Finding tessellations provides a context for students to
practise working out interior angles of polygons and angles around
a point, or to introduce these ideas. In order to completely solve
the problem, students will need to work systematically, test out
ideas and reason carefully.

Introduce the interactivity to the class, and ask them to
consider which shapes you might choose if you wanted to tile an
area (with just one type of shape) without leaving any gaps or
overlaps.

For each suggestion, use the interactivity to determine
whether it does or doesn't tessellate.

Ask the class to think about why some shapes tessellate but
others do not. If the class have only suggested triangles, squares
or hexagons so far, ask them to speculate on what would happen with
pentagons.

Click on the hint for some
methods for calculating interior angles of regular polygons, if
your students haven't met this before.

Once it is established that the interior angle needs to be a
factor of $360^{\circ}$, challenge the class to construct a
convincing argument that there are only three regular
tessellations.

Next, introduce the concept of semi-regular
tessellations:

"What if you were allowed to use
more than one type of shape to tile your area?"

Ask the students for suggestions or show them one of these
examples:

Then introduce the notation describing the polygons around
each vertex ({3, 3, 3, 4, 4} and {3, 6, 3, 6} for the examples
above).

NB in a semi-regular
tessellation, each vertex must have exactly the same shapes in
exactly the same order (but can be clockwise or anticlockwise). For
example, with triangles and hexagons, if you put {3, 6, 6, 3} round
one point, you can't have {3, 6, 3, 6} round another.

Set students the challenge of finding semi-regular
tessellations.

Encourage students to use the interior angles to calculate
combinations of shapes which might tessellate. Once they have made
a prediction, they can test it using the interactivity or shapes
printed off and cut out. (3,
4, 5, 6, 8, 9, 10, 12 sided regular polygons to
print.)

It is very important that students try sufficient shapes to
verify that the tessellation will work - in this example, the three
shapes fit together around a point but the pattern cannot be
continued.

Once students have had a chance to find one or two
semi-regular tessellations, challenge them to find them all, and to provide convincing
evidence that they have got the complete set.

Teachers who participated in an NRICH workshop produced some posters suggesting how they might use the tessellation interactivity in a range of situations. See Enriching Classrooms, Inspiring Learning.

Why do some shapes fit together and others don't?

Can you tell in advance whether shapes will tessellate,
without using the interactivity?

Can you organise your work in such a way as to convince
someone you have found all the semi-regular tessellations?

Can students explain why some patterns work around a point but
don't tile the area? For example, the decagons and pentagons shown
above. Challenge students to find other similar examples (which
could use polygons with more than 12 sides, such as {18, 9,
3}).

Suggest students read Shaping up with
Tessellations.

Move on to three dimensions: Which Solids Can We
Make?

Ask students to draw and cut out a quadrilateral on a piece of
card. By drawing round their tile, ask them to show that their
quadrilateral tessellates. After repeating several times with
different quadrilaterals, ask them to comment on what they notice
about the four angles that meet at each point.