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Problem Solving

You may have noticed that I write a lot of posts looking at my approach to solving maths puzzles. If not, you can see them here.

I think the skills involved in problem solving are very important in mathematics, and I’m general life. I feel that we, as maths teachers, need to be equipping our students with the tools to tackle these problems, not least due to the shift in focus on the new GCSE and A Level specifications towards problem solving. I think that by showing them these puzzles, and modelling our approach to solving them, we can make a great start.

We can model strategies for spotting when we use certain things, but the real tools we need to give then are the ones needed when they don’t have a clue where to start. I always advice starting with these three questions:

What does it look like?

Sketching is something that I find students very reluctant to do, and it infuriates me. Very often a puzzle or problem becomes much simpler once a sketch has been made. Take quadratic inequalities for example, once you have sketched it is is straightforward to choose which region you need. But still each year I find at least a few year 12 students who hand in work, mock papers even, without sketches. It’s not limited to quadratic inequalities though, any geometric puzzle, any algebraic puzzle and many stats, combinatorics, topological puzzles can be simplified with a sketch. So if you, SKETCH IT! It often shows you the way to solve it.

What do you know?

This is a question I find I ask a,lot during lessons when students are struggling to begin a question. C1 and C2 sequence questions give rise to it all the time. Students need to learn how to identify key information, and rewriting each bit of information away from the question can really help, especially if the question is wordy. For example a year 12 student this week was struggling to start a particular wordy arithmetic progression question on a past paper. I asked him what info he could gain front the question, once he wrote it out underneath he could see he had 2 simultaneous equations to solve for a and d.

This is a tactic students need to learn, and it ties to sketching too. I always tell my students Ince they’ve sketched they need to fill in any other info they know onto the sketch, it helps them clarify what they know and where they can go next. This is often enough to show a route to a solution.

What else can you deduce?

When I’m solving a puzzle this is the third question I ask myself. I’ve sketched if appropriate and I’ve specified what I know, next I look at what else I can deduce. In a geometric puzzle it could be angles, lengths or areas. In an algebraic puzzle it may be there are easy to calculate variables, if there are surds involved it may mean rationalising denominators. This step can be the key, it often gives rise to a hidden piece of information which makes the puzzle easy to solve. It could be the realisation that two triangles are similar or congruent, it could be a missing side, or angle, that allows you to calculate more. Or it could be a way to work out one variable, that would lead to finding the one you actually want.

These three key questions are part of the problem solving journey, and they are ones I try to encourage all my students to use for those problems where they don’t know where to start.

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