# Common Core State Standards for Mathematics

## Historical Overview for the Standards for Mathematical Practice

The Standards for Mathematical Practice describe varieties of expertise that mathematics educators at all levels should seek to develop in their students. These practices rest on important*processes and proficiencies*with longstanding importance in mathematics education. The first of these are the NCTM process standards of problem solving, reasoning and proof, communication, representation, and connections. The second are the strands of mathematical proficiency specified in the National Research Council’s report

*Adding It Up*: adaptive reasoning, strategic competence, conceptual understanding (comprehension of mathematical concepts, operations and relations), procedural fluency (skill in carrying out procedures flexibly, accurately, efficiently and appropriately), and productive disposition (habitual inclination to see mathematics as sensible, useful, and worthwhile, coupled with a belief in diligence and one’s own efficacy).

Standards in this domain:

## K-12 Mathematical Practices

K-12 Mathematical Practices (processes and proficiencies) ought to engage with the subject matter as they grow in mathematical maturity and expertise throughout the school years.

- Make sense of problems and persevere in solving them.
- Reason abstractly and quantitatively.
- Construct viable arguments and critique the reasoning of others.
- Model with mathematics.
- Use appropriate tools strategically.
- Attend to precision.
- Look for and make use of structure.
- Look for and express regularity in repeated reasoning.

## Common Core State Standards Mathematics Practice 1 (CCSS.Math.Practice.MP1)

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

Common Core State Standards Mathematics Practice 2 (CCSS.Math.Practice.MP2)

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

Common Core State Standards Mathematics Practice 3 (CCSS.Math.Practice.MP3)

*Construct viable arguments and critique the reasoning of others*. Mathematically proficient students understand and use stated assumptions, definitions, and previously established results in constructing arguments. They make conjectures and build a logical progression of statements to explore the truth of their conjectures. They are able to analyze situations by breaking them into cases, and can recognize and use counterexamples. They justify their conclusions, communicate them to others, and respond to the arguments of others. They reason inductively about data, making plausible arguments that take into account the context from which the data arose. Mathematically proficient students are also able to compare the effectiveness of two plausible arguments, distinguish correct logic or reasoning from that which is flawed, and—if there is a flaw in an argument—explain what it is. Elementary students can construct arguments using concrete referents such as objects, drawings, diagrams, and actions. Such arguments can make sense and be correct, even though they are not generalized or made formal until later grades. Later, students learn to determine domains to which an argument applies. Students at all grades can listen or read the arguments of others, decide whether they make sense, and ask useful questions to clarify or improve the arguments.

Common Core State Standards Mathematics Practice 4 (CCSS.Math.Practice.MP4)

*Model with mathematics*. Mathematically proficient students can apply the mathematics they know to solve problems arising in everyday life, society, and the workplace. In early grades, this might be as simple as writing an addition equation to describe a situation. In middle grades, a student might apply proportional reasoning to plan a school event or analyze a problem in the community. By high school, a student might use geometry to solve a design problem or use a function to describe how one quantity of interest depends on another. Mathematically proficient students who can apply what they know are comfortable making assumptions and approximations to simplify a complicated situation, realizing that these may need revision later. They are able to identify important quantities in a practical situation and map their relationships using such tools as diagrams, two-way tables, graphs, flowcharts and formulas. They can analyze those relationships mathematically to draw conclusions. They routinely interpret their mathematical results in the context of the situation and reflect on whether the results make sense, possibly improving the model if it has not served its purpose.

Common Core State Standards Mathematics Practice 5 (CCSS.Math.Practice.MP5)

*Use appropriate tools strategically*. Mathematically proficient students consider the available tools when solving a mathematical problem. These tools might include pencil and paper, concrete models, a ruler, a protractor, a calculator, a spreadsheet, a computer algebra system, a statistical package, or dynamic geometry software. Proficient students are sufficiently familiar with tools appropriate for their grade or course to make sound decisions about when each of these tools might be helpful, recognizing both the insight to be gained and their limitations. For example, mathematically proficient high school students analyze graphs of functions and solutions generated using a graphing calculator. They detect possible errors by strategically using estimation and other mathematical knowledge. When making mathematical models, they know that technology can enable them to visualize the results of varying assumptions, explore consequences, and compare predictions with data. Mathematically proficient students at various grade levels are able to identify relevant external mathematical resources, such as digital content located on a website, and use them to pose or solve problems. They are able to use technological tools to explore and deepen their understanding of concepts.

