MATHEMATICS

Grade 9

QUADRILATERALS, PERSPECTIVE DRAWING,TRANSFORMATIONS

Module 25

UNDERSTAND AND USE THE PRINCIPLE OF TRANSLATION, LEARNING SUITABLE NOTATIONS

Having fun with plane shapes

ACTIVITY 1

To understand and use the principle of translation, learning suitable notations

[LO 3.2, 3.7]

Transformation through translation

Figure 1

Above we have the first quadrant of a Cartesian plane. There are ten plane figures to be seen.

If you imagine that you cut out the shaded shapes above, and then move them to new positions (unshaded) by sliding them across the page, then you have translated them. Notice that they stay upright (they don’t change their orientation). These shapes have been transformed through translation.

If you label the vertices of the shape, then the new position has similar (but not the same) labels. You can see this on the rectangle above. From now on, you will use the same system of labels in your work. In the rectangle, position A moves to position A, B to B, etc.

We have different ways of describing translations. This is like giving someone instructions so that they can produce the result you want.

1. For instance, if I say, “Move the oval shape 4½ units right and 3 units down,” this gives the new position of the oval.

2. Translating the square:

Square ABCD → square ABCD means map square ABCD onto square ABCD. This is better said by specifying the positions: A (1 ; 9) → A (5 ; 8) and B(4 ; 9) → B(8 ; 8), etc.

3. We can also say how far the shape must move in a certain direction, which we can specify as a compass bearing. This says how many degrees (navigators normally use three figures) clockwise we turn from due north. Refer to the figure. You can see that east is at 090° and west is at 270°. The line is at approximately 200°. The triangle above is 5 units away on a bearing of 090°. In other words, if you are at the top vertex of the triangle, you can see the new position of the top vertex 5 units away if you look east.

Figure 2

A 21 units right and 3 units down

B 11 units on a bearing of 090°

C 20 units left and 6 units down

D (31 ; 4) → (11 ; 6), (34 ; 4) → (14 ; 6), (31 ; 1) → (11 ; 3) and (34 ; 1) → (14 ; 3)

E 7 units on a bearing of 270° followed by 4 units on a bearing of 180°

ACTIVITY 2

To understand and apply reflection

[LO 3.2, 3.7]

Transformation through reflection

Look again at the last problem (E) in the previous section. Can you see that it actually gives us two translations, one after the other? The descriptions for A and C do the same! This will happen again, as it is often the simplest way to describe a complicated transformation of a shape.

First plot the following points on the given Cartesian plane, connect them in order with straight lines to draw the shape, and then map the coordinates as given to transform the figures.

A(2 ; 2) , B(2 ; 4) , C(4 ; 4) , D(4 ; 6) , E(6 ; 6) , D(6 ; 2) , A(2 ; 2)

A(2 ; 2)→A(12 ; 2) ,

B(2 ; 4)→B(12 ; 4) ,

C(4 ; 4)→C(10 ; 4) ,

D(4 ; 6)→D(10 ; 6) ,

E(6 ; 6)→E(8 ; 6) ,

D(6 ; 2)→D(8 ; 2).

Can you see that the shape is reflected in the line on the grid? This means that if you were to fold the grid on the line, then the shape will fall on (coincide with) its reflection. In other words, the line of reflection is a line of symmetry for the shape and its reflection. We can also say we are flipping the shape, but this doesn’t tell us where it ends up.

Figure 4

We could say: “Flip the shape to the right and then move it two units to the right.”

Choose one of the shapes above and connect each point of the shape with its corresponding reflected point. Now take the centres of these lines and draw a line through the centres. This is the line of reflection.

On the grid below, draw the position of each shape once it has been reflected in the given line. Note that the line of reflection can go through the figure; it can touch the figure, or be outside it.

Figure 5

We often reflect figures in the x–axis or the y–axis.

Figure 6

You may colour the design in.

ACTIVITY 3

To learn how to transform by rotation, and put translations together

[LO 3.2, 3.7]

Rotation

In the diagram below, there is a point marked X on each shape. Imagine that the shaded shape was cut out and loose. A pin was stuck into the point X, and the shape was turned around the pin so that it fell over the unshaded shape. To specify how far we have to turn it, we have to use angles. For example, the triangle was turned (rotated) clockwise through 90°.

Below you have been given figure A. Draw figure B by reflecting figure A in the given line. Draw figure C by translating figure B 8 units right and 2 units down. Then rotate figure C 180 ° around the point marked X in figure A to give figure D. We can say that figure D is a complex transformation of figure A, as we needed several steps to draw it.

Figure 8

ACTIVITY 4

To enjoy transformations in the form of tilings and tessellations

[LO 3.2, 3.7]

The most remarkable and widely spread use of tessellations can be found in the decoration applied to buildings in the Islamic world. Islam forbids the making of images, so the builders concentrated on shapes. The Persians were competent mathematicians, and this helped to establish the rules of tessellation they used to such brilliant effect in the mosques and other important cultural centres. Even more interesting is the fact that the surfaces were often curved, not flat, which makes the principles of tessellation even trickier.

Figure 9

Figure 10

Figure 11

Figure 12

Figure 13

Figure 14

Assessment