Composing and decomposing shapes

When you reach for it

Shapes can be put together to make bigger shapes (composing) and taken apart into smaller ones (decomposing). Two triangles can make a square, two squares make a rectangle, and a square folded and cut makes triangles or smaller squares. Reach for this once a child can name circles, triangles, squares and rectangles; it shows that a shape is made of parts, which is where halves, fourths and area will later come from.

Each day takes ten to fifteen minutes. Pattern blocks are ideal if you have them; paper shapes cut from card work just as well. Let the child move the pieces with their own hands every time.

Day 1: Two triangles make a new shape

You need: several pairs of paper triangles, each pair cut from one paper square by cutting it corner to corner (cut these ahead of time).

Give the child one pair. Say, “Put these two triangles together so their sides touch all the way along. What shape did you make?” Then, “Can you make a different shape with the same two triangles?” Matching a different pair of sides, or turning one triangle around, gives a new shape. Ask each time, “How many sides? How many corners? What is its name?”

There is no single right answer here. The point is that the same two pieces can make several shapes, and that a shape’s name comes from its sides and corners, not from the pieces inside it.

Practice. For the first three, ask what shape the two triangles make.

1. Shape A. Answer: a square: 4 straight sides all the same length and 4 square corners.
2. Shape B. Answer: a triangle, bigger than either piece, with 3 sides and 3 corners.
3. Shape C. Answer: not a square, rectangle or triangle. It has 4 sides and 4 corners, but the corners are not square corners, so it leans like a pushed-over box.
4. Pull Shape A apart and build it again with one triangle flipped over. Answer: it still makes the same square, because the long sides still fit together.
5. Use 4 triangles to make one bigger square. Answer: put the square corners of all 4 triangles together in the middle; the long sides make the outside of the square.

Day 2: Squares make squares and rectangles

You need: about 6 paper squares, all the same size (or square pattern blocks).

Ask the child to put 2 squares side by side, edges touching all the way. “What shape is the whole thing?” It is a rectangle: four square corners, but two sides are long and two are short. Then hand over 4 squares: “Can you make a bigger square? Can you make a long rectangle? Can you make a shape that is neither?”

To decide, run a finger around the outside edge only, counting sides and checking corners. The lines inside do not count; they show the pieces, not the shape.

Practice. Name the whole shape.

1. Shape A. Answer: a rectangle, 2 squares long and 1 square tall.
2. Shape B. Answer: a square, 2 squares on each side.
3. Shape C. Answer: a rectangle, long and thin.
4. Shape D. Answer: not a square, rectangle or triangle. Going around the outside there are 6 sides and 6 corners, and one corner points inward.
5. How many small squares are in Shape B? Answer: 4.

Day 3: Fold and cut a shape apart

You need: paper squares and paper rectangles, child-safe scissors.

Now go the other way. Fold a paper square in half so the edges line up, open it, and ask, “If we cut on this fold, what shapes will we get? How many?” Let the child guess, then cut. Lay the pieces back together to show they still make the square. Try other folds on new paper: down and across the middle, or corner to corner. A long rectangle does not fold neatly corner to corner, so for Shape B draw that line with a ruler and cut along it.

Ask, “Can you put the pieces back to make the shape we started with? Can you make a new shape with them?”

Practice. Cut on the dashed lines. What pieces do you get?

1. Shape A. Answer: 2 rectangles, each half of the square.
2. Shape B. Answer: 2 triangles. The fold runs from one corner to the opposite corner.
3. Shape C. Answer: 4 squares, small ones.
4. Shape D. Answer: 4 triangles, all meeting at the middle of the square.
5. Lay the 2 rectangles from Shape A end to end instead of side by side. Answer: a long rectangle, twice as long as the square.

Day 4: One shape, many ways to fill it

You need: pattern blocks (the hexagon, the half-hexagon, the diamond and the triangle), or paper copies of them; a hexagon traced on paper.

Ask, “Can you cover this hexagon with other blocks, with no gaps and nothing hanging over the edge?” When it is covered, ask, “How many pieces did you use? Can you cover it a different way?” Using fewer, bigger pieces or more, smaller pieces are both fine. Many children notice that two triangles fit exactly on one diamond; let them test it.

Practice. How many pieces fill each hexagon?

1. Hexagon A. Answer: 6 pieces, all triangles.
2. Hexagon B. Answer: 2 pieces, two halves.
3. Hexagon C. Answer: 3 pieces, all diamonds.
4. Hexagon D. Answer: 4 pieces: 1 half piece and 3 triangles.
5. Cover one half-hexagon piece with triangles only. Answer: 3 triangles, just as the top half of Hexagon D shows.

Exercises

1. How many small triangles make the big shape in the picture, and what shape is the big one?
2. Give the child 3 paper squares, all the same size. Ask, “Make a rectangle using all 3. Now can you make a square using all 3?”

Where this goes next

Shapes made of squares lead to cutting a rectangle into equal squares, and pieces that all fit the same shape lead to covering a shape with squares, the start of area. Halves cut different ways that are still equal become Equal shares that look different. Much later, a hard figure is split into easy pieces to find the area of a composite figure.