Math // Fractions

Splitting shapes into equal parts (age 7+)

Partition circles and rectangles into two, three, or four equal shares; describe shares as halves, thirds, and fourths; recognise that equal shares of identical wholes need not have the same shape

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

Quick check

If you cut a rectangle into thirds in two different ways (three strips vs three squares), can the learner recognise that both give equal thirds — even though the pieces look different?

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.
  • Partition a circle into 3 equal parts and label each 'a third'
  • Partition a rectangle into 4 equal shares in more than one way
  • Explain that two different-looking shares can still be equal in size

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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Splitting shapes into equal parts (age 7+)

Ages 7 to 8 // Concept

Leads to

2 topics
  • Fractions of a wholeRequired

    Fractions // Ages 8 to 9

    Partition into equal shares is prerequisite to understanding unit fractions

  • Fractions as parts of shapesRequired

    Fractions // Ages 8 to 9

    Prior partitioning experience feeds into area-fraction work

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

Children learn that a fraction requires EQUAL parts — equal in AMOUNT (area), not necessarily equal in shape or appearance. Two load-bearing ideas: having the right NUMBER of pieces is not enough (unequal pieces are not halves/thirds/quarters), and two shares that look completely different can still be equal if they cover the same amount. Teach by partitioning the same shape several ways and by comparing equal-area-but-different-shape pieces, so children judge shares by amount, not by look. Thirds (and a circle into three) force children off repeated halving.

Common misconceptions

  • Equal shares must look the same / be congruent — rejects equal-area different-shape pieces as 'not fair'
  • Any N pieces count as Nths regardless of size (ignores the equal-parts requirement)
  • A shape can only be partitioned into a fraction one 'proper' way
  • Judges share size by one dimension (length) rather than area
  • Thinks halving repeatedly is the only way to make parts, so thirds feel impossible

Useful representations

  • Rectangle partitioned several ways into the same fraction (multiple valid cuts)
  • Circle partitioned into thirds (120° shares)
  • Equal-area different-shape pieces that rearrange onto one another
  • Faint unit-square grid under a shape for checking equal area

Teaching practices

  • Partition the SAME shape several different ways into the same fraction (a rectangle into quarters by vertical cuts, a 2×2 grid, two diagonals) so no single picture becomes 'the' way
  • Deliberately show equal-AREA but different-SHAPE pieces and have children verify equality by rearranging/overlaying or counting unit squares
  • Contrast a correct partition with an unequal one that has the right NUMBER of pieces, forcing the equal-parts criterion open
  • Move beyond halving to thirds (a circle into three) where children cannot rely on repeated halving
  • Check equality physically — cut and re-lay pieces, or tile them onto a grid — rather than trusting appearance

Activities to try

  1. 1

    Cut it, re-lay it: equal amount beats equal look

    Why it works: A quarter means each part is the same amount (area) of the whole, one of four equal parts — the pieces may look different as long as each covers the same amount; and the same shape can be partitioned into the same fraction in many valid ways.

    1. Partition a square into four parts where the pieces look different — e.g. one long thin strip plus some oddly-shaped bits — but each covers the same amount.
    2. Before checking, predict: is the long thin piece the same amount of the square as a fat corner piece? Say why.
    3. Check equality physically: cut the long strip and lay it over the fat piece to see it rearranges exactly onto it. For pieces whose edges follow a faint underlying grid you can also count the unit squares each covers; for diagonal or irregular pieces use cutting and rearrangement rather than counting, since squares get split.
    4. Decide whether all four pieces are quarters and give the reason (each is the same AMOUNT, one of four equal parts).
    5. Now quarter the SAME shape in a completely different way (e.g. three equally spaced vertical cuts giving four equal strips, then a 2x2 grid, then two diagonals) and argue each piece is still a quarter.
  2. 2

    Sort the partitions: quarters or not?

    Why it works: Whether a partition shows quarters depends on equal amount, not on the number of pieces or on whether the pieces look alike.

    1. Sort a set of partitioned shapes into 'shows quarters' and 'does not show quarters'.
    2. Include cases that are equal-area but different-shape (these ARE quarters) and cases with four pieces of unequal size (these are NOT quarters).
    3. For each 'does not' card, say what is wrong (pieces are not the same amount).
    4. For each tricky 'is' card where pieces look different, say how you could prove they are equal (cut and rearrange/recompose, or count squares where edges follow the grid).
  3. 3

    Which is the bigger share: a third or a quarter?

    Why it works: For the same whole, one of three equal parts is bigger than one of four equal parts — cutting the same whole into more equal parts makes each part smaller.

    1. Take two identical shapes; partition one into three equal parts and the other into four equal parts.
    2. Predict which single piece is bigger: one of three equal parts, or one of four equal parts, of the same shape.
    3. Check by overlaying one piece on the other when both wholes are cut the same way (e.g. both into strips the same direction, or both into slices from the centre of a circle) so the larger covers the smaller and sticks out; otherwise compare the area each covers.
    4. State the reasoning: cutting the same whole into more equal parts makes each part smaller.

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
2.G.3

Math // Fractions

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