The rectangle puzzle

Today began with me reading through some tweets I’d favourited, including the latest from @mathsjem (who if, for some ridiculous reason, you’re not already following, you should follow immediately). In the section on problem solving, she remarked the following:

“It’s a good idea for maths teachers to try to solve unfamiliar problems every now and then (like the example below from ‏@dannytybrown) to remind ourselves that mathematical problem solving often requires patience, creativity and multiple attempts. We all experience frustration in problem solving, just like our students do, but we know that the satisfaction of eventually finding the solution is well worth it.”

I wholeheartedly endorse this view. It is something I have become acutely aware of recently and, as I will be teaching top set for the first time in my career from next week, I find this an excellent way of testing my rustier skills. I blogged previously about the looped polygons puzzle, and today I decided to tackle another of @dannytybrown’s puzzles, namely this one:

rectangle puzzle

I must admit I huffed and puffed for a while on this, but in actual fact I was only really able to solve it once I’d had a look at another of Danny’s puzzles which was mentioned in @mathsjem’s post:

rectangle problem

This had me confused for ages. I just couldn’t find a way into it. So I looked at solutions proposed by others and tried to understand them. Eventually, and after a lot of confusion on my part, I got there (although I am awaiting a reply from Danny based on what I regard as a key aspect of the solution with which I am not overly happy at present). In the end, the solution boils down to using trigonometric ratios to make finding the answer much easier.

So back I went to the paper folding puzzle. I had already jotted down what I knew and had deduced – the length is rt 3 x w, the width is w. This means that the base of the right angled triangle can be written as (rt 3 x w)/w. During my first attempt at the problem I had calculated the hypotenuse at being w + ((rt 3 x w)/3). When I went back to it and tried again from scratch, however, my curiosity was stirred, and I worked through it slightly differently.

We know that hyp^2 = w^2 + ((rt 3 x w)/3)^2. Working this out leaves hyp^2 = w^2 + 3w^2/9. This adds up to 4/3w^2, which when you root it gives a hypotenuse equal to 2w^2/rt 3. And if we rationalise the denominator? Well then the hypotenuse of the triangle equals (2 x rt 3 x w)/2 – twice the length of the base. This then means that the angle between the base and the hypotenuse must be 60 degrees.

On my first look at the problem, I really struggled with identifying how much of the folded section would fall outside the rest of the sheet. In fact, I couldn’t comprehend how I would even begin to calculate this. Having gone away and come back to it, however, I realised the thing I was trying to grasp earlier was that the fold line acts as a line of symmetry. So if I doubled the angle, that would allow me to calculate how far the sheet would fold across the existing section. Except that it was now quite straightforward. The angle being 60 degrees, added to the original 60 degrees, creates an equilateral triangle. The section that will fold over neatly onto the existing paper can be shown by drawing a diagonal from the bottom line (where the diagonal already ends) to the top left corner. The length of the diagonal is (2 x  rt 3 x w)/2 which, believe it or not, is the same as the length of the sheet from the left hand corner to the point the diagonal meets it. This means that there is half of a small rectangle (or 1/6 of the total shape) that will overhang when folded.

The original question asks for the ratio of the area of the new shape : the area of the original shape. The shape is 1/2 + 1/6 of the original (1/3 is folded over), leaving 2/3 of the original, meaning the ratio is 2:3.

I loved doing this problem, and I think it also helped having a break and focusing on another problem….particularly when that problem and this shared a key, common, but overlooked (in my eyes at least) bit of mathematics!


Looped polygons (odd number sides)

So much of my Sunday was occupied with the looped pentagon problem. Having solved that, I was intrigued to see how the problem would play out for other polygons. I have spent most of my time since focusing on odd-sided polygons (because I think even-sided ones are relatively straightforward, although I need to check) and thought I might share what I had come up with so far.

Firstly, I have to say this has been a bit of an eye-opening experience for me. Not having a solution, or any real help, has added to the intrigue. I’ve extended the problem, I’ve made connections and generalisations, and I’ve refined my approach all the time until now, where I feel I have a method to share. It’s the kind of thing I’d love to spend time on with students….if only!

