Math // Fractions

Equivalent fractions on a number line

Understand two fractions as equivalent if they are the same size or the same point on a number line; recognise and show families of common equivalent fractions using diagrams

Ages
8 to 9
Type
Representation
Builds on
3 topics
Leads to
5 topics
Standards
2 codes

Quick check

If the learner sees that 2/4 and 1/2 land on the exact same point on a number line, can they explain in their own words why those two fractions are equal?

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 fraction strips to show 1/2 = 2/4 = 3/6
  • Verify on a number line that 2/3 and 4/6 land on the same point
  • Identify at least three fractions equivalent to 1/2 using 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

3 topics
  • Equivalent fractionsRequired

    Fractions // Ages 7 to 8

    Diagram-based equivalent fractions is prerequisite to formal equivalence understanding

  • Fractions of a wholeRequired

    Fractions // Ages 8 to 9

    Must understand unit fractions to reason about equivalence

  • Fractions on a number line (age 8+)Required

    Fractions // Ages 8 to 9

    Equivalent fractions as the same point on a number line directly uses the fraction number-line representation

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Equivalent fractions on a number line

Ages 8 to 9 // Representation

Leads to

5 topics
  • Equivalent fractions (age 8+)Required

    Fractions // Ages 8 to 9

    Must understand equivalence before generating equivalent fractions

  • Equivalent fractions (age 9+)Required

    Fractions // Ages 9 to 10

    Understanding equivalence conceptually is prerequisite to explaining algebraically

  • Fraction-Decimal EquivalentsHelpful

    Fractions // Ages 8 to 9

    Equivalent fractions understanding supports recognising decimal equivalents of common fractions

  • Choosing mathematical toolsHelpful

    Mathematical Thinking // Ages 8 to 9

    Equivalent fractions require selecting appropriate fraction models

  • Times tables (age 8+)Helpful

    Mathematical Thinking // Ages 8 to 9

    Equivalent fraction patterns exercise repeated reasoning

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

Two fractions are equivalent when they mark the SAME point (same magnitude) on a number line or the same length on a strip, even though they are written with different numerators and denominators. Families such as 1/2 = 2/4 = 3/6 arise because re-partitioning each unit part into finer equal parts leaves the position unchanged. Stacked/aligned 0–1 number lines, dual labels above and below one line, and length-matched fraction strips make the coincident points visible. The documented hard cases are (a) accepting two differently-written fractions as the same point and (b) finding an equivalent on a differently-partitioned line (e.g. locating 1/2 as 3/6 on a line divided into sixths). Re-partitioning only produces a new name when the finer parts subdivide evenly all the way to the point.

Common misconceptions

  • Cannot accept two differently-written fractions as the same point (IES Roadblock 2.3)
  • Cannot locate an equivalent on a differently-partitioned line (e.g. finding 1/2 as 3/6 on a sixths line)
  • 'Different numbers can't name the same amount' — denies equivalence altogether
  • Additive drift — thinks 1/2 and 2/3 are equivalent because you 'add 1 to each'
  • Whole-number bias — 2/4 must be bigger than 1/2 because the numbers are larger

Useful representations

  • Stacked / aligned 0–1 number lines (IES Figure 4)
  • One line with dual labels above and below (1/2 above, 4/8 below at one point)
  • Fraction strips of different unit sizes compared by length
  • Fraction wall

Teaching practices

  • Use stacked 0–1 number lines (halves, thirds, fourths … tenths — IES Figure 4) and read off the points that line up.
  • Label one set of fractions above a single line and another set below it, pointing to 1/2 and 4/8 at the same spot (IES fix for Roadblock 2.3).
  • Measure equal lengths with different fraction strips (IES Figure 5) to feel 'same size'.
  • Practise the documented hard case: find an equivalent on a differently-partitioned line (e.g. 1/2 on a line divided into sixths).
  • Insist that 'same point = same number', directly countering the belief that different symbols must mean different amounts.

Activities to try

  1. 1

    Plumb-line on stacked thirds and sixths lines

    Why it works: Equivalent fractions are different names for the SAME point on the number line; same position means same magnitude.

    1. Stack a 0–1 line divided into thirds directly above a 0–1 line divided into sixths, carefully aligned at 0 and 1.
    2. Point to 2/3 on the thirds line and drop a vertical plumb-line straight down to the sixths line.
    3. Name the fraction on the sixths line that sits at exactly the same point (4/6), and state in words: two different names, one position.
    4. Repeat for other target points and record each equivalent pair you find.
    5. Say aloud why the shared point means the two fractions are equal in size.
  2. 2

    Re-partition toggle: the point that does not move

    Why it works: Re-partitioning each unit part into finer equal parts preserves a fraction's position, which is why equivalent families exist. When the finer partition subdivides evenly all the way to the point, that same point simply gains a new name (1/2 becomes 3/6 on a sixths line); when the partition does not reach the point (e.g. fifths for 1/2), no such name exists.

    1. Start on a thirds line with 2/3 marked.
    2. Split each third into two equal parts to create a sixths line, keeping the 0–1 ends fixed.
    3. Observe whether the 2/3 mark moves, and add the new label 4/6 to that same point.
    4. Explain in your own words why subdividing each part does not shift the position, only adds a finer-grained name.
    5. Now locate 1/2 on a line divided into sixths by splitting each half into three (not two): count three sixths to the midpoint and mark 3/6 at the 1/2 point.
  3. 3

    Measuring equal lengths with different fraction strips

    Why it works: Equivalence can also be felt as equal length: differently-partitioned strips that reach the same point name the same amount.

    1. Lay a 1/2 strip, a 2/4 strip and a 3/6 strip along the same 0–1 length.
    2. Check by eye and by measuring that they reach the same endpoint (allowing for the slight thickness and cutting error of physical strips).
    3. State that 'same length = same number' even though the pieces and labels differ.
    4. Find and record at least three fractions equal to 1/2 using the strips and/or a number line (e.g. 2/4, 3/6, 5/10).

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
3.NF.3.a
The national curriculum in England: Key stages 1 and 2 framework document
Ma/KS2/Y4/F/1

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