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WHTW – Number puzzles, Pythagoras, Constructions and nice students!

Lots of ‘just getting on with things this week but, some interesting points came up.

Number puzzles

I spent lesson time allowing students (year 10 and year 9) to get stuck with this nrich maths puzzle. It was fascinating watching students really want to seek help from their friends despite me telling them that this was their opportunity to battle through something and demonstrate perseverance. What’s nice about the puzzle is that is completely understandable – a couple of clarifications means that everyone knows what they have to do, but it’s the how that’s the challenge. This is a classic example of maths where it isn’t the answer that’s interesting, it’s the puzzle of trying to find out how to get to it that’s the reward. I enjoyed having that conversation with students as it really doesn’t come up as much as it should.

Tilted squares

In the tilted squares investigation, I’ve got to the point where one of my classes has realised that it was actually pythagoras all along! It was great to see it dawn on them and alongside the “Couldn’t you just have told us that from the start” comments, there was also a sense of “Oh, so now we can get much further with this more quickly.”
I need to think of a more satisfactory way of working through the investigation backwards: If we are trying to find a square with an area of 21, how could we use pythagoras to do that, or, to show it’s not possible?


I’m not a fan of teaching constructions as it seems to take forever and there’s a lot of different techniques to remember. Having said that, this year, my year 11 students seem to be really good at using pairs of compasses. That’s practically unheard of but it did mean that the lesson went really well. I’ve had some of the best ‘centre of a circle’ constructions I’ve ever seen from this class and I might be coming round to liking constructions a bit more!

Odds and Ends

We had maths teacher interviews this week and have selected a great PGCE student from a strong field. One of the comments made was that it was obvious that the teachers in the department really loved maths. I already knew that was true and you might assume that all maths teachers love maths but I can assure you that it isn’t the case. I was pleased that this fact about our department comes through even to relatively brief visitors.

I marked the mechanics questions of my year 12 mocks this week. I’ve decided that I’m slightly warming to mechanics. If you know me and/or listen to the Wrong, but Useful podcast, you’ll know that’s quite a breakthrough.

It’s quite common at my school for students to say “Thank you” at the end of a lesson as they’re leaving. I was particularly struck by one student who, last lesson on Friday, left the room, realised they hadn’t said “Thank you” so made a point of coming back to say it. If you haven’t picked up on it yet, I’m in a rather nice school!


WHTW – Magic Gopher, Shape Sorting, Indices and Vectors

Busy week as always. Some nice finds and experiences this week.

Magic Gopher

In the 1089 project, we prove that the process always leads to 1089 for a 3-digit number. As a precursor to that, another teacher showed us all a nice website that has a simpler ‘think of a number’ effect that really impressed my students.


It was great to work through the algebra with them and for them to really ‘get’ why the trick was working. There were some actual jaw drop moments, especially when the gopher was right twice in a row with a different symbol each time!

As an aside, I kept catching myself referring to the gopher as ‘he’. Must stop doing that.

Shape Sorting

The battle to correct misconception about shape names continues.


shape sorting standards

Our head of maths showed us a nice standards unit that has a lot of scope for playing around with classifying shapes. I didn’t bother printing the grids out and some students thought to just use pens/pencils to recreate the grid on their page. There were good discussions such as:

  • Should the circle go in the ‘has no right angles’ group?
  • Does a square have two pairs of equal sides?
  • Can a shape have ‘one side equal’?

I incorrectly named an angle as obtuse when it was really reflex and a normally fairly quiet student felt happy to correct me. That was good to see.

I also used this site in a computer room and it does a lot to work on these sorts of categorisations.

shape sorting online

Students have suggested I should write a ‘My First Shapes Book – a Maths Teacher’s version’ and I honestly think there’s some scope in that. One for the ‘to do’ list.


When discussing trapeziums, two separate students said:

  • My primary school teacher said that’s the one that if you turn it upside down, it looks like a flower pot.
  • It’s the thing that elephants stand on.