The general problem is like this: a polygon is marked, with circles of 1cm radius placed around its vertices. A piece of string in looped around each vertex and pulled tight so there is no sag. The challenge is to calculate the length of string needed. In the original pentagon problem, the distance from vertex a (at the top) to vertex c (bottom left) is given as 5cm. I will refer to it simply as l.

The main issue I have found so far, and which I may well go back and revisit at some point, is deciding the order in which the string is looped. Once you move to a heptagon there are options. In all cases I present below, I have opted to go for the furthest vertex from the top vertex, which I have called the bottom left. The string then loops back up to the vertex immediately to the right of the top vertex, and so on.

I had, you will recall, noticed that l was parallel to the string – it must be so, as extending the string would create a tangent to the centre of the circle. My approach therefore involved a lot of angle theory. Beginning with the equilateral triangle, I split the interior angle in half to give 30 degrees, as this allowed me, by deduction, to calculate the angle from due North to the string. This done, and the two added together to give 150 degrees, I subtracted the tangential right angle to leave an angle of 60 degrees from due North to the tangent. Drawing a right angled triangle showed me that the angle from the centre of the circle to the tangent was 30 degrees from East, meaning that on both sides of the circle, the string was wrapped around an extra 30 degrees before moving towards the next triangle.

This gives a total contact with the circle of 180 + 30 + 30 = 240 degrees, or 2/3 of the circle’s circumference. The LENGTH of string needed can be calculated by 2/3 x 2 x pi for one circle, and then multiplied by 3 for all 3 circles. This gives a total of 4 pi for the circles, plus 3l for the straight bits, for a total of 3l + 4 pi. Knowing that the pentagon gave a solution of 5l + 6 pi for the length of string, I began to conjecture that for any s sided polygon, the length of string needed was sl + (s +1) pi.

I next tried a nonagon, being a shape with the number of sides being a factor of 360. And this was where I got stuck for a while. I guessed the answer was 9l + 10 pi, but this was based on me guessing the angle from due North to the bottom left circle. I eventually realised that the answer may lie inside the polygon – in the centre, in fact. By taking a line segment from the centre of the polygon to each vertex, I had the angles there, ready to go. So for the nonagon, each vertex was a movement of 40 degrees (360/9). Using angles on parallel lines, I could see that the angle created by drawing an isosceles triangle from the points due North, in the centre of the polygon and the bottom left vertex, gave me the angle I needed to subtract from 180 degrees. In this case that angle is 10 degrees, meaning the tangent/due North angle was 170. This, to cut things a little short, means that 10 degrees was the angle beyond 180 where the string was touching the circle on each side. Here, this resulted in 180 + (2 x 10) = 200 degrees, a total of 5/9 of the circle’s circumference. Again, the total curved contact can be found by calculating 5/9 x 2 x pi x 9, which equals 10 pi. So the total length is 9l + 10 pi.

Having found a shortcut, I then tried the heptagon. This was particularly brave given my lack of a scientific calculator, but I got there in the end. The large angle in the isosceles triangle I needed was 3/7 x 360, so the small angle worked out at 12 6/7 degrees. This was the angle beyond 180…..etc etc….and I eventually arrived at a length of 7l + 8 pi.

I had spotted another pattern here, and stopped to investigate. The proportion of the circle which was touching the string was 2/3, 3/5, 4/7, and 5/9, and can be expressed as (s-1)/s for a s-sided polygon. What I also found fascinating was angle from due North vertex to bottom left vertex as a proportion of 360: 1/3, 2/5, 3/7, 4/9…clearly this will gradually get closer to 1/2 without ever reaching it, and can be expressed as 1/2(s – 1) / s for an s sided shape. And if you add this angle to the sector touched by the string, you get 1 every time. Pretty neat, huh?

I realised now that this allows me to quickly work out the information I need without having to worry too about the angles – so for a 15-sided shape, we take 7/15 (168 degrees) as the largest angle in the isosceles triangle, meaning the angle beyond 180 degrees will be 4/15 (6 degrees) on each side, or 8/15 (12 degrees) in total. This will give total string contact of 15l + 16 pi.

