Gougu Rule or Pythagoras' Theorem

Session 1

1. Lead off with a question to the whole class.
   Why did Pythagoras kill 100 oxen?
   This might lead into a discussion of who Pythagoras was, when did he live, where did he live, what are oxen, and so on. Gradually reveal enough information to lead into the fact that he had just proved a theorem.
   What is a theorem?
2. We want to find out what Pythagoras’ Theorem is, how it can be justified, and what uses it has.
   Does anyone know what Pythagoras’ Theorem says? What objects does it deal with?
   If it looks as if someone knows all about the Theorem, then ask them to write it down on a piece of paper so that it can be looked at later. If no one does, then say that it has something to do with the lengths of the sides of a right angled triangle.
   Fine, so what is a right angled triangle?
   Revise the basic ideas, especially the word hypotenuse.
3. We are now going to collect some data so that we can conjecture the relationship between the side lengths of a right angled triangle. Show them a diagram. You may want to look at specific values of a, b, and h before you go to the general case.
   How could you collect this data?
4.  
   Lead them to the idea of drawing several triangles and measuring their sides. Help them to see that, by pooling their individual data, the class as a whole can collect a great deal of data even if each student only collects data from a few triangles.
   How could we do it systemically so that it will be easier to guess what will happen in the general case?
   Help them to see that they may get more insight into the problem by making small variations from triangle to triangle. So they might decide that this group of students should all start with a base length, a, of 3 but one student will    use b = 4 and 5, another student will use b = 6 and 7, and so on.
5. Now give them the chance to draw a couple of right angled triangles. This should be done as accurately as they are able to, so it is worthwhile for them to used rulers and compasses to construct their right angles. (Knowing how to do this construction will be assumed here. So they should have done it in a previous lesson. You might need to refresh their memory.) Tell them to be sure to measure the sides as accurately as possible.
6. Draw up a table on the board with all of the students’ results on it stating from smallest a and b upwards. Give them a chance to copy this table in their books. It might looks something like the one below.

6. This table seems very complicated. It may be difficult to see any pattern here at first glance.
   Can we say what patterns don’t hold? Is there a linear relation between a, b, and h?
7. It might be easier to see what happens if we compare situations where a and b are the same or similar.
   What do you have to multiply 3 by to get 4.2?
   What do you have to multiply 4 by to get 5.64?
   Is there a pattern here? Test it against other data on your table.
8. Try the same thing with 3 and 4, and 6 and 8, and 9 and 12. How does this connect to the last case where a and b were the same?
   Lead them to the well known:
                           h² = a² + b²    or    a² + b² = h².

9. What is the conjecture that we now have?
   Conjecture: If we have a right angled triangle with side lengths a, b, c, where c is the hypotenuse, then h² = a² + b². OR
   Conjecture: In a right angled triangle the square of the hypotenuse is equal to the sum of the squares on the other two sides. OR …
   Encourage them to say, and then write, the conjecture in as many different ways as they can.
10. Get them to test the Conjecture against various other values from the table. Because of rounding errors both in measurement and in calculation, they can’t expect to find that every piece of data fits exactly. However, the data should be a reasonable fit to the equation. The fit should be good enough to enable them to be confident that the equation is not too bad anyway.
11. It is possible that some piece of data doesn’t fit at all well. It might be worth checking the drawing and measurements for this case to see if there was an error here.
12. Finish the session by giving them time to write down the Conjecture and their comments on the Conjecture.

1. How can you make a right angle?
   They should recall how they made a right angle in the last session when they were making a right angled triangle.
   What if you wanted a right angle outside in the playground? What if you were marking out a soccer field.
   Let’s see how to tackle this problem.
2. Let the students work in pairs. Let them have a piece of string, a ruler, a pair of scissors, red ink, and a protractor. Leave them with the challenge of using only the pencil, the string (the scissors), drawing pen, red ink, and the ruler to make a right angle. (See Teachers’ Notes. Clearly some of this equipment is redundant.) Tell them they can check the accuracy of their right angle with the protractor.
3. Go round the class and check progress. Let them struggle with the problem for a while.
4. Have a reporting back session. If they can’t do the problem without help, discuss the problems that they are having and how these might be overcome.
5. Let the students work in pairs to implement one of the methods that have been discussed. Get them to check their angles with a protractor.
   Which of the various methods seem to be the most accurate?
   Is there a reason for this?
6. Give the students time to record their summary of the session.

