# Pentagon Triangles

### Task 81 ... Years 4 - 10

#### Summary

Pentagon Triangles were designed by Geoff Giles, a well known Scottish maths educator. The mathematics that derives from this pair of isosceles triangles is amazing. Students begin by exploring shapes and soon find themselves building a visual pattern. Wherever there is a visual pattern there will be a number pattern too and in this case it is the Fibonacci Numbers which appear. Beyond the task there are possibilities for exploring the angles in each triangle and the Golden Ratio.

#### Materials

• 10 each of 2 types of isosceles triangle; two colours of in each set

#### Content

• language and properties of regular polygons
• 2D spatial perception
• properties of isosceles triangles
• Fibonacci Sequence & the Golden Ratio
• angle sum of a triangle
• properties of similar triangles #### Iceberg

A task is the tip of a learning iceberg. There is always more to a task than is recorded on the card.

Question 2 shows us why these triangles are called Pentagon Triangles. It also suggests how we can calculate all the angles in the triangles if we know the angles of a regular polygon. Further, it shows that there are only two different side lengths involved.

To find the quadrilateral in Question 1, use the two acute angled isosceles triangles ('sharp' ones) and match their long sides.

• What name would you give this shape?
• How many quadrilaterals can you make with two triangles?
• What names would you give them?
• How many quadrilaterals could you make if you can use any number of the triangles?
• Are any of them similar to each other?
Create your own shapes with Pentagon Triangles. Photograph them to add to the class display board.

In Question 6, it doesn't matter which triangle you make they all seem to be either a sharp or blunt isosceles triangle the same shape as the originals.

• How can you convince me that all these triangles are the same shape as one or other of the originals?
Question 6 has left out some numbers.
• Can you make a triangle from 4 triangles? 6 triangles? 7 triangles? 9 triangles? ...
What happens if we start with a regular heptagon (7 sides) and divide it into triangles from one of the vertices?

#### Whole Class Investigation

Tasks are an invitation for two students to work like a mathematician. Tasks can also be modified to become whole class investigations which model how a mathematician works.

The whole class lesson based on this task requires sets of pentagon triangles. You can design and print these in a computer drawing program and the students can make their own sets from copies you make on card. However, these are less satisfying to explore than the foam Plastazote ones. Find out here about class sets of Pentagon Triangles.

Either way, experience suggests there is almost no way to prevent this becoming an open-ended investigation. Students seem to get hooked on their own questions as they play. Following the initial question (say Question 1 on the card), teachers find it useful to intermittently add a new challenge question to a growing list on the board. Sometimes these come from students. The artfulness in the lesson (lesson sequence) is to choose the appropriate time to work with each group, or to leave the students alone, or to draw the whole class together to explore a particular piece of mathematics.

For more ideas and discussion about this investigation, open a new browser tab (or page) and visit Maths300 Lesson 134, Pentagon Triangles, which includes cut-out triangles.

#### Is it in Maths With Attitude?

Maths With Attitude is a set of hands-on learning kits available from Years 3-10 which structure the use of tasks and whole class investigations into a week by week planner.

The Pentagon Triangles task is an integral part of:

• MWA Space & Logic Years 9 & 10

The Pentagon Triangles lesson is not part of any Maths With Attitude kit, but it can be used to enhance:

• MWA Space & Logic Years 3 & 4
• MWA Space & Logic Years 9 & 10 