More Isometric Areas

Why do this problem?

This problem, along with Isometric Area, invites students to look at area in a slightly different way from usual, using a triangle as the basic unit of area rather than a square, to reinforce the concept that area is about the space enclosed within a shape.

Along the way students have the opportunity to derive and justify a formula that they will not have met before.

Possible approach

This problem follows on from Isometric Areas

This dotty grid environment might be useful, and you can print off isometric paper for students. "On squared paper, it's easy to work out area by counting squares, but on isometric paper it's a bit different. If we want to compare areas of shapes drawn on isometric paper, we can use a small triangle as our unit of area, instead of a square!"

Show students this slide with the image from the problem. "These triangles all have at least two sides that have a whole number length. For each triangle, can you work out its area, measured in triangular units? How does the area relate to the length of the two sides that are whole numbers?"

Give students some time to work in pairs on the problem, sharing any useful strategies and conjectures that they come up with. 

Finish off by bringing the class together to share their thoughts, and make the connection between the formula for the area of triangles in this problem, and the formula for the area of parallelograms in Isometric Areas.

Students may comment on the pairs of triangles with the same area, which are formed from splitting a parallelogram in half in two different ways (along each diagonal).

Key question

Can you see how you could join another identical (congruent) triangle to each triangle to form parallelograms?

How does the area of each triangle relate to the area of the parallelogram?

Possible support

Torn Shapes works with areas based on counting squares, so would be a useful task for students who are not confident with area.

Possible extension

Of All the Areas looks at the area of equilateral triangles with sides that are not whole numbers.