Math // Ratio & Proportion

Understanding fractions

Solve problems involving unequal sharing and grouping using knowledge of fractions and multiples

Ages
10 to 11
Type
Method
Builds on
4 topics
Leads to
3 topics
Standards
1 code

Quick check

If three friends share a bag of 24 sweets so that one person gets twice as many as each of the others, can the learner work out how many each person receives?

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.
  • Share 40 sweets between two children in the ratio 3:5
  • Solve: 'Tom gets twice as many as Sam and Sam gets three times as many as Jo. If there are 30 altogether, how many does each get?'
  • Use fraction knowledge to explain why sharing in ratio 2:3 means one person gets 2/5 of the total

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

4 topics
  • Percentages (age 9+)Required

    Ratio & Proportion // Ages 9 to 11

    Unequal sharing problems require 'ratio', 'unequal', 'part-to-part', and 'proportion' vocabulary

  • Bar Models for RatiosRequired

    Ratio & Proportion // Ages 9 to 12

    Unequal sharing problems are the canonical use case for tape diagrams — the visual model makes the structure of the problem transparent

  • Simplifying FractionsRequired

    Fractions // Ages 10 to 11

    Understanding fraction of total requires simplification/common denominator skills

  • Ratio ProblemsRequired

    Ratio & Proportion // Ages 10 to 11

    Unequal sharing is ratio reasoning in context

You are here

Understanding fractions

Ages 10 to 11 // Method

Leads to

3 topics
  • Ratio NotationRequired

    Ratio & Proportion // Ages 11 to 12

    Ratio notation and simplification extends KS2 unequal sharing and grouping with ratios

  • Dividing Quantities by RatioRequired

    Ratio & Proportion // Ages 11 to 13

    Dividing quantities in a ratio extends KS2 unequal sharing problems

  • Advanced Multi-Step ProblemsHelpful

    Mathematical Thinking // Ages 10 to 11

    Ratio/proportion problems are key Y6 multi-step contexts

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

Unequal sharing splits a quantity so the parts follow a given ratio rather than being equal. The reliable routine: add the ratio numbers to find the TOTAL number of equal parts, divide the quantity by that total to find the value of ONE part, then multiply the value of one part by each person's number of parts to get their share. The core idea children most often get wrong is that sharing in the ratio 2:3 gives fractions of the WHOLE of 2/5 and 3/5 (denominator = sum of parts), NOT 2/3. Chained comparisons ('Tom gets twice as many as Sam') must first be turned into a single ratio before sharing. A bar/tape diagram cut into (a+b) equal segments makes the equal-parts structure visible and is the canonical representation.

Common misconceptions

  • Reading the ratio 2:3 as fractions 2/3 and 3/3 (or 2/3 of the total) instead of 2/5 and 3/5 — using one part as the denominator rather than the sum of parts.
  • Sharing equally, ignoring the ratio (40 ÷ 2 = 20 each).
  • Finding the value of one part but forgetting to multiply each share back (reporting the number of parts, or one-part, as the answer).
  • Unable to combine chained comparisons into a single ratio, so treating 'twice as many' and 'three times as many' as unrelated.

Useful representations

  • Bar model / tape diagram cut into a+b equal segments
  • The 'one part' unit (quantity ÷ total parts) as the building block
  • Part-to-whole fraction strip (2/5 and 3/5 of the whole for 2:3)
  • Ratio table linking parts to values

Teaching practices

  • Teach the explicit routine: total parts = sum of the ratio numbers → one part = quantity ÷ total parts → each share = its parts × one part.
  • Draw the bar model FIRST (a strip cut into a+b equal segments), label the whole, find one segment, then read off each share — diagram before arithmetic.
  • Make the part-to-whole fractions explicit: in 2:3, one child gets 2/5 and the other 3/5; contrast sharply with the wrong 2/3 to fence the misconception.
  • Convert chained/multiplicative language ('twice as many', 'three times as many') into a single ratio before sharing (Jo:Sam:Tom = 1:3:6), using the smallest quantity as one part.
  • Check by adding the shares back to the original total — a fast self-verification pupils can run every time.

Activities to try

  1. 1

    Cut the sharing bar into the right number of parts

    Why it works: The ratio a:b needs (a+b) equal parts, so each share is a/(a+b) and b/(a+b) of the whole — not a/b. One part is the building block; chained comparisons must become a single ratio before sharing.

    1. Take a ratio such as 3:5 and draw an empty strip (a tape/bar diagram) to represent the whole quantity.
    2. Decide how many EQUAL segments to cut the strip into so that both children's parts together fill it exactly (try 3, 5 or another number and check), then explain in your own words why the sum 3 + 5 = 8 is the simplest count that works.
    3. Place the total quantity (40) across the strip and work out the value of one segment: 40 ÷ 8 = 5.
    4. Colour 3 segments for the first child and 5 for the second, then read off 3 × 5 = 15 and 5 × 5 = 25.
    5. State what fraction of the WHOLE each child gets: 3/8 and 5/8, and check these sum to 1 (and that 15 + 25 = 40).
  2. 2

    Explain and check the shares

    Why it works: Adding the shares back to the original total is a fast self-check, and articulating why the denominator is the sum of parts secures the part-to-whole fraction and the handling of chained comparisons.

    1. Answer: how many EQUAL parts are there altogether in 3:5, and where does that number come from?
    2. Add your two shares back together — do they make 40? Say what it would tell you if they did not.
    3. You said the fraction is 3/8 — why is 8 on the bottom and not 5? Test the wrong rule on its own terms: if the 3-part child got 3/5, then by that same rule the 5-part child would get 5/5 (all of them) — do 3/5 and 5/5 fit inside one whole?
    4. Given 'Tom gets twice as many as Sam and Sam three times as many as Jo, 30 altogether', turn it into one ratio (Jo:Sam:Tom = 1:3:6) before sharing, and find each share.
  3. 3

    Short questions to consolidate the routine

    Why it works: Separating the share values from the part-to-whole fraction lets the learner spot and discard the 'equal share' and 'a/b of the total' errors.

    1. Share 40 in the ratio 3:5 and write the two shares.
    2. Choose what fraction of the whole the '3' child gets.
    3. Read a chained-comparison problem and state the combined single ratio before working out the shares.

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
Ma6/2.4d

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