## A lovely sequence proof problem

Another puzzle I found in Chris Smith’s (@aap03102) newsletter, and had fun doing is this one:

It’s a nice little puzzle that relies on ones ability to use algebra, reasoning and to understand what an arithmetic sequence is.

I considered the problem and thought that as I know the difference between each consecutive term in an arithmetic sequence is always the same I should first start by considering the differences. I looked at each one and expressed it as a single fraction:

When I had the two defences I knew that I could then equate them:

At this point I simplified the equation:

And was surprised it had fallen out so nicely . This obviously show that the differences between x^2, y^2 and z^2 are also equal so this is also and arithmetic sequence. I think this is a lovely result.

It occurred to me later that I could have set the mean of the 1st and 3rd term equal to the second and the algebra may have been a little easier. I’d love to hear your approach.

## Patient problem solving

I recently read a piece by D Pearcy called “Reflections on Patient Problem Solving”, from Mathematics Teaching 247. It was an interesting article that looks at how teachers need to allow time for students to try their own ideas out while problem solving, rather than just coax them along in a “this is how I would do it” kind of way.

Pearcy’s definition of problem solving is looking at something you have never encountered before that is difficult and frustrating at times, takes a reasonable amount of time, can be solved more than one way and can be altered or extended upon easily. He then goes on to ask whether this is actually happening in classes or if teachers are just walking students through problems, rather than allowing them to problem solve.

He quotes Lockhart (2009) – “A good problem is one you don’t know how to solve” and states that it follows that if you give hints then it defeats the point of setting problems. He goes on to say that maths advocates talk of the importance of maths as a tool to problem solving – but that this isn’t actually happening if students are not being allowed to get frustrated and struggle through to a solution.

He explains how he finds it difficult not to give hints when students are struggling, both because it is in most teachers nature to help, and because of the external pressure to get through the syllabus quickly. This is something I too have encountered and something I have become increasingly aware of as I try to allow time for struggle. Other factors at play are maintaining interest, and increasing confidence. If we let students struggle too much they may lose interest and confidence in their ability – thus it is important to strike a balance between allowing the struggle but not letting it go too far. This is certainly something I keep in mind during lessons, and I feel it is something that we all should be aware of when planning and teaching.

This is an interesting article that looks at a specific problem and allowing students time to struggle and persist. This importance of this is paramount, in my view, and this is also the view expressed by the author of the article. I find it very hard to not offer hints and guidance when students are struggling. One way I manage to combat this at times is by setting problems I haven’t solved yet, thus leaving me a task to complete at the same time. This can work well, particularly at A Level and Further Maths level as then I can take part in the discussion with the students almost as a peer. This is a technique I have used often with my post 16 classes this year.

I have been reading a lot about problem solving recently, and a recurring theme is that teachers can often stifle the problem solving they are hoping to encourage by not allowing it to take place. This is something we need to be aware of, we need to have the patience to allow students the time to try out their ideas and to come up with solutions or fall into misconceptions that can then be addressed.J

.

*Have you read this article? If so, what are your thoughts on it? Have you read anything else on problem solving recently? I’d love you to send be the links if you have and also send me your thoughts. Also, what does problem solving look like in your classroom? Do you find it a struggle not to help? I’d love to hear in the comments or via social media.*

**Further Reading on this topic from Cavmaths:**

Dialogic teaching and problem solving

**References:**

Pearcy.D. (2015). Reflections on patient problem solving. *Mathematics Teaching. ***247** pp 39-40

Lockhart, P. (2009). *A Mathematician’s Lament*. Retrieved from: https://www.maa.org/external_archive/devlin/LockhartsLament.pdf

## Understanding students’ ideas

I read a really interesting article today entitled “Teachers’ evolving understanding of their students’ mathematical ideas during and after classroom problem solving” by L.B. Warner and R.Y. Schorr. It is a great report that looks at three teacher’s responses to their students’ solutions to a problem, and it discusses in detail how the teachers reflected on them together. It is well worth a read for all maths teachers.

