What it shows
The seven tangram pieces are cut from one square. The two large triangles are each one quarter of the square, the medium triangle, the square and the parallelogram are each one eighth, and the two small triangles are each one sixteenth, so the fractions add up to one whole.
The science
Cutting a square along its diagonals and midpoints makes pieces whose areas are simple fractions of the whole. If the square is 16 cm by 16 cm its area is 256 square cm; each large triangle is 64, the medium triangle, square and parallelogram are 32 each, and each small triangle is 16. Adding 64 + 64 + 32 + 32 + 32 + 16 + 16 gives 256, so no area is lost when the pieces are rearranged into any picture. Shapes made from the same pieces always have the same area even if their perimeters differ.
Materials
Cost: free. Many items come from home or from waste; spare parts such as LEDs, motors and wire are in the store.
- A square of thick card or foam sheet 16 cm by 16 cm
- A ruler and pencil
- Scissors or a cutter
- Coloured paper or paint for the seven pieces
- A chart for the fraction table
Safety
For this project: An adult must use the cutter if the card is thick.
Never use mains electricity: power models with cells, batteries or a low-voltage adapter. Sharp and hot tools, flames and lab chemicals need an adult or your teacher beside you. Keep magnets, button cells and small parts away from young children, and wash your hands after making.
Step by step
7 steps. Do them in order, and test each part before you go on to the next. Write what you notice in a notebook as you go.
Draw the square and its diagonal from the top left to the bottom right corner.
Mark the midpoints of the top and right sides and join them; this line is parallel to the diagonal.
Join the midpoint of that line to the centre of the square. From that midpoint draw a line parallel to the top side across to the diagonal, and from the right-side midpoint draw a line parallel to the line you just joined, also across to the diagonal. Finally join the bottom-left corner to the centre of the square. This makes the standard seven pieces: five triangles, one square and one parallelogram.
Cut the pieces out and colour each one. Measure the sides of every piece and work out its area.
Make a table: piece, area in square cm, fraction of the square. Check that the fractions add to 1.
Rearrange the pieces into a rectangle, a triangle and a cat, and measure the perimeter of each with string.
Display the solved square beside the table and let visitors try to rebuild it from the loose pieces.
What you should see
The table shows areas 64, 64, 32, 32, 32, 16 and 16 square cm adding to 256, and every picture made from the pieces keeps the same area while its perimeter changes.
Viva questions
A teacher or a judge will ask how it works. Try to answer each question before you open it, then compare.
01What fraction of the square is one small triangle?
One sixteenth.
02Which pieces have the same area as the small square?
The medium triangle and the parallelogram, each one eighth.
03Does rearranging the pieces change the total area?
No. Area is conserved; only the perimeter changes.
04How many triangles are in a tangram?
Five: two large, one medium and two small.
Take it further
Cut a second set from a 32 cm square and show that doubling the side makes every area four times as big.
Change one thing, measure what happens, and add a table or chart of your readings: that is what turns a model into your own investigation.
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