hand; ????????

coherent representation

considering the ____ involved; ????????

units

attending to the____ of quantities, not just how to ____ them; and ????????

meaning   compute

knowing and flexibly using different ____ of operations and objects. ????????

properties

##### 3) Construct viable arguments and critique the reasoning of others.

Mathematically proficient students understand and use stated ____, definitions, and previously established results in constructing ____. ????????

assumptions    arguments

They make conjectures and build a logical ____ of statements to explore the truth of their conjectures. ????????

progression

They are able to analyze situations by breaking them into ____, and can recognize and use ____. ????????

cases     counterexamples

They ____ their conclusions, communicate them to others, and respond to the ____ of others. ????????

justify    arguments

They reason ____ about data, making plausible arguments that take into account the ____ from which the data arose. ????????

inductively    context

Mathematically proficient students are also able to compare the effectiveness of two ____ arguments, distinguish ____ logic or reasoning from that which is flawed, and—if there is a____ in an argument—explain what it is. ????????

plausible    correct   flaw

Elementary students can construct arguments using concrete____ such as objects, drawings, diagrams, and actions. ????????

referents

Such arguments can make sense and be correct, even though they are not ____ or made formal until later grades. ????????

generalized

Later, students learn to determine domains to which an ____ applies.????????

argument

Students at all grades can listen to or read the ____  of others, decide whether they make sense, and ask useful questions to clarify or improve the arguments. ????????

Students build proofs by ____  and proofs by ____. CA 3.1 (for higher mathematics only). ????????

##### 4) Model with mathematics.

Mathematically proficient students can ____ the mathematics they know to solve problems arising in everyday life, society, and the workplace. ????????

apply

In early grades, this might be as simple as ____  an addition equation to describe a situation. ????????

writing

In middle grades, a student might apply ____  reasoning to plan a school event or analyze a problem in the community. ????????

proportional

By high school, a student might use geometry to solve a ____  problem or use a function to describe how one quantity of interest ____ on another. ????????

design     depends

Mathematically proficient students who can apply what they know are comfortable making  ____ and approximations to simplify a complicated situation, realizing that these may need ____  later. ????????

assumptions    revision

They are able to identify important quantities in a ____  situation and map their relationships using such ____ as diagrams, two-way tables, graphs, flowcharts and formulas. ????????

practical    tools

They can analyze those relationships mathematically to draw ____. ????????

conclusions

They routinely interpret their mathematical results in the ____ of the situation and reflect on whether the results make sense, possibly improving the ____ if it has not served its purpose. ????????

context      model

##### 5) Use appropriate tools strategically.

Mathematically proficient students consider the available tools when ____  a mathematical problem.????????

solving

These ____ 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. ????????

tools

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 ____  to be gained and their ____. ????????

insight      limitations

For example, mathematically proficient high school students analyze graphs of ____ and ____  generated using a graphing calculator. ????????

functions       solutions

They detect possible errors by strategically using ____  and other mathematical knowledge. ????????

estimation

When making mathematical models, they know that technology can enable them to ____ the results of  varying assumptions, explore ____, and compare predictions with ____. ????????

visualize       consequences        data

Mathematically proficient students at various grade levels are able to identify relevant external mathematical resources, such as digital content located on a ____, and use them to ____ or solve problems. ????????

website     pose

They are able to use technological tools to ____  and ____  their understanding of concepts. ????????

explore      deepen

##### 6) Attend to precision.

Mathematically proficient students try to communicate ____ to others. ????????

precisely

They try to use clear definitions in ____ with others and in their own reasoning. ????????

discussion

They state the meaning of the ____ they choose, including using the equal sign consistently and appropriately. ????????

symbols

They are careful about specifying units of ____, and labeling axes to clarify the correspondence with quantities in a problem. ????????

measure

They calculate accurately and ____, express numerical answers with a degree of precision appropriate for the problem context. ????????

efficiently

In the elementary grades, students give carefully formulated ____  to each other. ????????

explanations

By the time they reach high school they have learned to examine claims and make ____ use of definitions. ????????

explicit

##### 7) Look for and make use of structure.

Mathematically proficient students look closely to discern a ____ or structure. ????????

pattern

Young students, for example, might notice that three and seven more is the ____ ____ as seven and three more, or they may sort a collection of shapes according to how many sides the shapes have. ????????

same amount

Later, students will see 7 × 8 equals the well-remembered 7 × 5 + 7 × 3, in preparation for learning about the ____ ____  . In the expression x2 + 9x + 14, older students can see the 14 as 2 × 7 and the 9 as 2 +7. ????????

distributive property

They recognize the significance of an existing line in a ____ ____ and can use the strategy of drawing an auxiliary line for solving problems.  ????????

geometric figure

They also can step back for an overview and ____  perspective. ????????

shift

They can see complicated things, such as some algebraic expressions, as single objects or as being ____  of several objects. ????????

composed

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 ____ cannot be more than 5 for any real numbers x and y. ????????

value

##### 8) Look for and express regularity in repeated reasoning.

Mathematically proficient students notice if calculations are ____, and look both for general methods and for shortcuts. ????????

repeated

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 ____ ____ whether points are on the line through (1, 2) with slope 3, middle school students might abstract the equation (y – 2)/(x – 1) = 3.  ????????

repeatedly check

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 ____ ____  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. ????????

general formula

They continually evaluate the reasonableness of their____  results. ????????

intermediate

##### 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 ________ as they grow in mathematical maturity and expertise throughout the elementary, middle and high school years. ????????

subject matter

Designers of curricula, assessments, and professional development should all attend to the need to connect the mathematical practices to mathematical content in ____ ___. ????????

subject matter

The Standards for Mathematical Content are a balanced combination of  and understanding. ????????

