Mathematics

Mathematics practices

Students will:

• make sense of problems and persevere in solving them;
• reason abstractly and quantitatively;
• construct viable arguments, and appreciate 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.