Math // Multiplication & Division

Understanding fractions

Solve problems involving scaling by simple fractions and problems involving simple rates

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

Quick check

If a recipe serves 4 and the learner wants to make it for 6, can they scale up each ingredient — for example, changing '2 cups of flour' to '3 cups'?

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.
  • A recipe for 4 people needs 200 g of flour — how much for 6 people?
  • If 3 kg costs £12, how much does 5 kg cost?
  • Scale a shape by a factor of 1/2 and find the new dimensions

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

Understanding fractions

Ages 9 to 10 // Method

Leads to

1 topic
  • Real-World Maths ModellingHelpful

    Mathematical Thinking // Ages 9 to 10

    Scaling problems exercise modelling with appropriate representations

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 9–10 year olds: solve problems involving scaling by simple fractions and simple rates (e.g. a recipe for 4→6 people; 3 kg costs £12 → cost of 5 kg; scale a shape by 1/2). The through-line is the UNITARY METHOD — find the value of one unit, then scale — supported by a double number line, ratio table and bar model. Learners should experience scaling as multiplicative (× a fraction or rate), explicitly contrasted with the common additive 'add the difference' error, across authentic contexts (recipes, prices, maps/shapes). Activities range from a manipulable double-number-line scaler that builds the proportional feel, through a Socratic unit-rate dialogue, a worked annotated exemplar, to checkable practice with targeted distractors.

Common misconceptions

  • Scaling additively — adding the difference in people/kg instead of multiplying by the ratio (recipe for 6 = recipe for 4 + 2 portions of everything, done wrongly).
  • Not finding the unit value, so unable to scale to a new amount.
  • Inverting the rate (£12 per 3 kg treated as £3 per kg).
  • Confusing scaling up with down, or mis-applying ×1/2.
  • Treating rate problems as unrelated to multiplicative comparison / fractions.

Useful representations

  • Double number line (two quantities scaling together) — core rate/scaling tool
  • Ratio table — unit value and scaled values
  • Bar model — fraction scaling of a quantity
  • Scale-factor arrow (×6/4, ×1/2) linking original to scaled

Teaching practices

  • Teach the unitary method — find one unit then scale — as the reliable general strategy, supported by a double number line and ratio table.
  • Represent rates on a double number line (kg vs £) so the proportional (multiplicative) relationship is visible and both quantities scale together.
  • Treat scaling as multiplicative (× a fraction/rate), explicitly contrasting with additive change, to prevent the classic 'add the difference' error.
  • Use both directions — scale up and scale down (×1/2) — and connect fraction-of-quantity skills to scaling.
  • Ground problems in authentic contexts (recipes, prices, maps/shapes) and check reasonableness.

Activities to try

  1. 1

    Double-number-line scaler: find one unit, then scale

    Why it works: Scaling and rates are proportional, multiplicative relationships: finding the value of one unit unlocks any target amount, and adding the same difference to both quantities does not preserve the relationship.

    1. On two parallel lines locked by a rate (e.g. kilograms against cost in pounds, £), slide to mark 3 kg opposite £12.
    2. Use a 'find 1 unit' action to collapse the lines to 1 kg = £4, then slide on to 5 kg and read off £20.
    3. Switch to a recipe mode: scale ingredients by ×6/4 (e.g. 200 g flour for 4 people → how much for 6?) shown on a bar model.
    4. Switch to a shape mode: halve every dimension (×1/2) and read the new measurements.
    5. Try the 'additive trap' toggle: add the same number to both lines (3 kg + 2 = 5 kg, so £12 + 2 = £14), then compare with the locked-rate result of £20. Notice that adding matching amounts instead (2 kg and its value, £8) keeps the lines aligned and gives £20.
  2. 2

    Unit-rate dialogue: 'what does one cost?'

    Why it works: Routing every proportional problem through 'the value of one, then times the target' is a reliable general habit that works even when the scale factor is not a whole number.

    1. Respond to a market-trader or chef character posing rate and recipe problems: 'If 3 kg costs £12, what does ONE kilogram cost? So what about five?'
    2. State the unit value aloud or in writing, then state how you scale it to the target amount.
    3. Answer the probe 'Someone said 6 people is 2 more than 4, so add 200 g more — what's wrong with that?' and explain the flaw.
    4. Tackle a stretch case ('a recipe for 4, but you need it for 10') and say whether the unit-value method still works.
  3. 3

    Worked exemplar: annotated double number line and ratio table

    Why it works: The unit value is the key column of a ratio table / anchor point of a double number line; scaled values are multiples of it.

    1. Study an annotated double number line (kg vs £) and a ratio table with a unit-value column.
    2. Reveal the unitary steps one at a time: 3 kg = £12 → 1 kg = £4 → 5 kg = £20.
    3. Read the ratio table to see the unit column (1 person = 50 g) and how scaled values follow from it.
    4. Use the exemplar as a reference, then solve a new, different rate/scaling problem yourself.

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
Ma5/2.3k

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