Probably some more to do here then.


Had a really solid lesson where students just practised the skills of multiplying, dividing and raising a power to a power.

Some year 9 students were able to get their head around rewriting powers in a different base. eg rewrite 8³ as a power of 2. This was a good stretch for those that had solidly understood the main points.


Year 11 students are finding this either easy or very hard. There isn’t a middle ground. I need to find some ‘prove that these are a straight line questions’ that are more structured.

Mock year 12 exams this week so that will be interesting to mark. More to come next week.

WHTW? 1089 and classifying quadrilaterals

Last week, I wrote about returning to the title of my blog site and being a ‘reflective maths teacher’. I haven’t figured out exactly what format I want this to take yet, but for now, I’m going with What Happened This Week? (WHTW?)


In year 7, our project this term is 1089 and there’s a very brief description here if you’re not familiar with it. It’s a nice little project and really flags up who isn’t comfortable with adding and subtracting.


Students are doing really well at making conjectures:

  • I think that after the subtraction, the middle digit is always 9
  • The difference between the first and third digit, multiplied by 99 is the answer to the first subtraction

And, after moving into 4 digits, students are productively amending previous conjectures:

  • Student A: If three digits are the same, you get 10989.
    • Student B: No, you only need two digits the same.
      • Student C: Not quite, the two the same have to be in the middle.

I’m really pleased about the conjecturing and that the students seem to be fairly comfortable with having their conjectures shown to be wrong. However, I think I still have a way to go with this as there are some students that are reluctant to share their thinking. I may well force this point and individually ask them for a conjecture 1-to-1, then put it on the board.

Two students particularly impressed me. Some of the three digit numbers lead to 99 after the subtraction and you have to discuss whether to treat that as 99 or 099. If you choose to treat it as 099, then you end up at 1089, if not, you get 198. As part of the class discussion, one student said that you’d like a reversed number to reverse back to itself (eg 192 to 291 to 192 to 291 etc) and for that to happen, if we start with 990, then we would need to keep the zero on 099 for it to self reverse. This is the nicest reason I’ve ever seen for why to keep that leading zero!

When we moved to 4 digit numbers, I asked whether we should keep the rule about ensuring our first digit is larger than the last digit. One student said, “I think we should keep that rule as we’ve already changed from 3 to 4 digits so we shouldn’t be changing more than one thing at a time.” This was absolutely the most mathematical thing that any of this class said last week and really showed strong progress in understanding how to do a maths investigation.


My weakest students are still really struggling with subtracting and adding consistently. I’m at a little bit of a loss as to what to do as they can do it when I talk them through 1-to-1 but then don’t seem able to carry on by themselves. I have contacted one parent and probably need to with the others.

The layout of the calculations is widely variable. I evidently didn’t model this as clearly as I could have. I also need to think more clearly about what to do with students that easily could lay it out better but are choosing not to. I think I might use a structured worksheet but I also don’t want to ‘do it for them’ and take the responsibility away.

Classifying Quadrilaterals


I haven’t really got very far into this yet but students are clearly (mostly) comfortable with things like right-angled and parallel.


Quite a few sadly.

The first one is that I realised this week that I have been teaching a vocabulary word incorrectly. There are a number of ‘named’ quadrilaterals and then, there are the ‘other’ ones. I’ve been calling those ones irregular and have even said “Irregular ones are the ones where none of the sides are the same length”. This is simply wrong. I don’t really know how I’ve gotten to this point but it is quite a weird feeling knowing that this relatively simple thing is something I’ve been doing wrong, probably for 16 years now! Obviously it’s good that I’ve spotted this and can correct it now but still…

I don’t seem to be able to get across the nested nature of quadrilaterals. Having gone through with a class what the properties of various quadrilaterals are, I asked whether rectangles were parallelograms, to be met with a slight majority vote of “no”. I’m prepared to accept that quadrilateral classification is a bit tricky/nuanced but it shouldn’t be that bad. Evidently, I need to reflect on how I teach this. Current thoughts include using venn diagrams in future.