With odd sided polygons, I think the next step is to investigate whether a different (but still repetitive) looping arrangement changes the total  amount of string needed. I’m not sure at the moment, but I do know I have some strategies which will hopefully allow me to find the answer pretty quickly!

Looped triangles – my solution

Up early on a Sunday for my long run (9 miles today, in case you wondered), I came across a tweet from @srcav:

As I had time to kill whilst my breakfast digested, I thought I’d have a go. Below is my solution. There has been some discussion as to whether this is right or not. I welcome any comments/corrections.

Initial thoughts – I looked at the site from which the puzzle was taken ( and saw the following information:

“All Five Triangles problems share a characteristic: however opaque they may appear initially, none requires more than common mathematics skills learnt in classrooms throughout the world.”

I decided to solve the problem using angles and circle theorems. First, I worked out the angles I needed. I took the approach that there are 5 segments of 5cm in the diagram which run parallel to the line segments AC, CE, EB, BD, and AD. My belief (and this is where I may have gone wrong) is that this is effectively the tangent, where to string leaves the circumference of the circle. Because the tangent is always at 90 degrees to the radius, I thought we might be able to use this to find the sector of the circle the string does not touch, and use this work out the circumference touched by the string, and therefore the solution.


Given that the interior angle of a pentagon is 108 degrees, the exterior must be 252 degrees. Because ABC is an isosceles triangle ( as AB = BC), angle BAC = 36 degrees. If we draw the line AD, we can see that angles CAD and DAE are also equal to 36 degrees. If we add in a point F so that segment AF bisects the reflex BAE, then FAE = 126 degrees, and FAD = 126 + 36 = 162 degrees. The same is true for FAC.

IMG_0003 (1)

I think a key part of the problem (and again, this may be a mistake in my thinking) is that AC is parallel to the segment of string running from the right hand side of the circle around point A to the right hand side of the angle around point C. The same is true for the other 4 5cm segments. I said above that FAC = 162 degrees. Drawing the tangent to the circle from point A gives a 90 degree angle, which leaves a remaining angle of 72 degrees. This means the difference between the angle from A to the tangent, and the angle created by drawing a perpendicular to FA, is 18 degrees. I therefore think that on each side of each circle, the string remains in contact for a further 18 degrees. This gives a total sector of 216 degrees.

(Interestingly enough, this means that the sector of the semi circle touched by the string is 108 degrees – I wonder if this is significant given the context of the problem?)

Taking this as a fraction of 360, and then multiplying by 5 (for there are 5 circles), gives an answer of 3. The circumference of each circle is 2 pi, so the total curved distance of the string is 6 pi. All that remains is to add the 25cm for each segment, giving a total of (25 + 6 pi)cm, or 43.85cm to 2 decimal places.

As I say, I may be wrong. Shoot away!

I can’t get no sleep…. (or what I’ve been up to these past months…)

“But there’s no release, no peace
I toss and turn without cease
Like a curse, open my eyes and rise like yeast”

Insomnia, Faithless, 1995

This is one of the all time classic dance tracks (indeed, it is my favourite dance track of the 1990s) and it has been quite prevalent in my mind recently. Because for the past five or six weeks, I have been suffering regular sleep loss. This blog is hopefully part of the process of dealing with the issues that have caused my depleted sleep.

I was always adamant that I would never change jobs in the middle of a school year. In 2010, however, I moved at May half term, and spent the last 6/7 weeks of the year at my new school, settling in with no real teaching timetable (in reality, I DID teach a few lessons, to a greater degree as the half term went on, but it wasn’t really full-time teaching). It was pretty easy for me.

After missing out on a couple of TLRs, I had decided this would be my last school year at my current school. However, a chance browsing of the TES jobs site one day caught my eye. A job was advertised at one of the fastest improving schools in Leeds. They held an open day which I attended with my wife (just as she had done 5 years earlier before I got my current job). We heard the executive headteacher speak. He was deeply impressive in his vision and his convictions. He spoke of mindset and a rigid positive discipline system. To put it into context, he was so impressive even my wife said she wouldn’t mind working in one of his schools – and she can’t stand children!

