Math // Fractions

Unit fractions

Recognise, find, and write fractions of a discrete set of objects: unit fractions and non-unit fractions with small denominators

Ages
7 to 8
Type
Concept
Builds on
2 topics
Leads to
4 topics
Standards
1 code

Quick check

If there are 15 books on a shelf, can the learner work out how many two-thirds of them is — and check by grouping the books into three equal piles?

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.
  • Find 1/4 of 12 objects by dividing into 4 equal groups
  • Find 3/4 of 12 objects
  • Write the fraction of a set that is shaded or selected

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
  • Fractions of amountsRequired

    Fractions // Ages 6 to 7

    Finding fractions of discrete sets extends finding fractions of shapes/quantities

  • Division as equal sharingHelpful

    Multiplication & Division // Ages 4 to 6

    Finding 1/4 of 12 objects connects to division as sharing equally

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Unit fractions

Ages 7 to 8 // Concept

Leads to

4 topics
  • Comparing fractions (age 7+)Required

    Fractions // Ages 7 to 8

    Fraction problems require finding fractions of quantities

  • Fractions of amounts (harder)Required

    Fractions // Ages 8 to 9

    Finding fractions of discrete sets is prerequisite to harder fraction-of-quantity problems

  • Probability as a FractionRequired

    Probability // Ages 9 to 10

    Expressing probabilities as unit fractions (1/6, 1/2, 1/3) requires prior knowledge of unit fractions from the Fractions domain

  • Egyptian Maths and EngineeringHelpful

    History // Ancient Egypt // Ages 11 to 13

    The Rhind Mathematical Papyrus works extensively with unit fractions; prior knowledge of unit fractions makes its mathematical significance tangible

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

Teaches fractions of a discrete set by treating the whole SET as the '1': the denominator tells how many equal groups to share the objects into, and the numerator tells how many groups to take. 1/4 of 12 means share 12 into 4 equal groups of 3 and take one group (3); 3/4 of 12 means take three groups (9). Because the whole is the set, the same fraction gives a different count for a different-sized set (1/4 of 12 = 3, but 1/4 of 8 = 2). Also targets naming a coloured part part-to-whole (3/12), not part-to-part (3/9).

Common misconceptions

  • Takes the denominator as the answer: 1/4 of 12 is '4' because the fraction has a 4
  • For a non-unit fraction, finds 1/4 (=3) and stops, giving 3 for 3/4 instead of 9
  • Names a coloured set part-to-part (3 coloured, 9 not → 3/9) instead of part-to-whole (3/12)
  • Divides by the numerator instead of the denominator

Useful representations

  • Counters shared into equal groups (12 → four groups of 3)
  • Array / grouping diagram
  • Bar model of the set partitioned into b parts with a shaded
  • Double number line (objects ↔ fraction of the set)

Teaching practices

  • Physically share the objects into 'denominator' equal groups first, then take 'numerator' groups — grounding the fraction in equal sharing/division
  • Stress that the whole is the whole SET, so 1/4 of a set is a number of objects, not the digit 4
  • Distinguish the numerator's job (how many groups to take) from the denominator's (how many groups to make), especially for non-unit fractions like 3/4
  • When naming the fraction of a set that is coloured, count coloured objects against the WHOLE set (part-to-whole, 3/12), not against the uncoloured ones (part-to-part, 3/9)
  • Vary the set size so children see 1/4 is a relationship, not a fixed number of things

Activities to try

  1. 1

    Share the set into equal groups, then take some

    Why it works: The whole is the whole SET; the denominator counts the equal groups to make and the numerator counts the groups to take. The same fraction names a different quantity when the set changes, so a fraction is a relationship, not a fixed number of objects.

    1. Start with 12 counters and the request 'show me 1/4'. Set the denominator to 4 and form four group-rings (or draw four rings/circles on paper).
    2. Deal the 12 counters out one at a time so each ring holds an equal share — 3 each. If any group is unequal, keep dealing until they match.
    3. Take one group for 1/4: say and record '1/4 of 12 = 3'.
    4. For 3/4, light up / shade / move three of the four groups and count 3 + 3 + 3 = 9; record '3/4 of 12 = 9'.
    5. Change the set size: re-share 8 counters into four equal groups and see each group is now 2, so 1/4 of 8 = 2. Compare with 1/4 of 12 = 3.
  2. 2

    Name the coloured fraction of a set — part to whole

    Why it works: Naming a fraction of a set is part-to-whole: compare the coloured objects to the whole set, not to the uncoloured remainder.

    1. Look at a set of 12 counters with 3 coloured. Decide what fraction is coloured.
    2. Count the coloured objects (3) and then count the WHOLE set (all 12), not the uncoloured ones.
    3. Write the fraction as coloured-over-whole: 3/12.
    4. Explain why it is 3/12 and not 3/9: the denominator is the whole set, not the leftover uncoloured counters.
    5. Say the fraction by name ('three twelfths') rather than 'three over twelve'.
  3. 3

    Predict before you re-share

    Why it works: A child who thinks a fraction names a fixed number predicts 3 for 1/4 of 8; sharing reveals 2, refuting the 'fixed count' idea and showing 1/4 is a relationship to whatever the whole set is.

    1. Share 12 counters and establish that 1/4 of 12 is 3.
    2. BEFORE sharing again, predict: 'What will 1/4 of these 8 counters be — the same 3, or different?' Give a reason.
    3. Now share the 8 into four equal groups — 2 each — and compare the observed 2 with the prediction.
    4. Answer follow-up prompts: Is someone right that '1/4 of 12 is 4 because there is a 4 in the fraction'? For 3/4 of 12, how many of the four groups of 3 do you take, and why three?
    5. Explain in your own words why 1/4 means making 4 equal groups, not just taking 4 counters.

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/2

Math // Fractions

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