Questions: Decomposing Shapes

Question 1

You cut a square with a straight line from one corner to the opposite corner. What two shapes are formed?

• A. Two smaller squares
• B. Two triangles
• C. Two rectangles
• D. A triangle and a rectangle

Answer: (B)

A diagonal cut from corner to corner divides a square into two triangles. The line connects two opposite corners, making two equal triangular halves. A cut straight across the middle (not diagonal) would make two rectangles. Which shapes you get depends on where the line goes, not just that the shape is being divided.

Question 2

A student says: 'I can only decompose this rectangle one way — down the middle to make two smaller rectangles.' Is the student right?

• A. Yes — rectangles can only be split into smaller rectangles
• B. Yes — every shape has exactly one correct way to be decomposed
• C. No — the same rectangle can also be cut diagonally to make two triangles
• D. No — rectangles cannot be decomposed at all

Answer: (C)

The same shape can be decomposed in multiple ways depending on where you draw the cut. A rectangle cut straight across makes two smaller rectangles. The same rectangle cut from corner to corner (diagonally) makes two triangles. The pieces you get depend entirely on where the line goes, not on the original shape alone.

Question 3

Two right triangles can be put together to form a rectangle.

• T. True
• F. False

Answer: True

Composing (putting shapes together) and decomposing (taking them apart) are reverse operations. Two congruent right triangles, placed with matching sides together, form a rectangle. This is exactly the reverse of decomposing a rectangle by cutting it corner to corner — the same relationship that addition and subtraction have with numbers.

Question 4

Decomposing a square usually produces the same pieces, no matter where you draw the cut.

• T. True
• F. False

Answer: False

The pieces you get depend entirely on where and how you draw the cut. Cutting a square from corner to corner produces two triangles. Cutting it straight across the middle produces two rectangles. Cutting it at a different angle produces different shapes altogether. The square contains many possible decompositions — you choose which one by choosing where to draw the line.

Question 5

How is decomposing a shape similar to subtraction? Use an example.

Model answer: Decomposing takes a whole shape apart into smaller pieces, just as subtraction takes a number apart. Composing puts pieces back together to make the original, just as addition reverses subtraction. For example, cutting a rectangle into two triangles (decomposing) is the reverse of placing two triangles together to make a rectangle (composing). They are opposite operations on shapes, the same way addition and subtraction are opposite operations on numbers.

The composing/decomposing relationship mirrors the addition/subtraction relationship. When you decompose a rectangle into two triangles, you can reverse the process by composing (combining) those two triangles back into the rectangle. Understanding this two-way relationship builds the flexible spatial thinking that connects geometry to later topics like fractions and area.