## Proving Products

Just now one of the great maths based pages I follow shared this:

So naturally I figured I would have a go. I thoughts just get stuck in with the algebra and see what happens, normally a good approach to these things.

My first thought was that if I use 2n – 3, 2n -1, 2n + 1 and 2n +3 then tgere would be less to simplify later. I know that (2n + 1)(2n – 1) = 4n^2 – 1 and (2n – 3)(2n + 3) = 4n^2 – 9 so I multiples these together.

(4n^2 – 1)(4n^2 -9) = 16n^4 – 40n^2 + 9

I thought the best next move would be to complete the square:

(4n^2 – 5)^2 – 16

This shows me that the product of 4 consecutive odd numbers is always 16 less than a perfect square and as such that the product of 4 consecutive odd numbers plus 16 is always a square.

(4n^2 – 5)^2 – 16 + 16 = (4n^2 – 5)^2

A nice little proof to try next time you teach it to your year 11s.

## Consultation time again

Is it cynical of me to question the DoE’s repeated tactic of releasing consultations either just before the summer, when most teachers are in the midst of high stakes exam testing, or over the summer when a lot of teachers are either away or spending time catching up with their families who they haven’t seen through the heavy term time?

Anyway, this year they have released another one. It focusses around the new GCSEs, and more specifically the awarding of grades. The consultation states that for the first award there will be a heavier reliance on statistical methods to set the grade boundaries, allowing the same proportion of grade 4s as we currently have of grade Cs, likewise similar proportions of 1s to Gs and of 7s to As. The rest will be split arithmetically ie the boundaries in between will be equally spread. From Year 2 onwards it will revert back to examiner judgement, but use the statistical analysis as a guide as well as the national reference tests.

This immediately raises questions – how do we know that the first year to sit it should have a similar proportion of 4s as Cs? It seems that this has been decided without much thought about the prior attainment; the consultation certainly doesn’t mention it for the first year. It does going forward, but that doesn’t really explain how this prior attainment will be measured. I have been under the impression that the KS2 SATs are moving from level based assessments to assessments where the students’ scores will be reported as percentiles – surely then comparisons of prior assessment will always be the same? “This year, bizarrely, we saw exactly 10% score above the 90th percentile, what’s more bizarre is that is exactly the same proportion as last year!”

It seems strange to me to put such a heavy reliance on these prior attainment targets anyhow. We live (for now) in a society that has a fairly fluid immigration system, so the students who get to year 11 haven’t always been through year 6 in this country. There is also a question of the validity of the assumption that every year group will progress over the 5 years of secondary at the same rate.

The obvious elephant in the room is floor targets. By setting the boundaries so the same proportion of students get above a grade 4 as get above a C, but switching the threshold to a grade 5 you immediately drop the results of a whole host of schools down, what happens then remains to be seen, but I can imagine lot of departments will become under pressure and scrutiny for something that is statistically inevitable given the new grading formula.

This is all interesting, but it’s not much different to previous announcements and consultations, what is different is the formula for awarding grades 8 and 9. The formula looks to be a fair way of doing it, but it seems strange to me to use this formula just for the first year. Why then revert to examiner judgement about the grade standard? The government seem to be happy to use statistical analysis and similar grade proportions in parts of their grading system, but not in all of it, and that seems odd to me.

Have you responded yet? If not you can here (but hurry, the consultation closes June 17th). I’d love to hear other people’s views either in the comments or via social media.

## Forming and Solving Equations

While checking the work of a year 11 student on Friday I came across a question that could have been a great one for the higher GCSE students to practice their skills together and also their selection of which mathematics to use.

The question was to find the area of this triangle:

A great question. One that to you or I is straightforward but that would take GCSE level students and below a bit of thinking and let’s them hone their skills.

The way to tackle it is to use Pythagoras’s Theorem to form an equation, solve for x then find the area. I feel is beneficial as it combines Pythagoras’s Theorem with a decent amount of algebra then includes the find the area bit at the end.

In this case though, that wasn’t the question. There was more information on offer and the question was:

Which is still a fairly nice form and solve an equation problem.

*3x + 1 + 3x + x – 1 = 56*

*7x = 56*

*x = 8*

*A = 0.5×7×24 = 84*

There is a niceness to this question that goes beyond the question itself. It shows us a great way of differentiating within lessons. Just be leaving out a tiny portion of the information, in this case the perimeter, we can make the question much harder. This idea is something I’ve been working on in various places. M1 questions can be made much easier by providing a diagram, for example.

*Have you used questions in a similar way? If so I’d love to see them, please do get in touch.*

*Cross-posted to Betterqs here.*

## Is one solution more elegant?

Earlier this week I wrote this post on mathematical elegance and whether or not it should have marks awarded to it in A level examinations, then bizarrely the next day in my GCSE class I came across a question that could be answered many ways. In fact it was answered in a few ways by my own students.

Here’s the question – it’s from the November Edexcel Non-calculator higher paper:

I like this question, and am going to look at the two ways students attempted it and a third way I think I would have gone for. Before you read in I’d love it if you have a think about how you would go about it and let me know.

**Method 1**

Before I go into this method I should state that the students weren’t working through the paper, they were completing some booklets I’d made based on questions taken from towards the end of recent exam papers q’s I wanted them to get some practice working on the harder stuff but still be coming at the quite cold (ie not “here’s a booklet on sine and cosine rule, here’s one on vectors,” etc). As these books were mixed the students had calculators and this student hadn’t noticed it was marked up as a non calculator question.

He handed me his worked and asked to check he’d got it right. I looked, first he’d used the equation to find points A (3,0) and D (0,6) by subbing 0 in for y and x respectively. He then used right angled triangle trigonometry to work out the angle OAD, then worked out OAP from 90 – OAD and used trig again to work out OP to be 1.5, thus getting the correct answer of 7.5. I didn’t think about the question too much and I didn’t notice that it was marked as non-calculator either. I just followed his working, saw that it was all correct and all followed itself fine and told him he’d got the correct answer.

**Method 2**

Literally 2 minutes later another student handed me her working for the same question and asked if it was right, I looked and it was full of algebra. As I looked I had the trigonometry based solution in my head so starter to say “No” but then saw she had the right answer so said “Hang on, maybe”.

I read the question fully then looked at her working. She had recognised D as the y intercept of the equation so written (0,6) for that point then had found A by subbing y=0 in to get (3,0). Next she had used the fact that the product of two perpendicular gradients is -1 to work put the gradient of the line through P and A is 1/2.

She then used y = x/2 + c and point A (3,0) to calculate c to be -1/2, which she recognised as the Y intercept, hence finding 5he point P (0,-1.5) it then followed that the answer was 7.5.

A lovely neat solution I thought, and it got me thinking as to which way was more elegant, and if marks for style would be awarded differently. I also thought about which way I would do it.

**Method 3**

I’m fairly sure that if I was looking at this for the first time I would have initially thought “Trigonometry”, then realised that I can essential bypass the trigonometry bit using similar triangles. As the axes are perpendicular and PAD is a right angle we can deduce that ODA = OAP and OPA = OAD. This gives us two similar triangles.

Using the equation as in both methods above we get the lengths OD = 6 and OA = 3. The length OD in triangle OAD corresponds to the OA in OAP, and OD on OAD corresponds to OP, this means that OP must be half of OA (as OA is half of OD) and is as such 1.5. Thus the length PD is 7.5.

**Method 4**

This question had me intrigued, so i considered other avenues and came up with Pythagoras’s Theorem.

Obviously AD^2 = 6^2 + 3^2 = 45 (from the top triangle). Then AP^2 = 3^2 + x^2 (where x = OP). And PD = 6 + x so we get:

*(6 + x)^2 = 45 + 9 + x^2 *

x^2 + 12x + 36 = 54 + x^2

12x = 18

*x = 1.5*

Leading to a final answer of 7.5 again.

