## AS Levels

We are now fully into “Exam season”, Year 11 have their GCSE exams, and Year 13 have their A Levels. Then Year 12 have AS Levels.

AS levels are a weird thing. They are no longer a component part of the A Level, they are very early in the exam session and it seems to me an unnecessary added pressure.

Last year we took a decision as an academy not to enter pur maths students for the AS exams. We did this to maximise pur teaching time and avoid unnecessary stress. This year the decision was taken at trust level to enter them in all subjects.

I can now see two sides of the argument. Last year our students focussed heavily on their other subjects and not maths as they had external exams for those subjects. This meant we lost teaching time and their homework suffered during exam season. This year we have not finished all the content early enough to really focus the revision. I really dont know whats best. I do think, however, that it is important to have a decision made for all subjects.

*Are you entering your students for AS Levels? I’d love to know if you are or not and why you made that decision. You can answer in thw comments or on social media.*

## Proof by markscheme

While marking my Y11 mocks this week I came across this nice algebraic proof question:

The first student had not attempted it. While looking at it I ran through it quickly in my head. Here is the method i used jotted down:

I thought, “what a nice simple proof”. Then I looked at the markscheme:

There seemed no provision made in the markscheme for what I had done. *(Edit: It is there, my brain obviously just skipped past it)* *How did you approach this question? Please let me know via the comm*ents *or social media.*

Anyway, some of my students gave some great answers. None of them took my approach, but some used the same as the markscheme:

And one daredevil even attempted a geometric proof…….

## Another Year Over

*So this is summer, and what have you done, another year over and a six week holiday just begun. – *What Lennon may have written had he been a teacher.

I know what you are thinking, “why are you up so early? It is sunday and it is summer!” And you are right to wonder. Usually its my body clock that makes it so, but this year my 6 year old daughter has taken on that responsibility. Argh.

This year has been a good one for me. Tough in places, but enjoyable over all. I work at a school where I like my colleagues, like the vast majority of the students, feel that the department I work in is strong and that the senior leadership know what they are doing and are making decisions that are pushing the school in the right direction. When I moved to my current school, which was in the process of academy conversion following a 4 Ofsted grading, part of the draw was the chance to be part of affecting a positive change and improving the chances of the students. In the 2 years I’ve been here I’ve seen massive improvements and can see the trajectory we are on.

There’s been some tough times, but there has been some good ones too and I look forward to next year and our next steps in the journey.

This year I’ve spent a lot of time improving subject knowledge amongst the department. I feel this is something that needs to continue. It was made necessary this year as we had a number of non specialists and trainees in the department and most of the experienced maths teachers had never taught the new content that is now on the GCSE. This is something that needs to conrinue next year. We have no non specialists next year, but do have NQTS, trainees and staff who still wont have taught the new content. These sessions allow not only for building content knowledge but also for discussing subject specific pedagogy and possible misconceptions.

I’ve also thought a lot about transition from KS2 to KS3, this has been driven in part by a need to improve this area and in part by a fascinating workshop we hosted led by the Bradford Research School. I hope to write more about the workshop and the fascinating findings I’ve had while looking at KS2 sats data, nationally and locally, and the KS2 curriculum. Suffice to say, if you are a secondary teacher who hasn’t looked, your year 7s probably know a considerable amount more than you think they do on arrival.

The KS2 sats provide some great data and there really is no need to retest students on entry. Except maybe the ones who have no data. I’ve always been averse to KS2 SATS but the data they produce is so rich I feel I’m coming round to them. Although I’m not sure I agree with the way they are currently reported and I certainly stand against the idea of school league tables.

I’ve not written as much as I would have liked on here this year, and I hope to change that going forward. I didn’t decide to blog less, it just sort of happened, so hopefully I can turn that around.

Now it’s summer, I’m looking to relax, have fun and to teach my daughter how to enjoy a lie in….

## Angle problem

Today has been quite a geometric based day for me. I spent a couple of hours solving non-RAT trigonometry problems with year 10 and then a while with year 11 looking at various algebra angle problems. Then I went on Twitter and saw this from Ed Southall (@solvemymaths):

A couple of nice parallel lines questions that I might grow at y11 tomorrow.

Both are fairly straight forward to solve. I looked at the first one, imagines a third parallel line through the join if x and saw x must be the sum of 40 and 60 hence 100.

The second I saw an alternate angle to the 50 in the top triangle and used angle sum of a triangle is 180 to spot that x is a right angle. I glanced down at the responses and saw the vast majority had the same answers as me. That would probably have been the end of it but then I noticed this response:

The same thought process for the first one, but a significantly different approach to the second.

It made me wonder what approach others would take, and which approach my students would take. I wondered if the first problem had led this respondent into this solutions the second, and if so why it hasn’t had the same effect as me.

I don’t know if either approach is better, I just thought the differences were interesting. I’d love to hear your thoughts on it and how you would approach it.

