5 Maths Gems #201

Welcome to my 201^st gems post. This is where I share some of the latest news, ideas and resources for maths teachers.

1. Volumes of Revolution

My excellent A level student Samarth made a website volumesofrevolution.co.uk. He has very kindly shared it so that A level Further Maths teachers can use it in lessons when teaching volumes of revolution.

The website is very easy to use. You can create piecewise functions to make some interesting solids, and there's transparency settings to allow two different functions to be seen one inside the other. You can play and pause an animation of the rotation being performed, and sign in with Google to save projects. 

To minimise Samarth's costs (he's heading off to university this September!), the website will only run on weekdays. He's also included a buymeacoffee link to help keep the website  running. I'm very proud of Samarth and hugely impressed by his level of skill and attention to detail in creating this website. 

2. Clumeral

Emma Lark showed me the lovely puzzle site clumeral.com made by David Lark and Jamie Evawin. You work out the three digit numbers using clues and a process of elimination. Once you do today's puzzle you get access to the archive. I think both maths teachers and students will really enjoy these puzzles!

3. A Level Resources

Because most of the maths teachers frequently posting on Bluesky are A level teachers, I always have loads of lovely A level resources to share in my gems posts these days! Here are some I've spotted recently:

• @mrlevmaths.bsky.social shared some ideas for introducing students to double angle formulae.

He also shared a resource for reciprocal trig functions. 

• @andrewmaths1.bsky.social found a great website with TMUA practice papers: jzmaths.com/papers. There's also a very cool Easter Egg hunt that students (or teachers!) might enjoy completing this summer. 
• @sxpmaths.bsky.social has shared a blog post about all the changes he's been making to the sequence in which he teaches A Level Maths. This post includes a link to one of Stuart's booklets - Unit 1 : Points and Lines.
• @markywillis63 on Twitter shared a new series of Year 1 A-level questions by topic with both answers and worked solutions. 

4. Riddle of the Day

Gabriel Sobrino emailed me about the website riddleoftheday.org. Riddle of the Day is an independent bilingual site publishing one logic or mathematical puzzle each day, with progressive hints and fully explained solutions. I thought this might be helpful for maths clubs and competitions in schools.

Riddles are classified by difficulty level. For example here's one of the easier ones:

5. Constructions
@giftedhko.bsky.social has started writing a monthly newsletter for her department. Her July 2026 issue focused on construction:

Helen has some fantastic construction resources on mathshko.com.

 

I also spotted some constructions tasks on Transum when reading their latest newsletter. Their updated constructions activities give students the chance to practise compass skills, with each task leading to a measurable result: students complete the construction on paper, measure a length or angle, and enter their answer online for immediate marking. 

Whilst on Transum I spotted this cheating poster, updated to mention AI - I think it's worth talking to students about this in September (particularly A level students).

Update
On the last two days of the summer term I had the privilege of attending an event hosted by XTX Markets and Purposeful Ventures. You may already be aware of XTX's generous charitable donations in the sector, supporting organisations such as UKMT, Axiom Maths, Maths Horizons and Dr Frost (if you're interested in finding out more, read mathsexcellencefund.org, and XTX's philanthropy page). Over the course of the residential event, which was held at the very quirky but beautiful Brockett Hall in Hertfordshire, we discussed how to build more future mathematicians. 

For me it was incredibly worthwhile to consider the whole 'talent pipeline' from primary school through to PhD. For example on Thursday we broke into small groups and I ended up spending most of the day in a pair with Tom Crawford (his YouTube channel is popular with my students!) discussing in great depth how we can get more A level students to choose maths degrees. I loved how intellectually challenging this event was for me - I had the opportunity to do a lot of deep thinking and generate solutions to big problems. I really enjoy this kind of thing. I also loved catching up with a number of people who I haven't seen in years, and meeting lots of new people with fascinating jobs. There were plenty of attendees who work closely with schools, including people who do some teaching in schools as part of their jobs, but I was the only classroom teacher at the event - I hope I did a good job of representing my profession. I'm very grateful to XTX for the invitation.

