Gougu Rule or Pythagoras' Theorem

Purpose

This unit introduces Pythagoras’ Theorem by getting the student to see the pattern linking the length of the hypotenuse of a right angled triangle and the lengths of the other two sides. Applications of the Theorem are considered, and students see that the Theorem only covers triangles that are right angled.

Achievement Objectives
GM5-10: Apply trigonometric ratios and Pythagoras' theorem in two dimensions.
Specific Learning Outcomes
  • find lengths of objects using Pythagoras’ Theorem
  • understand how similar triangles can be used to prove Pythagoras’ Theorem
  • understand that Pythagoras’ Theorem can be thought of in terms of areas on the sides of the triangle
Description of Mathematics

Against the background of Pythagoras’ Theorem, this unit explores two themes that run at two different levels. At one level this unit is about Pythagoras’ Theorem, its proof and its applications. At another level, the unit is using the Theorem as a case study in the development of mathematics. So, after some experimentation, we try to guess what the Theorem is and so produce a Conjecture. Then we test the Conjecture in a number of situations. We then prove the Conjecture and then check the Theorem to see if it applies to triangles other than right angled ones in attempt to extend or generalise the result.

Learning to ‘interrogate’ a piece of mathematics the way that we do here is a valuable skill of life long learning. By this we mean that it should be read and checked by looking at examples. It should also be applied to a new situation. The conditions of the Theorem should then be changed slightly to see what effect that has on the truth of the result. This process will help students to look at any piece of new mathematics, in a text book say, and have the confidence that they can find out what the mathematics is and how to apply it. It also provides a deeper understanding of what the result says and how it may connect with other material.

The title of the unit, the Gougu Rule, is the name that is used by the Chinese for what we know as Pythagoras’ Theorem.

Before doing this unit it is going to be useful for your students to have worked on the Construction unit, Level 5 and have met and used similar triangles.

Required Resource Materials

String

Scissors

Protractors

Compasses

Red ink

Pencils

Rulers

Drawing pins

Activity

Session 1

In this session the students are given the opportunity to experiment with right angled triangles and to conjecture a relation between the lengths of the various sides.
 
Teacher Notes
Pythagoras was a Greek mathematician who lived from about 569 BC to 475 BC. You can find more details about him on the St. Andrew’s www-groups.dcs.st-and.ac.uk/~history/Mathematicians. Actually this is a good site to find out about many famous mathematicians and we recommend that you encourage your students to use it. If you asked them to write a series of posters on Greek mathematicians, say, for display in the maths classroom, the St. Andrew’s site would be a good source.
One story says that Pythagoras killed 100 oxen when he discovered the proof of the Theorem, though the truth of this is hard to establish.
We are advocating a ‘play’ approach to this unit for at least three reasons. First it helps them to get a ‘feel’ for the main Theorem and secondly it will help them remember the result better. Finally, this is the way that mathematicians approach new situations.
In this unit they will be asked to construct a number of right angles triangles. To save time on their constructions they could draw several triangles with the one base size as we have done below.

diagram.

We use a, b, h here because h is suggestive of ‘hypotenuse’ but this does not mean that you can’t use a, b and c or whatever you are used to.
Teaching sequence
  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.  triangle.
    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.

 

a
b
h
3.0
3.0
4.2
3.0
4.0
5.0
3.0
5.0
5.78
4.0
4.0
5.64

 

