Wallpaper Patterns

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Acknowledgement Thanks to Ian Lowe, Mathematical Association of Victoria. for inspiration and assistance with this cameo. Depending on which tile is chosen and how it is oriented in the first square of question 1, results will vary. Two examples are:          As students explore the tiles they discover there are two sets - one a reflection of the other - and 16 tiles in each set. Translations (also called slides) and rotations (turns) of a tile produce a tile from the same set. But reflections result in a tile from the other set. Students might also discover that a vertical reflection followed by a horizontal reflection (or vice versa) produces the same result as a half turn rotation around the bottom right corner. Further they might realise that the tiles are special because each is non-symmetric. The objective of each transformation is to move the tile on to the next square. Since the tiles are non-symmetric, reflections are in a line through one of the sides of the square - the right side or bottom the way the grid is set up - so a new tile is 'created' in the next position. A simple example comes from our alphabet. If the letter 'd' is reflected (flipped) to make 'b' next to it, we get 'db' and together they display reflective symmetry. Reflection in a line through the centre wouldn't achieve movement to the next square. The only time centre line reflection is allowed is when combined with a slide to create a 'glide reflection', which is a slide to the next square while, at the same time, flipping the tile in a line through its centre. You have seen this effect many times in action films. The speeding car hits a ramp, becomes airborne, turns around an axis in the direction of travel and lands on its roof. In effect, this is a glide reflection. The task does not assume knowledge of glide reflections, but the door is opened to learning about these through the Investigation Guide. The Guide includes some glide reflections, but that term is not used. Instead students will read descriptions such as: Slide right and reflect in a horizontal line through the centre of the tile to make the second tile (two steps for one tile). When students have connected these words to the matching pattern, teachers can ask Would you like to learn the mathematician's name for this two step transformation? and hence introduce the language. The task is an opportunity to explore transformations in an open environment with encouragement to become ever more clear in providing descriptions for someone else to be able to recreate their patterns. Questions 2, 3, and 4 lead students further into the creation of wallpaper patterns and, again depending on the starting tile chosen and its orientation, possible results are shown here. Pattern A Pattern B Pattern C The Challenge invites students to create more patterns by placing a start tile and then using two rules - one that transforms the tile to the right one square and the other (which might be the same) that transforms the tile down one square. Free exploration is encouraged, along with recording. Digital cameras make recording easy. Perhaps your students have a place on the school server where they can store their photos, or even an electronic journal in which the photos can be embedded. The most important part of the challenge is the instruction to Write the rules for making it. A mathematician has to be able to explain their work to colleagues. Many transformations can be explained in more than one way (equivalent transformation expressions), so it becomes even more important for the student to be able to justify their descriptions. If students become interested in comparing their creations, they might find that apparently different wallpaper patterns are sometimes actually the same, just an upside down or reflected version. Beyond the Challenge, this Investigation Guide and companion Descriptions explores 12 possibilities for creating wallpaper patterns from squares. Students are challenged to match transformation rules to wallpaper patterns. There are six pages of patterns, but if you print on card and laminate, you will only need one set of these and one Descriptions page. Students record matches in their journal. Answers to which pattern matches which description are here. Interested students could now take on the challenge of designing their own non-symmetric tile with software and applying the rules to create their own unique wallpaper patterns. (Note: In mathematics the convention is to name a reflection as horizontal or vertical according to the 'lie' of the line of reflection. However, computer programs, including Word with which students are likely to be familiar, use a different convention. They name the reflection by the direction in which the transformed shape appears to move. Hence a mathematical vertical reflection results in a horizontal movement to the transformed shape and software would describe this as a horizontal reflection. Annoying, and possibly confusing for students doing mathematics with such software.)

Note: This investigation has been included in Maths At Home. In this form it has fresh context and purpose and, in some cases, additional resources. Maths At Home activity plans encourage independent investigation through guided 'homework', or, for the teacher, can be an outline of a class investigation.

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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 Wallpaper Patterns task is an integral part of: MWA Space & Logic Years 7 & 8

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