procedure

Expectations that begin with the word “understand” are often especially good opportunities to ____ the practices to the content. ????????

connect

Students who lack understanding of a topic may rely on ____ too heavily. ????????

procedures

Without a flexible base from which to work, they may be less likely to

consider ____ problems, ????????

analogous

represent problems ____, ????????

coherently

justify ____, ????????

conclusions

apply the mathematics to ____ situations, ????????

practical

use technology mindfully to ____ with the mathematics, ????????

work

explain the mathematics  to other students, ????????

accurately

step back for an , or ????????

overview

deviate from a known ____ to find a shortcut. ????????

procedure

In short, a lack of understanding effectively prevents a student from ____ in the mathematical practices. ????????

engaging

In this respect, those content standards which set an  of understanding are potential “points of intersection” between the Standards for Mathematical Content and the Standards for Mathematical Practice.????????

expectation

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 ____ ____ the curriculum, instruction, assessment, professional development, and student achievement in mathematics. ????????

qualitatively improve

## Mathematical Practices – Common Core CA

The Common Core State Standards for Mathematical Practice are expected to be integrated into every mathematics lesson for all students in Grades K-12.  Below are the standards and a few examples of how these Eight Practices may be integrated into problem solving tasks.

1. Make sense of problems and persevere in solving them.

Solve real world problems through the application of arithmetic, algebraic, statistical and geometric concepts. These problems involve addition, multiplication, subtraction, division, ratio, rate, area, volume and statistics.   —-   Seek the meaning of a problem and look for efficient ways to represent and solve it.  —  You may check your thinking by asking, “What is the most efficient way to solve the problem?”, “Does this make sense?”, and “Can I solve the problem in a different way?”.   —  Explain the relationships between equations, verbal descriptions, tables and graphs. —   When mathematically proficient check answers to problems using a different method.

 How can this information be used?
 What other information may be needed?
 Why did I choose that operation?
 What is another way to solve that problem?
 What did I do first? Why?
 What can I do if I don’t know how to solve a problem?
 Have I solved a problem similar to this one?
 When will I realize my first method may not work for this problem?
 How do I know my answer makes sense?

2. Reason abstractly and quantitatively.

Represent a wide variety of real world contexts through the use of real numbers and variables in mathematical expressions, equations, and inequalities.   —  Analyze to understand the meaning of the number or variable as related to the problem — Manipulate symbolic representations by applying properties of operations.

 What is a situation that could be represented by this equality?
 What operation can I use to represent the situation?
 Why does that operation represent the situation?
 What properties did I use to find the answer?
 How do I know my answer is reasonable?

3. Construct viable arguments and critique the reasoning of others.
Construct arguments using verbal or written explanations accompanied by expressions, equations, inequalities, models, and graphs, tables, and other data displays (i.e. box plots, dot plots, histograms, etc.). — Further refine mathematical communication skills through mathematical discussions in which I critically evaluate my own thinking and the thinking of other students.  —  Pose questions like “How did I get that?”, “Why is that true?” “Does that always work?” —  Explain my  thinking to others and respond to others’ thinking.

 Will that method always work?
 How do I know?
 What do I think about what was said?
 Who can tell me about a different method?
 What do I think will happen if …?
 When would that not be true?
 Why do I agree/disagree with what was said?
 How does that drawing support my work?

4. Model with mathematics.
Model problem situations symbolically, graphically, with tables, and contextually. —  You form expressions, equations, or inequalities from real world contexts and connect symbolic and graphical representations.  —  You begin to explore co-variance and represent two quantities simultaneously. —  You use number lines to compare numbers and represent inequalities.  —  You use measures of center and variability and data displays (i.e. box plots and histograms) to draw inferences about and make comparisons between data sets. —  You need many opportunities to connect and explain the connections between the different representations.  —  You should be able to use all of these representations as appropriate to a problem context.

 Why is that a good model for this problem?
 How can I use a simpler problem to help me find the answer?
 What conclusions can be made from this model?
 How would I change my model if…?

5. Use appropriate tools strategically.

Consider available tools (including estimation and technology) when solving a mathematical problem and decide when certain tools might be helpful. For instance, you may decide to represent figures on the coordinate plane to calculate area. —  Number lines are used to understand division and to create dot plots, histograms and box plots to visually compare the center and variability of the data. —  Additionally, you might use physical objects or applets to construct nets and calculate the surface area of three-dimensional figures.

 What could I use to help me solve this problem?
 What strategy could I use to make that calculation easier?
 How would estimation help me solve that problem?
 Why did I decide to use…?

6. Attend to precision.

In grade 6, you continue to refine their mathematical communication skills by using clear and precise language in your discussions with others and in their own reasoning. —  You use appropriate terminology when referring to rates, ratios, geometric figures, data displays, and components of expressions, equations or inequalities.

 How do I know my answer is reasonable?
 How can better I use math vocabulary in my explanation?
 How do I know those answers are equivalent?
 What does that mean?

7. Look for and make use of structure.

You routinely seek patterns or structures to model and solve problems. For instance, you recognize
patterns that exist in ratio tables recognizing both the additive and multiplicative properties. —  You apply properties to generate equivalent expressions (i.e. 6 + 2x = 3 (2 + x) by distributive property) and solve equations (i.e. 2c + 3 = 15, 2c = 12 by subtraction property of equality, c=6 by division property of equality). —  You compose and decompose two  and three-dimensional figures to solve real world problems involving area and volume.

 How did I discover that pattern?
 What other patterns can I find?
 What rule did I use to make this group?
 Why can I use that property in this problem?
 How is that like…?

