Math // Fractions

Dividing fractions

Divide a fraction by a fraction using the 'keep-change-flip' method and visual models; interpret and solve word problems involving division of fractions by fractions

Ages
11 to 12
Type
Method
Builds on
2 topics
Leads to
0 topics
Standards
2 codes

Quick check

If the learner has 3/4 of a litre of juice and each glass holds 3/8 of a litre, can they work out how many glasses they can fill — using fraction division?

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.
  • Divide a fraction by a fraction using the reciprocal method
  • Use visual fraction models to represent and explain fraction division
  • Create story contexts for fraction division problems and solve them

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

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Dividing fractions

Ages 11 to 12 // Method

Leads to

0 topics

Nothing in the catalogue builds on this topic yet. Browse Fractions

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

Dividing a fraction by a fraction is where invert-and-multiply ('keep-change-flip') completes: 2/3 ÷ 4/5 = 2/3 × 5/4 = 5/6. The risk is that it becomes a meaningless ritual — students fail to invert either fraction (→ 8/15), invert the wrong fraction, or invert both. The teaching goal is to bind the rule to meaning: read it as a measurement ('how many 4/5s fit into 2/3?' — a bit less than one), model that, contrast correct and misapplied methods, derive why only the divisor inverts, and use estimation to guard the answer.

Common misconceptions

  • Does not invert either fraction — just multiplies: 2/3 ÷ 4/5 = 8/15 (IES)
  • Inverts the wrong fraction — flips the dividend: 2/3 ÷ 4/5 = 3/2 × 4/5 (IES)
  • Inverts both fractions: 3/2 × 5/4 (IES)
  • Applies keep-change-flip correctly but as a rote rule — cannot model it, justify the flip, or judge if the answer is reasonable

Useful representations

  • Common-denominator bar/number line showing how many divisor-lengths fit in the dividend
  • The reciprocal-derivation chain: (2/3 × 5/4) ÷ (4/5 × 5/4) = (2/3 × 5/4) ÷ 1
  • Keep-change-flip written with only the SECOND fraction inverted, colour-coded
  • Estimation number line placing the quotient near 2/3

Teaching practices

  • Anchor the procedure in the measurement meaning — 'how many 4/5s fit in 2/3?' — modelled on a common-denominator bar or number line, so the flipped rule has something to explain
  • Derive invert-and-multiply rather than assert it (IES): multiply dividend and divisor by the reciprocal of the divisor; the divisor becomes 1, leaving dividend × reciprocal
  • Drill the STRUCTURE of the rule against its misapplications — flip ONLY the divisor (the second fraction), keep the first, change ÷ to × — by contrasting correct vs 8/15 / wrong-flip worked examples
  • Estimate first: 2/3 ÷ 4/5 is close to 2/3 ÷ 1, so a little more than 2/3 — catches an inverted-wrong-fraction blunder
  • Connect to the common-denominator method (2/3 = 10/15, 4/5 = 12/15, so 10 ÷ 12 = 10/12 = 5/6) so pupils see the rule and the meaning agree

Activities to try

  1. 1

    Measure it and run the rule side by side

    Why it works: Inverting the divisor is not an arbitrary trick: the rule's answer matches the measured 'how many 4/5s fit in 2/3', and flipping the dividend or both pulls the answer away from that measured amount.

    1. Take a bar representing 2/3 and lay 4/5-length bars into it; find that not even one whole 4/5 fits, and read how much of a 4/5 does fit (about 5/6 of one).
    2. Alongside, run keep-change-flip: keep 2/3, change ÷ to ×, flip ONLY the 4/5 to 5/4, and compute 2/3 × 5/4 = 10/12 = 5/6.
    3. Confirm the measured amount (5/6 of a 4/5 fits) and the rule's answer (5/6) agree.
    4. Deliberately flip the wrong fraction (3/2 × 4/5 = 6/5) or both (3/2 × 5/4 = 15/8) and compare each result to the measured 5/6 — notice how far each wrong flip lands from what you measured.
  2. 2

    Derive why only the divisor inverts

    Why it works: Invert-and-multiply can be derived rather than asserted: multiplying dividend and divisor by the reciprocal of the divisor makes the divisor 1, leaving dividend × reciprocal, which is why only the divisor flips.

    1. Start from 2/3 ÷ 4/5 and multiply both the dividend and the divisor by the reciprocal of the divisor, 5/4: (2/3 × 5/4) ÷ (4/5 × 5/4).
    2. Compute 4/5 × 5/4 = 1, so the expression becomes (2/3 × 5/4) ÷ 1 = 2/3 × 5/4.
    3. State in your own words why the dividend 2/3 is left untouched and only the divisor is turned into 1.
    4. Explain, using this derivation, why flipping both fractions (3/2 × 5/4) is wrong — the dividend was never inverted.
  3. 3

    Estimate before you compute

    Why it works: A quick estimate of the expected size catches inverted-wrong-fraction and no-flip blunders before they are accepted.

    1. Before calculating 2/3 ÷ 4/5, note that 4/5 is close to 1, so 2/3 ÷ 4/5 should be close to 2/3 ÷ 1 = 2/3, and a bit more than 2/3 because you are dividing by a little less than 1.
    2. After computing, check whether your answer (5/6) is more than 2/3 (2/3 = 10/15, 5/6 ≈ 12.5/15) — it is.
    3. Use the estimate to reject answers like 6/5 (too big) or 8/15 (less than 2/3, wrong direction).

Topic // Standards

Curriculum standards

Codes for the curriculum statements this topic maps to. Each publisher owns its standard text.
Common Core State Standards for Mathematics
6.NS.1
The national curriculum in England: Key stages 1 and 2 framework document
KS3.Maths.Num.4

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