Circle puzzle

​Here’s a lovely puzzle I saw on Brilliant.org this week:

It’s a nice little workout. I did it entirely in my head and that is my challenge to you. Do it, go on. Do it now….

Scroll down for my answer….

Have you done it? You better have…..
I looked at this picture and my frat thought was that the blue and gold areas are congruent. Thus the entire picture has an area of 70. There are 4 overlaps, each has an area of 5, so the total area of 5 circles is 90. Leaving each circle having an area of 18.

This is a nice mental work out and I feel it could build proprtional reasoning skills in my students. I am hoping to try it on some next week.

Did you manage the puzzle? Did you do it a different way?

This post was cross posted to cavmaths here.

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Differentiating Questions By Removing Information

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:

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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.

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

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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. Mechanics 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 Cavmaths here.

Is one solution more elegant?

Cross-posted to Cavmaths here.

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:

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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:

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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!

Lovely, simple, trigonometric puzzle

Sometimes a puzzle can look complicated,  but be rather simple (see this geometry puzzle). I love puzzles like this and I particularly like to test them out on classes to try and build their problem solving ability.

Just now, I saw the following trig puzzle from brilliant.org and I love it! It’s amazing!

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Have you done it yet?

How long did it take you to spot it?

My initial thought was, it’s got three terms,  it’s bound to be a disguised quadratic that will factorise. A few seconds later I realised that it wasn’t. I saw the – sin^4 and suspected a difference of two squares but then a few seconds later it became clear.

If you haven’t spotted it yet, have a look at the expression rearranged:

Sin^6 + sin^4 cos^2 – sin^4

See it now? What if I rewrite it as:

Sin^4 sin^2 + sin^4 cos^2 – sin^4

I’m sure you have seen it now, but to be complete,  take the common factor of the first two terms:

Sin^4 (sin^2 + cos^2) – sin^4

Obviously sin^2 + cos^2 = 1, so we’re left with:

Sin^4 – sin^4 = 0

A lovely, satisfying, simple answer to a little brain teaser. Hope you liked it as much as I did.

Cross-posted to Cavmaths here.

An interesting area puzzle

Here’s an interesting little question for you:

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Have you worked it out? How long did it take you to see it?

It took me a few seconds at least, I had screenshotted the picture and was reaching for the pencil when the penny dropped, and that’s why I thought it was an interesting question.

The answer is, of course,  100pi. This follows easily from the information you have as the diagonal of the rectangle is clearly a radius – the top left is on the circumference and the bottom right is on the centre.

So why didn’t I spot it immediately?

I think the reason for me not spotting it instantly might be the misdirection in the question, the needless info that the height of the rectangle had me thinking about 6, 8, 10 triangles before I had even discovered what the question was.

I see this in students quite often at exam time, they can get confused about what they’re doing and it links to this piece I wrote earlier about analogy mistakes. The difference is I wasn’t constrained by my first instinct but all too often students can be and they can worry that it must be solved in the manner they first thought of.

Earlier today a student was working on an FP1 paper and he was struggling with a parabola question, he had done exactly this, he had assumed one thing which wasn’t the right way and got hung up in it. When he showed me the problem my instinct was the same as his, but when I hit the same dead end he had I stepped back and said “what else do we know”, then saw the right answer. I’m hoping that by seeing me do this he will realise that first instincts aren’t always correct.

I’m going to try this puzzle on all my classes tomorrow and Friday and see if they can manage it!

How quickly did you see the answer? Do you experience this sort of thinking from your students? I’d love to hear any similar experiences.

Cross-posted to Cavmaths here.

Passivity in the maths classroom

Today I managed to find a few minutes to browse the latest issue of Maths Teaching, the ATM journal. One article that caught my eye was the “from the archive” section, where Danny Brown (@dannytybrown) introduced an article that was first published in 1957. The article was written by Ruben Schramm and is entitled “The student’s passive attitude towards mathematics and his activities.”

The article discusses mathematics teaching, particularly the nature of students who often, for whatever reason try to find an algorithmic method to follow to solve a problem, looking to recognise the problem and answer it in a similar way to how they have answered questions before. This is a problem that was obviously prevalent in the 1950s, as evidenced in the paper, but it is still prevalent now, and I feel the nature of our exam system must at least hold a portion of the blame. The questions on exams tend to be very similar and students will learn methods to answer them whether the teachers like it or not. This is one issue I hope will be dampened a little with the upcoming changes to the exams.

Schaum suggests that this passivity in maths, this tendency to look for algorithms, is in part down to how students see mathematics. He suggests that when they see teachers solve problems on the board by delivering a slick, scripted solution they can get a feeling that it is via “witchcraft” and see the whole process of uncoordinated steps, rather than a series of interconnected mathematical ideas. The latter would encourage the students to drive the mathematics from their internal ideas, and this would lead to them being more able to apply their knowledge in new contexts. If we can develop this at all levels then I feel we really would be educating mathematicians – ie giving students the skills to be able to apply their knowledge in new contexts, rather than teaching them to follow a recipe to answer a question.

Schaum goes on to discuss authority, the infallible authority that students see in their teachers and in the mathematical theorems and formulae. It is suggested that students see these theorems as infallible, and as such they reach out for them in their memories and try to apply them to problems. This can mean that the problem they are applying them too is only vaguely similar to the problem the theorem or method is actually there to solve. Schaum calls these “analogy mistakes”, and suggests that it is down to how comfortable with the content students feel that mean they revert to them. I feel that this is true in part, but that also the pressure of exams can lead students to confuse things in their head if they have opted to learn algorithms rather than looking to develop a deeper understanding.

I’ve had a couple of examples of these “analogy mistakes” in lessons and exams recently.  A year 12 student came to an afterschool elective as she was trying to solve some coordinate geometry problems involving tangents. She had gotten herself really confused because in her notes she had written tangent gradient is perpendicular (when discussing circles) but she didn’t think it should be perpendicular because a tangent at a point should have the same gradient as the curve. I spend a little time discussing where her misconception had come from (her notes should have said “perpendicular to the radius”) and discussed how she could remember this more easily if she has thought about the graphs and sketched them.

Another example was in a recent exam one of my students had answered part of a question on alternative from incorrectly, she had done the alternative form bit well and the answer was 25 Sin(x + a), but it then asked her for the maximum she had written -25. When I questioned her about this after it seems she had fallen victim to an “analogy mistake”, she had remembered that “maximum is positive” when discussing second derivatives and in the pressure of the exam this memory had taken over, rather than the rational thought process that should have flagged up that the maximum or the function would be 25, which is definitely bigger than -25.

In his preface Danny Brown suggested that one way to counteract this would be by questioning and discussion, if we remove the authority from the discussion and don’t validate the answers by issuing statements saying they are correct or incorrect, but rather open them as conjecture to the class who then can discuss this, then we can allow students to develop their own mathematical ideas. Lampert (2001) also discussed this idea and suggests that as teachers we need to be striking the right balance between allowing students to discuss and conjecture and ensuring they understand what is important and aren’t making mistakes. This is something I strive for in my own classroom, and something I am currently working on trying to improve.

This post was cross posted on Cavmaths 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:

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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:

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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.

Cross-posted to Cavmaths here.

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:

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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 Cavmaths here.

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:

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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:

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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:

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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:

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Then the gradient of PQ:

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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:

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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 Cavmaths and One Good Thing.

Interesting questions

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:

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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:

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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:

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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.

This was cross-posted to Cavmaths here.