So I applied, I heard back, I went for interview. The school was fantastic – not a new building, but one full of impressive displays, full of incredibly polite and attentive students. I could even have my own room! My lesson went pretty well, and I felt I nailed the interview. Hours later, the call came. I had been successful!

This was the Wednesday before half term. Since then, my head has been racing in my quieter times, including, clearly, last thing at night. I am racked with excitement, nervousness, and self doubt. I went for my day at my new school on Thursday and, although I know my timetable, I am still waiting for confirmation of topics to teach. And so my insomnia persists.

Let me make it clear: this is not a criticism of my new employers. I know what is expected of me in terms of end of the year grades. When I do receive the confirmation I am after, I shall be ready. Ready to begin intensively planning for this new and exciting chapter in my career. I chapter I WANTED to embark upon.

It’s just…..there seems so much to think about now:

  • what posters do I display?
  • what messages do I want to convey in my first week?
  • how can I help the students in making the transition from one member of staff to me as seamless as possible?
  • what happens if I go in too harshly?
  • and finally….what if I fail?

So now I have shared my worries, a) I hope I shall begin to sleep better, but b) I need your help. Those of you who have moved jobs mid-year – how did you cope? What advice would you share? Can you confirm that I will indeed be absolutely fine?


Negative numbers, Pythagoras, and automaticity in teaching…

A bit haphazard this post, but I was doing my usual Sunday evening thing of reviewing my Twitter favourites, catching up on blogs and resources, and came across a couple of things that I wanted to share some thoughts on. The first was this post from @cavmaths on the topic of damaging short cuts – or as I call it in my lessons, cheap and dirty maths. I have a few pet hates – but number one is teachers who encourage students to ‘just add a zero’ when multiplying by 10 (to which my immediate response is “so if I see something in a shop for £4.99, does that mean I can buy 10 of them for £4.990?”). But I digress – the reason for me discussing this post is not to turn it into a rant about my pet hates, but because the original post was about negative numbers, which I am half way through teaching with my year 7 groups.

Negative numbers is a really important concept to get a grasp of early – my year 11s have shown some difficulties early this term in the context of negative terms when expanding brackets, to name but one example. So, knowing I was due to teach it early on in the year, I was intrigued to see this explanation given to an answer on (the full thread can be found here):

“The key idea is that negative numbers represent changes, not amounts. It doesn’t make sense to say that you have -4 slices of bread. It does, however, make sense to say that you ate 4 slices of bread, and therefore the change in the number of slices you have is -4.” (My emphasis)

This was a revelation for me – a really clear and succinct example that I was desperate to use in my teaching. I was lucky I had an upcoming opportunity. I knew from their pretests (see my summer posts) that my year 7s were not confident on negative numbers, so I set up a slide to show a game scenario where I received a point each time I won, but my opponent gained a point when they won. I expressed this as my receiving -1 when my opponent won. We modelled a variety of situations, focussing on the total. The initial results seem encouraging, with the students telling me they have a much clearer understanding about negative numbers. Tomorrow we move on to multiplying and subtracting after a quick recap, but I now can’t wait to do the post-test in a couple of weeks to see if it really DID make a difference. I suspect it did, and I suspect I have a ‘banker’ method for teaching a key topic.

Pythagoras – even simpler than a, b, c!

Another of my pet hates is the teaching of Pythagoras’ Theorem. Our year 11s are taught identikit lessons put together by our SLT maths link, which states that a^2 + b^2 = c^2. WHY???? My immediate response (I always seem to have one….!) is “well how do you know which is side a, b and c?” Clearly this method is based on labelling the hypotenuse as c. Here’s a thought, though: if you know that c is the hypotenuse, why not call it, I don’t know, h for shorthand? Or hyp? After all, Pythagoras’ Theorem states that the square of the hypotenuse is equal to the sum of the squares of the other two sides. By using the a, b, c method, I fear students assume that the square of one side is equal to the sum of the squares of the other two sides. And if they DO know which side is c, surely it makes sense (and is more correct mathematically) to give that side its proper mathematical name!

