Paving Views

Task 157 ... Years 4 - 8

Summary

Within the story shell of an architect's work, students are asked to create a tiled courtyard with a rotationally symmetric pattern. There are many possibilities - just putting a tile on every square for example - but the challenge is to use the minimum number of tiles. Students don't have to know what rotational symmetry means to start with (although it is explained in part on the card), because the challenge makes sense in the context of the story. Once created, the design can be used to create more designs using transformation such as translation and reflections.
 

Materials

  • 8 x 8 board
  • about 40 tiles cut to fit the squares of the board

Content

  • patterns, colour
  • patterns, visual
  • spatial perception, 2D or 3D
  • symmetry, rotational
  • transformation experiences
  • transformations, reflection
  • transformations, rotation
  • transformations, translation
Paving Views

Iceberg

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

Rotational symmetry links this task to Knight Swap and Wallpaper Patterns. Perhaps this is an underrated aspect of mathematics learning, which is a pity since the tool is used in:

  • Alloy wheel & wheel trim design (see Extensions below)
  • Logos for businesses and organisations
  • Designing landscaped gardens
  • Designing jewellery
  • Patterns in nature
  • Wallpaper and tiling patterns
and even making fans and extractor fans more aesthetic and efficient, as in this example in the ceiling at at Trädgårdstadsskolan, Sweden.

Getting back to the task, the solution to Architect Sanja's puzzle is:

     

The second image shows the original tiles using white dots. Eighteen tiles have been added. This is the minimum solution. The courtyard can be viewed from all four sides and will look the same. But what clues could the students use to develop this solution?

  • Choose one side - say the left half.
  • Imagine this rotated about the centre to become the bottom half. (Or, if imagining is a bit hard, place a clear plastic sheet over the left half of the grid, outline the black squares, then rotate.)
  • Which squares need a black tile added? (Remember some of the black ones needed may already be there.)
  • Which squares are already black when you get to the bottom half? These will tell you which additional squares on the left half should have been black to start with.
  • Now rotate the new bottom half to become the right half. Fill in the blacks and also fill backwards again, this time into the bottom half first, then back again to the left half.
  • Rotate to the top half and repeat, filling back three times as necessary.
  • Check the whole visually from each side.
This approach combines the mathematician's strategies of breaking a problem into parts and working backwards. Of course, there will other ways students will examine this problem and reach the same conclusion.

Extensions

  • Use a drawing program or the table facilities of a word processor to make an electronic form of the courtyard as a record of the solution.
  • Students design their own rotationally symmetric courtyard.
  • What happens if the original courtyard is an equilateral triangle grid tiled with equilateral triangle tiles? Can we still create a rotationally symmetric tiling as viewed from the three buildings bordering the courtyard? Which such tiling uses the minimum number of tiles?
  • Use the electronic form as a tile to create new designs, eg:

    The 8x8 tile at the left end is the solution to the task. It has been reflected in a vertical line through its right side. This second tile has been reflected again the same way. Then the third has been reflected the same way to create the fourth. The whole might be a tiled section of a railway platform. Children enter at the left end and exit at the right, stepping only on black. How many different pathways can they take?

    In this next example, the original 8x8 solution tile is in the top left. It has been translated right, then the pair has been translated down. Use this grid to make a Crossnumber to demonstrate your knowledge of mathematics? (See Task 39, Criss-Cross Numbers which has a simpler rotationally symmetric playing board.)

    Of course, the same challenge could be offered using simply the original 8x8 solution tile.

  • This Investigation Guide is a double page of wheels which display point symmetry but do not also display line symmetry. It includes the challenges below (choose what is appropriate for your class) or use it as a starting point to generate interest in the world of rotation around us.
    1. Which wheels belong together? Why? Discuss.
    2. Invent your own car company, then design a rotationally symmetric logo for it.
    3. Design an alloy wheel, or wheel trim, which is rotationally symmetric.
    4. Design an alloy wheel, or wheel trim, which is rotationally symmetric even when your company logo is included.
    5. Most vehicle wheels are attached to the axle by 4 or 5 or 6 nuts (...and we are not talking about the mechanics!). Design an alloy wheel which is rotationally symmetric even when your company logo and the nut positions are included.
  • We would love to see the results of your students' 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.
   

Most schools have multi-coloured collections of tiles and the tiles are usually about 2.5cm square. Using the table facility of a word processor, it is easy to create an 8x8 grid for these tiles, although your printer would need to operate with 0.5cm left and right margins. Two centimetre wooden blocks are an alternative and a grid to take those would fit better on the width of a page. Each pair of students would need about 40 tiles of the one colour.

Hand out the grid sheets to each pair and display the equivalent on your electronic white board. Tell the story of Sanja the architect and fill in the starting point tiles on your display. Students match this on their grid. Now set the challenge and share ideas as groups work. End the lesson by reviewing the key steps in the process that led to the solution and highlighting the mathematics of rotational symmetry. Leave time for students to record the key steps and the answer in their journals. Use the lesson as a starting point for further investigations of rotation and other transformations. As suggested above there is room within this lesson to use drawing software or the drawing tools of a word processor. They all easily draw and colour grids which can be grouped as a unit and then rotated, reflected and translated.

At this stage, Paving Views 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 Paving Views task is an integral part of:

  • MWA Space & Logic Years 7 & 8

Green Line
Follow this link to Task Centre Home page.