Symmetric Tiles

Task 164 ... Years 4 - 8

Summary

Using tiles with a different colour on each side, the task encourages exploration of symmetry, where the line of symmetry will most likely be seen as diagonal. One of the shapes is the key to finding many solutions easily, and even when this is realised, the final challenge on the card is not easy to resolve. The task reinforces the principle that an object and its reflection are the same perpendicular distance on either side of the line of symmetry.
 

Materials

  • 28 two centimetre tiles with a different colour on each side
  • Recording sheet

Content

  • measurement, angle
  • measurement, length
  • patterns, visual
  • spatial perception, 2D or 3D
  • symmetry, line
  • tessellation
  • transformation experiences
  • transformations, reflection
  • transformations, rotation
  • transformations, translation
Symmetric Tiles

Iceberg

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

Symmetric Tiles can be used in a work station structure based around the content of symmetry (line and rotational), transformations and tessellations using these tasks.

A combination of this work station, whole class investigations such as Lesson 53, Spirolaterals, from Maths300, which includes software, and text work would make a strong Mixed Media unit in Shape, Space & Geometry.

The earlier form of Symmetric Tiles used two differently patterned cardboard tiles as shown in the photo. More recently 2cm tiles have become available which are raw wood on one side and white on the other. These make the task easier to visualise and easier to record.

Throughout the task the students must be sure that there are always 12 and only 12, tiles used brown side up. The other 16 must be used as white.

             
Given this, many symmetric designs can be created. Perhaps some students will begin by realising that the four extra white tiles could be placed on the line of symmetry, leaving equal numbers of brown and white for the other places. Two possible outcomes of this approach are:
        
Brown tiles can also be on the line of symmetry:
        
Using the Recording Sheet, it's quick and easy to record solutions and individual students, or the class as a whole, might like to make as many different symmetric tiles patterns as possible. A collection of different ones makes a great display. (Please send photos.)

Challenge

The challenge on the card is not one that should have its solution revealed, so it's not here. (It is right at the bottom of the page, so don't look unless you are so frustrated you are about to jump off a cliff.) Reasoning suggests that 6 lines of 4 implies 24 brown tiles, but only 12 are allowed. So, each tile must be in two lines. Now play with it.

Extensions

Using multiple copies of any of the design units new designs can be created by applying mathematical transformations. Students can make multiples on the Recording Sheet and cut them out, or, as has been done here, they can make a master of the playing board using software, colour using the software tools then use the transformation tools of the software to create extended designs. For example, the first design above can be translated to produce:

Then with one of the design elements rotated 180 and translated into the gap between them and another copy of the rotated design element translated onto the right hand end we get:

The next one is made from three upward pointing and one downward pointing element. Can you find them? The top one has also been made to overlap the top row of the downward pointing one - where the double diamonds are.

Perhaps this is how engineers design tiled forecourts and plazas between buildings?

Reflection can be used too. In the following example, using a different design element, the bottom section is a reflection of the top in a horizontal line through the middle. The two parts actually have no edges touching - only vertices (corners). Cover up the bottom half to reveal the original element if you have difficulty seeing this.

In the next, the bottom element of the one immediately above has been translated along an angle of 45 upwards to the right so that edges now touch and a rectangle appears.

And of course that rectangle can be rotated 45 about its centre to make this unit which can become a new design element in itself.

Using either colouring, cutting and pasting by hand, or software drawing tools, any student can become very creative and any classroom can become very attractive.

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.
   

The wood used in the recent version of the task is easily obtainable MDF. It may be easily obtainable, but it is not so easy to cut it into accurate 2cm squares. However, there are substitutes in school, for example:

  • larger coloured tiles (some of these are coloured differently on each side)
  • coloured wooden cubes
  • click together plastic cubes like Unifix.
You only need to prepare a playing board the right size for your material and make sets of tiles or cubes for each pair and the exploration can begin. The Recording Sheet above will work for whichever material you choose. It is also useful to have examples of the extension possibilities prepared in advance. (You could 'steal' the diagrams above and use them as slides.)

It helps too to begin the lesson on the floorboard (rather than blackboard or whiteboard) using large squares of card, say 20cm x 20cm, which are printed or coloured differently on each side. Hand a card to each student (you need 28) and organise them to produce the 'arrowhead' with the same colour showing up on all tiles.

  • Explain that the first challenge is to turn over exactly 12 cards (tiles) to make a symmetric design.
  • Explore possibilities and show how to record the outcomes, perhaps using an overhead projector version of the Recording Sheet. You could also prepare an Interactive White Board version of the Recording Sheet which could be coloured in an instant with the painting tools. The drawing tools in Word will work for this.
  • Set the challenge of finding and recording three more designs.
  • Gather the students back at the floorboard and ask some to show their designs.
  • Introduce the challenge as on the card above and invite pairs to search for the solution, then record it.
  • Show samples of patterns made by using these design elements and discuss the transformations within them.
  • Invite students to create their own Symmetric Tiles display.
  • Send photos of your display to record here.

At this stage, Symmetric Tiles 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.
   

Symmetric Tiles is not in any MWA kit. However it can be used to enrich the Space & Logic kits at Years 5/6 and 7/8.

Challenge Solution

Is each brown tile in two lines?

Green Line
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