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Algebra

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. 


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