4.4 Understand and use the principle of translation, learning  (Page 2/2)

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.

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

• The parallelogram has also been transformed by reflection. Draw the line of reflection.
• Draw the line of reflection for the circle.
• The circle can also be seen as having been slid . Describe in words how the circle was translated . What is it about the circle that makes it possible to see its transformation as either reflection or translation?

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.

• Find the line of reflection in this way for all three shapes above.

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.

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

• On the following Cartesian plane reflect each shape in the x –axis, then in the y –axis and again in the x –axis, so that you have four of them, each in a different quadrant.

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°.

• For each of the other shapes, say how many degrees, in which direction, it was rotated .

  

• Label the vertices of each of the three figures and describe each of the transformations in terms of coordinate mapping.
• Describe the transformation of the square as a translation (a) in terms of bearing and direction and (b) in words.
• Describe the transformation of the square as a reflection.

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.

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.

• When you can make tiles of a certain shape with the property that you can place them next to each other on a surface so that they don’t overlap, and don’t leave any gaps, then we call this a tessellation.
• You can experiment with this by cutting shapes carefully out of cardboard, and fitting them together.
• You can also do this as a drawing on paper, by combining the principles of transformation (translation, reflection and rotation) to a starting shape until you have tessellated the surface completely.
• The shapes can be simple, without any transformation except translation, or complicated with complex transformations. When you use more than one shape in a tessellation, you can produce some very beautiful designs.
• Below you can see a few tessellations. Discuss (in your group) what you see and then try to write down exactly what was done to each shape (translation, reflection and rotation), to produce the final result. Complete any incomplete ones.

Assessment