Tangent Half Angle Subsitution

I’ve always found integration by substitution satisfying, and the tangent half angle substitutions are particularly satisfying. These are also sometimes called “The Weierstrass Substitution” despite no evidence of it in his writings, and it have being used long before in the work of Euler.

It is guaranteed to transform a rational function of sine and cosine into an algebraic rational function that can be integrated (However, it should be noted that sometimes there will be neater, less laborious ways to evaluate a given integral).

I hadn’t spent too long on this during the course (it isn’t explicit in the AQA course), but contact with them has led me to believe that students could be guided through using these substitutions. Because of this I have produced a short guided work through sheet.

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My students enjoyed doing these which is nice 🙂

File for your use is here.

A-Level Mechanics Revision Clock

I’ve produced another revision clock activity for the new specification A-Level, this one is focussed solely on mechanics.

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Download the file here.

At some point I hope to get round to writing up worked solutions, but at the moment I am focussed on getting resources out there for students to use.

AQA Level 2 Further Mathematics Relay

I’ve recently shared an A-Level Further Maths vectors relay and thought I would make one for my AQA Level 2 Further Maths students.

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There are 16 questions in total, so 8 per person in a pair. The idea of these is like the UKMT Team Challenge relays. One person works on the even numbered questions and their partner works on the odd numbered questions. They have to pass values between them to enable them to complete all questions.

As ever, the file is available to download here – let me know if you spot any errors please 🙂

Geometric Interpretation of Simultaneous Equations

One of the topics requested by my current Year 13 Further Mathematicians in their revision lessons has been the geometric interpretation of the solution of simultaneous equations.

I hadn’t taught this at A-Level until the new specification so it was interesting to teach. I would always have done this using augmented matrices, performing row operations and then looking at the difference in rank between the matrix and the augmented matrix so the approach at A-Level was new to me.

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The idea of the activity is to sort the 12 given situations into the five geometric possibilities.

File for use available here.

Know-Why Tables

I’ve recently been listening to the latest episode of Craig Barton’s (@mrbartonmaths) always excellent podcast series. In it he discusses various things with Michael Pershan (@mpershan) but something that caught my ear in the first 15 minutes or so was a short discussion of “two column proofs”.

We don’t seem to have this formalism (actually “structure” is probably a better word here) much in the UK.

This post is to share a resource that I thought I had shared a couple of years ago (apologies if I did and I just can’t find my own post anymore….)

When I started teaching the new specification Mathematics A-Level; with its increased emphasis on proof, I decided to formalise some of the GCSE style number proofs and use these as an introduction to the topic at A-Level. To do this I used what I called “know-why” tables which seem very like this “two column” method Michael describes, so I suspect I originally saw the idea in an American textbook at some point.

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If you want to take a look at the whole worksheet – which includes definitions, examples and exercises – download it here.

I’d love to know what you think…

Sum-Products

This is a post mainly for me…. and a resource I wrote probably 5 years ago now.

I routinely want to use this resource and always have trouble finding it in my files so I hope that once it is up on here Google will be able to help me locate it in future!

As we all know when faced with a quadratic \(y = x^2+bx+c \) that has integer roots we can find the factors by looking for two numbers that multiply to give the value of \(c\) and add to give the value of \(b\). I often start this topic with getting students to complete a few of these kind of puzzles:

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I’m not sure where I saw this idea originally  – possibly Jo Morgan (@mathsjem) or Don Steward maybe.

Anyway if you fancy using my sheet with a lot of them download it here

Mike Lawler Inspired Integration

Back in March I saw the tweet shown below from Mike Lawler (@mikeandallie).

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I think I would default to substituting \( u = \mathrm{e}^x \) here but in the thread were a range of different approaches. Tim Gower’s (@wtgowers) was particularly inventive.

I’ve always enjoyed asking students to integrate things in different ways and then asking them to show their solutions are equivalent. A favourite one of mine is \(\int \sin(x) \cos(x) \ \mathrm{d}x \). Because of this I have made a sheet asking students to explore a few of the possible methods.

The sheet for printing to A3 can be downloaded here.