Exciting New Project

It was good to see Craig Barton at this event - we always have lots of ideas for how we can support secondary maths teachers, and this time we've come up with something which we think is going to be brilliant. I don't want to give any spoilers here, but watch this space because we'll be launching it really soon. It's aimed at Heads of Maths and aspiring Heads of Maths. I genuinely think it's going to be a game changer.

Future Events

La Salle's maths conference dates for 2026/27 are:

• 5th September 2026 - Maghull High School, Liverpool (tickets now on sale)
• 14th November 2026 - Trinity Academy, Barnsley
• 20th March 2027 - Penrice Academy, St. Austell
• 5th June 2027 - The Duston School, Northampton

I can't make the first one unfortunately but I hope to be at others!

I will definitely be at BCME, which falls in October half term for me. I hope to see lots of you there. Make sure you apply for a bursary!

As the weather has been so relentlessly hot this year, I will leave you with a great example of a misleading graph to share with students.

Whatever Happened to Vulgar Fractions?

 Back in 2018 I was asked to write a couple of guest blog posts for La Salle. I wrote this post about vulgar fractions and another post about factorising non-monics. These posts no longer appear on La Salle's website so I'm re-posting them here. 

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My Longman maths textbook from the 1960s tells me about the ‘Fraction family tree’. Its members are the brothers Vulgar Fraction, Decimal Fraction and Percentage, plus of course the illegitimate son Ratio who we don’t like to talk about.

Vulgar fractions are simply two integers (the numerator and denominator) placed above and below a fraction bar. They are also referred to as ‘common fractions’. I know the terms vulgar fraction and common fraction from my childhood but I never say them in my own classroom.

A Bostock and Chandler textbook from the 1980s tells us that “Usually we refer to common fractions simply as fractions and to decimal fractions simply as decimals”.

This alludes to the fact that during the latter part of the 20th Century the terms ‘vulgar fraction’ and ‘decimal fraction’ fell out of common usage in schools, leaving us with ‘fraction’ and ‘decimal’. With this change in vocabulary I don’t think we have lost any precision – the terms fraction and decimal are fit for purpose. So, although some lost mathematical vocabulary fills me with sadness, I don’t mourn the loss of the word vulgar.

The term vulgar fraction had been in use for centuries before it died out – it can be found in mathematics texts from all the way back in the 1500s. Here we can see it appear in a snappily titled textbook from 1741:

Textbooks from the 1700s and 1800s not only show that the term vulgar fraction was in common usage, but also provide some interesting definitions and explanations.

For example, in “A Treatise on Arithmetic, theoretical and practical” (Loomis E, 1856), we have this explanation:

It’s the last part that I particularly like: “This mode of expressing decimal fractions is used to avoid the inconvenience of writing the denominators’. I never thought of it like that. Just like the percentage sign is used as a shortcut to avoid writing a denominator of 100, decimal notation is just a shortcut.

All this talk of vulgar fractions made me wonder about the word vulgar. It’s a rather odd word to use in mathematics. A dictionary gives us the following definitions:

1. lacking sophistication or good taste.
2. making explicit and offensive reference to sex or bodily functions; coarse and rude.
3. [DATED] characteristic of or belonging to ordinary people.

Now I’m pretty sure that fractions don’t go round making offensive reference to sex and bodily functions. Certainly not in my classroom. Clearly the third definition is the relevant one here – the idea of being ordinary or common.

It turns out that the word vulgar comes from the Latin word vulgus meaning ‘the common people’, and it was first used in English in the fourteenth century. It then referred to something that was in common or general use or something customary or done as a matter of everyday practice.

The meaning of the word vulgar changed over time. It started as “relating to the ordinary people” but came to mean “commonplace”, and by the seventeenth century it had become “lacking sophistication or good taste”. One of the only places that the word continued to mean ‘ordinary’ was in mathematics. I think it’s a bit sad that a word meaning “of the common people” came to be associated with crude, uncouth and obnoxious. But I’m not sad that I don’t have to say the word ‘vulgar’ to a room full of teenagers anymore! That was asking for trouble.

In other news, did you know that it’s uncool to call a fraction improper these days? Apparently it’s a bit of an insult to fractions greater than one to suggest they’re not proper… Who knew.

A History of the Grouping Method

Back in 2018 I was asked to write a couple of guest blog posts for La Salle. I wrote this post about factorising non-monics and another post about vulgar fractions. These posts no longer appear on La Salle's website so I'm re-posting them here. 

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Factorising a quadratic when the coefficient of x² doesn't equal one (a 'non-monic') is apparently one of the more challenging skills that our pupils learn at GCSE. I've seen many pupils struggle with it, even those who achieve a grade 8 or 9 at GCSE and go onto take maths at A level. Interestingly, it doesn't seem like it should be challenging at all. I think it's way more straightforward than some of the harder reasoning questions that come up at GCSE - so why do pupils struggle with it so much?

When I first became a teacher, I taught my pupils to factorise harder quadratics in the way I've always done it: by inspection. Simply write out two empty pairs of brackets and try some numbers - thinking logically about what those numbers could be - until your terms expand correctly. It's very quick once you get the hang of it. In my NQT year this method was met with frustration by my Year 11s. They wanted a more defined procedure. I looked online to see if I was missing something and discovered 'the grouping method' which I then showed them as an alternative. I felt that it was an unnecessarily convoluted method but they clearly preferred having a set of rules to follow rather than having to reason for themselves. It made me a bit sad.

The grouping method, explained in a GCSE maths textbook published by OUP in 2016

This grouping method, particularly the last step where terms are gathered together, is a bit of a leap of faith for pupils who have never seen this kind of factorisation before. It kind of seems like magic.

Ten years on, I still prefer inspection (the 'guess and check' method) but I teach my pupils the grouping method as an alternative. I know they'll see it elsewhere even if I don't teach it to them - in textbooks, revision guides and online videos. I find that only the strongest pupils favour inspection - most pupils choose to use the grouping method but very often forget it. Days before the GCSE exam I hear cries of 'what's that thing you do to factorise hard quadratics? Something to do with the middle term...?'. The steps in the grouping method are not intuitive, and as a result it's difficult to remember.  

Given I had never heard of the grouping method before I started teaching, I was surprised to learn that it's actually an incredibly popular method. In fact, it appears to be the method that most maths teachers now use to teach non-monic factorisation.

How did I miss this method during my time at school? I did my maths GCSE in 90s. I have a couple of Bostock and Chandler textbooks from the 1990 - both only feature inspection, with no mention of grouping.

Grouping to factorise quadratics in the 1950s

This probably explains why I'd not seen it before. It wasn't in fashion when I was at school.

Looking further back I was surprised to see that older textbooks do feature the grouping method. Here, in New Algebra for Schools (Durell, 1953), we see an example of the grouping method. Durell recommends this method for both monic and non-monic quadratics.

The inspection method in use in the 1990s

Note though that this follows on from extensive use of grouping elsewhere. By this point in the textbook pupils have had considerable experience of factorising expressions like p(a + b) + q(a + b) and ax - ay + bx - by. This is absolutely key. There is a clear progression here that I feel is often missing from modern teaching of factorisation. I'm not sure it make much sense for pupils to only use the grouping method for non-monic quadratics, having never done any kind of factorising by grouping before.

 An exercise in factorising by grouping, to be completed prior to learning how to factorise quadratics by grouping

Later, this chapter tells us that simple quadratic functions can often be factorised at sight without using the grouping method. It says "use the grouping method whenever you are not able to obtain the factors by inspection, quickly".

Another textbook from eight years later (The Essentials of School Algebra, Mayne, 1961) also features both the grouping and inspection methods. Again, earlier in the chapter there is a considerable amount of work on the skills and understanding required for the grouping method. Of inspection it says,

"After a little practice, the pupil will be able to reject mentally the impossible pairs of factors... With simple numbers it is slightly quicker than the method of grouping terms, but the grouping method is the method to rely on. It should always be used whenever the pupil is not able to obtain the factors quickly be inspection."

