An idea for the classroom
Regions created by reflections
This is a task in which students create their own examples, and that they could tackle in pairs on computers using dynamic geometry software.
Introduce the task in the following way:
Using a large screen connected to one of the computers first draw a shape, such as an equilateral triangle, and a mirror line:
Challenge students to visualise the image of the triangle after reflection in the mirror line, allowing them plenty of time to establish images in their minds. Remind them that the whole diagram including the triangle, the mirror line and the image of the triangle will be composed of a number of regions. Ask: How many regions will there be?
Students are likely to conjecture that there will be five, or four or three regions. To test their conjectures show the reflection:
They can see that total number of regions in the diagram consisting of the object, the mirror line and the image, is actually four:
Now draw a diagram with the mirror line is in a different position in relation to the same shape. For example, like this:
Ask what will be the total number of regions in this case.
Students may conjecture correctly that in this example there will be six regions:
The students’ task is to investigate the numbers of regions in their own diagrams created by reflections. Tell them to start with any regular polygon, perhaps a square or a regular pentagon, and a mirror line. Then draw the reflection of the polygon in the mirror line, and count the number of regions in the whole diagram (excluding the exterior region).
- think about what you can change
- ask your own questions, and try to answer them.
Encourage students to explore their own examples by changing aspects of the situations that they create. For example, they may change the position of the mirror line but keep the same polygon. Or they might change the number of sides of the polygon while retaining the regularity.
From time to time, invite particular students to link their computer to the large screen, show their diagrams to the whole class, and talk about what they are finding. These brief whole-class episodes should generate discussion and prompt students to ask their own questions, such as:
- what are the maximum and minimum possible numbers of regions when the shape that is reflected is a square?
- what numbers of regions is it impossible to make?
- when you reflect a square, are there any numbers of regions that you can only make in one way?
- are the maximum and minimum possible numbers of regions the same for all regular polygons?
- is the maximum number of regions related to the number of sides of the polygon?
- how is the maximum number of regions related to the number of sides of the polygon?
In one lesson, the diagrams that students created and printed out to support their findings included these:
Encourage students to generalise and make conjectures.
In one lesson students conjectured that:
- when any regular polygon is reflected the minimum number of regions is always 2
- you cannot make three regions when you reflect a square
- when an equilateral triangle is reflected the number of regions is never more than 8
- you cannot make more than ten regions when you reflect a square
- you cannot make three regions with any regular polygon
- you cannot make an odd number of regions with any regular polygon
- whatever regular polygon is reflected the maximum number of regions is twice one more than the number of sides of the polygon.
A few students were able to express this last generalisation algebraically:
- the maximum number of regions is 2(n + 1), where n is the number of sides of the polygon that is reflected.
Page header - reflections, image by Lincolnian (Brian) some rights reserved