Task 164 ... Years 4 - 8
SummaryUsing 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.
IcebergA 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.
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.
ChallengeThe 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.
ExtensionsUsing 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 180º 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 45º upwards to the right so that edges now touch and a rectangle appears.
And of course that rectangle can be rotated 45º 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 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.
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:
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.
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.
Is each brown tile in two lines?