Mirror Patterns 2

Task 159 ... Years 4 - 10

Summary

Mirror Patterns 2 is simply intriguing. Just making polygons using the two mirrors captures student interest. After an initial play period it becomes clear that the rotation movement of the hand about the wrist is linked to the number of sides in the image polygon. Hence there is very likely to be a connect between the angles of the mirror and the created polygons. A search for numbers and patterns begins.

This cameo has a From The Classroom section which is an Investigation Guide and a Prompt Sheet designed by Damian Howison. The Prompt Sheet uses a clever way of using photos in student material.

Materials

  • 2 mirrors
  • 2 dumbbell cards - one marked in equal divisions
  • protractor
 

Content

  • algebra, generalisation in words & symbols
  • arithmetic, addition / subtraction
  • arithmetic, multiplication / division
  • equations, creating
  • equations, substitution & solution
  • factors, multiples & primes
  • fractions, calculations
  • graphical representation
  • measurement, angle
  • mental arithmetic
  • patterns, number
  • patterns, visual
  • shapes, recognition
  • shapes, properties
  • spatial perception, 2D or 3D
  • symmetry, line
  • transformation experiences
  • transformations, reflection
Mirror Patterns 2

Iceberg

A task is the tip of a learning iceberg. There is always more to a task than is recorded on the card.
   

This intriguing task makes links between geometry, number, pattern and algebra. It takes the students past the fun of creating mirror patterns by requiring them to measure the angle between the mirrors. Recording the measured angles in the table, as suggested, leads to the hypothesis that the angle A = 360 n, where n is the number of sides of the polygon.

It certainly takes co-operation between two people to complete, but the table in Question 2 eventually gives:

No. of sides (S) 3 4 5 6 7 8
Angle btn. mirrors (A) 120 90 72 60 513/7 45

Of course the result for 7 sides can't be measured as accurately as indicated. It has been calculated. The calculation for each of these angles is based on the realisation that:

  1. If one mirror is placed on the end of the dumbbell, the line is reflected in the mirror to make an angle that is bisected by the mirror.
  2. When making the polygons, the mirrors and the dumbbell line (base) always make a triangle.
  3. When making regular polygons, the mirror sections are the same length and form an isosceles triangle with the dumbbell line.
So, if we know the sum of the internal angles of a polygon we can work out the angles in the mirror triangle. For the case of the equilateral triangle made with the mirrors, the internal angles of the reflected triangle sum to 180, so 60 is in each corner at the base and half of that is inside the triangle made with the mirrors. So, for the isosceles triangle made by the mirrors, the base angles are 30, leaving 120 for the angle between the mirrors.

For the case of the seven sided polygon, the internal angles sum to 900, so 1284/7 is in each corner at the base and half of that is inside the triangle made with the mirrors. So, for the isosceles triangle made by the mirrors, the base angles are 642/7, leaving 513/7 for the angle between the mirrors.

Note

To find the angle sum of a polygon, put a dot in the middle and draw a line to each vertex. This makes a 'rotation' of triangles. Each triangle has an angle sum of 2 right angles, so multiply by the number of triangles to find the total. However this includes the angles at the dot, which are not required. These angles make a complete rotation around the dot, so they sum to 4 right angles. Subtract these four right angles from the previous total to get the sum of the internal angles of the polygon. If the polygon is regular, divide by the number of angles to find the value of each one.
Examining the pairs in the table leads to realising that the product of the sides and the angles in the table above is always 360, that is, S x A = 360.

The challenge on the card is an application of the mathematician's strategy of working backwards. Using S x A = 360, if we know one of the values, we can calculate the other. Therefore:

Angle btn. mirrors (A) 100 70 55 35 20 10
No. of sides (S) 36/10 51/7 66/11 102/7 18 36

Extensions

  • The tables pair the numbers in an order (S, A) and ordered pairs can be graphed. Students could predict and check the shape of the graph. It is, perhaps, a little surprising as this Excel chart version shows:
  • Ask the students if they are interested in exploring how children's Kaleidoscopes work.

Whole Class Investigation

Tasks are an invitation for two students to work like a mathematician. Tasks can also be modified to become whole class investigations which model how a mathematician works.
   

Obviously this task requires specific equipment and it may be difficult to provide sufficient for a whole class investigation. However, the task could be one of a menu of tasks in a unit of work in pattern & algebra or symmetry. As one of a limited number of choices more students, perhaps all students, would have an opportunity to explore it. The menu could also include investigation guides for each task and Maths With Attitude, Pattern & Algebra, Years 7 & 8, includes such a menu with Investigation Guides provided.

At this stage, Mirror Patterns 2 does not have a matching lesson on Maths300.

Is it in Maths With Attitude?

Maths With Attitude is a set of hands-on learning kits available from Years 3-10 which structure the use of tasks and whole class investigations into a week by week planner.
   

The Mirror Patterns 2 task is an integral part of:

  • MWA Pattern & Algebra Years 5 & 6
  • MWA Pattern & Algebra Years 7 & 8
This task is also included in the Secondary Library Kit. Solutions for tasks in the latter kit can be found here.

Green Line

From The Classroom

St. Mary MacKillop College, Victoria

Damian Howison
From the email:
Hi Doug,
I've made up a basic investigation guide for Mirror Patterns 2, and also a sheet of pictures to help think about the Why is that so? part of the investigation. You have already described the maths involved in your task cameo but you may wish to add either or both of these sheets.
Damian
Sure do. Thanks Damian.

Green Line
Follow this link to Task Centre Home page.