Grade by Grade
E. Spatial Sense · Shapes, Maps and Measurement

Putting Shapes Together and Taking Them Apart

E1.2 A1About 35 minutes
Learning goals
  • I can put smaller shapes together to make a bigger shape (compose).
  • I can break a shape into smaller shapes (decompose).
  • I can show that a shape covers the same amount of space (area) when I move its pieces around.
  • I can keep trying when a shape puzzle is tricky.

Words to know

compose
To put shapes together to make a new, bigger shape.
decompose
To break a shape apart into smaller shapes.
area
How much flat space a shape covers.
rhombus
A quadrilateral with 4 sides that are all the same length. A square is a special rhombus.
trapezoid
A quadrilateral with one pair of sides that are parallel (they go the same direction, like train tracks). The red pattern block is a trapezoid.

Shape puzzles

Have you ever done a puzzle? Little pieces fit together to make one big picture. Shapes can work the same way!

When we put shapes together to make a new shape, we compose. When we break a shape apart, we decompose.

2 triangles= rhombus3 triangles= trapezoid2 triangles= square
Two triangles make a rhombus. Three triangles make a trapezoid. Two triangles also make a square.

One hexagon, many ways

A hexagon can be made from different pieces. Each way covers the hexagon exactly, with no gaps and no overlaps.

1hexagon6triangles2trapezoids3rhombuses
A hexagon decomposed three ways: 6 triangles, 2 trapezoids, or 3 rhombuses.
Worked example

Example 1: How many triangles?

A trapezoid can be made from 3 triangles. How many triangles do you need to make a hexagon from 2 trapezoids?

  1. One trapezoid = 3 triangles.
  2. Two trapezoids = 3 + 3 triangles.
  3. 3 + 3 = 6.

6 triangles make the hexagon. That matches the picture above!

Area stays the same

Area is how much flat space a shape covers. If you cut a shape into pieces and move the pieces around, the shape looks different, but it covers the same amount of space. You didn't add any pieces and you didn't take any away.

One way to check area is to count squares on grid paper.

8 squaresstill 8 squaresMove the end column. The area stays the same.
A rectangle of 8 squares. Cut off the end column and move it to the top. The new shape still covers 8 squares.
Worked example

Example 2: Same pieces, new shape

Kai cuts a paper square in half from corner to corner. He makes a triangle with the two pieces. Does the triangle cover more space, less space, or the same space as the square?

  1. The square was cut into 2 pieces.
  2. Kai used both pieces to make the triangle. No pieces were added or thrown away.
  3. The pieces cover the same space as before. They are just arranged in a new way.

The same amount of space. The square and the triangle have the same area.

a squarea triangleSame 2 pieces, same area, new shape!
A square cut on the diagonal makes 2 triangles. Put together a new way, they make a big triangle.

Keep trying!

Shape puzzles can be tricky. If a piece doesn't fit, try turning it or flipping it. Mistakes help us learn where the piece doesn't go. Take a deep breath and try another way. Say to yourself: "I can't do it yet."

Just for fun

Why did the hexagon show up to the party in pieces?

It wanted to make a grand entrance: six triangles at a time!

Hands-on activity

Cut-and-move shapes

You need: paper (grid paper if you have it), scissors, a pencil. Crackers or sticky notes are optional.

  1. Draw a rectangle that is 4 squares long and 3 squares tall. Count the squares (12).
  2. Cut it into 3 or 4 pieces any way you like (straight cuts).
  3. Make a new shape with all the pieces. Trace it. Count the squares again. Is it still 12?
  4. Cut a paper square in half from corner to corner. How many different shapes can you make with the 2 triangles? (Try a big triangle, a parallelogram and the square again.)
  5. If you have pattern blocks or tangrams, build a hexagon in as many ways as you can.

Math talk: "Your shape looks different now. Did its area change? How do you know?"

Your turn

1

What does compose mean?

2

How many triangles make a rhombus (like the blue pattern block)?

3

How many trapezoids make a hexagon?

4

A shape covers 10 squares. You cut it apart and make a new shape with all the pieces. How many squares does the new shape cover?

5

Can you make a square from 2 triangles? How?

6

Mia says her new shape is bigger because it is longer. She used the same pieces. Is she right?

More practice: warm-up

1

What does decompose mean?

2

How many triangles make a hexagon?

3

How many triangles make a trapezoid (like the red pattern block)?

4

What is area?

5

You cut a sandwich in half and move the halves apart. Do you have more sandwich now?

More practice: on your own

1

How many rhombuses make a hexagon?

2

Build a hexagon from 1 trapezoid and some triangles. How many triangles do you need?

3

Shape A covers 6 squares. Shape B covers 8 squares. Can you rearrange the pieces of Shape A to make Shape B? Explain.

4

A rectangle is 2 squares tall and 5 squares long. How many squares does it cover? If you cut it and rearrange it into a different shape, how many squares will that cover?

5

Find two different ways to make a hexagon using rhombuses and triangles together.

Challenge

1

How many triangles would it take to make 2 hexagons?

2

Lee covered a shape with 1 hexagon and 2 trapezoids. Ana covered a shape using only triangles. Their shapes have the same area. How many triangles did Ana use?

3

A square is cut into 4 small matching squares. Can those 4 pieces make a long rectangle? Does the area change?

Check your understanding

Quiz

1.

Putting shapes together to make a new shape is called...

2.

How many triangles make a hexagon (pattern blocks)?

3.

How many triangles make a trapezoid (pattern blocks)?

4.

A shape covers 12 squares. You cut it and move the pieces to make a new shape. The new shape covers...

5.

What can 2 rhombuses and 2 triangles make?

6.

Which word means break a shape into smaller shapes?

For parents and teachers

Big idea: Shapes can be built from (composed) and broken into (decomposed) smaller shapes. When pieces are rearranged, the shape changes but its area stays the same. This idea is called conservation of area and is a foundation for later measurement work.

Common difficulties:

  • Thinking a longer or taller shape has more area. Rebuild the original shape from the pieces to show nothing was added.
  • Leaving gaps or overlaps. Area only matches if pieces fit edge to edge with no gaps and no overlapping.
  • Giving up on puzzles. Encourage turning and flipping pieces, and praise effort ("You tried three ways. That's how mathematicians work!").

How to help: Pattern blocks and tangrams are ideal, but paper shapes, crackers or toast cut into pieces work well. Pattern block pieces can be printed and cut out from paper if you don't have them.