My students still aren’t all convinced that squares are rectangles. I think this might be part of a bigger battle in society in general and maybe I worry about this more than I should but I’m wondering if I can find out a bit more about how quadrilaterals are introduced in nurseries and primary schools.

Index notation


One of my classes have picked up the three basic index laws (multiply, divide, power of power) just from seeing two examples of each. It just sort of ‘clicked’. I’m pleased that I (correctly) judged that a simple example would be enough for them to get it.

New year – new blog plan

Hi all. I’d imagine that many people have new year’s plans to blog more often/differently or something else and I’m well aware that it’s easy to make plans and not stick to them. I’ve clearly not been blogging that frequently recently and much less than in my early years:

All time blog views

I think this is probably to do with my change of approach. In the early days, I would send a quick email to Posterous (the host I used before WordPress) and it would be automatically uploaded, making it easy to put up quick thoughts after a lesson. More recently, I’ve been planning my posts more carefully and trying to think about creating more polished content. While I do like this approach, it does appear to mean that I just haven’t been posting anywhere near as much.

So, the plan is:

  • Post more frequent, short thoughts,
  • Focus more on things that happened in my lessons and/or lessons I’ve seen,
  • Record (in writing) some of the frankly, great conversations that happen amongst my maths team.

I think it’s time to go back to being a blog about a full time teacher with more day-to-day thoughts. At least, that’s the plan!

School Kindness Calendar – Dec 2017

There’s been quite a focus on kindness recently what with kindness week, my school having kindness as the ‘fortnight focus’, and this blog post from my headteacher (Chris Hildrew). It’s probably not a coincidence that my wife brought home a kindness calendar that she had seen on twitter from the website Action for happiness. We both really liked the idea and thought it was worth creating a similar one aimed at schools and for promotion during tutor time.

Here is our version. You’re welcome to use and/or adapt it as you wish.

School Kindness Calendar

pdf version School Kindness Calendar

pptx version School Kindness Calendar

We’d love to hear how you get on!

Maths Jam 2017 talks summaries

Here is a brief summary of individual talks from the Maths Jam annual conference. Thanks to Rob Lowe for writing the bulk of this and allowing me to share it.

There is a write up on its way from Maths Jam themselves and also this one from Chalkdust.

Colin Beveridge and I discuss it and have various guests comment on Wrong but Useful episode 50.

Saturday – Session 1a: 14:00 – 14:47

· Colin Wright, Matt Parker and Katie Steckles: Welcome to the MathsJam Gathering

Pretty much “what it says on the tin”

· Tom Button: All about the base (no trebles)

Using base 10 to prove a neat result about primes. (OK, that’s 10 in duodecimal.) Introduced the concepts of Threven for numbers in the three times tables.

· Matt Peperell: Logical deduction games

Introducing games where the aim of the game is to work out the rules. Sounds like a lot of fun and something I’ll be looking into more!

· Alison Kiddle: Alison talks crap

Alison presents and interprets the results of a fundamental poll: in fact, many fundaments were involved. Correlating number of sheets of toilet paper used with the Bristol stool scale. Used a linear correlation and log one too.

· TD Dang: The maths in Mean Girls

When is a limit not a limit? There’s no end to the possibilities. Looked at the mathlete’s activities in the film.

· Noel-Ann Bradshaw: Digesting the indigestible

On the importance of presenting data well and informatively. Wondering about the best way of presenting information about students to busy teachers.

Session 1b: 15:10 – 15:57

· Zoe Griffiths: A discourse on e

A heartfelt narration of the relationship between e and their x. You can listen to this on episode 50 of Wrong, but Useful.

· Phil Chaffe: Maths Jammin’ – Writing a song for the Maths Jam Jam

Hints and tips on how to write a song for maths jam. Essentially, a whole lot can be forgiven if there’s a really good line!

