# Mirror Patterns 3

### Task 168 ... Years 4 - 8

#### Summary

This task is a great opportunity to explore reflection which is one of the basic mathematical transformations. The starting point is an unusual design unit, chosen by a group of Year 8 students who had explored the partner tasks Mirror Patterns 1 and Mirror Patterns 2 and who asked What happens if ... we use a mirror on a different starting shape?.

This cameo has a From The Classroom section which shows an amazing number of mirror patterns made by one class of Year 8 students.

#### Content

• language of space, position and order
• measurement, angle
• measurement, length
• patterns, visual
• spatial perception, 2D or 3D
• symmetry, line
• transformation experiences
• transformations, reflection

#### Iceberg

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

This task is one of a number created to refresh and reinforce practical experience with symmetry. It was designed by members of a Year 8 class and, in the latter part of the card, it allows students to follow their own design directions. In order, the solutions to the first part of the card are shown here with the word MIRROR indicating the side looked into.

These can be drawn on the recording sheet but it would be worthwhile for students to use software drawing tools to reproduce the basic image and then mark the mirror on it as in these diagrams. The software might also be used in the creation of new designs since some of these tools allow the user to specify the reflection line.

#### Extensions

Any of the tiles created in this task, including the ones students create themselves can then be used to create tiling, or wallpaper, patterns by transforming and repeating the basic shape. For example, by simply translating the last tile above to the right to make a row and then translating the row downwards we make this pattern:

which could at least be described as 'striking' if used as wallpaper or as a tiled floor.

Patterns like the above arise from asking What happens if we repeat and transform the tile?, but we can also ask What happens if we design a different original tile?. The original tile only used the corners and the centre to design a non-symmetric starting point. The symmetry was created later by the mirror. So:

• What happens if any of the midpoints of the sides could also be used to create the original design?
• What happens if the original tile is designed using these starting points?

• What happens if the original tile begins as a hexagon rather than as a square?

#### 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.

Turning this task into a whole class investigation might be possible if you have the appropriate software drawing tools, but to do so as a real, rather than virtual, hands-on experience would require many mirrors. These might be available from the science department. An alternative approach to either of these is to have all the students engaged in the same sort of mathematics at the same time, rather than being simultaneously involved in the same task.

Several tasks in the collection involve the mathematics of transformations and they can be found using the Task Cameo Content Finder. Just scroll down to the sections on Symmetry, Tessellations and Transformations. A set of 15 tasks is plenty to provide a menu for a class of twenty students working in pairs and the collection can include multiple copies of some tasks. Another way is to provide a smaller set of tasks for half the class while the other half works on the transformation section of the textbook. Halves of the class swap over in the next lesson.

At this stage Mirror Patterns 3 does not have a matching Maths300 lesson, however Lesson 123, Mirror Bounce, investigates the use of mirrors to bring light deep into the pyramids.

#### 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 3 task is an integral part of:

• MWA Space & Logic Years 9 & 10

## From The Classroom

#### Thorne Grammar School Doncaster, England

Andy Martin
Year 8
Andy turned Mirror Patterns 3 into a whole class investigation. He simply challenged his students to find as many different mirror patterns as possible and record them. Every student was fascinated. Click the image to view a full size slideshow of just three journals from the class. Change slides with the arrow buttons. Save the show and use it to stimulate your own class if you wish. You could also use it as a stimulus few minutes in a faculty meeting.
What mathematics has been involved in the work these students have recorded? Is it just mirrors and reflection?
Sarah Hutchins' report, which is not included, has 120 images and she challenges anyone to find more than that.

Sorry it took so long to get this wonderful work on site Andy : )