8. Look for and express regularity in repeated reasoning.

.You use repeated reasoning to understand algorithms and make generalizations about patterns..  During multiple opportunities to solve and model problems, you may notice that a/b ÷ c/d = ad/bc and construct other examples and models that confirm your generalization. —  You  connect place value and your prior work with operations to understand algorithms to fluently divide multi-digit numbers and perform all operations with multi-digit decimals.  —  You informally begin to make connections between covariance, rates, and representations showing the relationships between quantities.

 What do I remember about…?
 What happens when…?
 What if I did …instead of …?
 What might be a shortcut for …?

Revised: 01/17/13
.

## Mathematical Practices

Mathematical Practices Make sense of problems and persevere in solving them.Reason abstractly and quantitatively
Summary of Standards for Mathematical Practice#1 Make sense of problems and persevere in solving them.#2 Reason abstractly and quantitatively
Make sense of problems and persevere in solving them.
• Interpret and make meaning of the problem looking for starting points. Analyze what is given to explain to themselves the meaning of the problem.
• Plan a solution pathway instead of jumping to a solution.
• Monitor the progress and change the approach if necessary.
• See relationships between various representations.
• Relate current situations to concepts or skills previously learned and connect mathematical ideas to one another.
• Students ask themselves, “Does this make sense?” and understand various approaches to solutions.
• Analyze problems and use stated mathematical assumptions, definitions, and established results in constructing arguments.
• Justify conclusions with mathematical ideas.
• Listen to the arguments of others and ask useful questions to determine if an argument makes sense.
• Ask clarifying questions or suggest ideas to improve/revise the argument.
• Compare two arguments and determine correct or flawed logic.
Questions to Develop Mathematical Thinking

How would you describe the problem in your own words?
How would you describe what you are trying to find?
What information is given in the problem?
Describe the relationship between the quantities.
Describe what you have already tried. What might you change?
Talk me through the steps you’ve used to this point.
What steps in the process are you most confident about?
What are some other strategies you might try?
What are some other problems that are similar to this one?
How else might you organize...represent...show...?
What mathematical evidence supports your solution?
How can you be sure that...? / How could you prove that...? Will it still work if...?
What were you considering when...?
How did you decide to try that strategy?
How did you test whether your approach worked?
How did you decide what the problem was asking you to find? (What was unknown?)
Did you try a method that did not work? Why didn’t it work? Would it ever work?
Why or why not?
What is the same and what is different about...?
How could you demonstrate a counter-example?
Task: elements to keep in mind when determining learning experiences

Requires students to engage with conceptual ideas that underlie the procedures to complete the task and develop understanding.

Requires cognitive effort - while procedures may be followed, the approach or pathway is not explicitly suggested by the task, or task instructions and multiple entry points are available. The problem focuses students’ attention on a mathematical idea and provides an opportunity to develop and/or use mathematical habits of mind.

Allows for multiple entry points and solution paths as well as, multiple representations, such as visual diagrams, manipulatives, symbols, and problem situations. Making connections among multiple representations to develop meaning.

Requires students to access relevant knowledge and experiences and make appropriate use of them in working through the task.

Requires students to defend and justify their solutions
Is structured to bring out multiple representations, approaches, or error analysis.

Embeds discussion and communication of reasoning and justification with others

Requires students to provide evidence to explain their thinking beyond merely using computational skills to find a solution.

Expects students to give feedback and ask questions of others’ solutions.
Teacher: actions that further the development of math practices within their studentsAllows students time to initiate a plan; uses question prompts as needed to assist students in developing a pathway.

Continually asks students if their plans and solutions make sense.

Questions students to see connections to previous solution attempts and/or tasks to make sense of current problem

Consistently asks to defend and justify their solution by comparing solution paths.

Differentiates to keep advanced students challenged during work time
Encourages students to use proven mathematical understandings, (definitions, properties, conventions, theorems, etc.), to support their reasoning

Questions students so they can tell the difference between assumptions and logical conjectures

Asks questions that require students to justify their solution and their solution pathway

Prompts students to respectfully evaluate peer arguments when solutions are shared.

Asks students to compare and contrast various solution methods.

Creates various instructional opportunities for students to engage in mathematical discussions (whole group, small group, partners, etc.).

## Lessons in Computation and Problem Solving

##### Basic operations of addition, subtraction, multiplication, and division

Whole Numbers – Factors, Divisors,  Multiples.Prime, Composite, LCM, GCF

Fractions – Mixed Numbers, Arithmetic, Ratio, Proportion

Decimals -Rounding Off, Arithmetic

Convert Fractions to Decimals and Vic Versa

Compare Fractions and Decimals

The Percent Proportion

Percentages

Solve Equations

Solve Inequalities

Backsolving

Identify Numbers Needed

What can be found

Alternative solution methods

## Lessons in Estimation, Measurement, and Statistical Principles

##### Basics Review

The Number Line

Places and Digitsa

Integer  Arithmetic

Arithmetic Operations

Factors and Multiples

Fraction  Arithmetic

Decimal Arithmetic

Percentage

##### Three Types of Questions

Questions of Process

Word Problems

Conceptual Understanding

##### Measurement: Standard units & conversions

Length

Temperature

Weight

Capacity

Time

Unit fractions and Conversions

##### Basic Geometry: Measure angles, length, perimeter and area

Triangles – Angles, Perimeter, Pythagorean Theorem and Area

Rectangles – Angles, Perimeter an Area

Circles – Degrees,  Diameters, Circumference, lines, segments and Area

Angle Measurement

Parallel Lines

##### Estimation: Find arithmetic results without complex computation

Rounding

Multiplication – Division

##### Statistical principles: Averages, ratios, and proportions

Levels of Data

Averages – Normal and Weighted

Ratios

Proportions

##### Probability

Defined

Expressed as a Fraction

##### Standardized test scores

Measures of Relative Standing

Percentile Scores

Stanine Scores

Relating Stanine with Percental

## Domain: Operations & Algebraic Thinking – CA Common Core

##### 1.OA  Operations and Algebraic Thinking

Students develop strategies for adding and subtracting whole numbers based on their prior work with small numbers.

