Starting with Bobby: Exploring Repeating Patterns in a Mathematical Context

Meet Bobby, a tile character I initially created this morning, to accompany me as I began to explore and think through some of the mathematical possibilities in using QCA ICT Unit 4b to explore repeating patterns. In a previous post I presented some ideas around how I have used these ideas and processes to develop a Design and Technology focussed project using MS Paint. Here I wanted to take a different tack, and expand on and explore another context where I found the process useful, in exploring the language of shape and space through a mathematical context.

Why MS Paint? Well since the PCs I use in school, all run Windows, and those of my students who have computers at home generally work within this environment, paint is a tool, that we have ready access to, frequently underused and exploited, there is the possibility that if excited and motivated by the tasks we develop, the students may choose to extend ideas away from school.

In beginning projects involving repeating patterns I favour an approach where my students begin by making their own tiles, using simple shapes, copy and paste, alongside flip and rotate, to develop more complex designs, that can eventually be used to develop their patterns. Repeating patterns activities such as the one I am thinking through in this post, have it seems to me enormous mathematical potential, an affordance we may not readily associate with graphic and painting packages.

Beginning with the development and creation of a set of tiles based on irregular polygons, such as an L shape, which we can describe as an irregular hexagon or hexagonal, and others such as these examples.

The sessions might be developed from the limiting of tool use, and copy and paste to develop initial tiles exploring different polygons that can made using only 2 rectangles, perhaps expanding to explore polygons that can be made using 3 and so on, and then using these tiles to support discussion around the properties of shapes. Perhaps we might explore the number of right angles they have? Are the angles always right angles, perhaps inviting reasoning about why the students think this might be the case? What happens if we overlap rectangles? How does this alter the possibilities for polygons we can make?

In developing these relatively simple "objects" we have begun to think mathematically about the activities we are going to engage in as the process unfolds, and this can be further built upon using the flip and rotate tools as we move to develop our tiles. In MS Paint the flip and rotate dialogue box offers options to flip horizontally or vertically and to rotate an object through 90, 180 or 270 degrees. These can be compared with 1/4, 1/2 and 3/4 turns, that our students may be familiar with through their use of tools such as the BeeBot floor turtle, and maybe even related to table top work related to using mirrors when exploring symmetry. Here we can begin to introduce or develop the use of numerical values for right angle turns with visual modeling and exploration of effect being developed as children create new tiles based on copy, paste and rotation of their original tiles. The idea of angle and a measurement of angles being related to rotation and turn around a point being introduced at the same time, through discussion and exploration of the effects applying these tools to the objects has.

This tile was created by copying and pasting one of my irregular polygons 4 times, rotating it each time progressively through, 90, 180 and then 270 degrees, and overlaying each newly pasted tile onto those pasted previously. From this context there is the possibility to expand on discussions begun around my Polygon activity, by exploring further, ideas around the properties of the tile and shape created. This shape possesses no line symmetry, as it was made by rotating tiles, it may be rotationally symmetrical however as all of my rotations were through right angles. We could test this together, using copy and paste, and the flip and rotate tool, with the object set as a transparent layer, and by dragging the tile over the original, to observe whether or not the tile is rotationally symetrical. We could also ask what the children notice about the angles? Again theyare all right angles, why might this be so? Using the tile, we pasted and the flip and rotate tool, we can make a tile or a shape that does possess line symmetry?

This tile was made using the "flip vertical" tool in MS Paint, by copying the tile, pasting it and then dragging to touch or tessalating the two tiles. What will happen if I copy this whole shape and join this to my other tile? How many lines of symmetry do I have? What effect will "flipping" this tile vertically have on the pasted shape and the one I make as a result of tessellating them?

Moving on from these simple tiles, the children could be challenged to make their own designs, firstly using rotation tools to make their own simple tiles and then flip tools to make symmetrical tiles, that they can tessellate, by repeated copy and paste.

Patterns are governed by rules, having produced their tesselating designs and saved, these they can then be coloured using flood fill tools, to follow either given rules or support reasoning and generalisation around visual lines of enquiry. Using save as in between each step of the activity, eg

• Using two colours, make a pattern where no two shapes adjacent to or next to each other are the same colour,
• Create a design where adjacent columns or rows are different colours?
• Use three colours to create a diagonal pattern?
• Use your design, save as and 2 different colours to investigate the different ways in which you can paint half, quarter, three quarters of your design.
• Use a multiples of 2, 3, 4, 5 or 6 pattern to colour your design.
• Colour your design using a multiples of 3 pattern, and then a multiples of 6 pattern. What happens to your design, when the new rule is used? What about a multiples of 2 and 4 pattern, or 5 and 10? What happens to my pattern if I use three colours and paint a multiple of 2 4 and 8 pattern?
  

These rules would obviously be varied and increase or decrease in complexity of language use and reasoning required according to needs of the students. The students might also be encouraged to design their own rules for patterns, and to apply these, or to challenge their friends to follow them.

As a bit of an afterthought a great follow up to these tasks would be to use a tool such as MS Photostory, to enable children to use and rehearse mathematical language and reasoning by including the digital outcomes of their tasks within a presentation. Dragging the images developed into Photostory, the students might use ideas from discussions they have engaged in during tasks, to record orally their work, ideas and findings as voice overs supporting their images, describing the processes they went through, and using and applying vocabulary developed during the sessions within the production of their presentation. The completed videos would make exciting evidence of learning while also supporting student understanding in using and applying mathematics activities and puzzles, as well as acting as a vehicle for sharing their work with others. These could be published for friends in school, but could also be added to blogs or VLEs for comments by visitors.

What happened to Bobby?

Well here he is and as I have been writing this he has not been far from my mind, in fact I have had a really challenging time working with my fussy friend, while in a mathematical frame of mind, to think about how I could use him to support systematic investigational work with a year2/3 Class or group. How about the possible colour combinations he could wear...
Despite his complexity, and his inability to tessellate, in his current form he does make some interesting designs and patterns, when used on his own initially to form part of an investigation. I wonder what combinations of trousers and jumpers are possible using only two colours, and what would happen if I had three Colours to choose from to make these combinations? How many different ways could he be dressed using four different colours? This as a simple pattern design would make a really attractive wall display!

Maybe extending this to make an initial tile, and using some of the rules from our investigation, we could systematically reapply the investigation to decorate these tiles, For this pattern I had three colours, and tried to find the different combinations I could use to colour opposite characters the same while rotating the colour fills between them.

As a discussion point, there other whole class possibilities for using some of my original more complex patterns using Bobby. Using an open ended question I might be able to use some of these following student engagement with tasks like those above.......
Perhaps embedding this pattern into a smart notebook, I could use a hide and reveal techniques to explore the patterns and shapes, asking questions such as What can you see? Is there anything else you can tell me? What shapes can you see? Is their anything special about the colours I have used, and the way I have used them? Encouraging responses and answers to be given in sentences, asking why do you think that? and seeking responses that require and support students as they use not only I think.. or I can see ... statements, but require reasoned "because" type responses to be formulated, perhaps through paired discussion teacher modeling and group rehearsal.