Folding a Square in Half

Take a square sheet of paper and fold it so that the shape produced is half the area of the original square. How many different shapes can you make that have this property? Can one of your final shapes also be a square? Can it be a parallelogram?

Comment

You will quickly find one or two simple ways to do this. In order to find more possibilities you could try combining some of the ‘basic’ methods you have found. (A useful reference is Franco, 1999).Think about how you can be sure that your final shape is indeed half the area of the original.

The comment about comparing areas may have reminded you of Task 1.4.1 from Chapter 1.There, the focus was on whether the folded parts of the square overlapped or left some uncovered space. Clearly, if they fit together exactly (without overlap or space) then you can be sure that the final figure is precisely one-half. However, you may find that you have produced some overlap and some uncovered space.Then you have to convince yourself that the amount of overlap is equal to the amount of space. This may not always be easy. Another approach is to consider the area outside the new shape formed (that is, imagining the sides of the original square are still present).

For example, suppose the quadrilateral ABCD in Figure 2.2f is formed by folding as shown on the left.The two shaded triangles in the middle diagram illustrate the over- lap and the uncovered areas. Take a moment to see why they must be equal in area. The third diagram illustrates the area outside ABCD.Take a moment to see why they, too, must account for half the area.

2.3 SIMILARITY AND RATIO

In many of the tasks in Chapter 1 you encountered similar shapes. In most cases the similarity was ‘perceptually obvious’ but, arising from Task 1.2.3, you were asked to determine whether two rectangles were similar in a situation where it was not obvi- ous. What test did you apply? An intuitive description of similarity is that the shapes are identical (which implies corresponding angles are equal) but their sizes may be different. Another way to put this is to use a transformation term and say that one is an enlargement of the other.The implication of the first description is that the ratio of any pair of lengths within one shape will be the same as the corresponding pair in the other shape. The implication of the second description is that the ratio of a corre- sponding pair of lengths between the two shapes will be the same as any other pair of corresponding lengths between the shapes.

Mathematically, these are in fact equivalent, but they represent two different ways of look- ing at similarity. For example, consider the two rectangles shown in Figure 2.3a.

Looking at the ratios within the rectangles,

the ratio of base: height for the smaller rectangle is 3:1 and for the larger one is 6:2 = 3:1. Looking at the ratios between corresponding sides of the rectangles, the heights are in the 28 BLOCK 1

Figure 2.2f

Reflection 2.2

What is the same and what is different about thinking and reasoning about the properties of shapes from a ‘conditions for uniqueness’ point of view, and a ‘paper folding’ point of view?

ratio 2:1 and the bases are in the ratio 6:3 = 2:1. If one of these lengths were unknown, but the rectangles were known to be similar, then either approach could be used to cal- culate the missing length.

Before considering ratios arising in triangles, here are some contexts that depend on ratios. Hans Freudenthal (1991) and his colleagues developed a very popular mini-project for mathematics classrooms in which the learners encounter a giant footprint 1.2 m long. They are then asked to find out all they can about the giant from this footprint.

This is a very open-ended task and in a classroom is likely to generate plenty of discussion about what kind of things can be determined and what kinds of assump- tions need to be made. It provides access to a wide range of ideas and concepts, especially once the giant starts leaving messages for the children!

Clearly there are qualitative aspects of the giant about which nothing can be deduced, for example, colour of hair, whether the giant is fat or thin, and so on. To make quantitative estimates such as height, weight, hand-span or finger length, you have to make the assumption that the giant’s body is similar in shape to an average human. This immediately raises questions about ‘average’ and, of course, a natural place for learners to start is with themselves. Here again, you can see the two differ- ent approaches concerning similarity that were identified earlier.You might measure your own footprint and compare it to your own height, then use this to estimate the giant’s height.This uses internal ratios of similar figures.You could also find the ratio of the giant’s footprint to your own and then multiply your own height by this value. This uses external ratios between the similar figures. Further mathematical issues develop when estimating the giant’s weight, food intake, and so on. Another important concept about similar figures will emerge if you try to find the weight of the giant.

Consider now a real-life situation. When a film is shown on television a problem immediately arises because, although television screens and film screens are both rec- tangular, in fact they are not similar. The standard television screen has a ratio 4:3 (approximately 1.33:1), and a wide-screen television has a ratio of 16:9 (approxi- mately 1.78:1). Film ratios vary considerably. Many films today have a 1.85:1 ratio but the Cinemascope ratio is about 2.35:1.The problem is, if the television and film ratios are different, how can a film be shown on the television screen? There are two ‘solu- tions’ to the problem.The first is to show the film in ‘letterbox’ format so that the film is seen in its original ratio but black strips appear at top and bottom of the television screen. So there is an unused part of the television screen.The second is to ‘chop’ the film so that it fits the television screen exactly. But this also results in unseen parts of the film (see Figure 2.3b).

LANGUAGE AND POINTS OF VIEW 29

Since the problem is concerned with the total unused or unseen in each case, where that portion is placed on the screen is irrelevant. It is easier, therefore, to do the calcu- lations by shifting all the wastage to one part of the diagrams; that is, all the unused television screen at the top (or bottom) of the screen and all the unseen film at the right (or left) of the film.You may be surprised at the result.The result could just be a coincidence that arises from the particular ratios used. But mathematicians are very suspicious of coincidences! This is why it is worth investigating other cases and trying to come to a general conclusion. Most people do find the general result here quite surprising. And even if they can confirm it by say, using algebra (for example, taking the television ratio to be a:b and the film ratio c:d ) it is still difficult to see why it is true. This is the difference between a proof that logically confirms a hypothesis and one that provides explanation. As Michael De Villiers (1990) puts it, explanation aims to offer understanding and insight into why something is true, whereas a proof may be content with verifiable logical steps that give no insight.

As with the television and film problem, you may be surprised by the answer to the next task.

You may have had the feeling that the relationship should be obvious just by looking at the diagram. However, if you solved the problem by drawing in some radii and then perhaps using Pythagoras’ theorem, you may be convinced by the logic of the argument without appreciating geometrically why the relationship holds. An insight appears when you change your viewpoint from seeing the inner square as initially 30 BLOCK 1