Math Rationale
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Math Standard 3
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Math Standard 5
Math Standard 6
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 ArizonaStandards

Mathematics Standards
(Adopted 8/26/96)

Standard 3: Patterns, Algebra And Functions

Students use algebraic methods to explore, model and describe patterns, relationships and functions involving numbers, shapes, data and graphs within a variety of real-world problem-solving situations.

Students know and are able to do the following:

READINESS (Kindergarten)

  • 3M-R1.Create, describe and extend a variety of patterns, using concrete objects
  • 3M-R2.Recognize that the same patterns can emerge from a variety of manipulatives and real-world situations

    • FOUNDATIONS (Grades 1-3)

  • 3M-F1.Create, describe and extend a variety of patterns using shapes, events, designs and numbers

    • Note: Types of patterns: manipulatives, symbols, words, numbers and pictures

    PO 1. Create a pattern using a model (e.g., symbolically: numbers or letters; visually: shapes, designs, numbers or pictures; auditorially: clapping, singing or listening: and kinesthetically: dancing, movement or tactile)

    PO 2. Communicate orally or in written form the repetition of objects in a pattern

    PO 3. Communicate orally or in written form a given pattern occurring in a sequence of numbers (e.g., counting by 10’s, 5’s, 3’s, 2’s, odd, even, forward and backward)

    PO 4. Extend patterns using a model

    PO 5. Extend a given pattern occurring in a sequence of numbers

  • 3M-F2.Formulate generalizations about patterns (e.g., color, shape, size, direction, orientation) to make predictions

    • PO 1. Make predictions based on a given pattern

  • 3M-F3.Represent and describe how changing the value of one variable results in a change in another

PO 1. Describe in a given situation how a change in one variable results in the change of another (e.g., if you have to share a batch of cookies with friends, the more friends you have, the fewer cookies you’ll each get) 

  • 3M-F4.Represent and describe mathematical relationships such as order, grouping, etc. (e.g., given a string of numbers, describe the pattern, define the relationship between the numbers and determine the next number in line)

PO 1. Identify the pattern in skip counting

PO 2. Determine the next number in a skip counting pattern

  • 3M-F5.Recognize the symbols of equality and inequality

PO 1. Use the symbols <, >, = to compare whole numbers

  • 3M-F6.Find missing elements in number sentences

PO 1. Find the missing number in addition and subtraction number sentences

ESSENTIALS (Grades 4-8)

  • 3M-E1.Use algebraic methods (write number sentences, in the form of expressions and equations) to explore, model and describe patterns and functions involving numbers, shapes, data, graphs and data plots

PO 1. Extend simple geometric and number patterns (e.g., 1, 1, 2, 1, 1, 3, 1, 1, 4 . . .) (Grades 4-5)

PO 2. Create simple geometric and number patterns (Grades 4-5)

PO 3. Describe a rule for a simple pattern (e.g., 5, 10, 15, 20 . . . rule = add five or count by fives) (Grades 4-5)

PO 4. Generate patterns using algebraic expressions (Grades 6-8)

  • 3M-E2.Describe, represent and analyze patterns and relationships using shapes, tables, graphs, data plots, verbal rules and standard algebraic notation (This is covered in 3ME1-PO1, PO2, PO3, PO4; and 3ME4-PO1, PO2, PO3, PO4
  • 3M-E3.Describe the concepts of variables, expressions, equations and inequalities

PO 1. Describe and use variables in a contextual situation (Grades 6-8)

PO 2. Evaluate an expression using substitution with four basic operations on whole numbers (Grades 6-8)

PO 3. Translate a written phrase to an algebraic expression and vice versa (words to symbols and symbols to words) (e.g., the quotient of x and y) (Grades 6-8)

PO 4. Express a simple inequality from a contextual situation (e.g., Joe earns more than $5.00 an hour: therefore, x > 5.) (Grades 6-8)

  • 3M-E4.Analyze functional relationships to explain how a change in one variable results in a change in another

PO 1. Describe a real-life situation in which a change in one variable results in the change of the other (e.g., temperature in the classroom goes up and the amount of clothing goes down) (Grades 4-5)

PO 2. Produce the rule (function) that explains the relationship (pattern) between the numbers when a change in the first variable affects the second variable (T-chart, two-row table, or input/output machine) (Grades 6-8)

PO 3. Compute an "output" for a given "input" in a function (Grades 4-5)

PO 4. Complete a T-chart for a given rule (Grades 6-8)

  • 3M-E5.Use patterns and functions to represent and solve problems both formally and informally (e.g., measuring the height a ball bounces by dropping different balls from different starting heights)

PO 1. Solve a problem given a pattern both formally and informally (e.g., "In a patterned necklace, how many red and green beads do you need for a 20-inch necklace?") (Grades 6-8)

  • 3M-E6.Distinguish between linear and nonlinear functions through investigations

PO 1. Distinguish between linear and nonlinear functions, given graphic examples (Grades 6-8)

  • 3M-E7.Solve simple linear equations and inequalities using a variety of methods (e.g., informal, formal, graphical) and a variety of manipulatives

PO 1. Solve equations using

  1. whole numbers with one variable-one step (Grades 4-5)
  2. whole numbers with one variable-multiple steps (Grades 6-8)

PO 2. Solve linear (first degree) equations using models/manipulatives, symbols and/or graphing in a one-step equation (Grades 6-8)

PO 3. Graph given data points to represent a linear equation

  1. on a coordinate grid with whole numbers (Grades 4-5)
  2. in (x, y) form using all four quadrants of a coordinate grid (Grades 6-8)
  • 3M-E8.Develop, analyze and explain methods for solving proportions

