Pentagon TrianglesTask 81 ... Years 4  10SummaryPentagon 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. 
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IcebergA 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. To find the quadrilateral in Question 1, use the two acute angled isosceles triangles ('sharp' ones) and match their long sides.
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.

Whole Class InvestigationTasks 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 openended 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 cutout triangles. 
Is it in Maths With Attitude?Maths With Attitude is a set of handson learning kits available from Years 310 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:
The Pentagon Triangles lesson is not part of any Maths With Attitude kit, but it can be used to enhance:
