Math // Geometry

Composing Shapes

Compose two-dimensional shapes (rectangles, squares, trapezoids, triangles, half-circles, quarter-circles) to create composite shapes, and compose new shapes from the composite shape

Ages
6 to 7
Type
Method
Builds on
2 topics
Leads to
0 topics
Standards
1 code

Quick check

Can the learner use flat shapes like triangles and rectangles to fit together and make a new larger shape — like a puzzle?

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.
  • Combine two triangles to form a rectangle or larger triangle
  • Put together half-circles and quarter-circles to form circles or new shapes
  • Decompose a composite shape and describe which simpler shapes make it up

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
  • 2-D shapesRequired

    Geometry // Ages 4 to 6

    Need to recognise 2-D shapes before composing them into composites

  • Combining Simple ShapesRequired

    Geometry // Ages 5 to 6

    Composing 2-D shapes at Grade 1 level extends the Kindergarten concept of composing simple shapes

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Composing Shapes

Ages 6 to 7 // Method

Leads to

0 topics

Nothing in the catalogue builds on this topic yet. Browse Geometry

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 compose 2-D shapes — combining triangles, squares, trapezoids, and half-/quarter-circles into composite shapes and finding new shapes within them. The goal is to move along Clements & Sarama's trajectory: from unsystematic trial-and-error assembly (Piece Assembler), through trial-and-error picture making, to composing WITH ANTICIPATION — choosing a piece because its angle and side length will fit, using turns and flips intentionally, and decomposing a composite back into the simpler shapes that make it up. The core teaching medium is manipulating flat pieces into outline (puzzle) tasks engineered so each successive level of thinking is required, with immediate fit/misfit feedback.

Common misconceptions

  • Assembles by trial and error with no advance plan and does not anticipate the new shape (an expected early trajectory level, not a fault).
  • Chooses pieces by general shape only, not by matching angles and side lengths, so pieces do not fit the gap.
  • Sees the composite as one whole and cannot decompose it into the simpler shapes it is made from.

Useful representations

  • Pattern blocks and tangram pieces
  • Outline puzzle cards of increasing difficulty
  • Two triangles composing a square or larger triangle; half-circles composing a circle
  • Angle-and-side-length matching between a gap and a piece

Teaching practices

  • Use outline puzzles engineered to require the next level of thinking (e.g. remove one internal guide line to push from Piece Assembler to Picture Maker).
  • Nudge from trial and error to anticipation: 'Can you see which shape will fit before you place it?'
  • Have children choose pieces by attributes — matching side lengths and angles — and use turns and flips intentionally.
  • Ask children to decompose the composite ('what smaller shapes make this up?') and to find new shapes within it (two triangles make a square; two half-circles make a circle).
  • Pitch tasks one level beyond the child's current level and fade support — this out-performed jumping straight to the target level (Clements et al. 2019, effect size .55).

Activities to try

  1. 1

    Fill the outline with the right pieces

    Why it works: A piece can be chosen in advance by matching its side length and angle to the gap, rather than by trying pieces at random until one happens to fit.

    1. Give the child flat shape pieces (for example, pattern blocks, tangram pieces, or cut-out triangles, squares, rectangles, trapezoids, half-circles and quarter-circles) and an outline puzzle card that those pieces can cover exactly. Standard pattern blocks and tangrams do not include half-circles or quarter-circles, and tangrams have no trapezoid, so add cut-out pieces for any shapes the set lacks.
    2. As they work, ask them to say which shape they are using and why.
    3. Before each placement, prompt: 'Will this piece fit this gap? How do you know?' — encouraging a prediction from the gap's edges and corners.
    4. When a piece doesn't fit, have them compare the gap's edge length and corner angle to the piece, then choose one that matches.
    5. Let them use turns and flips to make a piece fit, naming what they did ('I flipped it so the slanted edge lines up').
  2. 2

    Same pieces, different shapes

    Why it works: The same pieces can compose different shapes depending on how they are turned and joined — composing is a deliberate, reversible act.

    1. Using two congruent right-isosceles (tangram-style) triangles, have the child join them along their long (hypotenuse) edges to form a square. Then have them rejoin the same two triangles along one of the shorter equal edges, mirrored so the two right-angle corners sit side by side, to form a larger triangle instead. Joining them along a shorter edge with one triangle turned the other way makes a parallelogram, which is another new shape worth naming. (Note for the adult: two equilateral pattern-block triangles would instead make a rhombus, not a square, so choose right-isosceles triangles for this square example.)
    2. Have them put two half-circles together to form a circle, and explore what else the pieces can make.
    3. Ask: 'You made a square from two triangles — can you make a different shape with those same two pieces?'
  3. 3

    Find the shapes hiding inside

    Why it works: A composite shape is built from — and can be pulled apart into — simpler named shapes; seeing the whole does not mean you can only see the whole.

    1. Show the child a completed composite shape and ask 'Which smaller shapes are hiding inside this?'
    2. Have them lift and separate the pieces to reveal, for example, that the square is made of two right-angled (right-isosceles) triangles or the circle of two half-circles.
    3. Ask them to name each component shape as they take the composite apart, then rebuild it.

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.2

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