**Instructional programs from prekindergarten through grade 12**

should enable all students to—

**Understand patterns, relations, and functions **

**Pre-K–2 Expectations**: In pre-K through grade 2 all students should–

- sort, classify, and order objects by size, number, and other properties;
- recognize, describe, and extend patterns such as sequences of sounds and shapes or simple numeric patterns and translate from one representation to another;
- analyze how both repeating and growing patterns are generated.

**Grades 3–5 Expectations**: In grades 3–5 all students should–

- describe, extend, and make generalizations about geometric and numeric patterns;
- represent and analyze patterns and functions, using words, tables, and graphs.

**Grades 6–8 Expectations**: In grades 6–8 all students should–

- represent, analyze, and generalize a variety of patterns with tables, graphs, words, and, when possible, symbolic rules;
- relate and compare different forms of representation for a relationship;
- identify functions as linear or nonlinear and contrast their properties from tables, graphs, or equations.

**Grades 9–12 Expectations**: In grades 9–12 all students should–

- generalize patterns using explicitly defined and recursively defined functions;
- understand relations and functions and select, convert flexibly among, and use various representations for them;
- analyze functions of one variable by investigating rates of change, intercepts, zeros, asymptotes, and local and global behavior;
- understand and perform transformations such as arithmetically combining, composing, and inverting commonly used functions, using technology to perform such operations on more-complicated symbolic expressions;
- understand and compare the properties of classes of functions, including exponential, polynomial, rational, logarithmic, and periodic functions;
- interpret representations of functions of two variables

**Represent and analyze mathematical situations and structures using algebraic symbols **

**Pre-K–2 Expectations**: In pre-K through grade 2 all students should–

- illustrate general principles and properties of operations, such as commutativity, using specific numbers;
- use concrete, pictorial, and verbal representations to develop an understanding of invented and conventional symbolic notations.

**Grades 3–5 Expectations**: In grades 3–5 all students should–

- identify such properties as commutativity, associativity, and distributivity and use them to compute with whole numbers;
- represent the idea of a variable as an unknown quantity using a letter or a symbol;
- express mathematical relationships using equations.

**Grades 6–8 Expectations**: In grades 6–8 all students should–

- develop an initial conceptual understanding of different uses of variables;
- explore relationships between symbolic expressions and graphs of lines, paying particular attention to the meaning of intercept and slope;
- use symbolic algebra to represent situations and to solve problems, especially those that involve linear relationships;
- recognize and generate equivalent forms for simple algebraic expressions and solve linear equations

**Grades 9–12 Expectations**: In grades 9–12 all students should–

- understand the meaning of equivalent forms of expressions, equations, inequalities, and relations;
- write equivalent forms of equations, inequalities, and systems of equations and solve them with fluency—mentally or with paper and pencil in simple cases and using technology in all cases;
- use symbolic algebra to represent and explain mathematical relationships;
- use a variety of symbolic representations, including recursive and parametric equations, for functions and relations;
- judge the meaning, utility, and reasonableness of the results of symbol manipulations, including those carried out by technology.

**Use mathematical models to represent and understand quantitative relationships **

**Pre-K–2 Expectations**: In pre-K through grade 2 all students should–

- model situations that involve the addition and subtraction of whole numbers, using objects, pictures, and symbols.

**Grades 3–5 Expectations**: In grades 3–5 all students should–

- model problem situations with objects and use representations such as graphs, tables, and equations to draw conclusions.

**Grades 6–8 Expectations**: In grades 6–8 all students should–

- model and solve contextualized problems using various representations, such as graphs, tables, and equations.

**Grades 9–12 Expectations**: In grades 9–12 all students should–

- identify essential quantitative relationships in a situation and determine the class or classes of functions that might model the relationships;
- use symbolic expressions, including iterative and recursive forms, to represent relationships arising from various contexts;
- draw reasonable conclusions about a situation being modeled.

**Analyze change in various contexts **

**Pre-K–2 Expectations**: In pre-K through grade 2 all students should–

- describe qualitative change, such as a student's growing taller;
- describe quantitative change, such as a student's growing two inches in one year.

**Grades 3–5 Expectations**: In grades 3–5 all students should–

- investigate how a change in one variable relates to a change in a second variable;

- identify and describe situations with constant or varying rates of change and compare them.

**Grades 6–8 Expectations**: In grades 6–8 all students should–

- use graphs to analyze the nature of changes in quantities in linear relationships.

**Grades 9–12 Expectations**: In grades 9–12 all students should–

- approximate and interpret rates of change from graphical and numerical data.