Instructional programs from prekindergarten through grade 12 should enable each and every student to—

• Understand patterns, relations, and functions
• Represent and analyze mathematical situations and structures using algebraic symbols
• Use mathematical models to represent and understand quantitative relationships
• Analyze change in various contexts

Understand patterns, relations, and functions

Pre-K–2 Expectations: In pre-K through grade 2 each and every student 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 each and every student 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 each and every student 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 each and every student 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 each and every student 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 each and every student 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 each and every student 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 each and every student 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 each and every student 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 each and every student 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 each and every student should–

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

Grades 9–12 Expectations: In grades 9–12 each and every student 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 each and every student 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 each and every student 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 each and every student should–

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

Grades 9–12 Expectations: In grades 9–12 each and every student should–

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

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