So it seems that the grouping method may have been popular in the mid-20th Century.

Looking even further back, in Elementary Algebra for Schools (Hall & Knight, 1885) we are told that we should factorise non-monics by inspection:

"The beginner will frequently find that is it not easy to select the proper factors at the first trial. Practice alone will enable him to detect at a glance whether any pair he has chosen will combine so as to give the correct coefficients of the expression to be resolved".

There's no mention of any alternative methods here.

I haven't yet read enough old textbooks to track the full history of the grouping method, but from what I've seen it has come and gone over the years, and sadly seems to have become detached from prerequisite skills along the way.

If teachers are teaching the grouping method to factorise non-monics and have not already taught pupils how to factorise expressions like p(a + b) + q(a + b) and ax - ay + bx - by  then I think they may be trying to teach too many new skills in one go. It's something to think about.

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There are, of course, dozens of methods for factorising non-monic quadratics, some more 'tricksy' than others. Read my book A Compendium of Mathematical Methods for more on this.

5 Maths Gems #200

Welcome to my 200^th gems post. This is where I share some of the latest news, ideas and resources for maths teachers.

1. MathsPad
MathsPad's May and July updates includes lots of fantastic new resources.

Their Topic Key Facts Sheets are designed in a two-column question and answer format, so that pupils can self-test by covering up the second column. Very helpful for revision at Key Stage 3 and 4, they can be found at the top right of their curriculum pages.

MathsPad has also shared two packs of 'Grade 9 Club' Question Packs: Pack A and Pack B. These include worked solutions for each question. 

They have also published a series of Foundation One Markers containing non-calculator one mark questions for GCSE Foundation on a mixture of topics. The skills recur from one sheet to the next, so pupils get repeated practice of each skill.

There's other new resources too, including more in the Building Blocks range (aimed at secondary school students who need support with the basics), more Higher GCSE revision packs, and an inverse functions matching activity. We love MathsPad at my school - it's well worth the subscription.

2. X Why
Andy of m4th5.co.uk has shared a graphing tool for classroom use: X Why. It can be used to see graph transformations happen, draw instant inverses, put your own labels on curves and points, see integrals and areas under curves, draw tangents and normals and much more. Graphs can be exported to worksheets and presentations, and axes are auto-scaled for degrees and radians.

3. NCETM Professional Development Materials

The final set of the NCETM's Key Stage 4 Professional Development Materials is now available, completing the full suite of Secondary Mastery Professional Development Materials for both KS3 and KS4. This set covers Key Stage 4 Geometry including transformations, circle theorems, trigonometry and 3D shapes. As with the other sets, these materials contain lots of fantastic ideas for getting your students thinking. For example, there's a task where students are asked to identify which of the diagrams correctly show the alternate segment theorem.

I also like what they've done here with bearings diagrams in non-standard orientations...

... and I like relating vectors to throwing a ball.

4. A level 
In my last lesson with Year 12 this Friday I will be teaching Year 2 Binomial Expansion with Partial Fractions - I never understand how anyone manages to fit the entire A level curriculum into such limited time! I'm perpetually running behind with my A level classes. But if you have a spare lesson with Year 12 at the end of term, they might enjoy this murder mystery revision task from Danielle @PixiMaths. You can download it here.

Another A Level resource that caught my eye recently was a set of statistics tasks posted by MrLevMaths. Read his blog post for more.

5. Esheets and Whiteboard Workouts
I discovered two websites that aren't new, but I've not seen them before. 

esheets.io from Richard Linnington features self marking worksheets, maths puzzles and games (including end of term maths activities) and teacher visualisation tools. I particularly like the sheep tethered to a building and sheep with a sliding ring.

Whiteboard Workouts from southportmaths.co.uk are PowerPoints that are designed for mini whiteboard questioning.