They use a variety of models, including discrete objects and length-based models (e.g., cubes connected to form lengths), to model add-to, take-from, put-together, take-apart, and compare situations to develop meaning for the operations of addition and subtraction, and to develop strategies to solve arithmetic problems with these operations.

Students understand connections between counting and addition and subtraction (e.g., adding two is the same as counting on two).

They use properties of addition to add whole numbers and to create and use increasingly sophisticated strategies based on these properties (e.g., “making tens”) to solve addition and subtraction problems within 20.

By comparing a variety of solution strategies, students build their understanding of the relationship between addition and subtraction

##### Represent and solve problems involving addition and subtraction.

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.

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.

3. Apply properties of operations as strategies to add and subtract.3 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.)

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.

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

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.

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.

8. Determine the unknown whole number in an addition or subtraction equation relating 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 = .

##### 2.OA  Operations and Algebraic Thinking

Students use their understanding of addition to develop fluency with addition and subtraction within 100.

They solve problems within 1000 by applying their understanding of models for addition and subtraction, and they develop, discuss, and use efficient, accurate, and generalizable methods to compute sums and differences of whole numbers in base-ten notation, using their understanding of place value and the properties of operations.

They select and accurately apply methods that are appropriate for the context and the numbers involved to mentally calculate sums and differences for numbers with only tens or only hundreds.

##### Represent and solve problems involving addition and subtraction.

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.1 Add and subtract within 20.

2. Fluently add and subtract within 20 using mental strategies.2 By end of Grade 2, know from memory all sums of two one-digit numbers. Work with equal groups of objects to gain foundations for multiplication.

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.

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.

##### 3.OA  Operations and Algebraic Thinking

Students develop an understanding of the meanings of multiplication and division of whole numbers through activities and problems involving equal-sized groups, arrays, and area models; multiplication is finding an unknown product, and division is finding an unknown factor in these situations.

For equal-sized group situations, division can require finding the unknown number of groups or the unknown group size.

Students use properties of operations to calculate products of whole numbers, using increasingly sophisticated strategies based on these properties to solve multiplication and division problems involving
single-digit factors.

By comparing a variety of solution strategies, students learn the relationship between multiplication and
division.

##### Represent and solve problems involving multiplication and division.

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.

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.

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.1

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 = ?.

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

5. Apply properties of operations as strategies to multiply and divide.2 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.)

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.

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.

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.32

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 the four operations with whole numbers to solve problems.

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.

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.

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.

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.

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.

##### 5.OA   Operations and Algebraic Thinking

Students write and interpret numerical expressions.

They analyze patterns and relationships

##### Write and interpret numerical expressions.

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

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.

2.1 Express a whole number in the range 2–50 as a product of its prime factors. For example, find the prime factors of 24 and express 24 as 2 × 2 × 2 × 3. CA

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.

## Domain: Measurement and Data – CA Common Core

##### K.MD  Measurement and Data

Students describe and compare measurable attributes.

Students classify objects and count the number of objects in categories

##### Describe and compare measurable attributes.

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

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.

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

##### 1.MD  Measurement and Data

Students develop an understanding of the meaning and processes of measurement, including underlying concepts such as iterating (the mental activity of building up the length of an object with equal-sized units) and the transitivity principle for indirect measurement.1

##### Measure lengths indirectly and by iterating length units.

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

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.

##### Tell and write time.

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

##### Represent and interpret data.

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.

##### 2.MD  Measurement and Data

Students recognize the need for standard units of measure (centimeter and inch) and they use rulers and other measurement tools with the understanding that linear measure involves an iteration of units.

They recognize that the smaller the unit, the more iterations they need to cover a given length

##### Measure and estimate lengths in standard units.

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

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.

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

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.

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.

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.

##### Work with time and money.

7. Tell and write time from analog and digital clocks to the nearest five minutes, using a.m. and p.m. Know relationships of time (e.g., minutes in an hour, days in a month, weeks in a year). CA

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.

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.

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 problems4 using information presented in a bar graph.

##### 3.MD  Measurement and Data

Students recognize area as an attribute of two-dimensional regions. They measure the area of a shape by finding the total number of same-size units of area required to cover the shape without gaps or overlaps, a square with sides of unit length being the standard unit for measuring area.

Students understand that rectangular arrays can be decomposed into identical rows or into identical columns.

By decomposing rectangles into rectangular arrays of squares, students connect area to
multiplication, and justify using multiplication to determine the area of a rectangle.

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

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.

2. Measure and estimate liquid volumes and masses of objects using standard units of grams (g), kilograms (kg), and liters (l)

3. *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.

##### Represent and interpret data.

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.

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.

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.

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

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.

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.

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.

##### 4.MD  Measurement and Data

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

They represent and interpret data.

In geometric measurement students understand the concepts of angle and the measure of angles.

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

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), . . .

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.

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. Represent and interpret data.

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.

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.

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

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.

##### 5.MD  Measurement and Data

Students recognize volume as an attribute of three-dimensional space. They understand that volume can be measured by finding the total number of same-size units of volume required to fill the space without gaps or overlaps.

They understand that a 1-unit by 1-unit by 1-unit cube is the standard unit for measuring volume.

They select appropriate units, strategies, and tools for solving problems that involve estimating and measuring volume.

They decompose three-dimensional shapes and find volumes of right rectangular prisms by viewing them as decomposed into layers of arrays of cubes.

They measure necessary attributes of shapes in order to determine volumes to solve real-world and mathematical problems.

##### Convert like measurement units within a given measurement system.

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.

##### Represent and interpret data.

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 to addition.

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.

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.

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

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.

## Common Core Geometry

Geometry studies begin in early grades.

##### K.G  Geometry

Students describe their physical world using geometric ideas (e.g., shape, orientation, spatial relations) and vocabulary.

They identify, name, and describe basic two-dimensional shapes, such as squares, triangles, circles, rectangles, and hexagons, presented in a variety of ways (e.g., with different sizes and orientations), as well as three-dimensional shapes such as cubes, cones, cylinders, and spheres.

They use basic shapes and spatial reasoning to model objects in their environment and to construct more complex shapes.

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

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.

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

3. Identify shapes as two-dimensional (lying in a plane, “flat”) or three-dimensional (“solid”).
Analyze, compare, create, and compose shapes.

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).

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

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

##### 1.G  Geometry

Students compose and decompose plane or solid figures (e.g., put two triangles together to make a quadrilateral) and build understanding of part-whole relationships as well as the properties of the original and composite shapes.

As they combine shapes, they recognize them from different perspectives and orientations, describe their geometric attributes, and determine how they are alike and different, to develop the background for measurement and for initial understandings of properties such as congruence and symmetry.

##### Reason with shapes and their attributes.

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.

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.42

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.

##### 2.G  Geometry

Students describe and analyze shapes by examining their sides and angles.

Students investigate, describe, and reason about decomposing and combining shapes to make other shapes.

Through building, drawing, and analyzing two- and three-dimensional shapes, students develop a foundation for understanding area, volume, congruence, similarity, and symmetry in later grades.

##### Reason with shapes and their attributes.

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

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

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.

##### 3.G  Geometry

Students recognize area as an attribute of two-dimensional regions.

They measure the area of a shape by finding the total number of same-size units of area required to cover the shape without gaps or overlaps, a square with sides of unit length being the standard unit for measuring area.

Students understand that rectangular arrays can be decomposed into identical rows or into identical columns.

By decomposing rectangles into rectangular arrays of squares, students connect area to
multiplication, and justify using multiplication to determine the area of a rectangle.

Students describe, analyze, and compare properties of two-dimensional shapes.

They compare and classify shapes by their sides and angles, and connect these with definitions of shapes.

Students also relate their fraction work to geometry by expressing the area of part of a shape as a unit fraction of the whole.

##### Reason with shapes and their attributes.

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.

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.

##### 4.G  Geometry

Students describe, analyze, compare, and classify two-dimensional shapes.

Through building, drawing, and analyzing two-dimensional shapes, students deepen their understanding of properties of two-dimensional objects and the use of them to solve problems involving symmetry.

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

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

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. (Two-dimensional shapes should include special triangles, e.g., equilateral, isosceles, scalene, and special quadrilaterals, e.g., rhombus, square, rectangle, parallelogram, trapezoid.) CA

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.

##### 5.G  Geometry

Students recognize volume as an attribute of three-dimensional space. They understand that volume can be measured by finding the total number of same-size units of volume required to fill the space without gaps or overlaps.

They understand that a 1-unit by 1-unit by 1-unit cube is the standard unit for measuring volume.

They select appropriate units, strategies, and tools for solving problems that involve estimating and measuring volume.

They decompose three-dimensional shapes and find volumes of right rectangular prisms by viewing them as decomposed into layers of arrays of cubes.

They measure necessary attributes of shapes in order to determine volumes to solve real-world and mathematical problems.

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

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).

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.

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.

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

##### 6.G  Geometry

Students build on their work with area in elementary school by reasoning about relationships among shapes to determine area, surface area, and volume.

They find areas of right triangles, other triangles, and special quadrilaterals by decomposing these shapes, rearranging or removing pieces, and relating the shapes to rectangles.

Using these methods, students discuss, develop, and justify formulas for areas of triangles and parallelograms.

Students find areas of polygons and surface areas of prisms and pyramids by decomposing them into pieces whose area they can determine.

They reason about right rectangular prisms with fractional side lengths to extend formulas for the volume of a right rectangular prism to fractional side lengths.

They prepare for work on scale drawings and constructions in grade 7 by drawing polygons in the
coordinate plane.

##### Solve real-world and mathematical problems involving area, surface area, and volume.

1. Find the area of right triangles, other triangles, special quadrilaterals, and polygons by composing into rectangles or decomposing into triangles and other shapes; apply these techniques in the context of solving real-world and mathematical problems.

2. Find the volume of a right rectangular prism with fractional edge lengths by packing it with unit cubes of the appropriate unit fraction edge lengths, and show that the volume is the same as would be found by multiplying the edge lengths of the prism. Apply the formulas V = l w h and V = b h to find volumes of right rectangular prisms with fractional edge lengths in the context of solving real-world and mathematical problems.

3. Draw polygons in the coordinate plane given coordinates for the vertices; use coordinates to find the length of a side joining points with the same first coordinate or the same second coordinate. Apply these techniques in the context of solving real-world and mathematical problems.

4. Represent three-dimensional figures using nets made up of rectangles and triangles, and use the nets to find the surface area of these figures. Apply these techniques in the context of solving real-world and mathematical problems.

##### 7.G  Geometry

Students continue their work with area from grade 6, solving problems involving the area and circumference of a circle and surface area of three-dimensional objects.

In preparation for work on congruence and similarity in grade 8 they reason about relationships among two-dimensional figures using scale drawings and informal geometric constructions, and they
gain familiarity with the relationships between angles formed by intersecting lines.

Students work with three-dimensional figures, relating them to two-dimensional figures by examining cross-sections.