There is, of course, a question to be asked about how to label the other two sides. Not having an angle in Pythagoras situations (certainly not when introducing the topic), it would be unhelpful to use adjacent and opposite, or any shorthand thereof. In my own teaching, I refer to the other two sides simply as ss1 and ss2 (ss meaning ‘shorter side’). I explain that because the hypotenuse is the longest side, the other two sides are both shorter. It is irrelevant to distinguish further between the two.

I have yet to receive any negative feedback about this approach.

Automaticity in teaching

The second tweet was this from @oliverquinlan, which posed the question “How much can you systematise your teaching..?” The post is of little use to me (it is effectively an advert for a book), but the question resonated with me greatly. Systematising teaching – making it as predictable as possible is my interpretation – is something I have been trying to do with regard to my planning and my timetabling of my non-contact time, and which I wrote about during the summer. The reason is cognitive and expressed in ‘Why Don’t Students Like School?’ By getting into habits, reducing the number of decisions we make, and becoming robotic in certain aspects of our practice, we open ourselves up to creativity. As Willingham says, creative thoughts occur most when we think. By making things systematic, we think less, and our creativity increases. I can certainly point to many examples already this year of this in action. I can honestly say I feel like a different teacher. Systematising my teaching has definitely worked for me!

Week 1 reflections

I thought, having a bit of spare time this evening, I might post about my first week of the new term and how some of the things I have talked about over the summer have been going. Or not going.

1)  Reflection time and reflective journal – Not for the students, but for me! I have to say this has had a MASSIVE impact so far. Each day when I have thought about my day on the way home and written them down, I have been prompted immediately with things to do better/more effectively tomorrow. Not only that, but I feel I have more purpose, not only in lessons, but also in my non-contact time. And when I haven’t completed it, I’ve found a bit of a difference. I think the reason for this is that taking a step back and reflecting on things helps you to get out of the ‘daily grind’ of teaching. Which, after a full teaching day today for the first time, let me tell you, can be very easy to fall into! If I could make one recommendation, keeping a reflective journal would be it. I wish I’d done it years ago.

2) Switching off from the world – I wrote about this in my blog here, and I have to say it’s still going pretty well. I’m not running away from news, and in my down time I will still browse news sites, but unlike before, I’m not spending ages scrolling repeatedly and pointlessly looking for the latest micro-development. And, believe it or not, I don’t feel any poorer for it.

3) Teaching/- I don’t know why, but I have started this term taking a few more risks and doing activities I haven’t really done before. Already students have used their planners as mini whiteboards, and yesterday and today I used Diagnostic Questions with a couple of my classes. Things have changed so dramatically that I have even….wait for it…..CREATED SOME RESOURCES OF MY OWN TO SHARE!!! I spent a few minutes at the weekend creating a set of 10 Diagnostic Questions for the website of the same name and tonight, looking for a homework for my year 8s and being inspired by the PRET homeworks which can be found at, I decided to create and submit my own (well, I used existing formatting – one step at a time – but it’s a start). 

Of course, it’s never a case of having teaching ‘figured out’, and already there are challenges in the classroom this year. I guess for all the planning you do over the summer, the proof is only really ever in lessons. Some things work, some don’t. But I am encouraged by the start I’ve made (and, to be fair, most of my students have made – in particular my year 11s, who have been FAB so far) and I think I’ve managed to set homework for all of my classes thus far!

I have another post planned about a pedagogical issue which regularly gets under my skin, but that can wait for a while…..

My pedagogical plan for 2014-15 (or how I intend to teach…)

I’m feeling really excited and confident about the new term. Brimming with a whole new set of ideas I’ve got a grip on most of the issues I had identified within my practice as needing refinement. And I also feel, this year, that I have a solid structure for operating. It isn’t 100% complete, and I’m sure it will need refining once term starts, but allow me to present my grand pedagogical plan….

(Quick note before I begin: there are similarities between my ideas and those of David Fawcett (@davidfawcett27) in his blog which can be found here. Whilst these are unintentional, I would recommend you read David’s blog at some point. It’s an excellent read).

