Pythagoras 2Task 189 ... Years 6  10SummaryThe task provides a classic demonstration of Pythagoras' Theorem which links nicely to transformation geometry. The triangle pieces only have to be translated and in doing so produce two versions of the same uncovered space. The first, one square. The second, the sum of two smaller squares. The geometric representation is there in squares and it can also be represented by the two sides of the classic Pythagorean equation c^{2} = a^{2} + b^{2}. That is, Pythagoras is about shapes called squares before it is about numbers called squares which count the area of those shapes. 
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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. 
For each set of triangles, the same transformation changes an uncovered space from one square to two squares. The amount of uncovered space doesn't change. It is simply redistributed by the translation of the triangles. In Diagram 1 it is a square built on the hypotenuse c, so the uncovered space is c^{2}. In Diagram 2 it is two squares, one built on the side a, so this piece of uncovered space has an area of a^{2}, and one built on the side b, so this piece of uncovered space has an area of b^{2}. Note: Both diagrams show pieces fitting inside a greenedged square referred to below, which is the base board in the task.
DIAGRAM 2
So, the area of the square built on the hypotenuse of a right angle triangle is equal to the sum of the areas of the squares built on the other two sides. Or, c^{2} = a^{2} + b^{2} The three sets of triangles are provided to suggest that the same transformation will work for all all right angle triangles. Extensions
(a + b) (a + b) = c^{2} + 2abWhy doesn't it work to use Diagram 2 to guide this proof? Could Diagram 2 guide a different version of a proof? More Extensions
How does this creation relate to the Pythagoras challenge? It looks pretty neat when the same creation is done in each corner.

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. 
In advance, make copies of a 15cm square. Take the task into the room and use it in a fish bowl situation giving three students, one at a time, the opportunity to set up the task and try the translations. Each student uses a different set of the triangles supplied with the task. This happens to be a 15cm square and I have copies of it here for each pair. There is also some graph paper. Your challenge is to make your own set of four triangles, different in size to any of these, that will behave like this inside the 15cm square. I will put this one on the front table. You can come and look at it if you need to.When students can demonstrate that their triangles work, ask them to choose their own size square and find a set of triangles that will produce the same effect. So, it doesn't seem to matter which size square we start with or which size right angle triangle we use, as long as ... ?Draw attention to the areas of the squares as above and develop an oral, then worded form, then algebraic form of Pythagoras' Theorem. Connect to the history of mathematics in various cultures as appropriate. Continue from here to the algebraic proof and any of the extensions above. Ensure that students are expected to record and explain the investigation in their own words. At this stage, Pythagoras 2 does not have a matching lesson on Maths300. However, Lesson 157, Pythagoras & Other Polygons, investigates what happens when shapes other than squares are constructed on the sides of a right angled triangle. 
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 Pythagoras 2 task is an integral part of:
