*This post is cross – published and can be seen on Cavmaths here.*

I’ve written before about textbooks, they can be troublesome if used incorrectly but they can also provide a good amount of questions to allow students to get their teeth into without having a ridiculously high photocopying budget. A good textbook, in my opinion, is one with great questions covering a range of difficulties to allow stretch and differentiation when used properly.

Currently I have been struggling to find good textbooks for the A level syllabus, I think the majority of the textbooks out there for the current syllabus are a little rubbish and I hope that the ones being produced for the new board are better. *(If anyone wants to pay me a load of money I’ll write you a belter….)*

The ones we use are produced by our exam board and are one of the better ones out there but are still lacking. The questions tend to be straightforward, testing skills and not understanding and not really differing in difficultly. Imagine my surprise, then, when I came across this little beauty:

It was in an exercise on addition formulae and it really threw a number of my students. I loved the question and in the end we worked through it together on the board.

Once I’d railed at them about the importance of sketching they agreed that would be the best place to start. I made them sketch the triangle on mini whiteboards first to ensure they could then I sketched it in the board. They worked out the angle to be 60 minus theta and came up with the sine rule the selves, and then it was just a case of simplifying with the addition formulae.

This question doesn’t involve any overly taxing mathematics, but it does mix the skill being learned in with prior knowledge and as such serves to enhance the relational understanding of the students. They can see where the links are and they can build those links themselves. I think that this is a superb question and we need to be using questions like this regularly to build that relational understanding, helping our learners become mathematicians, and not just maths exam taking machines.

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Do you think that they would first think of doing a sketch/diagram/picture next time they get a problem?

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I hope so. I’ve been trying to hammer the importance of sketching home to them.

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It might be interesting to consider the solution strategy used, and then modify the problem slightly by changing the ‘120’ to ’90’. If you trace through the algebra, you end up with the correct solution; but if you draw the corresponding picture, then you can read 4/3 right off of it.

This suggest a way of writing the problem in the first place: Given legs of length 3 and 4, adjust them so that they form a right angle. Now denote one of the non-right angles as theta, and compute tan(theta).

Okay: That is a standard question for students beginning with trigonometry.

But what happens if you bend the legs of length 3 and 4 to some other value? For example, what if you bend them to form a 120 degree angle? Can you still put in a theta and compute tan(theta)? [Yes: This is exactly what your class did.]

Okay: What if you bent the legs to form a 30 degree angle? Is the problem still possible?

And what if you started with other Pythagorean numbers (e.g., 5 and 12)?

What if you started with entirely different numbers (e.g., 6 and 7)? Etc.

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Nice additional thoughts. I will investigate them with the class on Monday.

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