Questions: Building and Decomposing Shapes

Question 1

A student puts two right triangles together along their longest sides to form a rectangle. Then she draws a diagonal line across the rectangle, splitting it back into two triangles. What has she demonstrated?

• A. That triangles are always smaller than rectangles
• B. That shapes can be composed into a larger shape and then decomposed back into smaller parts
• C. That you can only decompose a shape the same way it was originally composed
• D. That rectangles and triangles are really the same shape viewed from different angles

Answer: (B)

Composing and decomposing are reverse operations. Building the rectangle from triangles is composition; drawing the diagonal to recover the triangles is decomposition. The student has traveled the two-way street: one direction makes a larger shape, the other breaks it back apart. The key insight is that these operations are always reversible.

Question 2

A hexagon can be split into 2 trapezoids, OR into 3 rhombuses, OR into 6 triangles. What does this demonstrate about decomposing shapes?

• A. Hexagons are fundamentally made of triangles; the other decompositions are just approximations
• B. There is exactly one correct way to decompose any shape
• C. The same shape can be decomposed in more than one valid way
• D. Only six-sided shapes can be broken into smaller shapes

Answer: (C)

A hexagon can be legitimately split multiple ways — each decomposition results in parts that perfectly tile the original shape without gaps or overlaps. There is no single 'correct' decomposition; the same shape contains multiple valid structures. This is a key insight: shapes are not locked into one set of parts.

Question 3

When you compose two smaller shapes into a larger shape, you can reverse the process and decompose the larger shape back into the original pieces.

• T. True
• F. False

Answer: True

Composition and decomposition are reverse operations. If two triangles combine to form a rectangle, that rectangle can always be split back into two triangles along the same line. The shapes don't 'fuse' — the parts remain identifiable, and the relationship is always reversible.

Question 4

Breaking a shape into smaller pieces increases the total amount of space the shape covers.

• T. True
• F. False

Answer: False

Decomposing a shape rearranges its parts but preserves the total area. If a square is cut into four smaller squares, those four pieces together cover exactly the same space as the original square. No space is added or lost — the pieces account for every part of the whole.

Question 5

If you split a rectangle into two triangles, does the rectangle 'disappear' — or is something preserved? What does this tell you about the relationship between composed and decomposed shapes?

Model answer: The total area (space covered) is preserved. The rectangle does not disappear — it is simply reorganized into two triangular parts that together cover the exact same space. Composing and decomposing shapes is a reversible process: the whole equals the sum of its parts, and those parts can always be reassembled into the whole.

This is the core insight of composition and decomposition: rearranging parts does not change the total. The big shape and its pieces are two ways of describing the same amount of space. Understanding this prepares students for fractions (parts of a whole) and area (measuring how much space a shape covers).