They solve real-world and mathematical problems involving area, surface area, and volume of two- and three-dimensional objects composed of triangles, quadrilaterals, polygons, cubes, and right prisms.

##### Draw, construct, and describe geometrical figures and describe the relationships between them.

1. Solve problems involving scale drawings of geometric figures, including computing actual lengths and areas from a scale drawing and reproducing a scale drawing at a different scale.

2. Draw (freehand, with ruler and protractor, and with technology) geometric shapes with given conditions. Focus on constructing triangles from three measures of angles or sides, noticing when the conditions determine a unique triangle, more than one triangle, or no triangle.

3. Describe the two-dimensional figures that result from slicing three-dimensional figures, as in plane sections of right rectangular prisms and right rectangular pyramids. Solve real-life and mathematical problems involving angle measure, area, surface area, and volume.

4. Know the formulas for the area and circumference of a circle and use them to solve problems; give an informal derivation of the relationship between the circumference and area of a circle.

5. Use facts about supplementary, complementary, vertical, and adjacent angles in a multi-step problem to write and solve simple equations for an unknown angle in a figure.

6. Solve real-world and mathematical problems involving area, volume and surface area of two- and three-dimensional objects composed of triangles, quadrilaterals, polygons, cubes, and right prisms.

##### 8.G  Geometry

Students use ideas about distance and angles, how they behave under translations, rotations, reflections, and dilations, and ideas about congruence and similarity to describe and analyze two-dimensional figures and to solve problems.

Students show that the sum of the angles in a triangle is the angle formed by a straight line, and that various configurations of lines give rise to similar triangles because of the angles created when a transversal cuts parallel lines.

Students understand the statement of the Pythagorean Theorem and its converse, and can explain why the Pythagorean Theorem holds, for example, by decomposing a square in two different ways.

They apply the Pythagorean Theorem to find distances between points on the coordinate plane, to find lengths, and to analyze polygons.

Students complete their work on volume by solving problems involving cones, cylinders, and spheres.

##### Understand congruence and similarity using physical models, transparencies, or geometry software.

1. Verify experimentally the properties of rotations, reflections, and translations:

a. Lines are taken to lines, and line segments to line segments of the same length.

b. Angles are taken to angles of the same measure.

c. Parallel lines are taken to parallel lines.

2. Understand that a two-dimensional figure is congruent to another if the second can be obtained from the first by a sequence of rotations, reflections, and translations; given two congruent figures, describe a sequence that exhibits the congruence between them.

3. Describe the effect of dilations, translations, rotations, and reflections on two-dimensional figures using coordinates.

4. Understand that a two-dimensional figure is similar to another if the second can be obtained from the first by a sequence of rotations, reflections, translations, and dilations; given two similar two-dimensional figures, describe a sequence that exhibits the similarity between them.

5. Use informal arguments to establish facts about the angle sum and exterior angle of triangles, about the angles created when parallel lines are cut by a transversal, and the angle-angle criterion for similarity of triangles. For example, arrange three copies of the same triangle so that the sum of the three angles appears to form a line, and give an argument in terms of transversals why this is so.

##### Understand and apply the Pythagorean Theorem.

6. Explain a proof of the Pythagorean Theorem and its converse.

7. Apply the Pythagorean Theorem to determine unknown side lengths in right triangles in real-world and mathematical problems in two and three dimensions.

8. Apply the Pythagorean Theorem to find the distance between two points in a coordinate system.

##### Solve real-world and mathematical problems involving volume of cylinders, cones, and spheres.

9. Know the formulas for the volumes of cones, cylinders, and spheres and use them to solve real-world and mathematical problems.

## Common Core Arithmetic

Arithmetic is the study of counting, comparison of quantities, number operations and properties of the rational number system.  These studies include applications to daily and professional life.

##### Know number names and the count sequence.

1. Count to 100 by ones and by tens.

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

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).

##### Count to tell the number of objects.

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.

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.

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

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

##### Extend the counting sequence.

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.

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).

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.

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.

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

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 between addition and subtraction; relate the strategy to a written method and explain the reasoning used.

##### Understand place value.

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).

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

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

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.

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

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

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.

7.1 Use estimation strategies to make reasonable estimates in problem solving. CA

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

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

##### 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.

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.

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.

##### Develop understanding of fractions as numbers.

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.

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.

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

##### Generalize place value understanding for multi-digit whole numbers.

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.

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.

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.

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

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.

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.2

##### Extend understanding of fraction equivalence and ordering.

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.

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.

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.

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.

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 100.4 For example, express 3/10 as 30/100, and add 3/10 + 4/100 = 34/100.2

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..

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 the number line or another visual model.

##### Understand the place value system.

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.

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.

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.

4. Use place value understanding to round decimals to any place. Perform operations with multi-digit whole numbers and with decimals to hundredths.

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

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.

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.

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

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.)

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 + 1/2 = 3/7, by observing that 3/7 < 1/2. Apply and extend previous understandings of multiplication and division to multiply and divide fractions.

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 3/4 as the result of dividing 3 by 4, noting that 3/4 multiplied by 4 equals 3, and that when 3 wholes are shared equally among 4 people each person has a share of size 3/4. 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?

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.

5. Interpret multiplication as scaling (resizing), by:

a. Comparing the size of a product to the size o 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 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.

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.

7. Apply and extend previous understandings of division to divide unit fractions by whole numbers and whole numbers by unit fractions.1

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?

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

1. Interpret and compute quotients of fractions, and solve word problems involving division of fractions by fractions, e.g., by using visual fraction models and equations to represent the problem. For example, create a story context for (2/3) ÷ (3/4) and use a visual fraction model to show the quotient; use the relationship between multiplication and division to explain that (2/3) ÷ (3/4) = 8/9 because 3/4 of 8/9 is 2/3. (In general, (a/b) ÷ (c/d) = ad/bc.) How much chocolate will each person get if 3 people share 1/2 lb of chocolate equally? How many 3/4-cup servings are in 2/3 of a cup of yogurt? How wide is a rectangular strip of land with length 3/4 mi and area 1/2 square mi?

