18 March 2023

Exact Trig Values in Depth

One of the most controversial topics to be added to GCSE in recent years is exact trig values. Although trigonometry has always been on Higher Tier GCSE, before 2017 it only ever appeared on the calculator paper. When the Department for Education did a big curriculum and assessment re-jig, exact trig values were moved from A level to GCSE - and, in what we can only assume was some kind of terrible admin error, somehow found their way onto the Foundation tier.

In the latest video from my 'Topics in Depth' project, Craig Barton and I discuss this surprising curriculum decision. We also examine how exact trig values are assessed at GCSE, and we take a look at both pedagogy and resources.

If you're a maths teacher with a spare hour, please watch this! If you enjoy it, also check out Area in Depth and Surds in Depth. Thank you to AQA for sponsoring these videos. 

2 March 2023

Surds in Depth

Everyone loves surds, right? They're so much fun...

Since 2017 I've produced a series of CPD sessions, each one covering an individual maths topic. I call this my 'Topics in Depth' project. These CPD sessions focus on curriculum, assessment, subject knowledge, pedagogy and resources.

Thanks to generous sponsorship from AQA, Craig Barton and I recently recorded a new set of Topics in Depth presentations. We released Area in Depth a couple of weeks ago - thanks for the lovely feedback on this. Today I have the pleasure of sharing Surds in Depth with you.

If you're a maths teacher with a spare hour, please watch this! I hope you find it enjoyable and helpful.

25 February 2023

5 Maths Gems #167

Welcome to my 167th gems post. This is where I share some of the latest news, ideas and resources for maths teachers.

1. Polypad
I've blogged about Polypad before, and used it myself in a lesson on nets last week. My students were suitably impressed! Recently there's been a series of tweets about some really cool stuff that Polypad can do - these are definitely worth a look. Click on the tweets below to see the animations in action, and check out @MathigonOrg's feed for more like this. I particularly like what you can do with factors.

2. Fingertip Knowledge
@mathsmrgordon has shared a set of fingertip knowledge documents for Foundation Tier GCSE. These are like self-quizzable knowledge organisers, similar to my Knoweldge Quiz books, but with calculations as well as factual recall. 

You can download these documents for algebra, number, ratio and proportion, charts and averages, probability and geometry and measure. Thanks Ben!

3. Standard Form
@nathanday314 continues to produce amazing resources. His Calculations with Standard Form Booklet is full of excellent tasks with loads of challenge. Here are a couple of examples:

Nathan also produced some Star Wars Pythagoras Challenges which I love:

4. Kinematic Graphs
I taught tangent to a curve and area under a graph to Year 11 a couple of weeks ago and spotted these two tweets about kinematic graphs. These are helpful for both maths and physics teachers.

5. Same and Different
Thank you to @KarenCampe for sharing her brilliant blog post 'Same and Different'. This post is about a thinking strategy Karen uses for discussion in her lessons in which she asks students to compare and analyse features of two mathematical situations. Here are a few examples.

The first shows two subtraction problems: 5 – 2 (top) and 2 – 5 (bottom).

And this one is for gradient and y intercept:

And here's one on simultaneous equations:

And one on notation:

There are loads more examples in Karen's blog post.

My school finally had its much-awaited first visit from Ofsted. They did a deep dive in maths. I'm really proud of my wonderful team - they're all superstars. Here's seven tenths of my department celebrating the end of a challenging week:

Over half-term I was busier than usual on my blog, publishing three posts:

I received two lovely parcels: David Acheson's excellent new book 'The Spirit of Mathematics' (with thanks to Oxford University Press) and a new Casio 991CW, engraved with my name (with thanks to Stephen at Science Studio).

