Tessellation, Shapes That Fit Together

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1 Getting Started
2 Watch and Notice
3 Try It Together
4 Fit the Shapes in Your Copy
5 Class Challenge
6 What Did We Notice?
7 What's Next
Pupil practice

1 - Getting Started ~4 mins

Illustration for Getting Started Look closely at these two photos. On the left, a tiled floor covers the whole surface with no space left over. On the right, five-sided paving stones have been laid, and little gaps have opened up between them.

Key point

Both are made of straight-edged shapes, so why does one leave no gaps while the other does?

Teacher Notes

Take two or three hands-up answers, not open call-outs. Listen for pupils noticing the square tiles meet edge to edge and leave no space, while the five-sided stones do not close up. Hold the word tessellation back for now, it lands in the next beat. Keep the pentagon puzzle open, the class returns to it in the closing maths-talk.

👇 Mark your steps as done as you complete them.

+5 XP

2 - Watch and Notice ~9 mins

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.

Teacher Notes

Squares — trace a finger along where four corners meet; nothing left over.
Hexagons — this is the honeycomb; ask what animal builds like this.
Circles — hold out for the class to spot the little gaps at the corners; a round edge can never sit flat against another round edge.

Then reach back to the Getting Started photo: the circles are round, so it is easy to see why they fail. But the five-sided stones had straight edges and still left gaps. Keep that puzzle in mind, we solve it at the end.

👇 Mark your steps as done as you complete them.

+10 XP

3 - Try It Together ~10 mins

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.

Key point

Before each new shape, everyone call out a prediction: will this one fit with no gaps? There is no wrong answer, it is just your best guess before we test it.

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.

Fit the copies edge to edge

Teacher Notes

Talk this through together, pupils take turns on the interactive and the class agrees or corrects out loud.

Interactive (explore): squares and triangles fill with no gaps. Hexagons need the honeycomb offset. Before each shape: predict, will this one fit with no gaps? Whole-class call-out.

Worked pentagon demo (board, you lead, one full pass):

Sketch one regular pentagon, then a second copy flush on one shared edge (straight edges meet, no gap).
At one shared corner, try to seat a third pentagon. Three corners meet but leave a clear wedge open; a fourth will not fit.
State the test result with the class: edges can meet flush, but corners do not close round the point, so a regular pentagon does not tessellate.

Slip to watch: pupils may say "straight edges mean it tiles." Revoice: both flush edges AND corners closing round a point are needed.

If a pupil notices hexagons leave triangle-shaped spaces, flag it for the Class Challenge two-shape round.

+15 XP

4 - Fit the Shapes in Your Copy ~3 mins

COPYBOOK MOMENT

In your maths copy, draw four squares fitting together with no gaps, so they make one bigger square. Then try to draw four regular hexagons fitting together the same way.

Teacher Notes

Walk the room glancing for edges that meet with no space — this is whole-class copybook practice, not marking. Some pupils will find the hexagons harder to draw; that noticing is part of the point.

+10 XP

5 - Class Challenge ~9 mins

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.

Tile the frame with no gaps

Teacher Notes

This is the practice round — pupils take turns at the board, check each answer, and the class confirms before moving on. Keep the board work brisk rather than over-explaining.

The two-shape rounds are the harder ones: pupils must rotate and interlock the shapes so no gap appears. On the last round, work it together first so pupils see how a ring of triangles fills the space around each hexagon, then let a pupil fill the outline with the same pair. Ask the pupil at the board to point out the join where the two shapes meet.

+15 XP

Pupil practice

Module 8 · Symmetry, Location and Transformation Algebra

Lesson 96 · Tessellation, Shapes That Fit Together

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This lesson is copyright of Coding Ireland 2017 - 2025. Unauthorised use, copying or distribution is not allowed.

Teacher Class Feed

1 - Getting Started ~4 mins

2 - Watch and Notice ~9 mins

3 - Try It Together ~10 mins

Fit the copies edge to edge

4 - Fit the Shapes in Your Copy ~3 mins

5 - Class Challenge ~9 mins

Tile the frame with no gaps

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