Another nice solution. I don’t know which I like best, to be honest. When I looked at the rest of the class’s work it appears that Pythagoras’s Theorem was the method that was most popular, followed by trigonometry then similar triangles. No other student had used the perpendicular gradients method.

I thought it might be interesting to check the mark scheme:

All three methods were there (obviously the trig method was missed due to it being a non calculator paper). I wondered if the ordering of the mark scheme suggested the preference of the exam board, and which solution they find more elegant. I love all the solutions, and although I think similar triangles is the way I’d go at it if OD not seen it, I think I prefer the perpendicular gradients method.

*Did you consider this? Which way would you do the question? Which way would your students? Do you tuink one is more elegant? Do you think that matters? I’d love to know, and you can tell me in the comments or via social media!*

Cross-posted to Betterqs here.

## Mathematics for all?

As part of George Osborne’s budget statement today he made some comments about mathematics education. He said that they would look into teaching mathematics to 18 for all pupils. This has caused a lot of discussion on twitter and the treasury have since clarified that by “looking into teaching mathematics to 18 for all” he actually meant “look to improve a level teaching” – why he didn’t just say that is beyond me….

The bigger debate that seems to have opened is whether mathematics should be taught to all. There seems to be people in both camps on this one, and it’s something I’ve thought about many times.

Some of the arguments for it that I read suggest that for non a level students this would be a great time to learn about the life skills. I would argue that that’s not actually mathematics, it’s more numeracy. And I’ve often thought that they should be taught as distinctly different subjects, with numeracy a core subject and mathematics one that is chosen as an option from KS4 onwards. I sometimes think this would be a great idea, strip back the core curriculum entirely to just numeracy, literacy and citizenship, leaving a wide range of options and a lot of time in the timetable to build truly bespoke schooling. Students could study academic or vocational qualifications and perhaps we could get both right. However I realise this would be a logistical nightmare, and I worry massively that 14 year olds would be picking things that defined the rest of their life, so the other part of me thinks actually we should be prescribing a broad curriculum giving everyone a fair grounding and allow them to choose at 18 what to specialise in.

**But what about in our current situation?**

Given the situation we have at pre 16, I started to think about the idea of compulsory maths to 18. Clearly making A level maths compulsory won’t work. I’m told that around 50 % of those who attempt it with a grade B fail in Y12, that’s a massive amount of students we would be setting up to fail, and that’s not counting the A grade students who can’t handle the step up or the C grade students who wouldn’t have a strong enough grounding in algebra to succeed.

**What about core maths?**

I’ve been teaching this as part of the early adopters programme and I am quite impressed by the qualification. We do the AQA version and I’ve found the specification has enough stuff that fits the “life skills” heading to cover that aim of it while also having some more mathematical elements. The optional papers give the option of creating a course that fits the needs of each student best, and I’m looking forward to continuing teaching it and seeing it develop.

**But should it be compulsory?**

Again, I’m torn on this, I can see that the life skills bits would be good for anyone to learn. On top of that the other bits offer help with a vast range of other subjects and future job roles and help build logical thought, all of which I feel would be a good argument for making it compulsory. But it eats into the time they could be spending working on the things that are really important to them and the qualifications that they directly need to move to the next stage of their lives plan.

One thing I find ill thought out about the qualification is the 2.5 hours a week for 2 years suggestion. The idea was that it was to ease the burden and to spread it out, however I found that students were disengaged around exam time as it was the only subject they weren’t examined in. We also lost a lot of candidates after year 1 as they secured apprenticeships and basically had a years working without any sort of credit. We think going forward that it is better suited as a 1 year 5 hours a week course, perhaps students could do core maths in Y12 followed by EPQ in Y13? This would mean, however that the objective of keeping students in maths education to 18 was no longer being met.

I certainly agree with the compulsory resitting of GCSEs up to 18, although the previous comments around Maths and numeracy are certainly highlighted in this issue too.