Here are some things you might have missed:

As we head into Spring, there's loads to look forward to. #mathsconf31 in Bracknell will be great - I'm very much looking forward to presenting on Pythagoras which is one of my favourite topics. La Salle is still looking for presenters for this conference and I strongly encourage teachers to submit workshop proposals. I'll also be taking forty Year 10 students to Maths in Action on 7 March which will be brilliant. And I'll be going to the Christopher Zeeman Medal Lectures and Ceremony at the Royal Society on 22 March. The Spring term ends with the Joint Conference of the Mathematics Subject Associations at Warwick University. I love a residential campus conference! So there's lots of fun stuff coming up. On the flip side, over the next two weeks I'll have 127 Year 11 mock papers to mark, which is a bit less fun... I'm sure lots of you are in the same boat.

I'll leave you with a set of GCSE revision memes (first shared by Paul Collins years ago - and I've added some more). We've put a few up in the maths corridor in the run up to mocks to lighten the mood and get people smiling. Don't forget, if you're looking for revision resources for Year 11, there's an extensive collection here.

17 February 2023


This is a contribution to the series of writings called 'Dose of Don', begun by Anne Watson, which delve into the collection of tasks on Don Steward’s blog. This piece is written by Sam Blatherwick (@blatherwick_sam). Many thanks to Sam for giving me permission to share his writing here. For the background on this series, please see the post Lines and Angles on Square Grids

Don was very generous with his tasks and it is hoped that you will return this generosity in the way he requested before he died, namely to donate to justgiving.com/fundraising/jessesteward.

Don Steward’s median blog was, and remains, a great resource for tasks and there are certain sets of questions that resonate when you come to teaching a topic. You’ll have your own soft spots, but exercises that come to mind for me are the recurring decimals calculations task and the small data set averages problems. Above all though one that stands out when I come to teaching ratio is the ‘Asterix and Cleopatra’ problem sheet. When I first saw it, it felt so modern, simply because we were suddenly encountering questions like this all over the GCSE papers but it didn’t feel like there were any questions in textbooks that were replicating them.

My initial reaction to teaching this was, how? I had my own elaborate algebraic procedures, but I had been chastised in the past by the students for reaching for algebra whenever the going got tough, it wasn’t something that was coming to naturally to students. And then, when you work this out by hammering it with algebra, you were kind of disappointed – like there must be a faster way.

My initial reaction to breaking this down then was to list equivalent ratios, and this is still something I recommend to students when they don’t know what to do. If you do this the answer drops out pretty fast:

What if they had 3:5 – you’d get 1 and 7 – this doesn’t work.
What if they had 6:10 – you’d get 4 and 12 – this does work.

Don crushes the idea that this will be a cinch with the next one:

This takes a lot longer. Fine – we need another method. This is where we hit upon the “what part of the ratio stays the same” approach. The number of marbles Kim has stays the same so we can compare the ratios like this:

Initially: 5:6 = 40:48

Now: 7:8 = 42:48

So we can now see those two marbles in action – Jan had 40 marbles and Kim has 42.

Later we come to a problem again – and if we’d looked back at Anthony and Cleopatra we’d have noted that a method of things being fixed wouldn’t really have worked for us.

What’s fixed here? Not the red socks, and not the white socks. For a while here I was under the impression we were in a pickle when it came to questions like this and just had to go back to listing. Alas, if we go back to listing here, we have to go a very long way.

Until I was introduced to what was actually fixed – and in this case our fixed part was our number of socks – the total of the ratio.

So initially we have: 9:5:14

And subsequently we have 3:2:5

Before: 9:5:14 = 45:25:70

After: 3:2:5 = 42:28:70

So each part of the original ratio is 3 pairs of socks, so we have 135 white pairs of socks and 75 pairs of red socks.

And this is where I thought the story ended, with a lovely straightforward universal method for these problems – decide what you want to be fixed and fix everything in line with it. Then I was in a training session and someone stuck up these questions and asked us to have a go at them, before sharing the algebraic methods that his students did.

And I scoffed and went “oh ho ho, how enlightened I am knowing a better way..!”