A Milestone
I started this blog whilst on maternity leave in April 2014. My first post was about a method for matrix multiplication. A lot of my earlier posts were about methods. Later I started writing more about resources. I wrote my first gems post in August 2014, when I realised that there were loads of great ideas and resources being shared on Twitter that needed to be shared more widely, so that all the maths teachers in the country (and beyond) could see them, not just the teachers using social media. Over the years I've been contacted by many teachers who tell me that they find these posts helpful, so I continue to write them. There are a number of authors and websites that have 'found fame' after they featured in my posts! 

Today I've published my 518th blog post, which is the 200th post in the Maths Gems series. I used to post gems posts every week, but it was impossible to continue posting so frequently as my daughters grew older, and my school workload increased. Now I try to write one gems post a month, but it can be difficult to find enough material since the mass exodus of maths teachers from Twitter. Back when the new GCSE was introduced in 2017, social media was packed full of teachers sharing resources and helping each other out - with another curriculum change on the horizon, maybe we will regain some of that magic once day. In the meantime, I will endeavour to continue posting once a month. If you're new to my blog then I recommend having a look at my back catalogue of gems posts. There's a lot of great stuff in them. 

Events
I had a fantastic time at the MEI Conference as always. I attended on Thursday evening for the dinner and drinks (free bar courtesy of Casio!) and all day on the Friday. I presented on methods to a lovely audience, and enjoyed trying some incredibly difficult geometry problems with Bernard Murphy. Sue Black's keynote was excellent. Shout out to Marc the tutor who sat with me and Megan Guinan after my session and showed us a fun method for factorising non-monic quadratics.

Looking ahead at events that are coming up...

• On the last two days of the summer term my headteacher has kindly given me permission to attend the Future of Maths Education event which is being held by XTX Markets. This is an invite-only event and I'm delighted to be on the guest list. I believe I'm the only teacher attending - in fact I have to go there directly after running a Year 7 trip so I'm expecting to be thoroughly exhausted... but I am really looking forward to it. 
• Mathsconf41 has been announced - it's taking place in Liverpool on 5th September. A conference at the very start of the academic year is a fantastic idea - the perfect timing to inspire and motivate. I'm not sure yet whether I'll attend - as both a mum of teenagers and Head of Department I have a feeling I'll have a million things to do that weekend. But I hate to miss a mathsconf so I will have to have a think about this one!
• BCME 10 is a big deal in the world of maths education, and it takes place in Nottingham on October 23rd and 24th. I loved the last BCME in 2018. To cut the cost for teachers, bursaries and early bird rates are available but you have to act now to take advantage of these opportunities (the deadline for bursaries is this week! And the early bird rate is valid until the end of July). Information about bursaries is here. I would apply for one myself but my school isn't in an area of high deprivation so I don't qualify. I really encourage teachers who meet the criteria to apply for a bursary to help cover the cost - it's so important that teachers attend this conference.

Update

What an insanely busy time at school! Our classrooms were sweltering when it was 37 degrees and we had a few early closures in response to parent concerns about students travelling home in the extreme weather. We also had new staff induction and Sports Day last week (Sports Day picture of my maths team below - minus four members who were busy elsewhere!). Plus I've been hosting lots of visitors in my department who come to see my curriculum in action and find out how we do things. This week is Open Evening and Year 10 are on work experience, and then the following week there are a whole load of trips to run. As I said in my post Tasks for the Summer Term, this time of year is always surprisingly busy! And the delay of SATs results will have me working in the summer holidays (even more so than usual) because I use them for Year 7 groupings.

Have a great week! Thanks for reading.

5 Math Gems #199

Welcome to my 199^th gems post. This is where I share some of the latest news, ideas and resources for maths teachers.

1. Trimension

Following on from the release of Vectorama (Gems 197) and Graphiti (Gems 196), Neil Kendall has completed a third project 'Trimension'. This tool allows the construction of 3D models to help illustrate 3D Pythagoras and trigonometry problems. The key feature is the ability to extract internal triangles into a flat diagram. Neil has included four pre-built examples from past GCSE papers to illustrate what it can do (just go to add > examples).

