Pythagoras 1

Task 175 ... Years 6 - 10

Summary

The task introduces Pythagoras' theorem through geometry checked by measurement. There are no As or Bs or Cs. Students are told Pythagoras' statement and then have to check it with the pieces. But that is only one example. They are then challenged to discover how the pieces have been created so they can create their own and check Pythagoras again.
 

Materials

  • 6 specially prepared shapes - 1 square, 1 triangle and 4 quadrilaterals

Content

  • concept of proof
  • history of mathematics
  • measurement, area
  • measurement, length
  • numbers, square
  • Pythagoras theorem
  • spatial perception, 2D or 3D
  • transformations, rotation
Pythagoras 1

Iceberg

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








   

This dissection proof of Pythagoras' Theorem is attributed to Henry Perigal who published it in 1891 when he was 90 years old. Perigal was a stockbroker by trade. For him mathematics was a hobby, but he was sufficiently serious about it to belong to the London Mathematical Society.

The spatial aspect of the problem is the challenge to make two squares, one with the small square and the four quadrilaterals and one with the just the four quadrilaterals. If students are having trouble with this part, help them identify that the quadrilaterals are congruent and that each contains two right angles. Right angles are what is needed to make squares, so perhaps in one of the squares the right angles formed by the longer sides make the corners of the square, while in the other square its the right angle formed by the shorter sides that makes the corners of the square. In either case, 'the other type of right angle' has to be 'hidden' inside somehow.

The Challenge invites students to try to recreate Perigal's dissection using a triangle they draw for themselves on grid paper. Little help is given because the intent is for students to discover the question:

Where do the cut lines go?
That's when the hint on the card about looking at the four piece square comes into play. However, it is important to look at those pieces in relation to the triangle. Students are fairly quick at answering this question with Through the middle of the square., but there are an infinite number of lines that go through the middle. Could it be that the required line is a special one and, if so, what is the criterion that defines it?

This is where putting the four piece square in its place on the triangle is necessary.

(Note: the triangle has to be oriented in a particular way to see the parallelism.)

It may take a while to see, but the critical observation is that the first cut line is parallel to the hypotenuse and the second is at right angles to that.

Now the students have what they need to try any right angle triangle. The card has been designed this way in the hope that in some way they have experienced something of the process Perigal must have gone through to develop the original proof.

It is worth taking time at this point to discuss the extent to which the 2 or 3 examples the students have tried constitute a proof and, perhaps, what would be needed to construct a full proof. Then, once the students have worked through the card, they should record what they have learned in their journal, in particular recording the statement of Pythagoras' Theorem as written on the card, with a diagram explaining it.

Extensions
It is likely that at some point the students have stacked the quadrilaterals, probably to check congruence. This can lead into a transformation investigation. It is possible to:

  • pick up the top three in the stack
  • rotate 90
  • slide and place the bottom one
  • rotate the remaining two another 90
  • slide place the bottom one
  • rotate the remaining one another 90
  • and slide and place it
to form the square.
Which square? The four piece one or the five piece one?

Either one actually depending on which way you rotate, but when you make the five piece one there is a hole in the middle into which the small square piece will exactly fit. Challenge the students to create both squares in this way and sketch how they do it.

Where is the centre of rotation in each case?
Another rotation challenge using these pieces is to begin with the five piece square and create the four piece square simply by rotating the quadrilaterals away from the small square. Give the students time to struggle with this one, then, if it seems like they need help, offer a hint from the sequence of photos above which indicates how it can be done.

You might also ask interested students to research Henry Perigal. They might find it interesting, for example, that he had a diagram of his proof engraved on his tombstone.

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.
   

With one copy of the task, grid paper, scissors and rulers it is easy to set up a whole class lesson which:

  • links to the history of mathematics by interacting with the work of a real mathematician,
  • introduces or revises Pythagoras' Theorem,
  • practises measurement skills.
Gather the students at a central table. Place the six task pieces on the table.
Some of you have used this task in other lessons. I would like you to show us all what you can remember about it.
You may have to question a little to draw out the construction but when it has been shown ask the students to try their own triangle.
Okay, so we can see this ... let's call it a jigsaw for now ... works for this particular right angled triangle. I would like you to draw your own right angle triangle on this graph paper and try it out. Make sure your triangle is a different size and shape from others around you.
As above, the question about where to put the cut lines will arise. Develop from here in a similar way to above until everyone has had success. Another question likely to arise in this exploration is:
How can I make sure my first cut line is parallel to the hypotenuse?
Encourage students to return to the actual task pieces to discuss and refine these points as necessary.

Note: It could be useful here to use software, if available, which allows students to create any shape right angle triangle and then at the click of button, draws the two smaller squares, the construction lines and rotates the pieces into the hypotenuse square. This would provide further evidence for the hypothesis that the demonstration works for every right angled triangle. Some students may be interested in designing a form of this software using software on their tablet devices.

Now we have shown that this 'jigsaw' works for many triangles. It's pretty good evidence, but not a complete proof, that it works for all right angle triangles. I would like you to work with a partner now to prepare the simplest statement you can that explains what is happening in this puzzle.
Work with the class's suggestions until they are refined to an agreed version which reflects the standard statement of Pythagoras' Theorem. Explain about Pythagoras as the recognised discoverer of this theorem (you might also acknowledge that his thinking was stimulated by knowledge shared by the Egyptians). Also explain about Perigal's discovery of this proof. Ask students to record Pythagoras' Theorem in their journal with an explanation of how Perigal demonstrates it.

Further work, depending on age level, might include:

  • Investigating why the cuts produce a square 'hole' when the pieces are transformed.
  • Examining ways to prove that Perigal's Proof works for all right angled triangles.
  • Asking what happens if the first cut line is not parallel to the hypotenuse?
  • Asking what happens if the triangle is not a right angled triangle?

At this stage, Pythagoras 1 does not have a matching lesson on Maths300.

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 1 task is an integral part of:

  • MWA Chance & Measurement Years 9 & 10

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