There was a discussion around the Sine Rule on Twitter recently where the opinion was aired that as maths teachers we have a tendency to “algebratise” everything too quickly. I’m not sure I saw it at the start of my career, but I think I tend to agree now. Isn’t there something to be said that maths teachers do “algebratise” though? And how do we develop that as a skill?

What about that ‘better way’ I was so smug about? It leads us to the answer, but is it just an endpoint? Where does this lead to? In fact I considered, what if in fact these questions are a lovely vehicle for also introducing some algebraic work? Even after the ‘universal’ method has been introduced there’s running to be had on playing around with algebra in these questions and playing around with how we introduce unknowns.

So going back to the original question:

There are so many ways we can introduce an unknown here. What if we said the number of shares that Asterix has is x?


What about if Cleopatra has x?

What about if the total was x?

A potential stepping stone from ratio to algebra depends on the models you use for ratio. I tend to build from a “boxes” method of ratio to demonstrate 3:5 as

Where the open box is a vessel that has to contain the same as all the other boxes.

If we use that open box as x then we are solving a question where x is one part of the original ratio, and we can apply that to the Asterix and Cleopatra problem.




It’s interesting to note that Don himself notes in his introduction to the tasks that the algebraic way is “relatively unthinking”!

In conclusion, not everything needs to be ‘algebratised’ and often other numerical ways of solving problems can give you solutions to problems in elegant ways. However, the answer need not be the end and exploring different algebraic ways of exploring these problems can help your students fluency and competency with algebra.

16 February 2023

Area in Depth

Since 2017 I've produced a series of CPD sessions, each one covering an individual maths topic. I call this my 'Topics in Depth' project. These CPD sessions focus on curriculum, assessment, subject knowledge, pedagogy and resources.

Maths teachers simply don't get enough time to research topics thoroughly when they plan lessons. In this project I aim to support teachers by doing the research for them. I've prepared a number presentations that teachers can look at individually or as a department. Some of these presentations have been recorded and you can find them on my Youtube Channel.

Thanks to generous sponsorship from AQA, Craig Barton and I recently recorded a new set of presentations which will be released over the coming months:

1. Area in Depth (below)
2. Surds in Depth
3. Exact Trig Values in Depth

I'm really proud of these. If they're well received, we'll record more videos like this in the near future.

I did intend to write a book to accompany my Topics in Depth project, but it's very difficult for a full-time teacher to find time to write a book! Maybe one day...

If you're a maths teacher with a spare hour, please watch Area in Depth and the two videos that follow. I hope you find them both enjoyable and helpful.

The slides for this Area in Depth video can be downloaded here.

12 February 2023

Ten Videos for Maths Lessons

I rarely show videos in maths lessons but every now and then I deviate from my usual lesson routine. Sometimes it's to help something stick in my students' memories, and sometimes it's part of the lesson's narrative. I thought it might be helpful to compile a 'top ten' short video clips for maths lessons. If you've been teaching for a while then you've probably seen them all before. But if you're new to teaching, you might find something you like here.

Clip 1
Source: Despicable Me
Used for: Introducing the concept and helping students remember the definition.
Duration: 40 seconds

Clip 2
Compound Interest
Source: Futurama
Used for: Introducing the concept (can also set a related task - see this post from @MathsEdIdeas)
Duration: 42 seconds

Clip 3
Topic: Pythagoras' Theorem
Source: QI
Used for: Demonstration (but not proof!) of the theorem
Duration: 90 seconds

Clip 4
Topic: Pythagoras' Theorem
Source: Wizard of Oz (and The Simpsons)
Used for: Asking students to correct the theorem
Duration: 44 seconds
Read Simon Singh's brilliant book The Simpsons and their Mathematical Secrets for more on this.

