## An Interesting Conics Question

My AS further maths class and I have finished the scheme for learning for the year, leaving oodles of time for review, recap and plenty of practice. Currently we are revisiting topics that they either didn’t score too well on the last mock in or that they have requested we look at again due to lacking confidence.

Today we were looking at conic sections, by request of the students, and this past paper question caught my eye:

We had recapped the topic quickly together then the students attempted so past paper questions. This one was one they rook to well, all competing the first 2 sections quickly and without trouble. Likewise, part c was mostly uneventful – bar one or two silly substitution errors and someone missing a difference of two squares factorisation.

Then they got to part d.

This caused them a little bit of worry so we worked together on it as a class, after we’d made sure everyone had a, b and c right.

Here’s what the board looked like when we’d done:

I asked the class what we should do to start, one suggested drawing a diagram *all this nagging about always drawing a diagram, especially if your stuck is paying off! *We drew it, but it didn’t help much:

Then one said, “if they’re parallel then one gradient is -1 over the other” – I refrained from scolding and calmly said “indeed, those gradients will be *negative reciprocals *of each other”. I think asked what the gradients were and quickly had “y1 – y2 over x1 minus x2” thrown at me.

So we worked out the gradient of the line join in n to the origin:

Then the gradient of PQ:

One then suggested we put the equal to each other, but he was corrected by another student before I could react. So we set one equal to the negative reciprocal of the other and solved:

A lovely question, with a lovely neat answer and a load of fun algebra on the way. I enjoyed watching the students tackle it and was glad I didn’t need to put too much input in myself.

We then discussed the answer and the class expressed surprise that 1 was the final answer and that a complicated algebraic journey could end so simply. It was a nice discussion and a nice way to start the day, and the week.

*This post was cross-posted to One Good Thing and Betterqs*

## Travelling Salesman

This year we are doing the module “Decision 2”, D2 for short. And I’m really, really enjoying teaching it. This week’s topic has been the Travelling Salesman problem, which is a fantastic springboard into a whole host of other areas of maths. When we looked at route inspection during the last module I made mention of the traveling salesman problem and the fact there is no known way to solve it in a reasonable amount of time and I briefly mentioned P vs NP and the millenium prize.

When we started this week with a slide that had the title on the class were automatically hooked. They had been eager to reach the traveling salesman and had even looked up P vs NP and the aforementioned millenium prizes. This meant before we even started d the lesson we had an awesome conversation about these amazing unsolved mathematical problems, with the class telling me what they had read and what they thought they understood of it and me filling in the gaps around it a bit and linking to other areas of maths.

Towards the end of this discussion one asked “but what are we going to study? We all already know it can’t be solved quickly enough for an exam!” Which led me onto the discussion of lower and upper bounds and optimal regions, and how we can find a good solution (within 1% of an optimal solution) within a reasonably short time.

This left around enough time to discuss least differences and tackle the nearest neighbour algorithm for an upper bound. The following lesson we looked at using minimum connectors for upprbounds and how we could identify the best upper bound. Then we looked at lower bounds, and how to identify of we had found an optimal solution or an optimal region. I do hope TSP makes it into the optional content of further maths when the new specification starts.

*This post was cross-posted to One Good Thing here.*

## A conic extention

This week has been a strange one, I’ve been trying to shake an illness but it keeps on getting worse, and my lessons have been disrupted a little bit by mock examinations. This, however, has given me a chance to work.woth small groups in some classes and to see how some classes are getting on with their courses.

In one of further maths lessons a student who takes maths but not further maths asked if he could.sit in the classroom and revise for his maths, asking if he needed help, I said it was fine and started the lesson,which was a review of the conic sections topic and an assessment sheet to identify any areas of weakness that may be apparent. I gave the none further maths student a spare copy of the assessment sheet, told him it was basically C1 skills but applied in a much more algebra heavy context, and asked him to have a crack.

