## A lovely angle puzzle

I’ve written before about the app “Brilliant“, which is well worth getting, and I also follow their Facebook page which provides me with a regular stream questions. Occasionally I have to think about how to tackle them, and they’re excellent. More often, a question comes up that I look at and think would be awesome to use in a lesson.

Earlier this week this question popped up:

What a lovely question that combines algebra and angle reasoning! I can’t wait to teach this next time, and I am planning on using this as a starter with my y11 class after the break.

The initial question looks simple, it appears you sum the angles and set it equal to 360 degrees, this is what I expect my class to do. If you do this you get:

*7x + 2y + 6z – 20 = 360*

*7x + 2y + 6z = 380 (1)*

I anticipate some will try to give up at this point, but hopefully the resilience I’ve been trying to build will kick in and they’ll see they need more equations. If any need a hint I will tell them to consider vertically opposite angles. They should then get:

*2x – 20 = 2y + 2z (2)*

*And*

*3x = 2x + 4z (3)*

I’m hoping they will now see that 3 equations and 3 unknowns is enough to solve. There are obviously a number of ways to go from here. I would rearrange equation 3 to get:

*x = 4z (4)*

Subbing into 2 we get:

*8z – 20 = 2y + 2z*

*6z = 2y + 20 (5)*

Subbing into 1

*28z + 2y + 6z = 380*

*34z = 380 – 2y (6)*

Add equation (5) to (6)

*40z = 400*

*z = 10 (7)*

Then equation 4 gives:

*x = 40*

And equation 2 gives:

*60 = 2y + 20*

*40 = 2y*

*y = 20.*

From here you can find the solution x + y + z = 40 + 20 + 10 = 70.

A lovely puzzle that combines a few areas and needs some resilience and perseverance to complete. I enjoyed working through it and I’m looking forward to testing it out on some students.

*Cross-posted to Betterqs here.*

*
March 23, 2016
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Categories: Commentary, Curriculum, Education Policy
Academies, Commentary, Education, Education Policy, White Paper
March 16, 2016
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Categories: #MTBoS, Assessment, Commentary, Curriculum, Education Policy, Maths, Teaching
Core Maths, Curriculum, Education, Education Policy, GCSE, Maths, Post 16, Teaching
March 14, 2016
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A level, Conic sections, Conics, Further Maths, Hyperbola, Maths, Teaching
March 12, 2016
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A level, Activity Networks, Core Maths, Critical Path Analysis, Decision, Graph Theory, Maths
March 9, 2016
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March 7, 2016
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A level, Exponentials, Logarithms, Maths, Teaching
March 1, 2016
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Categories: #MTBoS, Commentary, Teaching
Commentary, reflection, Teaching
*

## Academies, Local Authorities and a Research Based Profession

Today I finally had time to sit and look through the government white paper “Educational Excellence Everywhere”. A catchy title I thought, and I was interested to read what it actually said. I didn’t get chance to read all 150 pages – I will – but I did get to read the first chapter, and I thought I’d frame some initial thoughts.

**A fantastic aim**

In the foreword Nicky Morgan states that *‘Access to a great education is not a luxury, but a right for everyone.” *– Definitely a sentiment I agree with, and certainly ine James Kier Hardie would be proud to hear espoused by a conservative politician, but not one that has always been an obvious policy driver over the last six years.

**Academisation and Local Authorities**

The white paper continued in this way, setting out an idealistic vision, but in the early stages not much was said about how this would be achieved. There was a lot of talk on the forced academisation of all remaining local authority schools.

There were some qualifying statements about Local Authorities (LAs). The government are hoping to keep the current experience and envision those who run LAs to go and work for academy chains. This fits the Conservative ideology of small state, bigger private sector, and seems to hint that this ideology is the driving force.

They also claim that moving school control from LA control will give greater accountability, as those elected can’t are there to further the interests.pf their constituents and they apparently can’t do this when LAs control schools. This is a nonsensical argument and the reality is in fact the complete opposite. When schools are under local authority control they are run by officers of the local authority who are answerable to elected members. Thus they HAVE to respond swiftly and allay concerns. Academy chains have no such in built accountability to the elected members and hence the electorate.

LAs will focus their role on core functions. These will be – ensuring all have school places, acting as champions for children and families and ensuring the needs of the vulnerable are met. It’s the third one that worries me. Currently local authorities provide a great deal of support to vulnerable children through Ed Psychology, CAMHS, and a whole host of other services and agencies. In the new world of tiny LA budgets, how will they afford to keep up this level and meet this core function?

**Teachers, Training and Research**

The next section turned it’s attention to teachers. I was a little worried that this white paper seems to ignore the recruitment and retention crisis we are experiencing, and the idea of placing responsibility for accrediting teachers into heads hands worries me. I’m certain that for the vast majority this would be fine, but I have heard some terrible horror stories about bullying from heads, particularly in the primary sector, and to give more power to do this worries me.

There was an extremely positive line on ITT content though:

*“We’ll ensure discredited ideas unsupported by firm evidence are not promoted to new teachers.*

So no more VAK pushed on unsuspecting ITT students! This is part of a wider drive to get more teachers to engage with research and development a research based profession. This is an idea I am fully behind, but with the caveat that we need to include training on how to engage with research. Every class in every school is a different context. Just because research shows something works some places doesn’t mean it will work everywhere, there are no magic bullets, no snake oils. We can take ideas from research and try them, but we have to adapt them to our own contexts and be able to see when things are just not working.