##### Compute fluently with multi-digit numbers and find common factors and multiples.

2. Fluently divide multi-digit numbers using the standard algorithm.

3. Fluently add, subtract, multiply, and divide multi-digit decimals using the standard algorithm for each operation.

4. Find the greatest common factor of two whole numbers less than or equal to 100 and the least common multiple of two whole numbers less than or equal to 12. Use the distributive property to express a sum of two whole numbers 1–100 with a common factor as a multiple of a sum of two whole numbers with no common factor. For example, express 36 + 8 as 4 (9+ 2).

##### Apply and extend previous understandings of numbers to the system of rational numbers.

5. Understand that positive and negative numbers are used together to describe quantities having opposite directions or values (e.g., temperature above/below zero, elevation above/below sea level, credits/debits, positive/negative electric charge); use positive and negative numbers to represent quantities in real-world contexts, explaining the meaning of 0 in each situation.

6. Understand a rational number as a point on the number line. Extend number line diagrams and coordinate axes familiar from previous grades to represent points on the line and in the plane with negative number coordinates.

a. Recognize opposite signs of numbers as indicating locations on opposite sides of 0 on the number line; recognize that the opposite of the opposite of a number is the number itself, e.g., –(–3) = 3, and that 0 is its own opposite.

b. Understand signs of numbers in ordered pairs as indicating locations in quadrants of the coordinate plane; recognize red pairs differ only by signs, the locations of the points are related by reflections across one or both axes.

c. Find and position integers and other rational numbers on a horizontal or vertical number line diagram; find and position pairs of integers and other rational numbers on a coordinate plane.

7. Understand ordering and absolute value of rational numbers.

a. Interpret statements of inequality as statements about the relative position of two numbers on a number line diagram. For example, interpret –3 > –7 as a statement that –3 is located to the right of –7 on a number line oriented from left to right.

b. Write, interpret, and explain statements of order for rational numbers in real-world contexts. For example, write –3°C > –7°C to express the fact that –3°C is warmer than –7°C.

c. Understand the absolute value of a rational number as its distance from 0 on the number line; interpret absolute value as magnitude for a positive or negative quantity in a real-world situation. For example, for an account balance of –30 dollars, write |–30| = 30 to describe the size of the debt in dollars.

d. Distinguish comparisons of absolute value from statements about order. For example, recognize that an account balance less than –30 dollars represents a debt greater than 30 dollars.

8. Solve real-world and mathematical problems by graphing points in all four quadrants of the coordinate plane.

##### Understand ratio concepts and use ratio reasoning to solve problems.

1. Understand the concept of a ratio and use ratio language to describe a ratio relationship between two quantities. For example, “The ratio of wings to beaks in the bird house at the zoo was 2:1, because for every 2 wings there was 1 beak.” “For every vote candidate A received, candidate C received nearly three votes.”

2. Understand the concept of a unit rate a/b associated with a ratio a:b with b ≠ 0, and use rate language in the context of a ratio relationship. For example, “This recipe has a ratio of 3 cups of flour to 4 cups of sugar, so there is 3/4 cup of flour for each cup of sugar.” “We paid \$75 for 15 hamburgers, which is a rate of \$5 per hamburger.”1

3. Use ratio and rate reasoning to solve real-world and mathematical problems, e.g., by reasoning about tables of equivalent ratios, tape diagrams, double number line diagrams, or equations.

a. Make tables of equivalent ratios relating quantities with whole number measurements, find missing values in the tables, and plot the pairs of values on the coordinate plane. Use tables to compare ratios.

b. Solve unit rate problems including those involving unit pricing and constant speed. For example, if it took 7 hours to mow 4 lawns, then at that rate, how many lawns could be mowed in 35 hours? At what rate were lawns being mowed?

c. Find a percent of a quantity as a rate per 100 (e.g., 30% of a quantity means 30/100 times the quantity); solve problems involving finding the whole, given a part and the percent.

d. Use ratio reasoning to convert measurement units; manipulate and transform units appropriately when multiplying or dividing quantities.

##### Apply and extend previous understandings of operations with fractions to add, subtract, multiply, and divide rational numbers.

1. Apply and extend previous understandings of addition and subtraction to add and subtract rational numbers; represent addition and subtraction on a horizontal or vertical number line diagram.

a. Describe situations in which opposite quantities combine to make 0. For example, a hydrogen atom has 0 charge because its two constituents are oppositely charged.

b. Understand p + q as the number located a distance |q| from p, in the positive or negative direction depending on whether q is positive or negative. Show that a number and its opposite have a sum of 0 (are additive inverses). Interpret sums of rational numbers by describing real-world contexts.

c. Understand subtraction of rational numbers as adding the additive inverse, p – q = p + (–q). Show that the distance between two rational numbers on the number line is the absolute value of their difference, and apply this principle in real-world contexts.

d. Apply properties of operations as strategies to add and subtract rational numbers.

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

a. Understand that multiplication is extended from fractions to rational numbers by requiring that operations continue to satisfy the properties of operations, particularly the distributive property, leading to products such as (–1)(–1) = 1 and the rules for multiplying signed numbers. Interpret products of rational numbers by describing real-world contexts.

b. Understand that integers can be divided, provided that the divisor is not zero, and every quotient of integers (with non-zero divisor) is a rational number. If p and q are integers, then –(p/q) = (–p)/q = p/(–q). Interpret quotients of rational numbers by describing real-world contexts.

c. Apply properties of operations as strategies to multiply and divide rational numbers.

d. Convert a rational number to a decimal using long division; know that the decimal form of a rational number terminates in 0s or eventually repeats.

