Math // Multiplication & Division

Factors, multiples, and primes (age 9+)

Solve problems involving multiplication and division using knowledge of factors, multiples, squares, and cubes

Ages
9 to 10
Type
Method
Builds on
2 topics
Leads to
0 topics
Standards
1 code

Quick check

If the learner is asked to find all the square numbers less than 100, can they work through it systematically and give you the correct list: 1, 4, 9, 16, 25, 36, 49, 64, 81?

Ask this to see whether the topic is secure.

Topic // Mastery

What mastery looks like

What a learner can do once this topic is secure.
  • Is 156 a multiple of 6? Explain how you know
  • Find two square numbers that add to make 100
  • Use factor pairs to simplify 24 × 25 as 6 × (4 × 25) = 6 × 100 = 600

Topic // Map

Where this fits

What to learn first and what this topic opens up, with the reason for each link. Required links come first.
  • RequiredMust come first
  • HelpfulHelps, but not essential

Before this

2 topics
  • Square and cube numbersRequired

    Multiplication & Division // Ages 9 to 10

    Square/cube number knowledge is prerequisite to solving problems with them

  • Factors, multiples, and primesRequired

    Multiplication & Division // Ages 9 to 11

    Factor/multiple knowledge is prerequisite to solving problems with them

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Factors, multiples, and primes (age 9+)

Ages 9 to 10 // Method

Leads to

0 topics

Nothing in the catalogue builds on this topic yet. Browse Multiplication & Division

Topic // Research-informed

Teaching notes

From the Marble Skill Taxonomy. These notes have not been validated in classrooms, so treat them as a starting point for your own judgement.

The approach

For ages 9–10: solve multiplication and division problems by USING knowledge of factors, multiples, squares and cubes as strategic tools, rather than grinding a written algorithm. Three recurring moves: (1) use divisibility rules to reason about whether a number is a multiple (e.g. is 156 a multiple of 6?); (2) re-factor a hard product into a friendly one using associativity (24×25 = 6×(4×25) = 6×100 = 600); (3) recall or search squares and cubes to solve puzzles (two squares summing to 100 → 36+64). The central habit is to scan for exploitable structure before reaching for a procedure, and to justify answers rather than just state them.

Common misconceptions

  • Grinding the standard algorithm instead of spotting a factor-pair shortcut.
  • Misapplying divisibility rules (e.g. thinking any even number is a multiple of 6).
  • Re-factoring incorrectly (breaking associativity: 24×25 ≠ 24×4×25).
  • Confusing factors, multiples, squares and cubes when selecting a tool.
  • Not justifying membership/answers, so errors go unchecked.

Useful representations

  • Factor rainbow / factor pairs — the re-factoring toolkit
  • Divisibility-rule checklist — reasoning about multiples
  • Array/area re-partition — showing 24×25 as 6×(4×25)
  • Squares/cubes list — search space for sum/target puzzles

Teaching practices

  • Frame number facts as strategic tools — re-factor a product into a friendly form (×25→×100÷4) using associativity, rather than computing directly.
  • Use divisibility rules (2,3,4,5,6,9,10) to reason about multiples and justify membership ('156 = even and digit-sum 12 divisible by 3, so ÷6').
  • Draw on the factor rainbow and known squares/cubes as a search space for puzzle-type problems.
  • Prompt 'is there a smarter route?' before any long calculation, building the look-for-structure habit.
  • Require justification and checking, not just an answer, so reasoning is explicit.

Activities to try

  1. 1

    Re-factoring: turn an ugly product into a friendly one

    Why it works: A fearsome-looking product often never needs a written algorithm: spotting a friendly factor pair (like 4×25 = 100) and using associativity collapses it to a one-step mental calculation. Rearranging factors leaves the total unchanged.

    1. Face a product such as 24×25 and, before calculating, ask: which factors could pair up to make a round number?
    2. Split one factor and re-group: pull a 4 out of 24 (so 24 = 6×4) and pair it with 25 to make 4×25 = 100, leaving 6×100 = 600.
    3. Confirm the total is unchanged by the rearrangement — only the grouping changed, not the value (associativity).
    4. Try the same move on 16×25 (= 4×(4×25) = 4×100 = 400) and other products with a friendly hidden factor.
    5. State in words why the shortcut works and when it does NOT help (if no factor pairs to a round number).
  2. 2

    Divisibility detective: is it a multiple of 6?

    Why it works: Because 6 = 2×3, a number is a multiple of 6 only if it is divisible by BOTH 2 and 3. Divisibility rules let you reason about membership and justify it without dividing.

    1. Test whether 156 is a multiple of 6 WITHOUT long division, by checking the rule for 2 (even?) and the rule for 3 (digit sum a multiple of 3?).
    2. For 156: it is even (passes ÷2) and its digits sum to 1+5+6 = 12, a multiple of 3 (passes ÷3), so it passes BOTH and IS a multiple of 6 (check: 6×26 = 156).
    3. Explain why passing only one test is not enough, using a counterexample: 9 is divisible by 3 but is NOT a multiple of 6; 8 is even but NOT a multiple of 6.
    4. Repeat with 96 (even; digit sum 9+6 = 15 → multiple of 3; so yes, a multiple of 6) and a non-example such as 100 (even but digit sum 1, not a multiple of 3 → not a multiple of 6).
    5. Justify each verdict out loud or in writing, naming both conditions.
  3. 3

    Squares and cubes as a search space

    Why it works: Knowing squares and cubes turns puzzle problems into an organised search: you can systematically test candidates instead of guessing, and some numbers belong to more than one family.

    1. List the square numbers up to 100 (1, 4, 9, 16, 25, 36, 49, 64, 81, 100) as a search space.
    2. Find two square numbers that add to 100 by scanning the list: 36 + 64 = 100.
    3. Describe the search strategy used (e.g. pick a square, subtract from 100, check if the remainder is also a square) rather than guessing randomly.
    4. Extend to a cubes puzzle, e.g. list the cubes 1, 8, 27, 64 and find a cube greater than 1 that is also a square (64 = 8² = 4³; note that 1 = 1² = 1³ is also both, as a trivial case).

Topic // Standards

Curriculum standards

Codes for the curriculum statements this topic maps to. Each publisher owns its standard text.
The national curriculum in England: Key stages 1 and 2 framework document
Ma5/2.3i

Math // Multiplication & Division

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