Division as Unknown Factor

CONCEPTUAL

Mathematics›Multiplication & Division|Ages 8—9|ID: mt_GDG9_SZmsO

Understand division as an unknown-factor problem (e.g. find 32 ÷ 8 by finding the number that makes 32 when multiplied by 8)

Mastery Evidence

Explain that 32 ÷ 8 = ? is the same as 8 × ? = 32
Use a known multiplication fact to find a division answer
Describe the relationship between multiplication and division as inverse operations

Assessment Prompt

“Can [child] work out '32 ÷ 8' by asking 'what do I multiply 8 by to get 32?' — using times table knowledge rather than a separate division method?”

Curriculum Standards1 alignment

3.OA.6Common Core State Standards for Mathematics

Understand division as an unknown-factor problem

Understand division as an unknown-factor problem. For example, find 32 ÷ 8 by finding the number that makes 32 when multiplied by 8.

Prerequisites2

What Multiplication Means
hardAges 8—9
What Division Means
hardAges 8—9

Show full prerequisite tree

What Multiplication Means hard
Connecting division to multiplication requires understanding products
- 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'
What Division Means hard
Seeing division as unknown-factor requires understanding division first
- Division as equal sharing hard
  Interpreting quotients formalises equal sharing/grouping from Y1
  - 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'

Unlocks3

Arrays for multiplication (age 9+)
hardAges 9—10
Fluent multiplication and division facts
softAges 8—9
Unknown in Multiplication & Division
hardAges 8—9