Math Rationale
Math Standard 1
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 1: Number Sense

Throughout all of the following six standards, students will use reasoning, communication skills, and appropriate manipulatives (concrete materials) and technology within real-world problem-solving situations to make connections within mathematics and between mathematics and other disciplines.

STANDARD 1: NUMBER SENSE

Students develop number sense and use numbers and number relationships to acquire basic facts, to solve a wide variety of real-world problems, and to determine the reasonableness of results.

Students know and are able to do the following:

READINESS (Kindergarten)

  • 1M-R1.Develop an understanding of number meanings and relationships
  • 1M-R2.Demonstrate one-to-one correspondence between elements in collections (sets) (e.g., nine blocks is as many as nine ducks)
  • 1M-R3.Use manipulatives (concrete materials) to count, order and group
  • 1M-R4.Recognize relationships between concrete representations, number names, and symbolic representations of numbers (e.g., understand that three rocks can be represented as three circles, the numeral 3 and the word three)

FOUNDATIONS (Grades 1-3)

  • 1M-F1.Represent and use numbers in equivalent forms through the use of physical models, drawings, word names and symbols (e.g., using concrete materials and fraction equivalents to represent and compare halves, thirds, fourths, eighths and tenths)

PO 1. Make a model to represent a given whole number

PO 2. Identify a whole number represented by a model with a word name and symbol

PO 3. Construct equivalent forms of whole numbers (e.g., 15 + 5 = 10 + 10)

PO 4. Make a model to represent a given fraction (e.g., geometric model–shading a picture, set model–part of an egg carton) (halves, thirds and fourths)

PO 5. Identify the fraction represented by a model with a word name and symbol (halves, thirds and fourths)

PO 6. Identify a given model that is divided into equal fractional parts (halves, thirds and fourths)

  • 1M-F2.Relate counting, grouping and place-value concepts to whole numbers (e.g., reading and writing the number represented when objects are grouped by thousands, hundreds, tens and ones)

PO 1. Read whole numbers up to one thousand

PO 2. Write whole numbers up to one thousand

PO 3. Order whole numbers (e.g., smallest to largest, largest to smallest) up to one thousand

PO 4. Construct a model to represent place value concepts

PO 5. Write a whole number in expanded notation (e.g., 531 = 500 + 30 + 1)

PO 6. Read aloud a whole number with correct place value words (e.g., a student will read 5 2 1 as "five hundred twenty-one")

PO 7. Count money to $5.00 using bills and coins

  • 1M-F3.Understand the meaning for and application of the operations of addition, subtraction, multiplication and division

PO 1. Demonstrate with models to show the process used in addition (joins things together, increases)

PO 2. Demonstrate with models to show the process used in subtraction (takes away, compares, finds the difference, decreases)

PO 3. Demonstrate with models to show the process used in multiplication (uses repeated addition, counts by multiples, combines things that come in groups of equal size, makes arrays, uses area models)

PO 4. Demonstrate with models to show the process used in division (puts things into groups of equal size, shares equally, uses repeated subtraction)

PO 5. Demonstrate with models the operations of addition and subtraction up to two three-digit whole numbers

PO 6. Select appropriate operations to solve word problems

PO 7. Solve word problems using the appropriate operations

PO 8. Apply mathematical operations in everyday situations

  • 1M-F4.Demonstrate proficiency with the operations of addition and subtraction of whole numbers

Note: Proficiency–accurate and consistent solving of computational problems in a reasonable time, using self-checking skills

PO 1. Demonstrate proficiency with basic facts up to 20

PO 2. Add and subtract two three-digit whole numbers

PO 3. Solve problems using a variety of mental computations and estimation

  • 1M-F5.Demonstrate proficiency with the operations of multiplication and division of single-digit numbers

PO 1. Demonstrate proficiency with basic facts up to the fives

PO 2. Solve problems using a variety of mental computations and estimation

  • 1M-F6.Add and subtract commonly used fractions and decimals

PO 1. Demonstrate with models addition and subtraction of fractions with common denominators (halves, thirds and fourths)

PO 2. Add and subtract money up to $5.00

  • 1M-F7.Select and use appropriate techniques to facilitate computation (e.g., mental, estimation, paper-and-pencil, calculator and computer methods) while solving problems and determining the reasonableness of results

PO 1. Select a computational technique to solve a problem

PO 2. Solve a problem using the appropriate computational techniques

PO 3. Evaluate the reasonableness of results using a variety of mental computation and estimation techniques (e.g., compatible numbers, front-end, chunking)

PO 4. Use technology (e.g., calculators, computers, multimedia) to solve problems containing larger numbers

ESSENTIALS (Grades 4-8)

  • 1M-E1.Read, write and order integers, whole numbers and rational numbers

PO 1. Compare and order using concrete or illustrated models

  1. whole numbers (to millions) (Grades 4-5)
  2. common fractions (halves, thirds, fourths, eighths) (Grades 4-5)
  3. decimals (thousandths) (Grades 4-5)
  4. rational numbers (e.g., -5, 1.2, 1 3/4, square root of 16) (Grades 6-8)

PO 2. Represent place value using concrete or illustrated models

  1. whole numbers (millions), decimals (thousandths) (Grades 4-5)
  2. rational numbers (millions to millionths) (Grades 6-8)