This is a fantastic tool for teaching 3D Pythagoras and trignonomety - thank you Neil!

2. Interwoven Maths

Nathan Day (‪@nathanday) has made some updates to his excellent website interwovenmaths.com, including a new page of small tasks and a contents page showing all the resources and ideas now available on the site.

There's loads to explore on Interwoven Maths including this Quadratics Always Sometimes Never.

Nathan also shared Class Duels - a maths game featuring over fifty topics from across Key Stage 3 to 5. It can be played in two-player mode (split screen, which can be done on an interactive whiteboard) and single-player mode. 

3. Exam Countdown

Nathan Day also shared a newly updated Edexcel Exam Countdown page which includes both GCSE and A level (including Further Maths) and shows dates for 2027. Interestingly, there are two GCSE papers before half term next year, and A level Paper 1 is before half term too.

4. A level

@andrewmaths1 posted a link to a website that I've not seen before. mathstutorneri.com is great for students aiming for an A*. There's a booklet of challenge problems and a set of exam papers.

It's nice to have another website we can use for high challenge. We've always directed our A* students to MadasMaths which I first blogged about in 2016. This week's Edexcel A Level Paper 1 had two questions that were very similar to MadasMaths questions, so clearly it's a good idea to use this website! If you're not familiar with MadasMaths, there's IYGB practice papers for maths and further maths (don't ask what IYGB stands for...!) and there are also booklets by topic. I often use questions from these booklets in my maths and further maths lessons (tip for new users: zoom in on the PDFs to see full worked solutions hiding in the booklets). 

5. Tasks 
There's been a number of new tasks and resources shared on Bluesky and Twitter in the last few weeks.

Here's a nice task from ‪@4301maths on equations of parallel lines. 

@dalechapman shared a couple of tasks: one on simultaneous equations and one on constructions.

@canning_mrmaths shared a Quadratic Graphs Venn task designed for learners to think deeply about different features of quadratic functions. The task has four variations levelling up in difficulty.

@giftedHKO shared some distance time graph questions.

And @draustinmaths.com shared some new resources for simultaneous equations.

Thank you to everyone who shares resources!

Update

We're halfway through external exam season (for my department, eight exams down and eight more to go!). This is what we've had at my school so far:

• AQA GCSE: Paper 1 was nice! I thought Paper 2 was difficult on both Foundation and Higher Tier. On Higher there was a question where students had to draw a two way table and some of them said they didn't know what that was!  Eek, there goes four marks. Yet if they'd been given a two way table to interpret or fill in the gaps they would have had no problem. So we must get them drawing more two way tables in lessons! There were two questions on Foundation requiring equating coefficients - we need to change our Foundation scheme of work to ensure this is explicitly taught in future. One paper to go!
• Edexcel A Level: Paper 1 was difficult. There were some tough questions, fiddly answers and a few questions where students would be unable to do part b without managing part a which is frustrating. Some of our students have lost a bit of confidence but hopefully Paper 2 will give all students the chance to show what they know. The Further Maths (Core Pure) papers were both ok and I'm looking forward to Further Statistics 1 this Friday (my favourite module!).
• AQA GCSE Statistics: We switched to AQA this year and our students were pleased with Paper 1. We entered Year 10 for the first time (see my post about our Option Block Experiment) - they were terrified going into the exam, but all smiles coming out! 

For 'best guess' papers please continue to check Adam Creen's dropbox which includes GCSE (maths and statistics) and A level.

My recent post aimed at Heads of Maths Tasks for the Summer Term went down well - I've had lots of emails asking for advice and resources. I'm always happy to help but bear with me at this time of year when there's so much going on at school - I will always reply within a week!

I'm looking forward to the MEI Conference next month - always one of the highlights of the year! I hope to see lots of you there. I will leave you with this fantastic present gifted to Emily Rae by a student - I love this!