The teachers were middle school maths teacher and they were presented with a problem to solve by the researchers they then presented their classes with the problem and debriefed afterwards. It was clear that the teachers didn’t have the thorough subject knowledge of a high school maths specialist and this lead to them failing to pick up some misconceptions and not allowing students to explore their own methods if they didn’t understand it, rather moving them on to a method that was more familiar to the teacher. The reflections of the teachers are interesting, they all appear to become frustrated with themselves when analysing their responses and are able to reflect on this by offering alternatives. It does show that deeper subject knowledge is important to allow that exploration to take place. The study showed that in this context when the teachers just told students how to fix their mistakes, rather than question students as to why they had made them, this led to student confusion. This suggests that we should be striving to understand our students thinking whenever possible and using that to combat their misconceptions so they don’t fall into similar traps again. This will also allow students to see why they are coming up with these misconceptions.

There are many teachers who, at times, fail to understand the lines of mathematical thinking taken by their students when solving problems. This can lead to not giving the proper amount of credit to valid ideas and it can lead to teachers failing to spot misconceptions. Some students may have a perfectly valid method but as the teacher may not see where they are going they can sometimes block this route off. This has deep links to “Flowery math: a case for heterodiscoursia in mathematics problems solving in recognition of students’ authorial agency” by K. von Duyke and E Matusov , which I read recently (you can read my reflections here). I feel that it shows that deep subject knowledge is important, as is allowing students time and space to work through the problem on their own. Rather than saying, “No, do it this way” we should, be encouraging students to follow their nose, as it were, and see if they can get anywhere with it. It is always possible to show the students the more concise method when they have arrived at the answer to bui8ld their skill set.

Warner and Schorr believe that subject content, as well as pedagogical content is vitally important to teachers to enable than to know how to proceed when a student is attempting a problem. They look at relevant literature on this and quote Jacobs, Philip and Lamb (2010) who suggest that this is something that can be achieved over time and Schoenfield (2011) who says that teachers tend to be more focussed on students being engaged in mathematics and replicating the solutions of the teacher rather than allowing students to meander their own way through so the teacher scan identify their understanding and misconceptions. The latter would, in my opinion, be a much better way of developing, and I agree with JPL that this is a skill one can develop over time.

**References**

Jacobs, V. R., Lamb, L. L. C., and Philipp, R. A. (2010). Professional noticing of children’s mathematical thinking. *Journal for Research in Mathematics Education*, **41**, pp 169–202.

Schoenfeld, A. H. (2011). Toward Professional Development for Teachers Grounded in a Theory of Teachers’ Decision Making. *ZDM, The International Journal of Mathematics **Education*, **43 **pp 457–469.

Von Duyke, K. and Matasov, E. (2015). Flowery math: a case for heterodiscoursia in mathematics problems solving in recognition of students’ authorial agency.* Pedagogies: An International Journal.* **11**:1. pp 1-21

Warner, L.B. and Schorr, R.Y. (2014). Teachers’ evolving understanding of their students’ mathematical ideas during and after classroom problem solvin. *Proceedings of the 7th International **Conference of Education, Research and Innovation, *Seville, Spain, pp 669-677.

## Dialogic teaching and problem solving

I recently read an article entitled “Flowery math: a case for heterodiscoursia in mathematics problems solving in recognition of students’ authorial agency” by K. von Duyke and E Matusov. It was an interesting article that looked at some student teacher interaction in a lesson where students were asked to solve a mathematical problem dividing one dollar between three people. They had found an interesting exchange between the teacher (“John”) and a student who had approached the problem topologically and has a correct solution using physical coins but hadn’t calculated the amount each person had. This has irked and perplexed the teacher – seemingly because she hadn’t come up to the solution he had in mind. This is an interesting revelation and one that we, as maths teachers sometimes fall into. There can often be many ways to solve a problem in mathematics and all are equally valid. My view is that we need to be looking at solutions presented to us by our students with an open mind before telling them they are wrong. In this case the student had come up with her own approach and had the correct solution – an outcome that feel should be celebrated.