PO 3. Read and write whole numbers, integers, common fractions and decimals using real-world situations

  1. whole number (millions), decimals (thousandths), fractions (halves, thirds, fourths, eighths) (Grades 4-5)
  2. rational numbers (millions to millionths) (Grades 6-8)
  • 1M-E2.Relate the basic arithmetic operations to one another (e.g., multiplication and division are inverse operations)

PO 1. Represent the process of multiplication as repeated addition, using concrete or illustrative models

  1. whole numbers (Grades 4-5)
  2. fractions and decimals (Grades 6-8)

PO 2. Represent the process of division as repeated subtraction, partitioning a group and partitioning a whole, using concrete or illustrative models

  1. whole numbers (Grades 4-5)
  2. fractions and decimals (Grades 6-8)

PO 3. Write the family of equations using inverse operations for a given set of numbers

  1. whole numbers with addition/ subtraction [(4 + 5 = 9, 5 + 4 = 9, 9 – 4 = 5, 9 – 5 = 4) and multiplication/ division] (Grades 4-5)
  2. positive fractions and decimals, integers with addition/subtraction and multiplication/division (Grades 6-8)
  • 1M-E3.Demonstrate proficiency with the operations of multiplication and division of whole numbers

PO 1. Calculate multiplication/division

  1. three-digit by two-digit to find the product (Grades 4-5)
  2. facts through 12 (Grades 4-5)
  3. mental math and estimation with multiples of 10 (Grades 4-5)
  4. one-digit divisor to find quotient with remainder (Grades 4-5)
  5. two-digit divisor, with remainders and rounding in context (e.g., percentages and money) (Grades 6-8)

PO 2. Calculate multiplication and division problems using contextual situations
(Grades 4-5, 6-8)

  • 1M-E4.Develop and apply number theory concepts (e.g., primes, factors and multiples) to represent numbers in various ways

PO 1. State the factors for a given whole number (Grades 4-5)

PO 2. Factor a whole number into a product of its primes (prime factorization) (Grades 6-8)

PO 3. Identify greatest common factor and least common multiples for a set of whole numbers (Grades 6-8)

PO 4. Sort numbers by their properties

  1. odd, even (Grades 4-5)
  2. prime, composite, square, square root (Grades 6-8)

PO 5. Simplify numerical expressions using order of operations (Grades 6-8)

  • 1M-E5.Represent and use numbers in equivalent forms (integers, fractions, percent, decimals, exponents, scientific notation and square roots)

PO 1. Add, subtract, multiply and divide integers, positive fractions and decimals
(Grades 6-8)

PO 2. Demonstrate the relationship and equivalency among

  1. decimals, fractions and percents (e.g.,1/2 = .5 = 50% with halves, fourths and tenths) (Grades 4-5)
  2. decimals, fractions, ratios, percents (Grades 6-8)

PO 3. Factor numbers into prime form and express in exponential form (Grades 6-8)

PO 4. Convert standard notation to scientific notation and vice versa with positive exponents (Grades 6-8)

PO 5. Determine the square root of a perfect square (Grades 6-8)

  • 1M-E6.Recognize that the degree of precision needed in calculating a number depends on how the results will be used and the instruments used to generate the measurements

PO 1. Express answers to the appropriate place or degree of precision (e.g., time, money, pi) (Grades 6-8)

PO 2. Apply the appropriate strategy (e.g., estimation, approximation, rounding or exact numbers) when calculating to solve problems (Grades 4-5, 6-8)

PO 3. Demonstrate/describe the magnitude of

  1. whole numbers (e.g., "How many apples in the orchard?") (Grades 4-5)
  2. rational numbers (e.g., "How small is a bacterium?") (Grades 6-8)

Note: We recommend that this be assessed at the district level.

PO 4. Interpret calculations and calculator results within a contextual situation (Grades 4-5, 6-8)

PROFICIENCY (Grades 9-12)

  • 1M-P1.Compare and contrast the real number system and its various subsystems with regard to their structural characteristics

PO 1. Classify numbers as members of the sets (natural, whole, integers, rationals and irrationals)

PO 2. Compare subsets of the real number system with regard to their properties (commutative, associative, distributive, identity, inverse and closure properties)

PO 3. Compare subsets of the real numbers by determining which characteristics they have in common

PO 4. Identify whether a given set of numbers is finite or infinite

  • 1M-P2.Construct, interpret and demonstrate meaning for real numbers and absolute value in problem-solving situations

PO 1. Determine a rational estimate of an irrational number

PO 2. Define absolute value as the distance from the origin

PO 3. Solve real-world distance problems using absolute value

PO 4. Determine, among the solutions to a real-world problem, which, if any, is reasonable

PO 5. Define pi as the irrational ratio of circumference to diameter

PO 6. Choose the appropriate signed real number to represent a real-world value

PO 7. Use the appropriate form of a real number to express a real-world situation (e.g., choosing between a radical expression or rational approximation)

PO 8. Convert standard notation to scientific notation, including negative exponents, and vice versa

DISTINCTION (Honors)

  • 1M-D1.Develop conceptual understanding of the complex number system
  • 1M-D2.Demonstrate facility with operations in the complex number system