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Mathematics

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.