Not only did he have a good crack at it, he answered it near perfectly. I was extremely pleased with his resilience in working through a question that has way more algebra than anything he’s looked at before and was glad he could make the links to the C1. His thoughts on the question were interesting, and I think they allowed the rest of the class to see more clearly how the conic section of FP1 fits with, and build on, the coordinate geometry sections in C1 and 2.

I think that in future I will use these parabola and hyperbola questions with all my high attaining AS maths students.

*This was cross-posted to the “One Good Thing” blog here.*

## Newton Raphson

It’s Christmas! An important day because it marks the anniversary of the birth of a man who changed the world! That’s right, Isaac Newton! *(Ok, now I know at least one of you is about to comment with “but since the calendar changed from the Julian to the Gregorian his birthday should be moved to January 6th. I can see your point, but no one suggests moving Christmas so why would we move Newton’s birthday? Anyway, I tend to celebrate both.)*

Recently a couple of people have asked me about the Newton – Raphson method for finding roots of equations, and why it works, so in keeping with the festive spirit here is a brief overview of why. Incidentally, I’m working this up into a help sheet for my further maths class so any additional input would be great.

**Newton Raphson**

A nice little numerical method if finding the root of an equation. You start with an approximation (often referred to as Xo) and then you take away the ratio of f (Xo)/f’ (Xo) to get a better approximation. You keep going until you get an approximation which is correct to a suitable degree of accuracy.

**But why? What is this witchcraft and why does it work?!**

No, it’s not witchcraft, and it’s relatively simple and based on our old favourite “right angled triangle trigonometry” *see, I told you triangles were the saviours of everything….*

Let’s start with a sketch *(always a good start)*:

Here is a curve, as you can see I’ve drawn a tangent to it at C, a perpendicular from C to the x axis (which meets it at B) and labelled the point where the tangent intercepts the x axis as A. Already we have our right angled triangle!

Now we all know that f’ (Xo) or f'(B) will give us the gradient of the tangent to the curve at that point(C in this case). The gradient of that line is the same as the tangent ratio of the angle CAB, as you can see from the sketch (as the gradient is difference in y / difference in x). The opposite side in this case is f (Xo) so the adjacent side is f (Xo) / f’ (Xo) – ie is the opposite side over the tan ratio.

This shows us what’s going in here, we are taking away the adjacent side each time and getting closer to the actual root (when it does converge that is!).

So, a festive look at Newton Raphson. Merry Christmas.

## How are we questioning our students?

This month’s maths journal club is based in the article “Contrasts in mathematical challenges in A – level mathematics and further mathematics, and undergraduate mathematics examinations.” By Ellie Darlington

I found the article quite interesting overall. It looks at the differences in examination questions between A level mathematics and undergraduate mathematics. It starts off with the idea that A level mathematics is tested in a manner that involves routine questions and that as such this doesn’t prepare students for undergraduate mathematics, which it presumes is tested in a higher level. I think this is one of the issues with A level mathematics and I hope that when the new curriculum appears this will have been addressed. The problem is even worst at the transition point between GCSE and A level though, but again, I have hope that the new specification will address this.

Interestingly, my own experience of undergraduate mathematics was that there were a lot of courses that were tested in a routine manner, and that learning the lecture notes by rote and practicing the past papers for a course could allow people to score well despite not understanding what was going on and not being able to apply their knowledge in other contexts. There were some of my peers who had no conceptual understanding of some of the modules yet still scored high enough to achieve firsts.

That said, I still feel that the procedural nature of the GCSE and A level papers is a massive problem. In recent years we have seen a change in the A level papers towards questions that are not answerable in a routine manner, but it needs to go even further.

There are many problems with these procedural questions. My main issue is they allow students to score well without understanding the mathematics behind the questions. This in turn can allow teachers to skip teaching for a relational understanding and just teach an instrumental or procedural understanding, which lets down the learners, especially if they are hoping to go into mathematics or another mathematical based subject at higher education.