Clip 5
Topic: Lowest Common Multiple
Source: Father of the Bride
Used for: Starting a discussion about contextual LCM problems (can also set a related task - see this activity from mathsbits.com)
Duration: 127 seconds

Clip 6
Topic: Metric Units
Source: TED-Ed
Used for: Narrative, enrichment, historical context
Duration: 5 minutes

Clip 7
Topic: Multiplying Negatives
Source: Stand and Deliver (a classic 'must see' film for maths teachers)
Used for: Helping students remember, and also prompting the question 'why' (plus a great opportunity to tell them about Jaime Escalante)
Duration: 40 seconds

Clip 8
Topic: Loci
Source: Taken 2
Used for: Seeing loci and map scale in action 
Duration: 165 seconds

Clip 9
Topic: Polygons
Source: Ed Southall
Used for: Exploring the names of polygons 
Duration: 155 seconds

Clip 10
Topic: Area Scale Factor
Source: Big Bang Theory
Used for: Helping students remember the conversion between square units 
Duration: 80 seconds

Finally, two of my favourite gifs:

There are also loads of great discussion points in this Twitter thread of maths from quiz shows from @missradders. And loads of mathematical hooks on this Padlet from @tessmaths.

I should also mention the best maths video website in the world, Numberphile, which is packed full of fascinating maths for both teachers and students.

Do you recommend any videos that you show in maths lessons? Comment below.

29 January 2023

5 Maths Gems #166

Welcome to my 166th gems post. This is where I share some of the latest news, ideas and resources for maths teachers.

1. Mixed Interwoven Revision Tasks
@nathanday314 shared a set of interwoven revision tasks aimed at Year 11. These excellent tasks are editable and suitable for projecting on the screen. 

2. Completion Tables
@studymaths did a conference presentation on completion tables or 'fill in the blanks' activities. I've always been a big fan of this task format which features so much opportunity for challenge and reasoning. There are loads of lovely examples on @studymaths' Twitter feed. Here are a couple that I particularly like. First, one on standard form.

And second, one on functions - the idea for this one came from @PaulRodrigo2718.

3. Sparx Workbooks
Sparx Maths has shared a free set of Crossover Workbooks - these might be helpful for GCSE revision or intervention sessions.

Speaking of intervention, I've recently started running a Year 11 maths intervention in tutor time and I wrote a thread about how I'm structuring it. It includes a link to the diagnostic test that I used at the start of the intervention.

4. Formula Sheet Tasks

This year our students will receive formula sheets in their GCSE, and we need to make sure they know how to use them effectively. It's possible that they will forget they have this sheet in the exam because they're not used to using one. I've previously blogged about resources that students can use to practise using the formula sheet, including these 'Assessment Aid Tests' from @taylorda01 and these formula sheet tasks and PowerPoints from @Just_Maths. Thanks to @colleenyoung for drawing my attention to another resource: formula sheet tests from OCR. These are useful no matter what awarding body you use. OCR has also published some formula sheet top tips

5. Problem Solving Techniques
Thanks to @SarahFarrellKS2 for sharing a blog post about supporting students with problem solving techniques. This is aimed at primary teachers but there's lots of sensible advice for everyone. It features resources for teachers and maths leads to use. 

There are two things you only have one more day to do: 
  • Book a place at the Joint Conference of Mathematics Subject Associations and receive the early bird discount. This takes place at Warwick University in the Easter holidays. I've booked my place! I'm really looking forward to it.
  • Apply for the Key Stage 5 Maths Coordinator Role at my school which is a fantastic opportunity to be involved in the launch of a new Sixth Form. Alternatively, if you want to teach A level at my school but you don't fancy the TLR, please get in touch.

Here are some things you might have missed:

Yesterday, Craig Barton and I spent the day at a recording studio working on a project with AQA. Watch this space!

Finally, I'll leave you with this 'Spicy Question' from @1stclassmaths. You can find more spicy questions here - there's lots of great challenge for your students in these questions. 

8 January 2023

Tips for Teachers

Exciting news! Craig Barton has a new book out: Tips for Teachers. If you're a maths teacher who reads my blog then I'll be surprised if you haven't read at least one of Craig's other books, so I don't need to tell you how utterly engaging his style of writing is. You already know that this book will be a thoroughly enjoyable read, packed full of sensible and practical teaching ideas. It's classic Craig.

