Gougo rule or Pythagoras theorem

In this session, the students are given the opportunity to experiment with right-angled triangles and to conjecture a relationship 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 online. 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 advocate a ‘play’ approach to this unit for at least three reasons. First, it helps students get a ‘feel’ for the main theorem, and second, it will help students remember the result better. Finally, this is the way that mathematicians often approach new situations.

In this unit, students will be asked to construct a number of right-angled triangles. To save time on their construction, they could draw several triangles with the same base size, as we have done below.

We use a, b, and h here as symbols 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.

Pose this question to the whole class.

• Why did Pythagoras kill 100 oxen?

This might lead to a discussion of who Pythagoras was, when he lived, where he lived, what oxen were, and so on. Gradually reveal enough information to lead to the fact that he had just proved a theorem. 

2.

We want to find out what Pythagoras’ Theorem is, how it can be justified, and what uses it has.

• What is a theorem?
• Does anyone know what Pythagoras’ Theorem says?
• What objects does it deal with? 

3.

Encourage students to share their knowledge and tell them that the theorem has "something to do with the lengths of the sides of a right-angled triangle". 

4.

Revise the basic ideas, especially the word hypotenuse and the concept of a right-angle triangle. 

5.

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 students a diagram like the one below. 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?

6.

Lead students 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.
 

7.

Now give students 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 use rulers and compasses to construct their right angles. Ensure your students understand how to construct a right-angled triangle. You might model the processes involved initially and scaffold students to construct them independently. Emphasise that students need to measure the sides as accurately as possible.
 

8.

Construct 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 look something like the one below.

9.

Discuss the numbers in the table:
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 relationship between a, b, and h?

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.
 

10.

• Try the same thing with 3 and 4, 6 and 8, and 9 and 12. How does this connect to the last case where a and b were the same?

Lead students to the well-known formula:

                        h² = a² + b²    or    a² + b² = h².
 

11.

• What is the conjecture that we now have?

Conjecture: If we have a right-angled triangle with side lengths a, b, and 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 students to say, and then write, the conjecture in as many different ways as they can.
 

12.

Have students test the conjecture against various other values from the table. Because of rounding errors in measurement and 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, meaning it should enable them to be confident that the equation is not too bad anyway. It is possible that some pieces of data don’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.
 

13.

Finish the session by giving them time to write down the conjecture and their comments on it.

In this session, we use string as a practical way to make the right angle.

Teacher notes

The method for making a ‘string right angle’ is as follows:

Note: You might want to give students a bit of red ink to mark the lengths, or you could do this by tying knots at the appropriate parts of the sting.

1.

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.

2.

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 also be done on a tennis court with chalk. Get students to make a square; check by measuring the diagonal; then improve their first effort. Emphasise being as accurate as possible. Of course, you don’t have to use 3, 4, or 5. The numbers 5, 12, and 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 the right angle, please let us know.

Teaching sequence

1.

How can you make the right angle?

Students should recall how they made a right angle in the last session (i.e., by making a right-angled triangle).

• What if you wanted a right angle outside in the school yard?
• 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), the 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 around the class and check progress. Let them struggle with the problem for a while.
 

4.

Have a reporting-back session. If they can’t solve the problem without help, discuss the problems that they are having and how they 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.

In this session, we use the conjecture to find the lengths of a variety of geometric entities.

A rectangle can be constructed in a similar way as 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

8.

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

Teacher notes

There are an enormous number of proofs of Pythagoras’ Theorem. We have several on this site alone (see the unit 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:

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.

so using the letters,

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

so using the letters,

This gives b² = ct.

Combining these two we get a² + b² = cs + ct = c(s + t). But s + t = c, so

a² + b² = c².

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 or 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 to help you 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 in pairs. 

• Can they find any other equations?
• 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.

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.

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

The sum of the other two semi-circles is

But this last expression is

By Pythagoras, this equals

, 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 a semicircle, an 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 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 that “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 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 rectangles.

• Why is there a problem with the rectangle?

Discuss. 

6.

Let the students write up their findings in their books.