Math // Multiplication & Division

Multiply & Add Problems

Solve problems involving multiplying and adding, including using the distributive law, integer scaling problems, and harder correspondence problems

Ages
8 to 9
Type
Method
Builds on
3 topics
Leads to
1 topic
Standards
1 code

Quick check

If the learner needs to work out '5 × 23', can they split it into '5 × 20 + 5 × 3 = 100 + 15 = 115' — rather than just jumping straight to a written method?

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.
  • Use the distributive law to solve 14 × 6 as 10 × 6 + 4 × 6
  • Solve a scaling problem (e.g. 'A tower is 3 times as tall as a 15 m building')
  • Solve a correspondence problem (e.g. '3 types of bread, 4 types of filling — how many different sandwiches?')

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

3 topics
  • Multi-Step Multiply & DivideRequired

    Multiplication & Division // Ages 7 to 8

    Harder problems extend Y3 scaling/correspondence problems

  • Written MultiplicationRequired

    Multiplication & Division // Ages 8 to 9

    Problems require formal written multiplication skills

  • Properties of OperationsHelpful

    Multiplication & Division // Ages 8 to 9

    Distributive law is explicitly used in problem-solving

You are here

Multiply & Add Problems

Ages 8 to 9 // Method

Leads to

1 topic
  • Division with remaindersRequired

    Multiplication & Division // Ages 9 to 10

    Y4 M×D problem-solving is prerequisite to multi-step four-operation 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 8–9: solving multiply-and-add problems across three distinct multiplicative structures — (1) the distributive law as a calculation strategy (14×6 = 10×6 + 4×6 = 60+24 = 84); (2) scaling / multiplicative comparison ('3 times as tall as 15 m' = 45 m, NOT 15+3 = 18); and (3) correspondence / combinations (3 breads × 4 fillings = 12 sandwiches). The central aim is for learners to identify which structure a problem has and choose operations accordingly, rather than guessing from key words. Bar models and systematic tables/trees are the shared tools; the recurring pitfalls are reading scaling additively and adding the two counts in correspondence problems.

Common misconceptions

  • Reading scaling additively — 'three times as tall as 15 m' answered as 15+3 rather than 15×3.
  • Adding the two counts in a correspondence problem (3+4=7 sandwiches) instead of multiplying.
  • Unsystematic listing of combinations, so some pairings are missed or duplicated.
  • Applying the distributive split incorrectly (14×6 = 10×6 + 4, dropping the second multiplication).
  • Relying on key words instead of the problem's structure, choosing the wrong operation in multi-step problems.

Useful representations

  • Bar model — scaling and multiply-and-add (unit bar × scale factor; combine bars)
  • Systematic table / grid — correspondence combinations (rows×columns = pairings)
  • Tree diagram — correspondence (each bread branches to each filling)
  • Distributive split (area/partition) — 14×6 as 10×6 + 4×6

Teaching practices

  • Teach the three structures explicitly and by name — distributive calculation, scaling (times-as-many), correspondence (combinations) — so children recognise which they face rather than key-word guessing.
  • Use the bar model as the shared representation for scaling and multiply-and-add, making the multiplicative comparison visible against an additive one.
  • For correspondence, build systematic organisation — a grid/table or tree diagram — so children see why you multiply the two counts rather than add them.
  • Contrast scaling with addition directly ('3 times as tall' vs '3 m taller') using the same numbers so the multiplicative meaning is felt.
  • Progress from one-step to genuinely multi-step problems, requiring an estimate and an operation choice, and checking answers for reasonableness.

Activities to try

  1. 1

    Correspondence combinations: pair every bread with every filling

    Why it works: Multiplication counts pairings. When n objects each connect to m objects, the total number of combinations is n×m, not n+m. A systematic grid or tree makes every pairing visible so none are missed or duplicated.

    1. Work with a concrete set-up of 3 kinds of bread and 4 kinds of filling (cards, drawings, or a draggable grid).
    2. Systematically pair each bread with each filling, recording every sandwich in a table whose rows are breads and columns are fillings, or a tree where each bread branches to all 4 fillings.
    3. After completing the pairings, count the filled cells and state the total as a number sentence: 3 × 4 = 12.
    4. Explain in words why it is 3×4 and not 3+4: 'each of the 3 breads makes 4 sandwiches.'
    5. Predict then check: 'If there were 5 fillings instead of 4, how many sandwiches now?' (3 × 5 = 15).
  2. 2

    Scaling vs adding: 'times as tall' against 'taller by'

    Why it works: 'Times as many/tall/heavy' is a multiplicative comparison (copies of the original), fundamentally different from 'more than / taller by', which is additive. The same numbers give very different answers (45 vs 18).

    1. Draw or build a bar representing a 15 m building.
    2. Make the tower '3 times as tall' by building three copies of the 15 m bar stacked end to end, and compute 3 × 15 = 45 m.
    3. Alongside it, build a second comparison: '3 m taller than 15 m' = 15 + 3 = 18 m.
    4. Place the two results side by side and describe the difference between 'times as tall' (multiplicative) and 'taller by' (additive).
    5. State which phrase means multiply and which means add, using the same numbers so the contrast is felt.
  3. 3

    Distributive law: split, multiply both parts, recombine

    Why it works: The distributive law lets you multiply a two-digit number by splitting it, but every part of the split must be multiplied by the second factor and then added. Dropping a partial product gives a wrong answer.

    1. Split 14 into 10 + 4.
    2. Multiply BOTH parts by 6: 10 × 6 = 60 and 4 × 6 = 24.
    3. Add the two partial products: 60 + 24 = 84.
    4. Use an area/partition model — a rectangle split into a 10-strip and a 4-strip, both ×6 — to show both pieces are included.
    5. Check the result is reasonable (e.g. 14×6 is a bit more than 10×6 = 60, and less than 15×6 = 90).

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/Y4/MD/5

Math // Multiplication & Division

More in Multiplication & Division

All 57 topics