**So what can we do?**

Well, rather than waiting for the changes we can be implementing these questions in our classrooms, ensuring that we are teaching for relational, or conceptual, understanding rather than teaching purely procedures. Take the time to ask the questions that require application in new contexts. Take the time to teach the concepts, the why behind the what. Enrich the curriculum with tasks that involving thinking outside the box and questions framed in a way that the correct method isn’t always immediately obvious, perhaps try some of these puzzles?

**Other points in the article**

There was a lot early on that I thought I already knew, but it was nice, and useful, to see references and studies to back up some of the ideas.

The MATH taxonomy in this explicit form is new to me and I’m interested to look further into it and see how I can apply it myself.

I was a little purplexed to see that the article stated that questions can change their position on the MATH taxonomy with time, but then have no explanation of how these questions were classified in the research.

*All in all a very interesting read that I will re-read and digest in more detail later. I’d love to hear your thoughts on it also.*

*This post was cross posted to the BetterQs blog here *

## The small things

This week has been hellish. I’ve felt more ill than I can ever recall feeling but because we had an external review on Thursday I dragged myself in each day, compounding the tiredness and the run down feeling that goes hand in hand with feeling crap. Today was Friday, the end of the week and the chance to try sleep some of the grogginess off was in the offing so I felt positive about it. However a couple of incidents of poor in periods 2 and 4 managed to take the sheen off it a little.

But then period 5 arrived. Period 5 Friday is one of my favourites as I teach year 12 further maths, and today we were talking topology. We were looking at route inspection problems and discussing Euler in great depth. The discussion wandered to the traveling salesman problem and how it can be mapped do all NP problems, meaning that a solution to the traveling salesman problem able to sole any in polynomial time would be enough to prove that P = NP *ie that any problem that’s easy to verify can be solved quickly also* and earn the mathematician in question a million dollars.

This piqued their interest no end and we had a nice discussion around the millennium prize problemsand the other great unsolved maths problems. This also led onto a discussion of Fermat and his last Theorem. It was really great to see young people so enthused about mathematics and the different types of problems involved, and to see them trying to get their head round such complex ideas as the Riemann zeta function.

On top of this amazing fact they also completed some great work collaborating on a set of topological problems I’d set them. I ended the day felling extremely positive, and despite the week I’ve had I can’t wait to get back into it on Monday, although I do hope I feel better by then!

*This post was cross posted on the blog “One good thing”, here.*

## Flipping the classroom

So flipped classes are something I’ve read a lot about over the last few years, I’ve seen many people who do it claim fantastic results but I’ve always been a tad sceptical about the process. The reasons for this scepticism have been that often in advanced mathematics the topics are really hard, and that I work in a school where I often have to chase homework which could potentially derail the whole flipped class process.

During the half term just gone a colleague and I were discussing the merits and worries of flipped classes and decided to try it on the small KS5 classes we share. We provided them material to prepare for lessons. For Y13 the topic was Integration and for the Further maths class the topics covered were Traveling Salesman, Transportation problems and the hungarian algorithm.

The set up of the lesson was such, the class would arrive and complete a check in question based in the previous lesson, then we would discuss the preparatory material to draw out the understanding the class had manged to gain from it, answer as a group any questions that any had drawn put of it (with me or my colleague only inputting if no one could help) and then looking to apply these skills in an exam context.

My fears about the classes not doing the work have been unfounded, they all completed each bot of prep. Although these groups are small and all are very committed to doing well in maths, so I still have these concerns regarding this.

My fears about the difficulty were also unfounded. It’s true that, for the most part, the students would not have been able to go straight into answering questions on the skills the preparatory material covered, but they had gained enough of an understanding to discuss the topic and they had identified the areas they didn’t understand, allowing the lesson to focus on this, rather than cover everything. The only lesson that was met with entirely blank faces was the lesson on volumes of revolution, but through their misinterpreted ideas of it I was able to focus in on misconceptions that had arisen from prior knowledge and correct that as well as teaching them about solids of revolution.

At the end of the half term we checked the student voice and they were all positive about the process and wanted to continue in this manner for the rest of the year, we will be looking again at student voice at the end of the year and the results to determine whether we want to roll this out across the key stage, but so far the results look extremely positive.

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