Math // Fractions

Comparing fractions (age 9+)

Compare and order fractions with different numerators and denominators by creating common denominators/numerators or comparing to a benchmark such as 1/2; justify conclusions with visual models

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

Quick check

If the learner has eaten 2/3 of their lunch and a friend has eaten 3/4 of theirs, can they work out who's eaten more — by finding a common way to compare the fractions?

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.
  • Compare 3/8 and 5/12 by finding a common denominator of 24
  • Order 2/3, 3/5, and 7/10 from smallest to largest
  • Use the benchmark 1/2 to decide that 5/8 > 3/7

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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Comparing fractions (age 9+)

Ages 9 to 10 // Method

Leads to

2 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 compare and order fractions with different numerators and denominators using two complementary strategies: (1) renaming both fractions to a common denominator and comparing numerators, and (2) benchmark reasoning with 0, 1/2, and 1 (deciding which side of 1/2 each fraction sits on, or whether it is near 0 or 1). Benchmarks are quick and build number sense; common denominators are general-purpose. Conclusions are justified with visual models (partitioned bars, number lines). The core misconception to guard against is whole-number bias — ordering by numerator or denominator alone. This is a known hard skill: on the 2004 NAEP, 50% of 8th-graders could not order three fractions from least to greatest.

Common misconceptions

  • Whole-number bias — ordering by numerator or denominator alone (3/8 vs 5/12 → '5 > 3 so 5/12 is bigger', or '12 > 8 so smaller') (IES)
  • Comparing without a common denominator (cross-adding, or comparing tops and bottoms separately)
  • Renaming only one fraction, or using a non-common multiple
  • Benchmark misapplied — e.g. judging 3/7 > 1/2

Useful representations

  • Common denominator via equivalent-fraction renaming
  • Number line marked 0, 1/2, 1 (benchmark lane)
  • Dual-labelled number line for different denominators
  • Ratio table / times-tables for the common multiple

Teaching practices

  • Teach benchmark reasoning with 0, 1/2 and 1: decide which side of 1/2 each fraction sits on (5/8 > 1/2 because 5 is more than half of 8; 3/7 < 1/2) (IES Rec 3).
  • Teach the common-denominator method as renaming both fractions with equivalents, then comparing numerators.
  • Use minimally-labelled number lines (0, 1/2, 1) for landmark placement, and dual-labelled lines (one unit sequence above, another below) for different denominators (IES Rec 2).
  • Choose the strategy to fit the numbers — benchmarks when they straddle 1/2, common denominator when they are close.
  • Keep the same-numerator shortcut available (2/3 > 2/5).
  • Justify with a model and guard against whole-number bias.

Activities to try

  1. 1

    Strategy workbench: common denominator OR benchmark

    Why it works: Renaming two fractions to the same denominator makes them directly comparable (the comparison collapses to comparing numerators), and the 1/2 benchmark settles many comparisons at a glance without full computation. Choosing the strategy to fit the numbers is part of the skill.

    1. Take a pair such as 3/8 versus 5/12. Choose the common-denominator path: re-partition both bars (on paper or with a model) to the same fine grid — twenty-fourths — so 3/8 becomes 9/24 and 5/12 becomes 10/24, then compare the numerators 9 versus 10.
    2. Take a pair such as 5/8 versus 3/7. Choose the benchmark path: place each on a number line marked 0, 1/2, 1. Decide which side of 1/2 each sits on (5/8 is more than 1/2 because 5 is more than half of 8; 3/7 is less than 1/2 because 3 is less than half of 7).
    3. Switch into an ordering task: drag or write three or more fractions (e.g. 2/3, 3/5, 7/10) onto a line or into order, choosing whichever strategy fits the numbers.
    4. State, for each comparison, which strategy you chose and why it suited those particular numbers.
  2. 2

    Dual-labelled and benchmark number lines

    Why it works: A number line anchored to 0, 1/2 and 1 gives fractions a magnitude and a 'side of 1/2' (and a sense of nearness to 0 or 1), so many comparisons can be read off directly; dual labelling lets two different denominators be compared on one line.

    1. Use a number line marked only 0, 1/2, 1 and place common fractions where they belong relative to those landmarks; for each, say whether it is more than, equal to, or less than one half.
    2. Decide whether 4/9 is more or less than a half by comparing 4 with half of 9 (4.5): 4 is less than 4.5, so 4/9 < 1/2 — no common denominator needed.
    3. Order 1/5, 4/5, 5/9 using only benchmarks: 1/5 sits just above 0, 5/9 is just over 1/2 (5 is more than 4.5), and 4/5 is near 1, giving 1/5 < 5/9 < 4/5.
    4. For two different denominators, use a dual-labelled line (one unit sequence of marks above the line, the other below) to see where each fraction falls and which is larger.
    5. Explain how the landmarks 0, 1/2 and 1 let you decide a comparison without finding a common denominator.
  3. 3

    Check and consolidate ordering and benchmark decisions

    Why it works: Fluent comparison means ordering correctly, choosing an efficient strategy, and justifying with a model — not just getting a final symbol right. Any valid common denominator yields the same order.

    1. Order a set such as 2/3, 3/5, 7/10 from smallest to largest and show working (e.g. common denominator 30: 3/5 = 18/30, 2/3 = 20/30, 7/10 = 21/30, giving 3/5 < 2/3 < 7/10).
    2. Answer a quick benchmark question: 'Is 5/8 more or less than 1/2?' and justify.
    3. Rename both fractions of a pair to a shared common denominator before comparing, and check the order you get.
    4. Sort a set of fractions into 'less than 1/2' and 'more than 1/2'.
    5. Keep the same-numerator shortcut in play: with equal numerators, the larger denominator gives the smaller fraction (2/3 > 2/5).

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

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