Math // Ratio & Proportion

Ratio Notation

Use ratio notation to describe the relationship between two or more quantities, simplify ratios to their simplest form, and convert between ratio and fraction representations

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

Quick check

If a smoothie recipe uses orange juice and mango juice in the ratio 3:2, can the learner simplify that ratio, express it as a fraction, and work out how much of each is needed for a given total amount?

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.
  • Write a ratio from a word problem and simplify it (e.g. 12:8 = 3:2)
  • Convert a ratio to equivalent fractions of a whole (e.g. 3:2 means 3/5 and 2/5)
  • Simplify ratios involving decimals or fractions by finding a common multiplier

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
  • Factors, multiples, and primesRequired

    Multiplication & Division // Ages 9 to 11

    Simplifying ratios requires finding common factors — extends KS2 factor pairs and common factors

  • Understanding fractionsRequired

    Ratio & Proportion // Ages 10 to 11

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

  • Proportional Reasoning VocabularyRequired

    Ratio & Proportion // Ages 11 to 14

    Ratio notation (a:b) requires 'ratio', 'simplify', and 'equivalent' as defined terms

  • Proportion GraphsHelpful

    Ratio & Proportion // Ages 11 to 14

    Ratio notation and simplification is complemented by double number line representations

You are here

Ratio Notation

Ages 11 to 12 // Method

Leads to

3 topics
  • Dividing Quantities by RatioRequired

    Ratio & Proportion // Ages 11 to 13

    Dividing in a ratio requires understanding ratio notation and simplification first

  • Ratio Notation and RelationshipsRequired

    Ratio & Proportion // Ages 12 to 14

    Understanding multiplicative relationships requires fluent ratio notation

  • Similarity and enlargementHelpful

    Geometry // Ages 12 to 14

    Similar shapes have sides in a constant ratio — connects to ratio notation

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

Teach ratio notation as meaning before manipulation: a:b means 'for every a there are b', with order fixed to the wording (boys:girls written boys-first). Simplify by dividing BOTH parts by a common factor, reaching simplest form only when no common factor other than 1 remains (equivalently, once the parts have been divided by their highest common factor). Convert to fractions of the whole using the TOTAL of the parts as the denominator (3:2 means 3/5 and 2/5, not 3/2). Handle decimal or fraction ratios by first scaling up to whole numbers. The core insight is that simplifying keeps the SAME relationship with smaller numbers (an equivalent ratio).

Common misconceptions

  • Stops before simplest form — divides by a common factor but not the HCF (12:8 → 6:4 and leaves it)
  • Reverses the order relative to the wording (writes girls:boys)
  • Converts a:b to fractions using the other part as the denominator (3:2 → 3/2) instead of the whole (3/5)
  • Simplifies by subtracting a constant from both parts (12:8 → 4:0 or 11:7) rather than dividing — an additive slip

Useful representations

  • a:b notation anchored to 'for every a there are b'
  • Ratio table showing equivalent ratios (12:8 = 6:4 = 3:2)
  • Bar model of a+b equal blocks for the part-to-whole conversion (3:2 → 3/5, 2/5)
  • HCF/common-factor working shown beside the ratio (÷4)
  • Scaling-up step for decimal/fraction ratios (×2, ×10) to reach whole numbers

Teaching practices

  • Tie a:b to 'for every a there are b' and keep order fixed to the wording (boys:girls must be written boys-first) to prevent order reversal.
  • Teach simplifying as dividing BOTH parts by a common factor, and 'simplest form' as dividing by the highest common factor — explicitly check nothing divides both parts again (12:8 ÷4 = 3:2, and 3,2 share no factor).
  • Make equivalent ratios explicit: 12:8, 6:4, 3:2 all describe the same relationship — use a ratio table or scaled bars to show they are the same 'for every'.
  • Draw the part-to-whole conversion from a bar: 3:2 has 5 equal blocks, so 3/5 are the first part and 2/5 the second — the denominator is the total number of parts.
  • Handle decimal/fraction ratios by scaling up to whole numbers first (0.5:2 → 1:4; 1/2:1/3 → 3:2 via a common multiplier) before simplifying.
  • Contrast the ratio 3:2 with the fraction 3/2 aloud so children do not read a:b as a/b.

Activities to try

  1. 1

    Ratio simplifying bench with a shape-stays-the-same check

    Why it works: Simplifying a ratio keeps the same 'for every' relationship with smaller numbers; simplest form means no common factor remains; and converting to a fraction uses the whole (a+b) as the denominator.

    1. Enter a ratio such as 12:8 and simplify it by repeatedly choosing a number to divide BOTH parts by (e.g. ÷4 straight to 3:2, or ÷2 to 6:4 then ÷2 again).
    2. Watch two scaled bars drawn beside the numbers: as 12:8 becomes 6:4 becomes 3:2, the bars keep the same shape/proportion, showing the relationship is preserved.
    3. Use a 'can this go smaller?' indicator that stays red while a common factor greater than 1 still divides both parts and turns green only when none does, so 'simplest form' becomes a testable end-state.
    4. Then snap the 3 and 2 blocks together into one whole of 5 and read off the part-to-whole fractions 3/5 and 2/5 from the total number of blocks.
    5. For a decimal or fraction ratio (0.5:2, or 1/2:1/3), first apply a scale-up step (×2, ×10, or a common multiplier) to reach whole numbers (0.5:2 → 1:4; 1/2:1/3 → 3:2), then simplify.
  2. 2

    Scored practice set with targeted ratio errors

    Why it works: Fluent writing, simplifying and converting of ratios, with the common errors made explicit through the wrong-answer options offered.

    1. Fill in the blanks to simplify: 12:8 = ☐:☐ (expecting 3:2 in simplest form).
    2. Match several ratios (12:8, 6:4, 9:6) to their shared simplest form 3:2.
    3. Choose what fraction of the whole the first part of 3:2 represents, from options including 3/5 (correct), 3/2 (the part-to-part comparison read as a fraction) and 2/5 (the other part's share of the whole).
  3. 3

    Verbal coaching of the simplest-form check

    Why it works: Simplest form is reached only when no factor divides both parts, and simplifying must preserve the 'for every' relationship (divide, do not subtract).

    1. Work through one ratio aloud with a prompt such as 'Can both numbers be divided again?' at each step until no common factor remains.
    2. State why 4:0 (from subtracting) cannot be a valid simplification: it would mean 'no girls', which changes the relationship.

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.RP.1
The national curriculum in England: Key stages 1 and 2 framework document
KS3.Maths.Ratio.4

Math // Ratio & Proportion

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