PO 1. Describe how to solve a problem in context using a proportion (Grades 6-8)

PO 2. Compare quantities using ratios (Grades 6-8)

PO 3. Solve proportions using formal (e.g., cross product) or informal methods (e.g., diagrams, geometric models) (Grades 6-8)

PROFICIENCY (Grades 9-12)

  • 3M-P1.Model real-world phenomena (e.g., compound interest or the flight of a ball) using functions and relations (e.g., linear, quadratic, sine and cosine, and exponential)

PO 1. Identify the independent and dependent variables from a real-world situation

PO 2. Describe a real-world situation that is depicted by a given graph

PO 3. Sketch a graph that models a given real-world situation

  • 3M-P2.Represent and analyze relationships using written and verbal explanations, tables, equations, graphs and matrices and describe the connections among those representations

PO 1. Express the relationship between two variables using a table, equation, graph and matrix

PO 2. Describe the relationship suggested by two or more graphs of related real-world situations

PO 3. Determine whether a relation is a function, given the graphical representation

  • 3M-P3.Analyze the effects of parameter changes on functions (e.g., linear, quadratic and trigonometric) using calculators and/or computers

PO 1. Use technology to determine changes in the shape and behavior of polynomial functions (of degree 2 or less) when constants and coefficients are varied

  • 3M-P4.Interpret algebraic equations and inequalities geometrically and describe geometric relationships algebraically

PO 1. Graph a linear equation in two variables

PO 2. Graph a linear inequality in two variables

PO 3. Determine slope and intercepts of a linear equation

PO 4. Write an equation of the line that passes through two given points

PO 5. Determine from two linear equations whether the lines are parallel, are perpendicular or coincide

  • 3M-P5.Apply trigonometry to real-life problem situations (e.g., investigate how to find the distance across a river using similar triangles and trigonometric ratios; compare the sine and cosine curves to the curves of sound waves and tide variations)

PO 1. Use the definitions of trigonometric functions to find the sine, cosine and tangent of the acute angles of a right triangle

PO 2. Solve simple right-triangle trigonometric equations involving sine, cosine and tangent

PO 3. Use an appropriate right-triangle trigonometric model to solve a real-life problem

  • 3M-P6.Perform mathematical operations on expressions and matrices, and solve equations and inequalities

PO 1. Simplify numerical expressions using the order of operations, including exponents

PO 2. Evaluate algebraic expressions using substitution

PO 3. Simplify algebraic expressions using distributive property

PO 4. Simplify square roots and cube roots with monomial radicands that are perfect squares or perfect cubes

PO 5. Calculate powers and roots of real numbers, both rational and irrational, using technology

PO 6. Evaluate numerical and algebraic absolute value expressions

PO 7. Multiply and divide monomial expressions with integer exponents

PO 8. Add, subtract and perform scalar multiplication with matrices

PO 9. Solve linear equations and inequalities in one variable

PO 10. Solve formulas for specified variables

PO 11. Solve quadratic equations

PO 12. Solve radical equations involving one radical (restrict to square roots)

PO 13. Solve proportions which generate linear or quadratic equations

PO 14. Solve absolute value equations containing a single absolute value expression

PO 15. Solve systems of linear equations in two variables

  • 3M-P7.Translate among tabular, symbolic and graphical representations of functions

PO 1. Create a linear equation from a table of values

PO 2. Create a graph from a table of values

PO 3. Determine the solution to a system of equations in two variables, from a given graph

PO 4. Determine the solution to a system of inequalities in two variables, from a given graph (e.g., "Which of the shaded regions represents the solution to the system?")

  • 3M-P8.Use the power of mathematical abstraction and algebraic symbolism to represent various situations

PO 1. Translate verbal expressions and sentences to mathematical expressions and sentences

PO 2. Generate an algebraic sentence to model real-life situations, given a data set (limited to linear relationships)

  • 3M-P9.Determine maximum and minimum points of a graph and interpret results in problem situations

PO 1. Identify the maximum or minimum point from the graph of a quadratic function

PO 2. Determine domain and range of a relation, given the graph or a set of points

PO 3. Determine the solution to a real-world maximum/minimum problem, given the graphical representation (e.g., given the graph of the path of a ball, determine its maximum height, when it will reach its maximum height, when it will reach ground level)

  • 3M-P10.Investigate the limiting process by examining infinite sequences and series and areas under curves

PO 1. Compare the estimates of the area under a curve over a bounded interval, using progressively smaller rectangles (not using calculus)

PO 2. Estimate the limit of a given infinite sequence (e.g., given the sequence 1/n, as n gets larger) (not using calculus)

DISTINCTION (Honors)

  • 3M-D1.Use matrices to solve linear systems
  • 3M-D2.Demonstrate technical facility with algebraic transformations, including techniques based on the theory of equations
  • 3M-D3.Understand operations on, and the general principles and behavior of, classes of functions (including logarithmic functions)
  • 3M-D4.Apply general graphing techniques to trigonometric functions
  • 3M-D5.Solve trigonometric equations and verify trigonometric identities
  • 3M-D6.Understand the connections between trigonometric functions and polar coordinates, complex numbers and series
  • 3M-D7.Understand the conceptual foundations of limits, the area under a curve, the rate of change, and the slope of a tangent line, and their applications in other disciplines
  • 3M-D8.Analyze the graphs of polynomial, rational, radical and transcendental functions