The authors use this as a starting point for a discussion on various pedagogies, suggesting that to really allow this sort of maths to thrive in the classroom teachers need to take a dialogic approach – to discuss with the students where their thinking has come from and help them refine their models. They also suggest that the reason John was keen to dismiss this valid reasoning in this case was due to his favouring of a more rigid pedagogical structure. I tend to agree with the researchers. We are there to help students make their own meaning, their own links, in mathematics. Obviously we need to pass on the relevant subject content, but in an open ended task like this it is important to ensure all solutions are explored and refined.

This leads me back into a discussion I had recently regarding the purposes of assessment in mathematics which came about from this blog that I wrote on a question with multiple solutions. John R Walkup (@jwalkup) said that we should be assessing all methods to ensure that students can do it. I think that to an extent he has a point. We do need to test that our students can complete the content, and we should be doing this with low order questions where they are directed to practice and recreate skills. However, maths is about making links, making your own links, and solving problems that are unfamiliar – trying the methods you know to see if you can find a solution to a problem, you have never seen.

It is the latter that is increasingly being tested in our terminal external exams in the UK as we move to the new specification GCSE and A Level tests, so we need to be preparing our students to be successful in this type of question. I think that the dialogic approach mentioned here is an extremely powerful tool in this quest. It allows us to help students explore their thinking and create their own links. I heard a colleague recently explain to a student that maths was about “finding shortcuts, and finding tricks” this worried me a little at first but then he continued “we all have hundreds of tricks and shortcuts that we have developed over years of doing maths. If Mr Cavadino and I were to teach you our tricks they wouldn’t make sense to you and it would overwhelm you.” I can understand this point – if a student notices that d = s x t can be rearranged simply in a triangle because they understand how to rearrange that equation then they will save themselves time. If they learn the technique without understanding what is taking place they open themselves up to the possibility of more errors.

In the article the authors use the term heterodiscoursia, which means legitimate simultaneous diverse discourse. The suggestion is that as part of the dialogic teaching teachers should be allowing discussions and methods to abound and thrive in the classroom. They suggest that this mix of discourses allows students to bounce ideas, allows the teacher to correct any misconceptions and helps build meaning making and engagement. Their suggestions are certainly in line with my observations from my own lessons that have allowed these types of discussion to develop and I think that it would be beneficial to explore how this can be allowed to grow with my other classes.

The authors have some practical suggestions for us maths teachers. They suggest that we need to be familiar with the fact that there are often different solutions and be able to develop them. We need to allow the students to frame the question into a context that works for them then use that context to find a solution which is salient, and we need to be able to question our preconceived notion of the solution. This sounds like sound thinking, I feel that these are things we should all do while we are trying to build the problem solving capacity of our students.

The authors go on to discuss dialogic pedagogy and how this can in effect allow teachers to discuss the problem almost as a peer – I find that this is a great tool when working with A Level students. If i can find problems that I haven’t done before then I can share my thinking with them and we can work through them together. This has been very successful when developing problem solving strategies.

The authors go on to discuss dialogic pedagogy and how this can in effect allow teachers to discuss the problem almost as a peer – I find that this is a great tool when working with A Level students. If i can find problems that I haven’t done before then I can share my thinking with them and we can work through them together. This has been very successful when developing problem solving strategies.

**Reference:**

Von Duyke, K. and Matasov, E. 2015. Flowery math: a case for heterodiscoursia in mathematics problems solving in recognition of students’ authorial agency.* Pedagogies: An International Journal.* **11**:1. pp 1-21. [accessed 23/5/2016] http://dx.doi.org/10.1080/1554480X.2015.1090904

## 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 to Betterqs here.*

## Playing with jigsaws is the way forward

*This is a guest post written by my brother Andy (@andycav_25). Andy is a primary teacher in West Yorkshire and currently teachers Year 5.*

Fractions Jigsaw from __NRICH__

In the 2014 Primary National Curriculum, there’s much more emphasis on problem solving in maths. Henceforth, we’ve had a few staff meetings and twilights on this recently. We’ve been encouraged to use ‘rich mathematical tasks’, and some colleagues (myself included) expressed sheer horror at some of the ideas. However, I did embrace it and took a bit of a risk with my Year 5 class last Friday. If an OFSTED inspector had walked into the first 30 minutes of that session, I would’ve been mortified. If they’d walked in at the end, I would’ve been ecstatic.