· Matthew Scroggs: Big Ben Strikes Again

How can you hear Big Ben strike 13? Why does Scroggs like Captain Scarlet so much?

· Andrew Russell: Diabolo as a picture

The title was an outright lie. Instead, he explained why balloon animals were semi-Eulerian graphs, and provided a counter-example. This was a talk that appealed to my inner clown.

· Angela Brett: Mathematical poetry

In which were reminded that it isn’t just giants whose shoulders we stand on. Available on Etsy here.

· Adam Townsend: Stop! (or, using maths to pass your driving test)

Where did those weird stopping distances come from? Are they even correct? (no) What should they be?

· Elizabeth and Zeke: rat with an e

What is 3^(1/ln 3)? More to the point, why?

Session 1c: 16:20 – 17:07

· Rob Eastaway: Thinking Outside the Outside of the Box

On drawing lines through all the dots in a grid: how far can you go? The classic puzzle of 9 dots in a 3-by-3 array. Can you join them with 4 straight lines? What about if it’s a 4-by-4 array with 6 lines? How far ‘outside the box’ can you go?

· Rachel Wright: In A Spin

Between sheep’s back and your back, the wool undergoes various transformations. Some of it’s a bit confusing.

· Alex Burlton: Bags of Palindromes

What are the chances of getting palindromes out of bags of numbers?

· Alexander Bolton: Winning the Chalkdust Coin Game

Why it’s a good idea not to have too similar a name to any of your peers, and why it’s a good idea to take a risk if you think you’re going to lose.

· Vincent Van Pelt: Thank you, Mrs Holcombe

Homophones. And, by a happy accident, an introduction to mediaeval French poetry.

Session 1d: 17:30 – 18:17

· Dan Hagon: Double Negation and the Excluded Middle

Some people aren’t happy with ¬¬P=P. Dan gives a constructive explanation of how logic works – or ought to.

· Ben Pace: Building Successful Intellectual Communities

Working towards a way of ranking web pages by reliability, using the Page rank algorithm (named after Larry Page and not because it ranks pages) and karma.

· Alison Clarke: Stupid Units

Pressure in mmHg? Volume in Acrefeet? Temperature in Fahrenheit? What were they thinking?

· Belgin Seymenoglu: Donald in Mathmagic Land

Half a century ago, Donad Duck starred in a cartoon introducing some mathsy fun.

· Douglas Buchanan: Lowering the Tone

Puzzles. I nearly fell of my perch at the resolution of the parrot puzzle. Possible the worst pun of the weekend, and there was no lack of competition.

· David Mitchell: The Thereom of Trythagoras (Pythagoras is for Squares)

Extending Pythagorean triples in triangular ways.

· Dave Gale: Catchphrase and Coffee

What do the ‘strength’ indicators on ground coffee mean? Why do some go from 3 to 7?Also, an (as yet) unsuccessful attempt to get the host of Catchphrase to stop calling rectangles “squares”.

Sunday – Session 2a: 08:50 – 09:37

· Joel Haddley: Angle Trisection

What are the rules? What counts as a trisection?

· Katie Steckles: Sheeran Numbers

What numbers can you make using (all) the operators used as titles of Ed Sheeran albums, and numbers used as titles of albums released during Ed Sheeran’s lifetime.

· Ken McKelvie: A little ado about ‘nothing’

Looking at where zeros occur in decimal expansions of certain numbers.

· Tony Mann: The mathematics of competition

Where on the beach should you put your ice cream van to maximize profit?

· Will Kirkby: Life Beyond Binary

Generalizing cellular automata, and some very pretty pictures.

· Peter Rowlett: Fermi problems

The kinds of estimating you have to do (in ‘real’ life), and the Approximate Geometric Mean as a useful tool crying out for a better understanding.

· Kathryn Taylor: Adventures in modular origami

How to make wonderful models, and why not to take them on the train.