3. Solve real-world and mathematical problems involving the four operations with rational numbers.1

##### Analyze proportional relationships and use them to solve real-world and mathematical problems.

1. Compute unit rates associated with ratios of fractions, including ratios of lengths, areas and other quantities measured in like or different units. For example, if a person walks 1/2 mile in each 1/4 hour, compute the unit rate as the complex fraction ½/¼ miles per hour, equivalently 2 miles per hour.

2. Recognize and represent proportional relationships between quantities.

a. Decide whether two quantities are in a proportional relationship, e.g., by testing for equivalent ratios in a table or graphing on a coordinate plane and observing whether the graph is a straight line through the origin.

b. Identify the constant of proportionality (unit rate) in tables, graphs, equations, diagrams, and verbal descriptions of proportional relationships.

c. Represent proportional relationships by equations. For example, if total cost t is proportional to the number n of items purchased at a constant price p, the relationship between the total cost and the number of items can be expressed
as t = pn.

d. Explain what a point (x, y) on the graph of a proportional relationship means in terms of the situation, with specialattention to the points (0, 0) and (1, r) where r is the unit rate.

3. Use proportional relationships to solve multistep ratio and percent problems. Examples: simple interest, tax, markups and markdowns, gratuities and commissions, fees, percent increase and decrease, percent error.

##### Know that there are numbers that are not rational, and approximate them by rational numbers.

1. Know that numbers that are not rational are called irrational. Understand informally that every number has a decimal expansion; for rational numbers show that the decimal expansion repeats eventually, and convert a decimal expansion which repeats eventually into a rational number.

2. Use rational approximations of irrational numbers to compare the size of irrational numbers, locate them approximately on a number line diagram, and estimate the value of expressions (e.g.,π 2). For example, by truncating the decimal expansion of √2, show that √2 is between 1 and 2, then between 1.4 and 1.5, and explain how to continue on to get better approximations.

## THIRD GRADE 3.NF Number and Operation: Fractions

### FOURTH GRADE 4.NF Number and Operations | Fractions

#### Understand decimal notation for fractions, and compare decimal fractions.

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
100.4 For example, express 3=10 as 30=100, and add 3=10 + 4=100 = 34=100.
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.
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
4Students who can generate equivalent fractions can develop strategies for adding fractions with
unlike denominators in general. But addition and subtraction with un-like denominators in general
is not a requirement at this grade.

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

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 equiv-
alent sum or di erence 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.)
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 + 1=2 = 3=7, by observing that 3=7 < 1=2.

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

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 re-
lationship between addition and subtraction; relate the strategy to a written method and
explain the reasoning used.
Apply and extend previous understandings of multiplication and division to
multiply and divide fractions.
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 3/4 as the result of dividing 3 by 4, noting that 3=4
multiplied by 4 equals 3, and that when 3 wholes are shared equally among 4 people each

person has a share of size 3=4. 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
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
rectangles6 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 nd areas of rectangles, and represent fraction products as rectangular areas.
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 num-
bers greater than 1 as a familiar case); explaining why multiplying a given number by a
fraction less than 1 results in a product smaller than the given number; and relating the
principle of fraction equivalence a=b = (na)=(nb) to the e ect of multiplying a=b by 1.
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.
7. Apply and extend previous understandings of division to divide unit fractions by
whole numbers and whole numbers by unit fractions.7
a. Interpret division of a unit fraction by a non-zero whole number, and compute
6In the original, it is incorrectly stated as \squares”.
7Students 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.

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?

##### Understand ratio concepts and use ratio reasoning to solve problems.

1. Understand the concept of a ratio and use ratio language to describe a ratio rela-
tionship between two quantities. For example, \The ratio of wings to beaks in the bird
house at the zoo was 2 : 1, because for every 2 wings there was 1 beak. \For every vote
2. Understand the concept of a unit rate a=b associated with a ratio a : b with b 6= 0,
and use rate language in the context of a ratio relationship. For example, \This recipe
has a ratio of 3 cups of
our to 4 cups of sugar, so there is 3=4 cup of
our for each cup
of sugar. \We paid \$75 for 15 hamburgers, which is a rate of \$5 per hamburger.”11
3. Use ratio and rate reasoning to solve real-world and mathematical problems, e.g., by
reasoning about tables of equivalent ratios, tape diagrams, double number line diagrams,
or equations.
b. Solve unit rate problems including those involving unit pricing and constant
speed. For example, if it took 7 hours to mow 4 lawns, then at that rate, how many lawns
could be mowed in 35 hours? At what rate were lawns being mowed?
c. Find a percent of a quantity as a rate per 100 (e.g., 30% of a quantity means
30/100 times the quantity); solve problems involving nding the whole, given a part and
the percent.

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

11Expectations for unit rates in this grade are limited to non-complex fractions.
1. Interpret and compute quotients of fractions, and solve word problems involving
division of fractions by fractions, e.g., by using visual fraction models and equations to
represent the problem. For example, create a story context for (2=3)  (3=4) and use
a visual fraction model to show the quotient; use the relationship between multiplica-
tion and division to explain that (2=3)  (3=4) = 8=9 because 3=4 of 8=9 is 2=3. (In
general, (a=b)  (c=d) = ad=bc:) How much chocolate will each person get if 3 people
share 1=2 lb of chocolate equally? How many 3=4-cup servings are in 2=3 of a cup of
yogurt? How wide is a rectangular strip of land with length 3=4 mi and area 1=2 square mi?
3. Fluently add, subtract, multiply, and divide multi-digit decimals using the standard
algorithm for each operation.