*As you can probably tell, I have conflicting views on a lot of this, and I’m still trying to.make sense of them. I’d love to hear your views on this. Do you thing all students should have to do maths to 18? Do you think they even need to do it to 16 or should we split maths and numeracy? What are your views on the idea of a stripped bare curriculum where students build their own? Would you have the same 3 core subjects as me, or different ones? Or would you prefer my other idea of a broader curriculum where students are a bit older by the time they need to make those massive decisions? Please let me know in the comments, via social media or email.*

## Some interesting questions on the new maths GCSE

I’ve written before about the SAMs (Sample Assessment Materials) for the new GCSE, and currently we are swaying towards Edexcel. We have recently given year ten the SAMs to see how they got on with them and a couple of questions that stood out for me. First was this one:

Students need to find change, fair enough, but them part b seems to be purely testing their understanding of the word “expensive”. This seems a really bizarre question in my opinion, and I’m not sure it fits well on a maths exam. It’s not even really a mathematical term.

Another that stood out was this:

I think this one is a great question that approaches the assessment of fractions knowledge in a new way, it requires a deeper analysis but I do think there is a limitation to it if it isn’t thought out. If this type of question is regularly asked about fractions, then it becomes a “when they ask this you say this” sort of question. This could be combated by asking this type of question about different topics. It’s certainly a question I enjoyed seeing, and is much better to assess deeper learning than he current GCSE, particularly the foundation tier.

The final question that caught my eye was this one:

It’s similar to the type of question on forming and solving equations we see how, but the interesting bit is the additional bit of reasoning students need to apply at the end, ie to work out if the amount of marbles that Dan and Becky have together is odd or even to work out if they can have the same amount.

*Have you noticed any interesting questions cropping up? Have your students attempted the SAMs? If so, how did they get on? I’d love to hear.*

*Cross-posted to Betterqs here.*

## Isosceles triangles and deeper understanding

When marking paper 3 of the Edexcel foundation Sample Assessment Materials recently I came across this question that I found interesting:

It’s a question my year tens struggled with, and I think it is a clear marker to show the difference between the current specification foundation teir and the new spec.

The current spec tends to test knowledge of isosceles triangles by giving a diagram showing one, giving an angle and asking students to calculate a missing angle. This question requires a bit of thinking.

To me, all three answers are obvious, but clearly not to my year 10s who do understand isosceles triangles. The majority of my class put 70, 70 and 40. Which shows they have understood what an isosceles is, even if they haven’t fully understood the question. They have clearly mentally constructed an isoceles triangle with 70 as one of the base angles and written all three angles out.

What they seem to have missed was that 70 could also be the single angle, which would, of course, lead to 55 being the other possible answer for B. One student did write 55 55 70, so showed a similar thought process to most but assumed a different position for the 70.

I already liked this question, and then I read part b:

Now students are asked to explain why there can only be one other angle when A = 120. Thus they need to understand that this must be the biggest angle as you can’t have 2 angles both equal to 120 in a triangle (as 240 > 180), thus the others must be equal as it’s an isoceles triangle.

The whole question requires a higher level of thinking and understanding than the questions we currently see at foundation level.

In order to prepare our students for these new examinations, we need to be thinking about how we can increase their ability to think about problems like this. I think building in more thinking time to lessons, and more time for students to discuss their approaches and ideas when presented with questions like this. The new specification is going to require a deeper, relational, understanding rather than just a procedural surface understanding and we need to be building that from a young age. This is something I’ve already been trying to do, but it is now of paramount importance.

There is a challenge too for the exam boards, they need to be able to keep on presenting questions that require the relational understanding and require candidates to think. If they just repeat this question but with different numbers than it becomes instead a question testing recall ability – testing who remembers how they were told to solve it, and thus we return to the status quo of came playing and teaching for instrumental understanding, rather than teaching mathematics.

What do you think of these questions? Have you thought about the effects on your teaching that the new specification may have? Have you any tried and tested methods, or new ideas, as to how we can build this deeper understanding? I’d love to hear in the comments or social media if you do.Further Reading:Teaching to understand – for there thoughts in relational vs instrumental understanding

More thoughts on the Sample assessment materials available here and here.

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