Math Rationale
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Math Standard 2
Math Standard 3
Math Standard 4
Math Standard 5
Math Standard 6
Math Glossary
 Arizona Standards

Mathematics Standards
(Adopted 8/26/96)

Standard 6: Mathematical Structure/Logic

Students use both inductive and deductive reasoning as they make conjectures and test the validity of arguments.

Students know and are able to do the following:

READINESS (Kindergarten)

  • 6M-R1.Sort and classify objects according to observable attributes
  • 6M-R2.Justify their answers and reasoning processes

FOUNDATIONS (Grades 1-3)

  • 6M-F1.Recognize that numbers are used for different purposes in the world and a variety of mathematical notations represent these situations

PO 1. Formulate mathematical problems from everyday situations

  • 6M-F2.Draw inductive and deductive conclusions about mathematics

PO 1. Extend a pattern using inductive reasoning (e.g., "What is the next number after 2, 4, 6, 8?")

PO 2. Make a prediction based on existing information (e.g., "All the students in a 3rd grade class are under 10 years old. How old will the next new student probably be?")

  • 6M-F3.Distinguish between relevant and irrelevant information

PO 1. Select the information necessary to solve a given problem

  • 6M-F4.Interpret statements made with precise language of logic (e.g., all, every, none, some, or, many)

PO 1. Use words such as all, every, none, some and many to make reasonable conclusions about situations

ESSENTIALS (Grades 4-8)

  • 6M-E1.Use models to explain how ratios, proportions and percents can be used to solve problems and apply reasoning processes, such as spatial reasoning and reasoning with proportions and graphs

PO 1. Communicate how to solve problems involving ratios, proportions and percents using concrete and illustrative models (Grades 6-8)

  • 6M-E2.Construct, use and explain algorithmic procedures for computing and estimating with whole numbers, fractions, decimals and integers

PO 1. Design a method with a series of defined steps for solving a problem; justify the method

    1. whole numbers (Grades 4-5)
    2. fractions, decimals and integers (Grades 6-8)
  • 6M-E3.Use if . . . then statements to construct simple valid arguments

PO 1. Construct simple valid arguments using if . . . then statements based on

  1. graphic organizers (e.g., Venn diagrams and pictures . . .) (Grades 4-5)
  2. geometric shapes (Grades 4-5, 6-8)
  3. proportional reasoning in probability (Grades 6-8)
  4. syllogism (Grades 6-8)

PO 2. Solve problems using deductive reasoning (Grades 6-8)

PROFICIENCY (Grades 9-12)

  • 6M-P1.Use inductive and deductive logic to construct simple valid arguments

PO 1. Construct a simple informal deductive proof (e.g., write a proof of the statement: "You can fly from Bombay to Mexico City, given an airline schedule.")

PO 2. Produce a valid conjecture using inductive reasoning by generalizing from a pattern of observations (e.g., if 101 = 10, 102 = 100, 103= 1000, make a conjecture)

  • 6M-P2.Determine the validity of arguments

PO 1. Determine if the converse of a given statement is true or false

PO 2. Draw a simple valid conclusion from a given if . . . then statement and a minor premise

PO 3. Distinguish valid arguments from invalid arguments

PO 4. List related if . . . then statements in logical order

PO 5. Use Venn diagrams to determine the validity of an argument

PO 6. Analyze assertions about everyday life by using principles of logic (e.g., examine the fallacies of advertising)

PO 7. Recognize the difference between a statement verified by mathematical proof (i.e., a theorem) and one verified by empirical data (e.g., women score higher than men on vocabulary tests)

  • 6M-P3.Formulate counterexamples and use indirect proof

PO 1. Construct a counterexample to show that a given invalid conjecture is false (e.g., Nina makes a conjecture that x3 > x 2 for all values of x. Find a counterexample.)

  • 6M-P4.Make and test conjectures

PO 1. Write an appropriate conjecture given a certain set of circumstances

PO 2. Test a conjecture by constructing a logical argument or a counterexample

  • 6M-P5.Understand the logic of algebraic procedures

PO 1. Determine whether a given algebraic expression and a possible simplified form are equivalent (e.g., show that (x + y)2 = x2 + y2 is invalid)

PO 2. Determine whether a given procedure for solving an equation is valid

DISTINCTION (Honors)

  • 6M-D1.Prove elementary theorems within various mathematical structures
  • 6M-D2.Develop an understanding of the nature and purpose of axiomatic systems
  • 6M-D3.Construct proofs for mathematical assertions, including indirect proofs and proofs by mathematical induction