Ejemplos de aplicaciones

Figure 6.1e (i) is based on a right-angled triangle. Describe to yourself what you see. Try to work out how to construct it.

Figure 6.1e (ii) uses the vertex of the right angle and the point on the hypotenuse as diameter for yet another circle. Convince yourself, and someone else, that the new circle is necessarily tangential to the hypotenuse.

Which looks larger in each case, the area shaded light grey or the area shaded dark grey?

Comment

The largest semicircle has the hypotenuse of the triangle as diameter. Two circles have as diameters the other two sides of the triangle and these circles intersect twice, once on the hypotenuse. The two points of intersection are two end points of an altitude of the right- angled triangle. It is possible to use Pythagoras’ theorem with semicircles to show that, in fact, the shaded areas are the same area in each case.

(i) (ii) Figure 6.1e

6.2 LINKING TWO AND THREE DIMENSIONS

The general problem to be investigated in this section is to find the shortest distance between two points on a surface. Many three-dimensional objects can be thought of as made up of a number of two-dimensional faces. This means that the surface of a solid can be formed by a net of joined planar regions that are cut out, folded up and glued along edges to make the ‘solid’ shape.This way of looking at a solid helps with the shortest path question.The first thing to do is to gain familiarity with nets.

Learners who have not previously encountered nets of surfaces gain a great deal by being challenged to start from a surface and construct a net, and vice versa. Polygonal shapes such as ATM mats (ATM webref), Polydron and so on are ideal for this pur- pose. Some learners will benefit from working physically, others will benefit from trying to do it mentally. It is through manipulation of familiar objects that people get a sense of relationships and properties that they then can articulate, reaching a point, for example, where they can describe in words how to work out a net for a surface.

One of the many uses of nets is in finding shortest distances on surfaces. As a spe- cial case, billiard and snooker players can use an imagined ‘net’ made of copies of the table to work out how to do banked shots by aiming along straight lines.

98 BLOCK 2

Reflection 6.1

Most of this section is about areas of parts of circles. The underpinning pedagogy used here is to ask you to work on harder problems while the underpinning formula for the area of a circle is still relatively new. Has this been a useful strategy for you?

Task 6.2.1 Netted

Which of the nets in Figure 6.2a can fold to make a cube? How many different nets can you find for a cube? For a cuboid? What shape do you get when

you fold up the net in Figure 6.2b? If you make three identical ones, you will find that they fit together.

Comment

The principal value in such a task is to develop powers of visualisation, concentration, dis- cernment and mental agility. Did you take time to allow yourself to manipulate the nets mentally to see if they would make a cube?

Notice the difference in effect of the two types of task: ‘which of these has … ?’ and ‘in how many ways can you … ?’. In the first you seed an idea, while in the other you set a challenge. Learners constructing their own nets can challenge each other as to whether a diagram really is a net.

Imagining different nets for surfaces you encounter, such as prisms (including cylinders) and pyramids, exercises your powers to imagine creatively. Note that a net for a cylinder is forced to have a disc attached ‘at a point’ to a rectangle.

Shortest Paths

Imagine a piece of paper with two distinct points A and B marked on it. Now imag- ine scrunching up the paper in some way. How could you find the shortest path which stays on the surface of the paper between A and B?

At first it seems as though you might be able to make use of the scrunching in some way, but that is due to interference between shortest distance through space, and shortest distance staying on the paper. If you flatten the paper out again you do not change the length of the path. Draw the straight line between A and B, and then re- scrunch the paper. The line segment still gives the shortest distance staying on the paper, as distinct from moving through space to get from A to B.The next task looks at shortest paths on surfaces.

There are other dimensions of possible variation in this task, even when you restrict your attention to cuboids. For example, points A and B might not be at vertices.The variation of just one dimension in the three cuboids of Figure 6.2c might suggest a further task of finding a rule of thumb for deciding which route is the shortest for any cuboid when the ant goes between diagonally opposite corners.Would it be pos- sible for a shortest path on the surface of a cuboid to involve travel on more than three faces?

It seems a short step to move from cuboids to cylinders, since both can be seen as examples of prisms.

A cylinder can be thought of as a prism with a circular cross-section rather than a polygonal one.

MEASURING AREAS AND VOLUMES 99