In this book Craig takes things that many experienced teachers already do, and tells us how to do them considerably better. And for people new to teaching, highly effective classroom practices are broken down and explained in such a way that they are super easy to implement for the first time.

For example, Craig tells us how to Cold Call, how to use mini-whiteboards, how to do exit tickets and what to do if a student says 'I don't know'. He tells us what we're doing wrong and how to get better. I've been teaching a long time now, but when I read this book it made me acutely aware that I can improve in a number of areas. All teachers can always improve. It forced me to reflect on my embedded routines and wonder if a few tweaks here and there might benefit my students.

My favourite bits of advice are those regarding questioning and checking for understanding. I thought I was pretty good at questioning until I read these sections! In Craig's advice on improving Cold Calling, he gives detailed explanations and advice on:

1. Telling students why you're doing Cold Call
2. Asking the question, then saying the name
3. Giving adequate wait time before taking the answer
4. Asking students to respond using full sentences
5. Giving adequate wait time after hearing the response
6. Managing your tell
7. Promoting active listening

This takes the reader from this starting point (i.e. not very good use of Cold Call):

Teacher: Harry, can you tell me Pythagoras' Theorem?
Harry: c2 = a2 + b2
Teacher: Fantastic. Okay, next question...

right through to this:

Teacher: Okay, no shouting out, everyone silent thinking (wait)...  What is Pythagoras' Thereom?... (wait) ... Harry?
Harry: c2 = a2 + b2
Teacher: Full sentence please...
Harry: Pythagoras' Theorem is c2 = a2 + b2
Teacher: (wait) Thank you, Harry. Kyle, what do you think of what Harry just said?

There are further examples, including those that show a sequence of Cold Calls. 

What I like about Craig's books is that he explains things incredibly well, because he's an outstanding teacher himself. He practises what he preaches - his book is full of very clear modelling, making it easy for teachers to learn techniques and apply them in their own classroom.

In terms of checking for understanding and addressing misconceptions - something that is vitally important in the maths classroom but maths teachers often struggle with - I love Tom Sherrington's '8 out of 10' tip on Page 189 (read the book for more on this!).

Rounding Up
The other section that really struck a chord with me is about 'Rounding Up'. 

Rounding up, as it's described by Doug Lemov, is when a teacher responds to a partially or nearly correct answer by affirming it and adding critical detail to make the answer fully correct. Like this: 

Teacher: What is a good first step when trying to solve 6x + 5 = 29... Molly?
Molly: You move the add 5 to the other side, and it becomes a takeaway 5.
Teacher: Nice! So you are saying subtract 5 from both sides of the equation, because subtracting 5 is the inverse operation of adding 5.

I do this all the time, and I kind of feel like it's important to do it to ensure my students are always exposed to clear and correct explanations. But this book made me question my thinking on this.

Craig explains why we shouldn't round up, and gives examples of more effective exchanges between student and teacher. Since I read this, I've been working on improving my questioning and cutting down the frequency of my 'rounding up'.


Just like Craig's other books, Tips for Teachers is incredibly useful for both new and experienced teachers. If you're a reflective practitioner who is always striving to improve, then read it - I think you'll be surprised by the sheer volume of good ideas. 

This excellent book again shows us that Craig Barton is one of the UK's leading experts in teacher development.

31 December 2022

5 Maths Gems #165

Welcome to my 165th gems post. This is where I share some of the latest news, ideas and resources for maths teachers.

1. GCSE Revision
On The House Maths (@OTHMaths) shared a new set of GCSE revision resources. Their 'Aim for' revision grids are designed for both Higher and Foundation tiers.

I have added these to my GCSE Revision Resources page.