Definition from __Simon Borget__

We’d been doing fractions, decimals and percentages all week and, in my timetable, I have a 2 hour session every Friday morning. I normally do 2 lessons in this time unless it can be used for a science experiment, art or D&T depending on the long term plans. Last Friday, maths was the winner. The NRICH website is great for providing these rich mathematical tasks and I found __this one__, which I really liked the look of. Children had to complete a jigsaw on a 5×5 grid. Each square is cut into 4 triangles that were either blank (these would eventually form the outside of the jigsaw but there were a few extras thrown in) or they had a calculation using fractions. Obviously, the children had to match up the triangles to create a jigsaw.

I thought that it had to be a pair or small group activity, so do you group them as high/low ability or assign a high ability with a low ability? I kept the high achievers together this time, as an experiment in itself as much as anything.

What unfolded in front of me was just amazing. Every single kid in my class engaged. I don’t think I’ve ever achieved that before in 5 years as a teacher (and another few before that as a TA/HLTA).

One child in my class is ridiculously talented in maths (earlier in the term, it had taken him about 5 minutes, without a calculator, to find all of the factors of 256. He’s 9!). He absolutely loved this jigsaw task. He thrived on the challenge but was also made to reassess a few times when things didn’t appear to be working. It took his group about 45 minutes to complete the jigsaw, but when I spoke to them about how they did it, it was one of the lesser but still more able kids who had spotted the pattern in the jigsaw. They had taken on different roles in the group with child A quickly calculating in his head and child B spotting patterns to assist child A in finding the correct pieces.

Solution

That group also complained to me at one point that they had 6 ‘top’ pieces, which doesn’t work on a 25 square. They told me that it was wrong. ‘Look again,’ I said, numerous times. Eventually, they did realise that there were two blank areas inside the grid too.

The middle ability groups really struggled at first. As had been the plan at the offset, I shared a few answers and gave them a starting point after about 20 minutes. Bang. They started working through that grid as fast as I’ve ever seen them work in the 4 weeks they’ve been in my class. Not everybody finished (in fact probably only about half did, if that) but that didn’t matter either to me or them. They had all achieved something and we all knew it.

Once the HA group solved it, I gave them a template of the solution to make their own – they had fun testing each other. At that point, I also shared that template with the rest of the class as an extra piece of scaffolding. That made some of them look and immediately say ‘we need that shape here so which piece fits?’ Another great use of mathematical thinking. Key question: what’s the same and what’s different about the pieces and what does this tell us?

At the end of the session, I did two plenaries. First, what skills have we used this morning? Answers: ‘fraction skills’, ‘teamwork’, ‘spotting patterns’. It took a little bit of guidance but that’s fine.

I would’ve liked ‘problem solving’ to be an answer but alas…

We’ll get there in the end!

The second plenary was simple: ‘who has been completely confused and felt like your head is about to explode at least once this morning?’ (62 thumbs up – there’s 31 children in the class).

Who thinks you’ve made progress with your maths this morning? (again, 62 thumbs up). I think this risk definitely paid off! I will be trying to do at least one of this type of task or investigation for every topic in maths from now on. The skills and achievement shown by every child at their own level was amazing and they really enjoyed it too! Plus, there’s no marking to do… Bonus!

*I enjoyed reading this post. It’s nice to get a fuller view of what maths is taught at primary and how it is taught. I love the resources on nrich, I’ve not used this specific one, but I certainly would if I have a class that it fits too. Have you used anything similar? What are your views on this task/these tasks?*

## Nice summation puzzle

Chris Smith (@aap03102) runs a weekly maths newsletter during termtime. If you’re not on the list, I’d advise you to get on it (drop him a line in twitter). It has some great stuff in it, such as this fraction:

How ace is that?