*There are my initial thoughts on the first chapter. Some positives, some worries, and some signs that we are in the process of full privatising our education system. What are your thoughts on the ideas mentioned 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.*

## 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*

## Activity Networks and Critical Paths

When I first came to teach Activity Networks and Critical Paths I couldn’t really remember being taught them. I knew I had studied critical path analysis, but my memories of them were no less vague than that. In order to teach them I refreshed my memory using an edexcel textbook, as it was the board we used, and I got to grips with the “on arc” method.

This year I’ve been teaching a class as part of the core maths early adopters programme and we have selected the AQA qualification. For our optional content we chose the critical path option (option 2b). When browsing the mark scheme for the specimen paper I saw this:

Apparently the “on node” method. When I first looked at it it seemed strange. I had a good think about it and once I’d got my head past the fact that the “late time” was the latest an activity could finish, rather than the latest an activity could start – as in the “on arc” method, I think I actually prefer it.

The thing my students find hardest every year is drawing an activity network, especially where dummies are involved, and it seemed to me that this “on node” method would be much easier in this respect, and that dummies wouldn’t need to be considered. I did worry that actually identifying critical activities, and hence the critical path, might be a little more difficult, but I still didn’t think it would be too taxing.

So when I came to teach my year 12 core maths class activity Networks I went with this “on node” method. I’ve never known a class get the hang of drawing activity networks so quickly, and as this was a core maths class, rather than an a level class, their GCSE grades are much lower – mainly Cs compared to mainly As.

They also didn’t have a problem with the early and late times or identifying critical activities. I much prefer this method.

It got me thinking about the validity of each method and whether either would be allowable in an exam. I feel that both methods are valid and that we should be teaching how to solve these problems using maths, so each method should be allowable, but I’m not sure whether the exam boards agree.

Close inspection of the specimen mark scheme for AQA core maths paper 2b certainly implies that either method is allowable:

However a look at the Edexcel markscheme doesn’t:

If you look at the question paper and answer book there is in fact the heavy implication that only one method will suffice, as there is an explicit mention of dummies and the network is already set up for this method to be filled in:

*Do you have a strong preference to either method? Do you think exam boards should be prescribing which method to use? Do you have any further insights into which method the other exam boards favour or prescribe? If so, I’d love to hear about it.*

## 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.*

## An interesting approach to Logarithms

I’ve just been marking some C4 past papers and one students response to part a of this one caught my eye:

The young lady in question had shown real resilience when a number of approaches failed, she then got to 4x = xe^×/2 and took natural logarithms of both sides. At first I thought she had stopped there, but I turned over and found this:

Not the neatest approach, certainly not the fastest way to reach the answer, but valid mathematical reasoning leading to the correct answer. It shows resilience, perseverance and a real understanding of exponentials and logs. I love this answer and think that it shows a really interesting approach to the question.

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

## Questioning Authority

*This post was first published here, on Labour Teachers, on 28th February 2016.*

In life there are often many ways to reach a certain place. This is true in all aspects. You can take different routes when you travel to get to the same destination, you can solve trigonometric equations a number of ways, you can take many different routes into becoming a teacher, I’m sure there are plenty of examples from other subjects too.

Recently I was reflecting on the way I teach certain topics, and which method to complete them I think is best. This reflection led to a wider discussion and many of the contributers from that wider discussion stated that they preferred the method that they were taught at school. I found this interesting and it got me thinking about things.

I think there are a lot of instances where I am guilty of this. There are certainly topics that I have a preferred method for completing that stems back to the way I was taught, I’ve often suggested that I feel the PGCE is the best route into teaching – and it’s the route I took. I remember discussing with a friend the make up of exams when I was at uni and professing the belief that I thought a linear course was preferable at GCSE and a modular approach was preferable at A level – perhaps this discussion should have suggested we’d both end up.as maths teachers! There are even examples in wider life when I drive home from a friend who lives near my old school’s house I always take a particular route that passes my parents, and I think it’s because that’s the route my mum always drove me when I was a kid.

I have questioned all of these ideas to some extent. I now believe that different routes into teaching will suit different people and each has its merits and downsides. I now belive that a linear approach is preferable at GCSE and A level. I’m starting to use a different route from friends house that is actually quicker and my reflections gave led me to begin to question the methods I prefer.

It struck me as a blindingly obvious thing to do, but one that I had missed. I’ve become very good at challenging things I read and critically evaluating new things, it’s something I believe is vital and I’ve written on the topic before. But until recently I’d not thought to question some of the other things that stem from childhood – bizarre, perhaps, as I have questioned and left the religion I was brought up to be part of.

Young people take in things they are told by authority figures as hard truths. When we, as teachers, express the view that method A is preferable for solving trigonometric equations then those in our charge will follow that. We all know this, and often we need to ensure we are presenting an unbiased view on things (although trigonometry probably isn’t one of these…). What we sometimes miss, though, is that some of the things we learned as children may have been the opinion of our teachers, and we need to critically evaluate those things too.

*I intend to reflect on the topics I teach and the methods I prefer to ensure that I am teaching the what my students need to see.*

## 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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