Mirror Patterns 2Task 159 ... Years 4  10SummaryMirror 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 reveals a clever way of using photos to cue students. Materials

Content

IcebergA 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 cooperation between two people to complete, but the table in Question 2 eventually gives:
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:
For the case of the seven sided polygon, the internal angles sum to 900°, so 128^{4}/_{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 64^{2}/_{7}°, leaving 51^{3}/_{7}° for the angle between the mirrors. 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:
Extensions

Whole Class InvestigationTasks 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 handson learning kits available from Years 310 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:

St. Mary MacKillop College, VictoriaDamian Howison 
From the email:
Hi Doug,Sure do. Thanks Damian. Colour matters in the Prompt Sheet. You might print it in colour, or you could display it full screen in Acrobat. (Ctrl L toggles full screen in Acrobat.) 