He also has a puzzle of the week feature. This week’s looked cool, so I had a crack at it:

I thought about it a little, and saw that it boiled down to this:

I had the general term: *1/((n^1/2)+(n+1)^1/2)* but I wasn’t sure where that would get me. I tried rationalising the denominator of the first two terms:

Now I could see where I was getting to, I thought I’d check the general term, to be sure:

Then it was a case of looking at the sum:

*The sum of (n+1)^1/2 -n^1/2 for n=1 to n=91808*

Then I cancelled:

Leaving just

*91809^1/2 -1^1/2*

Which is

*303-1 =302*

A lovely neat solution to a lovely summation problem which I hope to try in my sixth formers soon, and which I hope will work well with the new GCSE specifications.

## Math(s) Teachers At Play #75

Hello, and welcome to the 75th issue of the Math(s) Teachers at Play Blog Carvinal! *For those of you who are unaware, a “blog carnival” is a periodic post that travels from blog to blog and has a collection of posts on a certain topic. This is one of two Maths Carnivals, the other being the Carnival of Mathematics, the current edition can be found here.*

75, 75, 75 – What is there to say about the number 75? It is a nice round integer, forming three quarters of 100. It is the 4th “Ordered Bell Number“, and more impressively is a “Keith Number“.

It is the amount of uniform polyhedral that exist (when coinciding edges are excluded), it is the atomic number of rhenium, the age limit for Canadian senators and , perhaps most impressively, has only two distinct prime factors, and they are the two lowest odd primes!

This is the first time I’ve hosted a carnival and there were some excellent submissions. I enjoyed reading them all and have discovered some new blogs. I have also input some posts I’ve seen this month which I thought were excellent too.

On to the carnival:

*First up, some posts aimed at those teaching maths to the little ones:*

Crystal Wagner in this post discusses some ideas for teaching maths to those under 6. She urges us not to rush into formal academics programmes, but to spend more time exploring the subject.

This post from Pradeep Kumar discusses finger counting and suggests methods to help children move on from it.

Margo Gentile give us a nice little activity for those first learning their times tables.

Lisa Swaboda asks “Do vegetables have OCD?” – This is a nice post that looks at the symmetry in the world around us and asks philosophical questions about its nature.

Yelena at Mobius Noodles asks “What maths do you have in your house?” and tells us about the maths house she created at the mini maker faire.

*For those a little older we have:*

Sarah Hagan writes about an investigation she has complete with one of her classes, which looks at linear functions and asks “How many stacking cups would equal your teachers height?”

Dave Gale discusses percentages, and helps me feel sane by proving I’m not the only one who sees maths everywhere.

*Here we have some ideas using modern technology:*

Penny Ryder has written this piece about fractions, and how they can be explored using 21st century technology. Interesting stuff for those with access to iPads.

Bryan Anderson gives us a fantastic arty maths project that uses DESMOS.

*For the more advanced mathematicians:*

Manan Shah has given us another great post as part of his series on teaching calculus

Colin Beveridge gives us this nice post about arithmetic sequences

And I discuss a further maths lesson which degenerated into a lengthy exploration of Pythagorean Triples!

*More Generally we have these on maths teaching:*

In this post , my star post of the carnival, Christopher Danielson explores the purpose of posing mathematical questions, and gives us all food for thought on the way we question children.

This nice post from fMaths discusses marking in maths.

You can follow Craig Barton and his team as they voyage through the journey of writing a new Scheme of Work.

*Also this month:*

Festival Founder Denise Gaskins has written a post talking about the brand new book, Playing with Math edited by Sue Van Huttam!

There is a suberb episode of Wrong, but useful.

while one of the hosts, Colin Beveridge, tells us why he loves maths, and I join in the love in.