This can be done by applying Pythagoras’ Theorem first to a right angled triangle formed by two sides and then to a right angled triangle formed by the last triangle’s hypotenuse and the height of the box. The answer that you get should be cm.

1. Problem: A spider wants to make a web in a shoe box with dimensions 30 cm by 20 cm by 20 cm. What is the shortest length of web she can string from one corner of the box to the opposite corner?
   Pose the problem. Say that it is probably a little hard to tackle at the moment so let’s work up to it.
2. Get the students to work in pairs to construct squares with side lengths 5 cm, 8 cm and 10 cm.
   Can you find the length of the diagonals of those squares?
   Is their another way to do this? Use it to check your first answer.
3. Have a reporting back session to check that everyone is on top of the problem. Now repeat step 2 using at least three rectangles. Specify whatever side lengths you think best.
4. Have a reporting back session. Now repeat step 2 asking them to find the heights (altitudes) of at least three equilateral triangles. Specify whatever side lengths you think best.
5. Now go back to the original problem. Show a model of the problem. Discuss ways that this might be tackled. Can you solve this problem by measuring?
   Let them solve the problem. You might let them work on constructing a box so that they can measure the diagonal, either in class or at home.
6. Have a reporting back session. Get them to write up their experiences.
7. If there is time, you might ask them to find the height of the point B above the line in the diagram below. Here the circles have a radius of 5 cm.
    

8. Remember there have to be two distinct ways of doing this. One is clearly measuring. One is not. Discuss their methods.

, so using the letters, . This gives a² = cs.

, so using the letters, . This gives b² = ct.

1. So far we really only have a Conjecture so we can’t fully believe it. Certainly it seems to give us the right answer every time we use it but in maths we need to be able to prove/justify everything before we can use it with confidence. So in this session we look at the proof of the Conjecture. This will enable us to believe that Pythagoras’ Theorem is true.
2. How can we prove something like this?
   Actually there are literally hundreds of proofs.
   Do you have any suggestions? (They might remember a proof from Pythagoras’ Theorem, Measurement, Level 5.)
3. Take them through the proof given in the Teacher Notes. This can be done by giving them specific examples of right angled triangles and getting them to show that the appropriate triangles are similar and that a calculation will show the required squares satisfy the conjecture. Then you might like to take them step by step through the proof that uses similar triangles. Ask them help you to explain why each step holds.
4. Let’s now, as they say, interrogate the Theorem.
   What are the key points of the Theorem statement? (right angled triangle; side lengths; sums of squares.) Each of the key points is needed in the statement.
   Will any other equation link a, b, and h?
   Can we get away without the right angle in the triangle?
5. Get the students to work their way through these two questions working in pairs. Can they find any other equation?
   Does a² + b² equal h² in any other triangle?
   This can be done by looking for other ways to link the lengths of the sides and by drawing other triangles where h is not a hypotenuse to see if the known equation holds.
   Let the students report back. It is not possible to find any other equation linking a, b, and h. If we don’t have a right angle in the triangle, then we don’t have
   a² + b² = h².
   This exercise shows that the Theorem has no fat in it. There are no pieces that can be thrown away.
6. Actually if there is no right angle we can still get an equation but it’s called the Cosine Rule.
7. Give the students time to write notes about what they have done in their note books.

But we can replace squares by semicircles. Note that, the area of a semi-circle on the hypotenuse = .

The sum of the other two semicircles is .

But this last expression is . By Pythagoras, this equals , which is just what we were trying to show.

1. Discuss the area nature of Pythagoras’ Theorem.
   Does the shape on each side have to be a square? Are there other shapes that could be used?
2. Send the class off in pairs to look at semi-circles. The Conjecture that they are pursuing may be “The area of the semi-circle on the hypotenuse of a right angled triangle is equal to the sum of the areas of the semi-circles on the other two sides”. Let them do this by first looking at specific examples. If the examples work they should then by try to prove it in general.
3. When the students report back, they should see that the Conjecture is true. The easiest way to prove this is to use Pythagoras’ Theorem (for squares).
4. Get them to go back into their pairs to look at whether the statement is true if we replace square by equilateral triangle, regular hexagon, and rectangle. They should know to experiment with particular examples first and then try to prove it in general. This proof will rely on the statement of Pythagoras’ Theorem for squares.
5. When the students report back, they should see that the Conjectures are true for regular shapes but not for the rectangle.
   Why is there a problem with the rectangle?
   Discuss.
6. Let the students write up their findings in their books.