Math // Ratio & Proportion

Percentages (age 9+)

Know and use the vocabulary of ratio and proportion — ratio, proportion, percentage, scale, equivalent, unequal, relative size, part-to-part, part-to-whole, and out of — and understand the difference between ratio (comparing parts to parts) and proportion (comparing a part to the whole)

Ages
9 to 11
Type
Language
Builds on
0 topics
Leads to
4 topics

Quick check

If the learner is sharing something 3 ways and 2 ways, can they explain the difference between a ratio (like 3:2) and a proportion (like 3 out of 5) — showing they know what each word means?

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.
  • Explain the difference between a 'ratio' and a 'proportion' using a concrete example like mixing paint
  • Use 'per cent' correctly and convert between fractions, decimals, and percentages in context
  • Define 'scale factor' and use it to describe how a shape has been enlarged or reduced

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

0 topics

Nothing in the catalogue comes before this topic, so it is a good place to start.

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Percentages (age 9+)

Ages 9 to 11 // Language

Leads to

4 topics
  • Calculating PercentagesRequired

    Ratio & Proportion // Ages 10 to 11

    Calculating percentages of amounts requires 'percentage', 'proportion', and 'out of' vocabulary

  • Scale and similar shapesRequired

    Ratio & Proportion // Ages 10 to 11

    Similar shapes and scale requires 'scale', 'equivalent', and 'proportion' as defined terms

  • Ratio ProblemsRequired

    Ratio & Proportion // Ages 10 to 11

    Solving relative sizes problems requires 'ratio', 'proportion', and 'relative size' as understood terms

  • Understanding fractionsRequired

    Ratio & Proportion // Ages 10 to 11

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

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

Language foundation for ratio and proportion (ages 9–11): children must use ratio, proportion, percentage, scale, equivalent, relative size, part-to-part, part-to-whole and 'out of' precisely, and grasp the central distinction that a RATIO compares part to part (3 red : 2 blue) while a PROPORTION compares a part to the whole (3 out of 5 are red = 3/5 = 60%). Because these words are used loosely in everyday speech and are often taught as the inseparable phrase 'ratio and proportion', the objective is to differentiate them through repeated use in concrete mixing/sharing/recipe contexts rather than through memorised definitions. The payoff is preventing the classic error of reading a part-to-part ratio as if it were a part-to-whole fraction.

Common misconceptions

  • Uses 'ratio' and 'proportion' interchangeably, as if they mean the same thing
  • Reads a part-to-part ratio as a part-to-whole fraction: hearing 'red to blue is 2:3' and saying '2/3 are red' (should be 2/5)
  • Treats 'proportion' only as everyday 'a large proportion' (a vague amount) rather than a comparison of part to whole
  • Confuses 'scale' / 'scale factor' with a difference (adds a fixed amount) rather than a multiplicative enlargement

Useful representations

  • Two side-by-side coloured bars (part-to-part) contrasted with one part-shaded whole bar (part-to-whole / proportion)
  • 'For every ___ there are ___' sentence frame linking language to the ratio
  • Squash/paint mixing jugs or coloured counters (concrete part-to-part)
  • A double-headed 'same proportion' triple: fraction 3/5 = 60% = 3 out of 5
  • An enlargement pair of shapes labelled with the scale factor

Teaching practices

  • Introduce terms through a concrete mixing/sharing context (e.g. squash: '1 part concentrate to 4 parts water') and the 'for every' phrasing before any notation — 'for every 1 concentrate there are 4 water' makes part-to-part audible.
  • Contrast ratio and proportion side by side on the SAME situation: 'the ratio of concentrate to water is 1:4' vs 'the proportion that is concentrate is 1 out of 5' — so children hear part-to-part against part-to-whole.
  • Attach each term to a representation: ratio → two coloured bars side by side; proportion/percentage → one bar (the whole) with a shaded part; scale → an enlargement pair. Language and picture reinforce each other.
  • Use 'out of' deliberately as the bridge to fractions and percentages: 3 out of 5 = 3/5 = 60%, all describing the same proportion.
  • Elicit and require the vocabulary in pupils' own explanations (talk moves) rather than accepting a right numeric answer with vague words — the objective IS the language.
  • Draw scale/equivalent language from enlargement examples ('every length is 3 times as big — a scale factor of 3') so 'scale' and 'equivalent' are grounded in action.

Activities to try

  1. 1

    Squash-mixing bench: one mixture, two descriptions

    Why it works: The same mixture is described two different ways depending on which picture you read: for 1 concentrate to 4 water, the 1 sits against the 4 in the twin-bar (ratio 1:4) and against the whole of 5 parts in the single bar (proportion 1 out of 5). Ratio compares part to part; proportion compares part to whole; the whole = the sum of the parts.

    1. Build a mixture by adding whole 'parts' of concentrate and water into a jug (for example 1 part concentrate to 4 parts water).
    2. As each part is added, read two pictures that update together: a twin-bar showing concentrate beside water (part to part) and a single stacked bar showing the whole mixture with the concentrate shaded (part to whole).
    3. Complete two sentence frames about the SAME mixture: 'the RATIO of concentrate to water is __ : __' and 'the PROPORTION that is concentrate is __ out of __'.
    4. Then 'say it another way': express the proportion as a fraction, a percentage and an 'out of' phrase (for 1 part in 5: 1/5 = 20%).
    5. Change the recipe and say whether the ratio and the proportion change.
  2. 2

    'For every' spoken frame before notation

    Why it works: 'For every ___ there are ___' makes part-to-part audible before symbols are introduced, so the ratio notation is read as a comparison of parts rather than copied as a rule.

    1. Describe a mixture aloud using 'for every' before any colon notation: 'for every 1 concentrate there are 4 water'.
    2. Then translate the spoken frame into the ratio 1:4.
    3. State separately how much of the WHOLE drink is concentrate: 1 out of 5.
    4. Repeat with coloured counters or paint instead of squash to hear the same structure in a new context.
  3. 3

    In-role recipe talk that requires precise words

    Why it works: Correct USE of the vocabulary in unscripted dialogue is the actual objective; being asked to clarify loose usage forces the part-to-part vs part-to-whole reference to become explicit.

    1. Describe recipes and drink orders to an in-role partner (for example a smoothie-bar owner), using 'ratio' and 'proportion' correctly.
    2. When the partner challenges a loose statement — 'you said proportion; did you mean part to part or part to whole?' — re-say it with the matching word.
    3. Explain a part-to-part comparison and a part-to-whole comparison for the same order.

Math // Ratio & Proportion

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