Session 2b: 10:00 – 10:47

· Marcin Konowalczyk: Unrolling the rolling shutter

Trying to train a neural net to recover the original image. The dangers involved in choosing the training data.

· Miles Gould: How Mountaineering is like Mathematics

In every possible respect, it turns out. You can watch a video of the talk here.

· Samuel Ball: Fake It Till You Make It

Markov processes for constructing tweets. Also, helping people learn to code.

· Wendy Foad: Context vs content

Transferable skills?

· Nicholas Korpelainen: A production line may need an arbitrarily large number of machines

Satisfying constraints is sometimes hard.

· Robert Woolley: Making board games fit – Numbers & Space

How to fit several board games into one box. Clever use of the space available on playing cards.

Session 2c: 11:15 – 11:48

· Glen Whitney: The Hole Truth

Holes in a handlebody, Euler characteristic, and why a topologist is somebody who can’t tell a steering wheel from a T-shirt.

· Sue de Pomerai: The life and times of Ada Lovelace

The briefest of introductions: and a class in delivering a one hour talk in four minutes. Sue will hopefully be appearing in Wrong, but Useful episode 51.

· Pedro Freitas: A programmed deck

Programming a deck of cards to solve simultaneous equations.

· Matthew and John Bibby: Boring log and geometrical tables

A father and son team show us where woodwork and seaside rock meet mathematics.

· Geoff Morley: Irrational Bases

Looking at some of the richness of the behaviour when an irrational number is used as a base instead of an integer.

· Adam Atkinson: Mathematics and Art: A Real-World Problem

On the challenges facing a sculptor who wants to put a statue on top of the nearby mountain for residents of Catania to enjoy.

Session 2d: 12:06 – 12:39

· Elaine Smith & Lynda Goldenberg: Multiplication: Magic or Madness

Multiplication methods, and the ‘new’ grid method is several centuries older than the ‘traditional’ column method. Also Napier’s bones.

· Robert W. Vallin: Maverick Solitaire and Three-Card Poker

Probability inspired by an episode of Maverick.

· Robert Low: Why knot?

What do you mean, I can’t tie a knot in a piece of string without letting go of the ends? You’re not the boss of me!

· Philipp Reinhard: From a tweet to Langnaus 4th problem in < 5min

How to get from a relatively innocuous looking puzzle to some really deep stuff in a few steps.

· Oliver Masters: The Fibonacci Matrix

Coding the Fibonacci sequence in powers of a matrix, and some surprises arising.

Whew – that’s all there was. Just a mere 49 talks!

Thanks again to Rob for the comments. I’ll expand on some of the talks in future posts.

Mathsjam 2017 summary

*nb I have undoubtedly missed some things out. I’ll remedy that in future posts.*

This weekend saw the annual Big Maths Jam in Staffordshire and once again, it was an excellent and invigorating injection of maths.

I’ll need more time to recall all the maths talks I saw and parts of the many discussions I had this weekend but I want to post a few thanks and highlights. I’ll expand on all of these in later blog posts but, for now…

Some highlights:

  • Meeting up in person with so many people I have twitter conversations, including wrong but useful podcast cohost – Colin
  • The massive range of topics in the 49 talks
  • All the toys, puzzles and 3D printed goodies
  • Zoe‘s poem about e
  • Peter’s talk about Fermi problems
  • Sam talking with me about coding and giving practical advice about getting started
  • Rob giving me useful advice about juggling four balls
  • Taking part in multiple competitions (and winning one)


  • Colin for organising the event
  • Katie for also organising the event
  • Matt for (apparently) not really organising the event but still taking up to 1/3 of the credit
  • The bake off and competition competition entrants
  • Everyone that did a talk
  • Anyone that laughed in the right places during my talk
  • Everyone that brought an interesting thing to look at/play with

Ok. That’ll do for now (more to come I promise). If you’ve never been before, seriously consider planning to go next year. You won’t regret it!