Math // Fractions

Comparing fractions

Compare and order unit fractions, and fractions with the same denominator

Ages
7 to 8
Type
Method
Builds on
2 topics
Leads to
3 topics
Standards
1 code

Quick check

If the learner has eaten 3/8 of a cake and a friend has eaten 5/8 of the same cake, can they tell you who has eaten more — and put those fractions in order from smallest to biggest?

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.
  • Order 1/2, 1/3, 1/4, 1/5 from largest to smallest
  • Explain that a larger denominator means smaller unit fractions
  • Compare 2/5 and 4/5 and explain that 4/5 is larger because it has more fifths

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

You are here

Comparing fractions

Ages 7 to 8 // Method

Leads to

3 topics
  • Comparing fractions (age 8+)Required

    Fractions // Ages 8 to 9

    Same-denominator comparison experience is prerequisite to same-numerator comparison

  • Comparing fractions (age 7+)Helpful

    Fractions // Ages 7 to 8

    Some problems require comparing fractions

  • The 0-to-1 Probability ScaleHelpful

    Probability // Ages 10 to 11

    Expressing probability as fractions, decimals, and percentages requires comparing and ordering fractions — a skill built in the Fractions domain

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

Year 3 learners (age 7-8) compare and order unit fractions (1/2, 1/3, 1/4, 1/5) and same-denominator fractions (1/5...4/5). Two directions that feel contradictory must be held together: for unit fractions a LARGER denominator means a SMALLER fraction (the whole is cut into more, smaller pieces), while for same-denominator fractions a LARGER numerator means a LARGER fraction (more of the same-sized piece). These are not one rule reversing; they compare different varying quantities. Ground both in piece size using aligned bars and a shared number line so children do not over-generalise a blanket 'bigger bottom = smaller'.

Common misconceptions

  • Whole-number bias — orders unit fractions 1/5, 1/4, 1/3, 1/2 (bigger denominator judged bigger)
  • Over-correction — after learning 'bigger bottom = smaller', reverses same-denominator fractions too (1/5 > 2/5 > 3/5)
  • Compares numerators and denominators independently rather than as one quantity
  • Cannot place the fractions consistently on a number line to check the order
  • Ignores that the whole must be the same when comparing

Useful representations

  • Aligned equal-length bars for unit fractions (a fraction wall column)
  • Number line locating the fractions being compared
  • Same-denominator strip where only the count of shaded parts changes
  • 'Cut into more' animation shrinking each unit fraction

Teaching practices

  • Order unit fractions against a common whole using aligned bars so 1/5 is visibly the smallest and 1/2 the largest
  • Use a 'cut into more pieces' action so children feel each piece shrink as the denominator grows
  • Order same-denominator fractions by counting how many of the SAME piece, keeping the piece size fixed
  • Place fractions on a shared number line so both comparisons are read as position
  • Make explicit WHY the denominator direction flips between the two cases (piece size vs how many pieces)

Activities to try

  1. 1

    Order unit fractions under one shared whole

    Why it works: When the whole is the same, a unit fraction with a larger denominator is smaller because the whole is split into more, smaller pieces.

    1. Take fraction cards 1/2, 1/3, 1/4, 1/5 and arrange them from largest to smallest.
    2. Represent each card as a bar the same length as a single common '1 whole' bar, split into that many equal parts with one part shaded, and line the bars up beneath that '1 whole' so the shaded pieces can be compared by length.
    3. State and check the order: 1/2, 1/3, 1/4, 1/5 from largest to smallest, confirming 1/5 is the shortest and 1/2 the longest.
    4. Explain out loud why 1/2 is bigger than 1/5 by pointing at the piece lengths against the same whole.
  2. 2

    Cut into more pieces and feel each one shrink

    Why it works: Cutting a fixed whole into more equal parts makes each part smaller, so more pieces means a smaller unit fraction.

    1. Start with one whole and partition it into 2, then 3, then 4, then 5 equal pieces, each time comparing the size of a single piece to the previous one.
    2. Describe what happens to the size of each piece as more cuts are added.
    3. Connect the action back to the cards: more pieces (bigger denominator) means each piece, and so each unit fraction, is smaller.
  3. 3

    Order same-denominator fractions by counting the same-sized piece

    Why it works: When the denominator is fixed, the numerator counts how many of the same-sized piece, so a larger numerator means a larger fraction.

    1. Take fifths fractions 1/5, 2/5, 3/5, 4/5 and order them from largest to smallest, keeping the piece size (one fifth) fixed.
    2. Represent each as a bar where only the number of shaded fifths changes, and check that 4/5 covers more of the bar than 2/5.
    3. State the order 4/5, 3/5, 2/5, 1/5 and explain that more fifths means more because each piece is the same size.

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
Ma/KS2/Y3/F/6

Math // Fractions

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