Common Core State Standards Mathematics Practice 6 (CCSS.Math.Practice.MP6)

*Attend to precision*. Mathematically proficient students try to communicate precisely to others. They try to use clear definitions in discussion with others and in their own reasoning. They state the meaning of the symbols they choose, including using the equal sign consistently and appropriately. They are careful about specifying units of measure, and labeling axes to clarify the correspondence with quantities in a problem. They calculate accurately and efficiently, express numerical answers with a degree of precision appropriate for the problem context. In the elementary grades, students give carefully formulated explanations to each other. By the time they reach high school they have learned to examine claims and make explicit use of definitions.

Common Core State Standards Mathematics Practice 7 (CCSS.Math.Practice.MP7)

*Look for and make use of structure*. Mathematically proficient students look closely to discern a pattern or structure. Young students, for example, might notice that three and seven more is the same amount as seven and three more, or they may sort a collection of shapes according to how many sides the shapes have. Later, students will see 7 × 8 equals the well remembered 7 × 5 + 7 × 3, in preparation for learning about the distributive property. In the expression x2 + 9x + 14, older students can see the 14 as 2 × 7 and the 9 as 2 + 7. They recognize the significance of an existing line in a geometric figure and can use the strategy of drawing an auxiliary line for solving problems. They also can step back for an overview and shift perspective. They can see complicated things, such as some algebraic expressions, as single objects or as being composed of several objects. For example, they can see 5 – 3(x – y)2 as 5 minus a positive number times a square and use that to realize that its value cannot be more than 5 for any real numbers x and y.

## Common Core State Standards Mathematics Practice 7 (CCSS.Math.Practice.MP8)

*Look for and express regularity in repeated reasoning. * Mathematically proficient students notice if calculations are repeated, and look both for general methods and for shortcuts. Upper elementary students might notice when dividing 25 by 11 that they are repeating the same calculations over and over again, and conclude they have a repeating decimal. By paying attention to the calculation of slope as they repeatedly check whether points are on the line through (1, 2) with slope 3, middle school students might abstract the equation (y – 2)/(x – 1) = 3. Noticing the regularity in the way terms cancel when expanding (x – 1)(x + 1), (x – 1)(x2 + x + 1), and (x – 1)(x3 + x2 + x + 1) might lead them to the general formula for the sum of a geometric series. As they work to solve a problem, mathematically proficient students maintain oversight of the process, while attending to the details. They continually evaluate the reasonableness of their intermediate results.

## Connecting the Standards for Mathematical Practice to the Standards for Mathematical Content

The Standards for Mathematical Practice describe ways in which developing student practitioners of the discipline of mathematics increasingly ought to engage with the subject matter as they grow in mathematical maturity and expertise throughout the elementary, middle and high school years. Designers of curricula, assessments, and professional development should all attend to the need to connect the mathematical practices to mathematical content in mathematics instruction.

The Standards for Mathematical Content are a balanced combination of procedure and understanding. Expectations that begin with the word “understand” are often especially good opportunities to connect the practices to the content. Students who lack understanding of a topic may rely on procedures too heavily. Without a flexible base from which to work, they may be less likely to consider analogous problems, represent problems coherently, justify conclusions, apply the mathematics to practical situations, use technology mindfully to work with the mathematics, explain the mathematics accurately to other students, step back for an overview, or deviate from a known procedure to find a shortcut. In short, a lack of understanding effectively prevents a student from engaging in the mathematical practices.

In this respect, those content standards which set an expectation of understanding are potential “points of intersection” between the Standards for Mathematical Content and the Standards for Mathematical Practice. These points of intersection are intended to be weighted toward central and generative concepts in the school mathematics curriculum that most merit the time, resources, innovative energies, and focus necessary to qualitatively improve the curriculum, instruction, assessment, professional development, and student achievement in mathematics.

## Kindergarten Overview

*Counting and Cardinality*

• Know number names and the count sequence.

• Count to tell the number of objects.

• Compare numbers.

*Operations and Algebraic Thinking*

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

*Number and Operations in Base Ten*

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

*Measurement and Data*

• Describe and compare measurable attributes.

• Classify objects and count the number of objects in categories.

*Geometry*

• Identify and describe shapes.

• Analyze, compare, create, and compose shapes.

## Grade 1 Overview

*Operations and Algebraic Thinking*

• Represent and solve problems involving addition and subtraction.

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

• Add and subtract within 20.

• Work with addition and subtraction equations.

*Number and Operations in Base Ten*

• Extend the counting sequence.

• Understand place value.

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

*Measurement and Data*

• Measure lengths indirectly and by iterating length units.

• Tell and write time.

• Represent and interpret data.

*Geometry*

• Reason with shapes and their attributes.

