## A lovely old problem

Recently Ed Southall shared this problem from 1976:

I’m not entirely sure if it is from an A level or and O level paper. It covers topics that currently sit on the A level, but I think calculus was on the O level at some point. *Edit: it’s O level* I saw the question and couldn’t help but have a try at it.

First, I drew the diagram – of course:

I have the coordinates of P, and hence N so I needed to work out the coordinates of Q. To do this I differentiated to get the gradient of a tangent and followed to get the gradient of a tangent at P.

Next I found the equation, and hence the X intercept.

And then, because I’m am idiot, I decided to work out the Y coordinate I already knew and had used!

The word in brackets is duh…..

Now I had all three point.

It was a simple division to find the tangent ratio of the angle.

The next 2 parts were trivial:

And then I misread the question and assumed I’d been asked to find the shaded region (actually part d).

Because I decided calculators were probably not widely available in 1976 I did it without one:

I thought it was quite a lot of complicated simplifying, but then I saw part c and the nice answer it gives:

Which makes the simplifying in part d simpler:

*I thought this was a lovely question and I found it enjoyable to do. It tests a number of skills together and although it is scaffolded it still requires a little bit of thinking. I hope to see some nice big questions like this on the new specification.*

*Edit: The front cover of the paper:*

## The long way round

Today one of my Y12s was looking through a C3 paper he found on my desk. *(For those unaware A level maths, studied in Y12 and Y13, ie from 16-18, is currently modular. There are 4 Core Pure modules known as C1, C2, C3 and C4. The first two are studied in Y12 and the second two in Y13.) *He came across this question:

While looking at it he said, “are you sure this is a C3 question?” I told him it was and he then said “But I can answer it.”

I looked at the question, all the main skills it tests are taught at C1 and C2, but the chain rule for differentiation isn’t taught til C3. I thought about it and realised that yes, with the application of the binomial expansion (a C2 skill), or indeed a long winded brackets expansion, it would give him a polynomial he could differentiate.

Then it occurred to me that it was in fact a brilliant question to set my Y12s as revision. It allows them to see links between the things they’ve learned, allows them to practice important skills from C1 and C2, namely the differentiation, the coordinate geometry involved finding te equation of a tangent and the binomial expansion, and to solve a problem using those skills.

It took them longer than it would have taken someone who knew about the chain rule, but it was time we’ll spent and I got some perfect answers from them. I didn’t tell any of them how to do it, they managed to talk each other through it, and I only had to pick up on one slight error when one of them had a slight hiccup with a power. I think I need to have a good look through some more higher level papers to see if I can find any other gens to test the earlier skills.

*This post has been cross-posted to Betterqs here.*

## Carnival of Mathematics #116

Hello, and welcome to the 116th edition of the Carnival of Mathematics *For those of you who are unaware, a “blog carnival” is a periodic post that travels from blog to blog and has a collection of posts on a certain topic. This is one of two Maths Carnivals, the other being **Math(s) Teachers at Play**, the current edition can be found here.*

Tradition dictates that we begin this round up with a few facts around the number 116. It’s a fine number, as far as even numbers go. It’s a noncototient, which is pretty cool. There are 116 irreducible polynomials of order 6 over a three element field, it is 38 in base thirtysix. 116 years is the length of the 100 years war and 116 is the record number of wins per season in major league baseball, set by the Cubs in 1906 and equalled by the mariners in 2001. It’s prime factorisation is 2^2 x 29, in roman numerals it is CXVI and it can be represented as the sum of two squares! (4^2 + 10^2)

Now, onto the carnival:

**In the news**

Katie (@stecks) submitted a number of fantastic news articles to this carnival. The first of which is this piece discussing Simon Beck and his geometric snow art. Ed Southall (@solvemymaths) also wrote about geometric art on a similar scale, sharing this about Jim Denevan.

The next article Katie shared was this amazing piece from Alex Bellos (@alexbellos) informing us of the amazingly geeky things Macau have done with their new Magic Square stamps! It almost makes me want to take up stamp collecting!

**Maths Applications**

Grace, over at “My math-y adventures” tells us how she’s put her undergraduate mathematical studies to good use by creating a model to help her with a real life decision.

And Andrea Hawksley writes this phenomenal piece exploring the maths of dancing.

But where there’s good, there’s bad, and here you can see the sort of fake real life contexts that irk Dave (@reflectivemaths) and myself.

**A bit of calculus **

John D Cook as submitted a lovely integration trick which I’d not seen before.

Augustus Van Dusen gives us part four of his review of “Inside interesting integrals” by Paul Nahin.

*Interesting numbers*

3010 tangents have been exploring the infinite with this superb post: “my infinity is bigger than yours”.

Over at the mathematical mystery tour they have shown us how to write pi using primes!

Fawn Nguyen has this great exploration of finding the greatest product.

**Geometric thinking**

Jo (@mathsjem) has written this excellent piece on circle theorems.

@dragon_dodo has produced this cartoon which illustrates Colin’s (@icecolbeveridge) position on radians.

**A bit of fun**

The folks over at futility closet have shared this idea about creating an indoor boomerang!

This week saw the UKMT Senior Maths Challenge, and here is my favourite question.

I spent some time at the children’s playground with my daughter, and found this mathematics there.

Manan (@shahlock) over at maths misery turns infuriating into fun talking about fractions and algebra.

**Behind the mathematician **

Behind the mathematician is a great series from Amir (@workedgechaos) who submitted Martin Noon’s (@letsgetmathing) edition for the carnival! (Mine is available here.)

**Podcast**

This month saw the annual maths jam conference, which brought with it the exciting prospect of a live special of “Wrong, But useful” my favourite maths podcast!

*And that rounds up the 116the edition of the Carnival of Mathematics. I hope you enjoyed reading all these wonderful posts as much as I did. If you have a post for the next carnival submit it here.*

## 20 Questions about C4 integration – A Book Review

A while a go I got a copy of this fantastic little ebook authoured by Colin Beveridge

(@icecolbeveridge). The book is great and written in Beveridge’s usual style- accessible, witty and very informative.

The book covers integration. IT is based on the current Edexcel A Level spec and covers all the integration you need to know for that specification, not just the bits that are solely in that module. There are some handy mnemonics, some really clear and concise explanations and some very funny quips.

The book would work really well as revision guide, and is something students can dip in and out of if they are having trouble with a particular aspect of integration. I think the section on which integration to use is perhaps the most handy bit of the book.

I would wholeheartedly recommend this book as a companion to anyone studying integration, but it should be used as that, a companion. It is written in a way to review stuff already learned and add clarity to areas you are struggling on, rather than as a book for the original teaching of the subject.

I hope Colin is planning an update for the new syllabus, and I would also love to see print copies available.

## Half term revision

Today I spent a few hours at school leading Y13 on a half term revision session. It’s something that I feel I would have shunned in my days as a student, but my students were keen and who am I to quash that enthusiasm.

We spent the time looking at problem topics from C3 and C4, mainly Trigonometric Identities and Integration. They are two of the topics that students struggle with most, but they are favourite two subjects too.

The students did some recalling of facts they needed to know, and i filled in aome gaps, they applied their knowledge to challenging exam questions, both individually and as a class and I modelled some answers on the questions they were really struggling with.

It was an enjoyable day, spending a few hours talking about some of my favourite areas of maths with some great young people who are eager to do.well in the subject. Let’s hope it has a positive effect on their scores.

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