*If anyone else has written, or would like to write, a post on why they love maths, I’d love to read them!*

## A sphere and a frustum

*A sphere and a conical frustum have the same volume, the frustum has a base radius which is twice the radius of its top. The sphere has a radius which is equal to the base radius of the frustrum. What is the ratio of the diameter of sphere to the height of the frustum?*

Recently the podcast “Wrong, but Useful“, (@wrongbutuseful) celebrated its 1 year anniversary. The hosts Dave (@reflectivemaths) and Colin (@icecolbeveridge) asked listeners to send in messages to be played on the anniversary show.

I thoroughly enjoy the podcast, and particularly the part where the hosts set puzzles for the listeners to have a go at. With that in mind, I decided to set a puzzle for them as part of my messages.

I know a number of puzzles, but I didn’t want this to be a problem either of them had heard before, so I set about devising one. I thought about the problems I like to discuss, and remembered that I’ve always liked the algebra based geometry problems that occur at the back of GCSE papers, you know the ones: *“Three tennis balls radius r fill a cylinder… What’s proportion of the cylinder is empty space?” etc.*

I thought a problem like this would be nice. I thought about it and decided I wanted to use a sphere and a frustum with the same volume. I decided that the frustums base radius should be twice the radius if the top and that I would ask them to find the ratio of diameter of the sphere to height of the frustum. I needed to think about the radius of the sphere. At first I thought it might be nice to have the sphere have the same radius as the top face of the frustum, but I reckoned that wouldn’t give a very satisfying answer as it would have hardly any height to speak of. So I tested (on a magna doodle no less) to see what it would be if I used the base radius as the same as the sphere:

*Volume of a sphere: (4(pi)(r^3))/3*

Volume of a frustum: ((pi)(h)(r^2 + rR + R^2))/3

Sub r=r, R=r/2

Equate equations

Cancel pi/3 from each side

4r^3 = (7/8)(h)(r^2)

16r = 7h

8d = 7h

*Ratio d:r is 7:8*

What a nice, simple, solution we get from quite a bit if algebra. I thought that as it fell out so nicely I would set it!

I realised that for Dave, Colin their listeners and the majority of people I posed this question to would not really have had too much of a problem solving it, but hopefully will have enjoyed it. I did think, however, that it would be an interesting problem to pose my yr11s (and my sixth formers). They have all the skills and knowledge required, so it will be interesting to see how they get on. I will pose it to them after the holidays!

## A surprising fraction?

The other day I saw this tweet from John Rowe (@MrJohnRowe):

I looked at the picture and decided the answer was probably a half. And thought about it a bit more. The 4 small circles in the top left should be the same as the white quarter circle below, and the 4 quarter circles in the top right should fit over the white circle below them. This was interesting to me, and I thought I’d look at the algebra.

Looking at the top left corner, the circles involved there have the smallest radius, so we will call that radius r. That means each circle has a radius of pi r².

Below it we have a square the same size and a quarter circle radius 4r, so the white area is (16pi r²)/4 or 4pi r² this is the same as the pink area in the square above. We can see from this that half of the left rectangle is pink. Or we can continue with our algebra. The area of each square is 16r² (its sidelength is 4r) so the pink but here is 16r² – 4pi r², so if we add this to the bit above it we have 16r² shaded (and 16r² white).

The top right has 4 shaded quarter circles, each with radius 2r, so the total shaded area is 4pi r². Below it the white circle has the same radius so same area 4pi r², again that makes the shaded area 16r² – 4pi r² so the shaded area in the right rectangle 16r², and in the big square 32r². The total area of the big square if 64r² (radius is 8r) so the shaded fraction is a half. Which is nice.

Now I know I started by saying I thought it looked like a half, which is what I did think at the time I first saw it. But I still think it’s a surprise fraction. I’ve done a lot of geometry puzzles. And many have included shapes like this, so when I look at this that knowledge helps me see. Most students would not have done anywhere near the amount of puzzles I have so wont have that foresight, and I think it would be a very surprising result that could open many up to the wow factor. I also think that it might be a good starting point for some rich class discussions.

I think it’s a great visual and a lovely answer.

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