2. GCSE Further Maths
Thank you to @1stclassmaths who has now completed the Level 2 Further Maths page of his excellent website. The page features videos and topic booklets for all Further Maths topics beyond the regular GCSE specification. These are really high quality resources - I've been using them with my Further Maths class.

3. New Resources and Websites
Thanks again to @draustinmaths for continuing to generously share her resources. Her latest batch of functions resources were really helpful for my Year 11s in December. There are some great tasks on composite and inverse functions. 

@ChrisMcGrane84 shared a practice task on finding where lines and quadratics intersect the x and y axes

@markywillis63 has updated his A level website. Each topic now has a set of questions with gaps for students to fill in whilst either trying the questions or watching the videos. 

I was emailed by maths teacher Robyn who told me about the website physicsclassroom.com which is helpful for A level mechanics teachers. They have printable worksheets, online concept checkers, interactive tools, videos with questions and so on. One example is a self checking quiz about free body diagrams, with an accompanying video and a worksheet.

I also received an email from Sam Kordan who created a website with a colleague when they worked together at a SEND school: www.adamupmaths.com. It's a collection of songs, games, printable teaching models and more.

My readers will be pleased to hear that onmaths.com has started doing A level papers, and that gcsemathsquestions.co.uk has been updated with questions from November 2021. 

Thank you to all these generous maths teachers who have been working hard to create and share content in recent months.

4. Infinite Whiteboard
I've blogged before about the fantastic tools shared by @JakeGMaths at mathsuniverse.com. This includes a huge collection of worked examples and a skills grid generator. Jake has recently been working on his Infinity Whiteboard which is fantastic. You can paste an image of a task or student's work and then annotate it. It's also really easy to paste a PDF which comes up as an image. It's great for live modelling and you'll never run out of space! One thing I'll use it for is to review exam papers - it takes seconds to set up:

Previously we've copied and pasted every question into PowerPoint to review an exam paper, which I now realise wasn't a good use of our time!

The infinite whiteboard is incredibly easy to use. Do check it out.

5. All Ten
In the run up to Christmas I massively enjoyed completing the nerdle advent calendar. I've completed a nerdle puzzle every single day since February. It brings a little bit of joy to my mornings. When I recently tweeted to sing nerdle's praises, I had a reply from @druinok about All Ten which I hadn't seen before.

All Ten is a lovely game which gives your four different numbers every day, and you have to use those four numbers to make each number from one to ten. The interface is great - it's very engaging and easy to use. As well as being an enjoyable game for teachers (I now do it every morning, as soon as I've completed my daily nerdle!) it's also a good one to use in lessons - particularly when teaching the order of operations.

It reminds me a bit of the classic 24 Game.

I've been super quiet on Twitter and my blog lately because, like so many others, I've been really unwell. But, just in time for the start of term, I'm starting to feel better now, and am looking forward to an exciting term ahead. At school we'll be running the Intermediate Maths Challenge, plus we're taking forty Year 10 students on a trip to see GCSE Maths in Action - and I'm delighted to have this trip funded by a grant from Jack Petchey

I'm looking forward to the conferences that are coming up. At La Salle's conference in Bracknell on 11th March I'll be speaking about Pythagoras. And at the Joint Conference of the Maths Subject Associations I'll be speaking about improving students' calculator fluency. This two day conference takes place at the University of Warwick on 3 and 4 April - I've you've not been to one of the residential Easter conferences before, I strongly encourage you to come along! Book by 31 January to receive the Early Bird discount.

My lovely team will be recruiting this term - we need a Key Stage 5 Coordinator to run A level maths, which we're launching for the first time this year. If you have A level teaching experience and you're looking for a new challenge in September, please contact me (resourceaholic@gmail.com). The job advert will be out soon.

I enjoyed attending Dr Frost's triannual drinks this week. It's always a pleasure to spend time with maths teachers.

Finally, check out my fantastic stickers! I searched high and low for some nice personalised stickers to buy my colleagues for Christmas - and when I found these, I bought some for myself too!

Happy New Year, and all the best for 2023.