The Triangle Tiling

The problem

You have an unlimited supply of equilateral triangles, and you may choose their sizes freely: all the same size, or many different sizes, as you like. Your job is to tile a quadrilateral completely, with no gaps and no overlaps. Try a square, a rectangle, a rhombus, a parallelogram, a trapezium and a kite.

Which of these shapes can be tiled with equilateral triangles, and which refuse? What do the ones that refuse have in common?

Before you begin

Don't look for a formula. Cut a handful of triangles out of paper, or draw on a triangular dot grid, and simply try. Start with the square, and pay close attention to what happens in its corners.

1. Put a tile into one corner of the square. What angle does the tile bring to that corner, and how much angle does the corner have room for?
2. Every corner of every tile is $60^\circ$. So which corner angles can a shape have, if its corners are going to be filled?
3. The four angles of a quadrilateral add up to $360^\circ$. Which combinations of $60^\circ$ and $120^\circ$ are possible? Draw a shape for each combination and tile it.

Dissect the task

1. Where is the floor of this task (scissors and paper) and where is its ceiling (an argument about corners that settles the square for ever)? Is there anything above the ceiling?
2. While a class works on it, which behaviours would you watch for: who keeps trying the square, who first says the word corner, who moves from "this one didn't work" to "no shape with a right angle can ever work"?
3. This is a shapes problem. Which children in your class does it reach that a numbers problem would not?