## Grade 2 Overview

*Operations and Algebraic Thinking*

• Represent and solve problems involving addition and subtraction.

• Add and subtract within 20.

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

*Number and Operations in Base Ten*

• Understand place value.

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

*Measurement and Data*

• Measure and estimate lengths in standard units.

• Relate addition and subtraction to length.

• Work with time and money.

• Represent and interpret data.

*Geometry*

• Reason with shapes and their attributes

## Grade 3 Overview

*Operations and Algebraic Thinking*

• Represent and solve problems involving multiplication and division.

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

• Multiply and divide within 100.

• Solve problems involving the four operations, and identify and explain patterns in arithmetic.

*Number and Operations in Base Ten*

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

*Number and Operations—Fractions*

• Develop understanding of fractions as numbers.

*Measurement and Data*

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

• Represent and interpret data.

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

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

*Geometry*

• Reason with shapes and their attributes

## Grade 4 Overview

*Operations and Algebraic Thinking*

• Use the four operations with whole numbers to solve problems.

• Gain familiarity with factors and multiples.

• Generate and analyze patterns.

*Number and Operations in Base Ten*

• Generalize place value understanding for multidigit whole numbers.

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

*Number and Operations—Fractions*

• Extend understanding of fraction equivalence and ordering.

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

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

*Measurement and Data*

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

• Represent and interpret data.

• Geometric measurement: understand concepts of angle and measure angles.

*Geometry*

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

## Grade 5 Overview

*Operations and Algebraic Thinking*

• Write and interpret numerical expressions.

• Analyze patterns and relationships.

*Number and Operations in Base Ten*

• Understand the place value system.

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

*Number and Operations—Fractions*

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

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

*Measurement and Data*

• Convert like measurement units within a given measurement system.

• Represent and interpret data.

• Geometric measurement: understand concepts of volume and relate volume to multiplication and to addition.

*Geometry*

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

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

## Grade 6 Overview

*Ratios and Proportional Relationships*

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

*The Number System*

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

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

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

*Expressions and Equations*

• Apply and extend previous understandings of arithmetic to algebraic expressions.

• Reason about and solve one-variable equations and inequalities.

• Represent and analyze quantitative relationships between dependent and independent variables.

*Geometry*

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

*Statistics and Probability*

• Develop understanding of statistical variability.

• Summarize and describe distributions.

## Grade 7 Overview

*Ratios and Proportional Relationships*

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

*The Number System*

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

*Expressions and Equations*

• Use properties of operations to generate equivalent expressions.

• Solve real-life and mathematical problems
using numerical and algebraic expressions and equations.

*Geometry*

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

• Solve real-life and mathematical problems involving angle measure, area, surface area, and volume.

*Statistics and Probability*

• Use random sampling to draw inferences about a population.

• Draw informal comparative inferences about two populations.

• Investigate chance processes and develop, use, and evaluate probability models.

## Grade 8 Overview

*The Number System*

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

*Expressions and Equations*

• Work with radicals and integer exponents.

• Understand the connections between proportional relationships, lines, and linear equations.

• Analyze and solve linear equations and pairs of simultaneous linear equations.

*Functions*

• Define, evaluate, and compare functions.

• Use functions to model relationships between quantities.

*Geometry*

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

• Understand and apply the Pythagorean Theorem.

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

*Statistics and Probability*

• Investigate patterns of association in bivariate data.

## Mathematics Standards for High School

The high school standards are listed in conceptual categories:

• Number and Quantity

• Algebra

• Functions

• Modeling

• Geometry

• Statistics and Probability

(+) Represent complex numbers on the complex plane in rectangular and polar form (including real and imaginary numbers) for advanced mathematics.

### Summary

*K-12 Mathematical Practices* (processes and proficiencies) ought to engage with the subject matter as they grow in mathematical maturity and expertise throughout the school years.

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

2. Reason abstractly and quantitatively.

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

4. Model with mathematics.

5. Use appropriate tools strategically.

6. Attend to precision.

7. Look for and make use of structure.

8. Look for and express regularity in repeated reasoning.

*Overview mathematical conten*t

Counting and Cardinality (K)

Operations and Algebraic Thinking (K, 1, 2, 3, 4, 5)

Number and Operations in Base Ten (1, 2, 3, 4, 5)

Number and Operations—Fractions (3, 4, 5)

The Number System (7, 8)

Measurement and Data (K, 1, 2, 3, 4, 5)

Geometry (K, 1, 2, 3, 4, 5, 6, 7, 8)

Ratios and Proportional Relationships (6, 7)

Expressions and Equations (6, 7, 8)

Statistics and Probability (6, 7, 8)

Functions (8)