This week I was planning to cover upper and lower bounds with year 11 as on the last mock a lot of them made mistakes so I felt it would be a good topic to revise. As part of the planning process I had a look through the higher textbooks our department has bought for the new specification GCSE *(we bought the Pearson ones, the full suite at KS3 and 4. Some great questions in them and the online version, activeteach, is great to take questions and place into your lessons. I’d definitely recommend it, if used correctly, but I will admit to being disappointed to see a formula triangle being advised…)* to see if there were any good questions I could pilfer, and I came across the section on using upper and lower bounds in trigonometry.

My first thought was, “that’s a nice topic”, and then the full spectrum of the topic began to unfold.

Initially, I had like the idea that students would be required to think about the fraction, and how minimising the denominator actually maximises it, but thin I remembered the nature of the cosine function! This example shows what excited:

Not only would students be required to understand the nature of a fraction, they’d also need a deep understanding of the cosine function itself, to understand that the bigger cos x is, the smaller x is, and vice versa (where x is between 0 and 90 of course). This could be a real deep understanding of the graph, or the unit circle, or just the geometry of a right angled triangle.

The example itself is very procedural based, which is a shame, but it does give a teacher a good frame to start discussions. I wouldn’t use textbook example as teaching anyhow, just as an additional example to talk through one on one with students who were still struggling.

The textbook goes on to pose this awesome discussion question:

A real nice prompt to get an in depth discussion around the trig ratios going. I often use similar prompts when looking at maximum values for sine and cosine “what’s the biggest opp/hyp can ever be?” for example. This often gives a nice discussion focus.

I think that this topic shows how different the new specification will be. Students are going to need a much deeper relational understanding if they are to achieve the top grades with questions like this being posed.

*What do you think of bounds being questioned in relation trigonometry? Have you used prompts like this before? How have you found them?*

This post was cross-posted to Cavmaths here.

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