Look at these two surfaces. On one, a tiled floor covers the whole surface with no space left over. On the other, five-sided paving stones have been laid, and little gaps have opened up between them.

Here are three shapes, each one copied again and again across the board. Watch where the copies meet, and check for any gaps or overlaps. First squares, then hexagons.

Now watch circles, and check for any gaps or overlaps.

Today we test which shapes tessellate. First we use the interactive together: squares, then equilateral triangles (all three sides the same length), then regular hexagons. Some of you will come up to fit the copies while the rest watch and agree or correct out loud. Fit each copy edge to edge and check for any gap or overlap before we decide whether it tiles.

Then we test a regular pentagon together on the board (five equal sides). Watch the full method once: fit copies edge to edge, then check whether the corners close all the way round a point with nothing left over. Decide together whether the corners close with nothing left over and whether a regular pentagon tessellates.

Today we work through these tilings together: first fill the frame with squares, then with triangles, then combine triangles and squares into one repeating tiling. For the last round we first work together to see how triangles and hexagons interlock, then you fill an outline using triangles and hexagons with no gaps left anywhere.