Properties of Operations

CONCEPTUAL

Mathematics›Multiplication & Division|Ages 8—9|ID: mt_Lb2ZnMdkYR

Apply properties of operations (commutative, associative, distributive) as strategies to multiply and divide

Mastery Evidence

Use commutativity: if 6 × 4 = 24 then 4 × 6 = 24
Use the distributive property: 8 × 7 = 8 × 5 + 8 × 2 = 40 + 16 = 56
Use associativity to multiply three numbers: 2 × 3 × 5 = 6 × 5 = 30

Assessment Prompt

“Can [child] use tricks like breaking up a number — working out '6 × 14' as '6 × 10 + 6 × 4 = 60 + 24 = 84' — to make multiplication easier?”

Curriculum Standards1 alignment

3.OA.5Common Core State Standards for Mathematics

Apply properties of operations as strategies to multiply and divide

Apply properties of operations as strategies to multiply and divide. Examples: If 6 × 4 = 24 is known, then 4 × 6 = 24 is also known. (Commutative property of multiplication.) 3 × 5 × 2 can be found by 3 × 5 = 15, then 15 × 2 = 30, or by 5 × 2 = 10, then 3 × 10 = 30. (Associative property of multiplication.) Knowing that 8 × 5 = 40 and 8 × 2 = 16, one can find 8 × 7 as 8 × (5 + 2) = (8 × 5) + (8 × 2) = 40 + 16 = 56. (Distributive property.)

Prerequisites2

Commutative Multiplication
hardAges 6—7
What Multiplication Means
hardAges 8—9

Show full prerequisite tree

Commutative Multiplication hard
Applying all three properties extends Y2 commutativity understanding
- Arrays for multiplication soft
  Arrays help demonstrate commutativity (rotate the array)
  - Division as equal sharing hard
    Using arrays for division requires understanding division as grouping
    - Subtraction as taking away or separating hard
      Division as equal sharing/grouping requires understanding subtraction as taking away/separating
      - How Many in Total? hard
        Understanding subtraction as taking away requires knowing numbers represent quantities (cardinality)
        
        One-to-one counting hard
        Cardinality principle builds on one-to-one correspondence — you must count correctly to know the last number tells 'how many'
  - Multiplication as repeated addition hard
    Using arrays requires understanding what multiplication means
    - Counting in 2s soft
      Skip counting supports understanding equal groups (2s, 5s, 10s)
      - Rote counting to 100 hard
        Skip counting in 2s, 5s, 10s requires fluent rote counting by ones first
        
        One-to-one counting hard
        One-to-one correspondence requires knowing the number name sequence
    - Addition as combining or putting together two hard
      Multiplication as repeated addition requires understanding addition as combining groups
      - How Many in Total? hard
        Understanding addition as combining groups requires knowing numbers represent quantities (cardinality)
        
        One-to-one counting hard
        Cardinality principle builds on one-to-one correspondence — you must count correctly to know the last number tells 'how many'
- Multiplication as repeated addition hard
  Commutativity of multiplication requires understanding multiplication
  - Counting in 2s soft
    Skip counting supports understanding equal groups (2s, 5s, 10s)
    - Rote counting to 100 hard
      Skip counting in 2s, 5s, 10s requires fluent rote counting by ones first
      - One-to-one counting hard
        One-to-one correspondence requires knowing the number name sequence
  - Addition as combining or putting together two hard
    Multiplication as repeated addition requires understanding addition as combining groups
    - How Many in Total? hard
      Understanding addition as combining groups requires knowing numbers represent quantities (cardinality)
      - One-to-one counting hard
        Cardinality principle builds on one-to-one correspondence — you must count correctly to know the last number tells 'how many'
What Multiplication Means hard
Using properties requires understanding what multiplication means
- Arrays for multiplication (age 7+) hard
  Extends array-based repeated addition to formal multiplication interpretation
  - Arrays for multiplication hard
    Rectangular arrays with repeated addition extends array representation from Y2
    - Division as equal sharing hard
      Using arrays for division requires understanding division as grouping
      - Subtraction as taking away or separating hard
        Division as equal sharing/grouping requires understanding subtraction as taking away/separating
        
        How Many in Total? hard
        Understanding subtraction as taking away requires knowing numbers represent quantities (cardinality)
        
        One-to-one counting hard
        Cardinality principle builds on one-to-one correspondence — you must count correctly to know the last number tells 'how many'
    - Multiplication as repeated addition hard
      Using arrays requires understanding what multiplication means
      - Counting in 2s soft
        Skip counting supports understanding equal groups (2s, 5s, 10s)
        
        Rote counting to 100 hard
        Skip counting in 2s, 5s, 10s requires fluent rote counting by ones first
        
        One-to-one counting hard
        One-to-one correspondence requires knowing the number name sequence
      - Addition as combining or putting together two hard
        Multiplication as repeated addition requires understanding addition as combining groups
        
        How Many in Total? hard
        Understanding addition as combining groups requires knowing numbers represent quantities (cardinality)
        
        One-to-one counting hard
        Cardinality principle builds on one-to-one correspondence — you must count correctly to know the last number tells 'how many'
  - Multiplication as repeated addition hard
    Expressing array totals as sums of equal addends requires understanding multiplication as repeated addition
    - Counting in 2s soft
      Skip counting supports understanding equal groups (2s, 5s, 10s)
      - Rote counting to 100 hard
        Skip counting in 2s, 5s, 10s requires fluent rote counting by ones first
        
        One-to-one counting hard
        One-to-one correspondence requires knowing the number name sequence
    - Addition as combining or putting together two hard
      Multiplication as repeated addition requires understanding addition as combining groups
      - How Many in Total? hard
        Understanding addition as combining groups requires knowing numbers represent quantities (cardinality)
        
        One-to-one counting hard
        Cardinality principle builds on one-to-one correspondence — you must count correctly to know the last number tells 'how many'
- Multiplication as repeated addition hard
  Interpreting products formalises repeated addition/equal groups from Y1
  - Counting in 2s soft
    Skip counting supports understanding equal groups (2s, 5s, 10s)
    - Rote counting to 100 hard
      Skip counting in 2s, 5s, 10s requires fluent rote counting by ones first
      - One-to-one counting hard
        One-to-one correspondence requires knowing the number name sequence
  - Addition as combining or putting together two hard
    Multiplication as repeated addition requires understanding addition as combining groups
    - How Many in Total? hard
      Understanding addition as combining groups requires knowing numbers represent quantities (cardinality)
      - One-to-one counting hard
        Cardinality principle builds on one-to-one correspondence — you must count correctly to know the last number tells 'how many'

Unlocks3

Multiply & Add Problems
softAges 8—9
Factor Pairs & Commutativity
softAges 8—9
Patterns in Times Tables
hardAges 8—9