Pythagoras 2

Task 189 ... Years 6 - 10

Summary

The 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 c2 = a2 + b2.
 

Materials

  • 3 sets of 4 right angle triangles

Content

  • algebra, generalisation in words & symbols
  • algebra, like & unlike terms
  • concept of proof
  • equations, creating
  • history of mathematics
  • measurement, area
  • multiplication, array concept
  • multiplication, calculations / times tables
  • numbers, square
  • Pythagoras theorem
  • transformations, translation
Pythagoras 2

Iceberg

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

The same transformation in each case 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 c2. In Diagram 2 it is two squares, one built on the side a, so this piece of uncovered space has an area of a2, and one built on the side b, so this piece of uncovered space has an area of b2.

DIAGRAM 1

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, c2 = a2 + b2

The three sets of triangles are provided to suggest that the same transformation will work for all all right angle triangles.

Extensions

  1. Use graph paper to make your own set of triangles to fit into the base board. Demonstrate that they also slide to demonstrate Pythagoras' Theorem.
  2. Use graph paper to make a different size square and then make your own set of triangles for this square and see if you can demonstrate Pythagoras' Theorem.
  3. Use the diagram, to guide an algebraic proof of Pythagoras' Theorem.
Using Diagram 1:
  • The area of the large green square is (a + b) (a + b).
  • The area of one triangle is Żab.
  • The area of four triangles is 4 x Żab = 2ab.
  • The area of the uncovered section is c2.
  • So, the area of the large green square is also c2 + 2ab.
We now have two ways of calculating the area of the large green square. The answers must be equal, so:
  • (a + b) (a + b) = c2 + 2ab
  • a2 + b2 + 2ab = c2 + 2ab
  • Therefore it must be true that a2 + b2 = c2
Why doesn't it work to use Diagram 2 to guide this proof?

More Extensions

  • Choose any two whole numbers p and q, p > q. Calculate p2 + q2, p2 - q2 and 2pq. Use the two smaller answers to draw the two shorter sides of a right angled triangle. What do you discover? Try another two whole numbers? Why is it so?
    HInt: Choose simple numbers for your first experiment.
  • What happens if we try to use triangles that are not right angled in the puzzle?
  • What happens if we start with a right angle triangle and build an equilateral triangle, semi-circle, regular hexagon ... on each side?
  • What happens if we start with a right angle triangle and build a cube on each side?

How does this creation relate to the Pythagoras challenge?

It looks pretty neat when the same creation is done in each corner.

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.
   

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.

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 ... ? ..., we can always start with one large square in the centre and split it into two smaller ones in the same way. Now let's see how we can use that.
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 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 Pythagoras 2 task is an integral part of:

  • MWA Chance & Measurement Years 9 & 10

Green Line
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