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# Adding life to old problems!

Tuesday 25 June 2013

I recently went to an International School in Geneva to work with their mathematics department for a day as part of their adoption of a 'bring your own device' scheme. It is always a pleasure to do such things, but this one provided an extra challenge! The brief for this day was to focus on how things change when you have computers at your disposal during every lesson. I could talk about technology and mathematics teaching all day long, but this refinement made me reflect a lot on when I started working at the International School of Toulouse where students have computers with them all the time. There are so many considerations that I can't possibly cover them all in one blog post, but I want to just show a quick example. In short, having technology available at any given moment multiplies the possible directions any given lesson can take an untold number of times and the important thing is to be aware of as many of these possible directions as possible. One thing that often happens is that you stop and think about what technology might bring to activities you have done for years without technology - not that it is always better of course. I am going to site the example of 'congruent halves'. This is a great example of a task I have known about since I started teaching, but have never known where it came from. I have a set of them in my file of 'Gems' that I bring out year after year. If you are not familiar, then the premise is simple. Take the shape in the image above and ask - can you split the shape into two halves that are congruent? There are some great examples of these like those shown below....

A good deal of thought and speculation is required and it is already a very rich mathematical activity. I love it, and even here with all our technology I have continued to do this on paper. This year, however, we had a vist from Douglas Butler, author of the mathematics software Autograph *(see more below). Douglas wanted to come and see how we worked with our access to technology. So I looked at what I was planning to teach and wondered if I could take the 'congruent halves' activity and use technology (namely autograph) to turn it in to an activity to help students explore transformations. Here is the result Congruent halves and transformations. Of course this could also be done using other dynamic geometry software.

I desperately want to avoid using the phrase 'breathing life in to old problems' beacuse that would imply that the old problem had lost its magic. That is certainly not the case. It is just important to speculate about how a task can be changed and developed and it is the great pleasure of mathematics teaching that this is always possible! Anyway, the task is outlined at the link above, but I will add a short summary here.....

Adding to the existing problem, can you recreate the shape by drawing one of the congruent halves and then transforming it in to the correct position so that the two halves together make the original shape. See the example below,

There are so many lovely hidden challenges like 'finding the centre of rotation' in these puzzles. The puzzles themselves provide the motivation for students to pursue solutions. I am a great advocate of the notion that 'practice can be found in rich tasks' and here is a prime example of that as well as 'Adding life to old problems'.

*Autograph - The landscape for mathematics education software is rapidly changing with Geogebra calling the shots, but there are still numerous occasions when I reach out for Autograph and I would not want to work in a department without it!