## Reverse percentages and compound interest

The other day a discussion arose in my year 10 class that I found rather interesting. There was a question on interest which incorporated compound interest and reverse percentages. One student was telling the other how to find the answer to the reverse part, “you need to divide it, because it was that amount times by the multiplier to get this amount and divide is the inverse of times.” All good so far, then they discussed how to complete it if it was a reverse of more than one year, “so in that case it’s the new amount dived by the multiplier to the power of how many years.” I was pleased at the discussion so I didn’t really interject.

Then one of them aid, “if I’m looking for two years ago, can’t I just times it by the multiplier to the power -2? Wouldn’t that work.” I thought this was an excellent thought process. The other student disagreed though, sating “no, it has to be divide.” So I thought at this point I’d better interject a little.

“Does it give you the same answer?” I asked. They both thought about it and tried it and discussed it and said yes. So I asked “does it ALWAYS give the same number?” they tried a number of scenarios using different amounts, different interest rates and different numbers of years. Eventually they had convinced themselves. “Yes, yes it is always the same.”

“So is it a valid method then?” I probed. Some more discussion, then one ventured “yes. It must be.”

“Why does it work?”, I then asked. And left them discussing it.

When I came back to the pair I asked if they could explain why it works and one of them said, “we think that it’s because multiplying by a negative power is the same as dividing by the positive version.”

## Oblongs

Last week while we were waiting for a swimming lesson to start my daughter told me that one of her teachers had got “higgledy piggledy” about oblongs. I asked what she meant and she said that she’d accidentally called one a rectangle and had to correct herself and had informed the class that at her last school she’d had to call them rectangles but at this school had to call them oblongs and sometimes got higgledy piggledy about this. I asked my daughter why they couldn’t call them rectangles and she said that it was because squares can be rectangles too.

This set off a lengthy chain of thoughts in my head. Firstly, I was quite impressed by the fact a 5 year old could articulate all this about knowledge about shapes so well. Then I thought, does it really matter whether they call them oblongs or rectangles? Then I thought, wait a minute, why are we prohibiting the use of rectangle because it can also mean a square, but we are not prohibiting the use of oblong when it can also mean an ellipse? My chain of thought then jumped down a rabbit hole questioning whether we should actually be referring to regular or equilateral rectangular parallelograms and non – regular/equilateral parallelograms. Why are we allowing children to call a shape a triangle, when it is one possible type of triangle in a family of triangles, but not allowing them to call a shape a rectangle when it is only one possible rectangle in a family of rectangles. These thoughts stewed around in my head for a while and I thought I’d ask the twittersphere for their opinions on the matter.

These opinions fell into a couple of camps. The first cam thought that oblong was a nice enough word and they didn’t mind others using it but preferred not to themselves. The second camp felt that it was important to distinguish between an oblong and a square so important to use oblong not rectangle and the third camp thought that actually it was better to use rectangles due to the elliptical oblongs. I questioned some of the respondents from the second two groups a little further to see why they fell into these groups. Those in the second seemed unaware that the word oblong also meant ellipse and those in the third thought it was more important to excluded ellipses than squares. Stating that it was easy enough to explain away the special case that is the square.

I’ve spend rather a lot of time considering this, and am now not really sure what I think on the issue. I can’t see a problem with using a rectangle and explaining away the square as a special case. We call all triangles triangles and expand as and when required. No one bothers about calling a non-rectangular parallelogram a parallelogram, despite the fact that that could mean a rectangle. But again I’m not sure I’m massively strongly against the term oblong either. It could open up a good discussion about the term and how it could apply to ellipses, although this probably is a little too much for a year 1 classroom. I think I’m leaning towards rectangle as a preference though, as explaining away a special case is, for me, much more preferable than ignoring a whole class of oblongs.

*If you have views on this, whichever way you lean, I’d love to hear them, either in the comments or via social media.*

## Simultaneous Equations

It’s been a while since i last wrote anything here. Which says more about how busy I’ve been than my desire to write, but I hope to start writing more regularly.

This week I was teaching simultaneous equations and a student asked a question that made me think about things so I thought i would share.

I was teaching elimination method and I had done some examples with the coefficients of y having different signs and I put one on the board with the same signs and asked the class to think how we may go about solving. One of the students in the class put uo his hand after a while and said he thought he had solved it.

5x + 4y = 13

2x + 2y = 6

I asked hime to talk us through his thinking and he said “first I multipled the bottom equation by -2”

5x + 4y = 13

-4x – 4y = -12

“then I added the equations as before”

x = 1

“Then I subbed in and solved.”

2 + 2y = 6

2y = 4

y = 2

“so the point of intersection is (1,2)”.

This wasn’t what I was expecting. I was expecting him to have spotted we could subtract instead, but this method was clearly just as correct. It wasn’t something I had considered as a method before this, but I actually really liked it as a method and it led to a good discussion with the class after another student interjected with her solution which was what I expected, to multiply by 2 and subtract.

It was a great start point to a discussion where the students were looking at the two methods, and understanding why they both worked, the link between addition of a negative and subtracting a positive and many more.

I was wondering, does anyone teach this as a method? Have you had similar discussions in your lessons? What do you think of it?## Share this via:

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