Math // Multiplication & Division

Multi-Step Multiply & Divide

Solve problems involving multiplication and division, including scaling problems and correspondence problems where n objects are connected to m objects

Ages
7 to 8
Type
Method
Builds on
2 topics
Leads to
1 topic
Standards
1 code

Quick check

If a recipe for 4 people needs 3 eggs, can the learner work out how many eggs are needed for 8 people — recognising it as a doubling or scaling problem?

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.
  • Solve a scaling problem (e.g. 'the ribbon is 3 times as long')
  • Solve a correspondence problem (e.g. '4 shirts and 3 trousers — how many outfits?')
  • Solve a missing-number multiplication or division problem

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

You are here

Multi-Step Multiply & Divide

Ages 7 to 8 // Method

Leads to

1 topic
  • Multiply & Add ProblemsRequired

    Multiplication & Division // Ages 8 to 9

    Harder problems extend Y3 scaling/correspondence problems

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 7–8: solve multiplication/division problems involving SCALING ('n times as big/many', a multiplicative enlargement distinct from additive 'n more') and CORRESPONDENCE (n objects each paired with m objects, e.g. 3 tops × 4 bottoms = 12 outfits, or 4 shirts × 3 trousers = 12), plus the inverse missing-number/division versions. Learners manipulate bars to see scaling as multiplying not adding, build organised pairings or arrays for correspondence, and work backward from a scaled quantity to the original.

Common misconceptions

  • Treating scaling additively: reading '3 times as tall' as '3 cm/units taller' rather than ×3.
  • Undercounting correspondence combinations by not being systematic (missing pairings).
  • Not recognising a correspondence problem as multiplication at all.
  • Confusing which quantity is scaled and by how much in comparison problems.
  • Ignoring the inverse: unable to work backward from the scaled quantity to the original (the division version).

Useful representations

  • Bar model with one bar a multiple of the other (scaling/comparison)
  • Array or organised list for correspondence/combinations
  • Correspondence diagram (each of n linked to each of m)
  • Missing-number equations for the inverse (division) version

Teaching practices

  • Distinguish scaling ('3 times as tall/heavy/many' — multiplicative) from additive change ('3 more') explicitly, using bar models where one bar is literally three times the length of the other.
  • Represent correspondence problems as an array or an organised list so all combinations are found systematically — 3 tops × 4 bottoms fills a 3×4 array of outfits — preventing undercounting.
  • Use bar models for scaling and comparison so the multiplicative relationship is seen, not just calculated.
  • Include missing-number and division versions (if the tower is 3 times as tall and 18 cm, how tall is the original?) so scaling and its inverse are both practised.
  • Connect correspondence to the array/commutativity work so children see it as the same multiplication structure in a new context.

Activities to try

  1. 1

    Stretch the bar: 'times as tall' versus 'taller'

    Why it works: 'n times as big' is multiplicative — it makes that many copies of the quantity — so for this original of 8 cm the ×3 bar (24 cm) ends up well beyond the +3 bar (11 cm); the size of that gap depends on the original (it is not universally true that 'times' exceeds 'plus' for every original and factor), and the inverse of scaling is division.

    1. Draw or stretch a bar to make it a chosen number of times as long as an 8 cm original bar (e.g. ×3).
    2. Alongside, place a second 'additive' comparison bar that is only 3 units longer than the original, and compare the two end points.
    3. State and solve: 'A blue ribbon is 3 times as long as a red ribbon that is 8 cm — how long is the blue one?' (answer 24 cm).
    4. Then reverse it: given a bar that is 3 times as long and measures 18 cm, find the original length (18 ÷ 3 = 6 cm).
    5. Say aloud why '3 times as long' means three copies, not 'longer by 3'.
  2. 2

    Every outfit: building correspondence as an array

    Why it works: Correspondence (each of n paired with each of m) is the same multiplication structure as an array; working systematically finds all combinations and prevents undercounting.

    1. Solve: 'You have 4 shirts and 3 pairs of trousers — how many different outfits can you make?'
    2. Make an organised list or draw lines pairing each shirt with each of the 3 pairs of trousers, so no pairing is missed.
    3. Arrange the completed pairings into a 4×3 (or 3×4) array and count the cells to confirm 12 outfits.
    4. Change one quantity (e.g. 5 pairs of trousers, giving 4 × 5 = 20) and rebuild to see how the total changes.
    5. Explain why the answer is 4 × 3, not 4 + 3.
  3. 3

    Fill the box: missing factors and the inverse

    Why it works: A missing factor is found with a known fact or by division, not by adding or subtracting; multiplication and division are inverses.

    1. Solve the missing-number problem 6 × □ = 42 by asking 'six times WHAT is 42?' and using a known fact or 42 ÷ 6.
    2. Check the answer (□ = 7) by multiplying back: 6 × 7 = 42.
    3. Create and solve a missing-number scaling problem, e.g. 'The tower is 3 times as tall and 18 cm — how tall is the original?' (6 cm).
    4. Explain how multiplication and division undo each other here.

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
Ma/KS2/Y3/MD/3

Math // Multiplication & Division

More in Multiplication & Division

All 57 topics