  1. 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?
  2. 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.
  3. 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.
  1. 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.
  2. 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.
  3. 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.
  4. Finish the session by giving them time to write down the Conjecture and their comments on the Conjecture.
Session 2
In this session we use string as a practical way to make a right angle.
Teacher Notes
Here’s how to make a ‘string right angle’ – you might want to give them a bit of red ink to mark the lengths or you could do this by tying knots at the appropriate parts of the sting.
(i) Use a string line whose length is a convenient multiple of 12 cm; mark a point on the string the same multiple of 4cm from one end and another point the multiple of 5cm from the other end on the string.
(ii) Pin down both ends of the string at the same point. Then pull the string taut so that the string makes a 3, 4, 5 triangle. You can then see where the right angle is.
This exercise can be done on a tennis court with chalk. Get them to make a square; then check by measuring the diagonal; then let them improve their first effort. Get them to be as accurate as they can.
Of course you don’t have to use 3, 4, 5. The numbers 5, 12, 13 will work just as well! Some students might want to try this. Which method is more accurate and why? Maybe the bigger number leads to a more accurate right angle?
If they come up with another way to find a right angle, please let us know.
Teaching sequence
  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.
Session 3
In this session we use the Conjecture to find the lengths of a variety of geometric entities.
Teacher Notes
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?
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 equation.cm.
Of course this problem could be solved by direct measurement if you first construct an appropriate box. You can do this by making a net. Remove squares of side length 20 cm from a 70 cm by 60 cm rectangle. If this is too big, then scale the box to fit the rectangular paper that you have or change the dimensions of the box.
To construct a square, make a line of the appropriate length. Then construct a right angle at both ends. Mark off the length of the first side on each of the vertical lines. Join the two points that have just been marked.
They may not be able to see how to do this without some help.
The diagonals can be found by direct measurement or by using Pythagoras’ Theorem.
Don’t waste time making a new square each time. Use the idea from the Teacher Notes of session 1 to make squares that sit inside larger ones (as in the diagram).

diagram.

A rectangle can be constructed in a similar way to a square.
The equilateral triangle is easier to construct. Use a radius the same length as the side of the triangle. Then put the point of the compasses on one end of the first side and draw an arc. Repeat this at the other end of the first side. Where the two arcs cross is the third vertex of the triangle.
The height of the triangle can be found by Pythagoras or by constructing the perpendicular bisector of one of the sides and measuring.
To solve the three circle problem, notice that the centres of the circles form an equilateral triangle. So the height of B is two radii plus the height of this triangle above the line.
Teaching sequence
  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.
     
diagram.
  1. Remember there have to be two distinct ways of doing this. One is clearly measuring. One is not. Discuss their methods.
Session 4
In this session we prove Pythagoras’ Theorem. Then we check that there are no unused hypotheses in the result by looking at non-right angled triangles to see if the Theorem still holds.
Teacher Notes
There are an enormous number of proofs of Pythagoras’ Theorem. We have several on this site alone (see Pythagoras’ Theorem). Here we give another one that uses the concept of similar triangles.
Look at the diagram below. First we see that triangle ABD is similar to triangle ACB (just check the common angle and the right angle). From this we have that
equation., so using the letters, equation.. This gives a2 = cs.
Similarly, triangle CBD is similar to triangle CAB (just check the common angle and the right angle). From this we have that
equation., so using the letters, equation.. This gives b2 = ct.
Combining these two we get a2 + b2 = cs + ct = c(s + t). But s + t = c, so
a2 + b2 = c2.

 triangle.

Teaching sequence
  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.
Session 5 (extension)
Teacher Notes
One way to look at the equation a2 + b2 = h2 is to think of it as saying that the area of the square on the side of length a plus the area of the square on the side of length b, add up to the area of the square on the hypotenuse, h. This can be seen in the diagram below.
Actually, many proofs of Pythagoras’ Theorem rely on this idea.

diagram.

But we can replace squares by semicircles. Note that, the area of a semi-circle on the hypotenuse = equation..
The sum of the other two semicircles is equation..
But this last expression is equation.. By Pythagoras, this equals equation., which is just what we were trying to show.
Note that we can replace ‘square’ by any other figure provided that the figure is completed as soon as we know one side. A square is completely known when one side is known. This is also true for semicircle, equilateral triangle and any regular polygon. It is not true for rectangles since there are many rectangles with one side of given length.
This area idea gives another dimension to the Theorem and hopefully helps students to remember it more easily.
Teaching sequence
  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.

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