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### Why do this problem?

This
problem challenges understandings concerning area on the way to
working in a context which leads to non-linear sequences that need
explanation. Learners will need to use their visualising skills to
help them to create equilateral triangles and find their
areas.

### Possible approach

You may need isometric paper, which can be found

here.

Without speaking, draw equilateral triangles on the board,
starting as shown in the problem. Write the areas of the first two
or three triangles and place question marks next to the rest.

Allow time for reflection and discussion drawing out ideas
such as the use of non-standard units and the interesting result of
square numbers. You might want to spend some time asking learners
to try to explain why (see

Picturing Square Numbers ).

Present the idea of tilted triangles, discussing how this
might be defined before setting the challenge posed in the second
part of the problem. A good point to discuss is how we know the
triangles are equilateral - those who are convinced that the
triangles are equilateral could explain their reasoning to those
who aren't.

It is worthwhile giving the class some time to draw out the
diagrams and try to come up with their own methods for finding the
areas of the tilted triangles, and to share the methods that they
find, but if they are struggling to find an efficient way, the
pictures in the hints section might be useful as prompts.

Once a few areas have been found, encourage the learners to
make conjectures about the areas of much larger triangles with tilt
1, and to justify their ideas.

### Key Questions

How do you know the tilted triangles are equilateral?

How can you find the area of a tilted triangle in terms of the
unit equilateral triangle?

Can you find a generalisation for the area of a tilted
triangle?

### Possible Extension

Can you find a general rule for finding the area of any sized
triangle with any tilt?

Are there any areas that it's impossible to make with a tilted
triangle?

### Possible Support

Use the problem

Tilted
Squares first.

Focus on the justification of equilateral triangles and the
calculation of areas rather than seeking generalisations.