### Mathematics practices

#### Students will:

- 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.

### Mathematics contents

#### Kindergarten

- Counting and Cardinality
- Know number names and the count sequence.
- 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 each category.

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

#### Grade 1

- 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

- 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

- 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

- Operations and Algebraic Thinking
- Use the four operations with whole numbers to solve problems.
- Gain familiarity with factors and multiples.
- Generate and analyze patterns.
- Multiply and divide within 100.

- Number and Operations in Base Ten
- Generalize place value understanding for multi-digit 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

- 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

- 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.
- Flexibly and efficiently compute 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

- 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

- 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.

**Grades 9-12 – Number and Quantity Overview**

- The Real Number System
- Extend the properties of exponents to rational exponents.
- Use properties of rational and irrational numbers.

- Quantities
- Reason quantitatively and use units to solve problems.

- The Complex Number System
- Perform arithmetic operations with complex numbers.
- Represent complex numbers and their operations on the complex plane.
- Use complex numbers in polynomial identities and equations.

- Vector and Matrix Quantities
- Represent and model with vector quantities.
- Perform operations on vectors.
- Perform operations on matrices and use matrices in applications.

**Grades 9-12 – Algebra**

- Seeing Structure in Expressions
- Interpret the structure of expressions.
- Write expressions in equivalent forms to solve problems.

- Arithmetic with Polynomials and Rational Expressions
- Perform arithmetic operations on polynomials.
- Understand the relationship between zeros and factors of polynomials.
- Use polynomial identities to solve problems.
- Rewrite rational expressions.

- Creating Equations
- Create equations that describe numbers or relationships.

- Reasoning with Equations and Inequalities
- Understand solving equations as a process of reasoning and explain the reasoning.
- Solve equations and inequalities in one variable.
- Solve systems of equations.
- Represent and solve equations and inequalities graphically.

**Grades 9-12 – Functions**

- Interpreting Functions
- Understand the concept of a function and use function notation.
- Intercept functions that arise in applications in terms of the context.
- Analyze functions using different representations.

- Building Functions
- Build a function that models a relationship between two quantities.
- Build new functions from existing functions.

- Linear, Quadratic, and Exponential Models
- Construct and compare linear, quadratic, and exponential models and solve problems.
- Interpret expressions for functions in terms of the situation they model.

- Trigonometric Functions
- Extend the domain of trigonometric functions using the unit circle.
- Model periodic phenomena with trigonometric functions.
- Prove and apply trigonometric identities.

**Grades 9-12 – Geometry**

- Congruence
- Experiment with transformations in the plane.
- Understand congruence in terms of rigid motions.
- Prove geometric theorems.
- Make geometric constructions.

- Similarity, Right Triangles, and Trigonometry
- Understand similarity in terms of similarity transformations.
- Prove theorems involving similarity.
- Define trigonometric ratios and solve problems involving right triangles.
- Apply trigonometry to general triangles.

- Circles
- Understand and apply theorems about circles.
- Find arc lengths and areas of sectors of circles.

- Expressing Geometric Properties with Equations
- Translate between the geometric description and the equation for a conic section.
- Use coordinates to prove simple geometric theorems algebraically.

- Geometric Measurement and Dimension
- Explain volume formulas and use them to solve problems.
- Visualize relationships between two-dimensional and three-dimensional objects.

- Modeling with Geometry
- Apply geometric concepts in modeling situations.

**Grades 9-12 – Statistics and Probability**

- Interpreting Categorical and Quantitative Data
- Summarize, represent, and interpret data on a single count or measurement variable.
- Summarize, represent, and interpret data on two categorical and quantitative variables.
- Interpret linear models.

- Making Inferences and Justifying Conclusions
- Understand and evaluate random processes underlying statistical experiments.
- Make inferences and justify conclusions from sample surveys, experiments, and observational studies.

- Conditional Probability and the Rules of Probability
- Understand independence and conditional probability and use them to interpret data.
- Use the rules of probability to compute probabilities of compound events in a uniform probability model.

- Using Probability to Make Decisions
- Calculate expected values and use them to solve problems.
- Use probability to evaluate outcomes of decisions.