Math // Fractions

Equivalent fractions (age 9+)

Explain why a fraction a/b is equivalent to (n×a)/(n×b) using visual models; use this principle to recognise and generate equivalent fractions, including tenths and hundredths

Ages
9 to 10
Type
Concept
Builds on
2 topics
Leads to
6 topics
Standards
2 codes

Quick check

If the learner needs to make 2/3 look different without changing its value, can they multiply the top and bottom by the same number and explain why the fraction stays the same?

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.
  • Use a fraction strip to show 2/3 = 4/6 = 6/9
  • Explain that multiplying numerator and denominator by the same number gives an equivalent fraction because the size of the whole is unchanged
  • Generate three fractions equivalent to 3/5 and verify with diagrams

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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Equivalent fractions (age 9+)

Ages 9 to 10 // Concept

Leads to

6 topics

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

Learners understand WHY multiplying numerator and denominator by the same whole number n (n ≥ 1) gives an equivalent fraction: splitting each of the b parts into n equal pieces makes n×b pieces in the whole and n×a shaded, so the SAME magnitude is renamed with a finer counting-unit. This is the same as multiplying by n/n, a name for 1, so the value cannot change. It extends to tenths and hundredths (3/10 = 30/100). The central contrast is with the additive error: adding a constant to top and bottom is not re-partitioning and in general changes the amount (the only exception is a fraction equal to 1, such as 2/2 → 3/3); for example, adding 1 to the numerator and 1 to the denominator turns 2/3 into 3/4, which is larger, not equal.

Common misconceptions

  • Additive equivalence — adding the same number to top and bottom keeps it equal (IES whole-number thinking)
  • Whole-number bias — (n×a)/(n×b) 'must be bigger' because the numbers are bigger (4/6 > 2/3)
  • Multiplying only one part
  • Believing the value must change because 'you did something to it'

Useful representations

  • Area model re-partitioned (each part → n parts)
  • Stacked / dual-labelled number lines
  • Multiplying by n/n (a name for 1)
  • Ratio table of numerators and denominators
  • Tenths ↔ hundredths grid

Teaching practices

  • Justify with re-partition: each part becomes n parts, so both counts grow ×n while the magnitude is fixed — the point does not move.
  • Generalise from many worked cases to the statement a/b = (n×a)/(n×b) (IES: any fraction can be written infinitely many equivalent ways).
  • Frame the move as multiplying by n/n = 1, so the value cannot change.
  • Apply directly to tenths and hundredths (3/10 = 30/100) as a bridge to decimals.
  • Confront the additive error head-on — multiplying is repeated re-partitioning; adding a constant is not.

Activities to try

  1. 1

    Rename the same amount: fix the shaded length, change the labels

    Why it works: Multiplying numerator and denominator by a whole number n (n ≥ 1) re-partitions the same whole into finer equal parts, renaming a fixed magnitude; adding the same number to each is not re-partitioning and changes the amount (unless the fraction equals 1). Multiplying by n/n is multiplying by 1.

    1. Shade 2/3 of a bar (or mark 2/3 as a point on a number line) and keep that shaded length physically fixed — e.g. with a strip of coloured paper laid over the bar.
    2. Subdivide every third into 2 equal pieces, then 3, then 4, and each time count the total pieces and the shaded pieces.
    3. Record the sequence of names for the SAME shaded length: 2/3 → 4/6 → 6/9 → 8/12, and alongside each write the equation ×2, ×3, ×4 applied to both numerator and denominator.
    4. State in your own words why the shaded amount never moved: each part was re-cut into more, smaller pieces, so both counts grew together while the whole and the shaded magnitude stayed the same.
    5. Now test the additive move: change 2/3 by adding 1 to the numerator and 1 to the denominator to get 3/4. Shade 3/4 on an identical bar and lay it against the 2/3 strip. Observe that 3/4 reaches further — it is a different, larger amount.
  2. 2

    Overlay check: is 4/6 bigger than 2/3?

    Why it works: A larger numerator and denominator do not imply a bigger amount when both were scaled by the same factor — the pieces are correspondingly smaller, so the magnitude is unchanged.

    1. Place 2/3 and 4/6 on a shared number line (or overlay two identical strips, one cut into thirds with 2 shaded, one cut into sixths with 4 shaded).
    2. Decide whether 4/6 lands on the same point as 2/3 or further along.
    3. Explain why 4/6 is NOT bigger even though 4>2 and 6>3: there are more pieces, but each is smaller, so the total shaded amount is the same.
    4. Connect this to multiplying by 2/2: state what 2/2 equals and why multiplying by it cannot change the value.
  3. 3

    Generalise and bridge to tenths and hundredths

    Why it works: The equivalence rule generalises to any whole-number factor n ≥ 1 and to the tenths↔hundredths renaming that underpins decimals; the justification (re-partition = multiply by n/n = 1) is the same every time.

    1. Generate three fractions equivalent to 3/5 and verify each with a strip or number-line diagram.
    2. State the general rule in words and symbols: a/b = (n×a)/(n×b) for any whole number n ≥ 1, and explain why it holds (re-partition each part into n).
    3. Write 3/10 as hundredths: 3/10 = 30/100. State what you multiplied by (10/10) and why the value does not change.
    4. Explain how this renaming into hundredths helps when working with decimals.

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
4.NF.1
The national curriculum in England: Key stages 1 and 2 framework document
Ma5/2.4b

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