1) It starts with ‘So that….’ and extends into visible success: This will surprise nobody. I have blogged extensively on this through the holidays and have had dialogue via this blog and Twitter with a couple of people. By taking the time, using Bloom’s Taxonomy, to think through why we are tackling a topic, and how we will know we have succeeded in that topic, I have a really clearly focused plan in the medium term which will allow me to plan day to day lessons easier. As a slight extension to this, I have revisited my medium term plans and started to add in key questions for each topic. David discusses this in his blog, and I have also seen the idea come up in Willingham’s ‘Why Don’t Students Like School?’ and Lemov’s ‘Teach Like a Champion’

2) The ‘So that…’ forms the basis for a pretest: I can honestly say that of all the blogs I have read this summer, the one I keep going back to the most is William Emeny’s Experiments with Visible Learning (this is the second part of a two part blog by William, so you might like to read part 1 for the context). One of the key issues I had from my teaching last year was about showing parents how much progress their child had made, particularly with the removal of levels. Well, now I have a ready made answer! Students will complete the pretest at the start of the block, I will then teach them the content, provide them with opportunities for revision through homework, and after a couple of weeks post-completion of the content, they will sit the pretest as a post-test. This will allow me, my line manager, parents, and most importantly the students, to see not what they have simply remembered, but what they understand and have committed to long term memory. I will also have the evidence right before my eyes! 

3) Day to day teaching: This is where I get to put all of my summer reading into action. I’ve already mentioned the key questions I will be planning into each topic, but I will also be making much greater use this year of Diagnostic Questions for hinge questions in class. I have created some answer packs (basically 4 cards, each labelled A, B, C and D, paper clipped together) for students to use and will be planning them into lessons on a much greater scale. The beauty of these hinge questions, particularly in Maths is that it is quite easy in many cases to provide convincing incorrect answers, revealing student misconceptions.

I also intend to implement a number of rules taken from Lemov’s ‘Practice Perfect’ – breaking down skills into smaller steps, and having 20% more practice than students need being just two examples. There are a range of resources out there (10 Ticks being one, although there are also a number of websites producing randomly generated worksheets) for this purpose. I also intend to continue, as I did last year, pushing students to use precise and technical language at all times. Teach Like A Champion provides some excellent examples of this.

Most importantly, however, is the need for factual knowledge. I wholeheartedly agree with Willingham that factual knowledge is the basis for skill, and so I aim to include factual questions in starters, in general questioning, and in homeworks, as well as in the pre- and post-tests, obviously. 


4) Making the maths explicit: Why Don’t Students Like School? was is an invaluable tool and has had a huge impact on my thinking for the new year. I had begun to consider at the end of last term the need to ‘interleave’ from some of the blogs I had read, and I am glad a now have a scientific back up for this. But equally powerful is the need to make the content explicit. ‘Memory is the residue of thought’ according to Willingham, and I intend to make a huge effort to lead students through at least one practical application of topics where applicable, highlighting where the maths occurs and showing that in all cases, questions can be thought of in purely mathematical terms. 

5) Reflections: This is, as I discussed in my previous post, a key aim for me outside of school (it is something I simply must practice, to avoid falling into the trap of merely being carried along by the busy school term), but I want it to be a huge part of my students’ learning, too. Whilst teaching I intend to ask students to consider why they have chosen a particular method for solving a question, whether there are any alternatives, and so on. And of course, I fully intend to continue using RAG123 with comments, to encourage students to let me know about any issues they may have had at the end of each lesson. 

6) Homework: Alongside assessment, homework is probably one of my biggest weaknesses. I have rarely set it in the past, and I want this year to be different. But I want it to be purposeful, too, not simply for the sake of setting it. A colleague in our department last year gave students an A1 sheet of paper and asked them to complete it on a weekly basis showing the work they had covered in class. This resulted in some excellent examples of student work and is something I intend to try with at least some of my students. In addition, I propose to use homeworks to encourage students to reflect on their learning. I also have a plan to encourage some students to teach topics to their parents, and then invite feedback as to how they did! 

I feel better prepared than perhaps ever before. I have a clear vision of how I intend this year to go. All I need now are some students!