Math // Fractions

Decomposing a shape into more equal shares

Understand that decomposing a shape into more equal shares creates smaller shares

Ages
6 to 7
Type
Concept
Builds on
1 topic
Leads to
2 topics
Standards
1 code

Quick check

If a pizza is shared between 2 people and then the same pizza is shared between 8 people, can the learner explain why each slice is smaller when more people share it?

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.
  • Explain that a quarter of a pizza is smaller than a half of the same pizza
  • Demonstrate that fourths are smaller pieces than halves
  • Compare the size of halves and quarters of the same shape

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

1 topic
  • Halves & Quarters of ShapesRequired

    Fractions // Ages 6 to 7

    Comparing share sizes requires experience partitioning into halves and quarters

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Decomposing a shape into more equal shares

Ages 6 to 7 // Concept

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

Help 6-7 year olds grasp the inverse-size relationship: cutting the SAME whole into more equal shares makes each share smaller. A quarter of a pizza is a smaller amount than a half of the same pizza, and an eighth smaller still. This directly confronts the whole-number bias by which children read a bigger denominator as a bigger amount ('4 > 2 so a quarter is bigger'). When comparing the actual amounts, the comparison relies on the wholes being identical, and it seeds later unit-fraction comparison (1/3 > 1/4).

Common misconceptions

  • Larger denominator = larger fraction / whole-number bias ('a quarter is bigger than a half because 4 is more than 2'; 1/8 > 1/3) - documented in IES
  • Comparing fractions of DIFFERENT-sized wholes without noticing the whole matters
  • Confusing the number of pieces with the amount in one piece ('4 pieces is more than 2 pieces')

Useful representations

  • Two identical circles/bars, one halved and one quartered, pieces overlaid
  • A share-counter with a slider for number of shares, one piece highlighted shrinking
  • A fraction wall (1 whole, halves, quarters, eighths) aligned to the same length

Teaching practices

  • Compare unit fractions of IDENTICAL wholes (same pizza, same bar) so the only thing changing is the number of shares
  • Use predict-then-check: ask which piece is bigger before cutting, then overlay to test (surfaces the whole-number-bias reason)
  • Increase the number of sharers and have children watch each share shrink (IES: link to names, 1/3 of it > 1/4 of it)
  • Name the pattern explicitly: more equal shares of the same whole = smaller pieces
  • Stress that the comparison only holds when the wholes are the same size

Activities to try

  1. 1

    Predict-then-check with two identical pizzas

    Why it works: A single share gets smaller as the same whole is split among more sharers, so a half of a pizza is a bigger amount than a quarter of the same pizza even though 4 is more than 2.

    1. Give the child two identical paper 'pizzas' of exactly the same size (the paper stands in for the flat region being shared - what matters is the amount each piece represents, not the paper itself).
    2. Before cutting, ask them to predict: if one pizza is cut into 2 equal pieces and the other into 4 equal pieces, whose single piece will be bigger, and why? Record the stated reason.
    3. Cut both pizzas and lay one half on top of one quarter so the child can see the half blankets the quarter with room to spare.
    4. Then cut (or imagine cutting) the same-size whole into 8, then more, and have the child watch a single piece keep getting smaller.
    5. Name the pattern out loud together: 'more equal shares of the same whole means smaller pieces.'
  2. 2

    Live share-slider: watch one piece shrink

    Why it works: As the number of equal shares rises, one share visibly shrinks, making the inverse relationship between number of shares and size of each share concrete.

    1. Show two identical circular pizzas side by side, each with a control that sets the number of equal shares.
    2. Have the child set one pizza to 2 shares and the other to 4, highlight one piece on each, then drag the highlighted pieces together to overlay them.
    3. Ask the child to describe what happens to the overlapped pieces: the half covers the quarter with space left over.
    4. Have the child push the share control higher (8, 10, 12) on one pizza and describe how a single highlighted piece keeps shrinking.
  3. 3

    Order and compare unit fractions of the same whole

    Why it works: Unit fractions of the same whole can be ordered by size (1/2 > 1/3 > 1/4 > 1/8); when comparing the actual amounts in the pieces, the comparison relies on the wholes being the same size.

    1. Show pictures of 1/2, 1/3, 1/4 and 1/8 of the SAME cake and ask the child to order them largest to smallest.
    2. Ask the child to explain why the order goes that way in terms of sharing.
    3. Ask the child directly: 'Why doesn't the biggest number on the bottom mean the biggest piece?'
    4. Pose a comparison question such as: 'Who has more, someone with a half of a cake or someone with a quarter of the same cake?' and ask for the reason.
    5. Ask the trap question: 'Could a quarter ever be a bigger amount than a half? What would have to be true about the two cakes?'

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
1.G.3

Math // Fractions

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