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!

Tasks for the Summer Term

I like the summer term.  

At my school Year 11 are in lessons until half term, but Year 13 went on study leave on 8th May, which means my teaching timetable has already lightened and will soon reduce further (this year I teach both Year 11 Maths and Year 11 Further Maths so I'll gain a lot of time when Year 11 leave). With all the gained hours in a day you would have thought the summer term would be relatively easy. But, surprisingly, it always feels like the busiest time of year. School work still extends into evenings and weekends in June and July in the same way it does in every other month. But, with my Head of Maths hat on, the summer term workload is distinctly different – fewer lessons to plan, but far more administrative tasks. A lot of the job becomes about preparing for the next school year.

As I’ve been Head-of-Mathsing for a good few years, I thought it might be helpful for inexperienced Heads of Maths if I share some of the things that they should be thinking about this term to make sure their department hits the ground running in September. If you’re an experienced Head of Maths then you will already have all of this on your radar, and perhaps other things I’ve missed. The idea of this post is just to give some ideas to Heads of Maths who are new to the role.

Curriculum Development
It’s a good idea to ask your team for feedback on whether your curriculum is working well. Perhaps there are times teachers feel rushed or the order of topics doesn’t quite work. Get this feedback now – if your curriculum needs a re-write it’s a big job. Also consider exams analyses from previous years. If your school’s A level or GCSE results showed underperformance in certain topics (compared to national performance) then you might need to make some changes. For example, one of the topics we were weakest on in a previous year’s GCSE was product rule for counting, so this year I’ve reviewed when we teach that and how long we give it, and I’ve done some department professional development on how to teach this topic.

If your scheme of work is working well then the only thing you need to do is adapt the timings to fit next year’s calendar. I usually need to move a couple of topics around to make things work, but changing the placement of topics has knock-on effects on resources and assessments which is a pain. The timing of the 2026 August Bank Holiday means a lot of schools are going back later than usual this September - for us, the first half of the autumn term is going to be a week shorter than it was this year, and the first half of the summer term is a week longer.  So everything still fits but some rearrangement is needed. My schemes of work look like this (each link on the schedule takes you to a topic page where a possible lesson breakdown is provided, but teachers don't have to follow this exactly - they adapt it for their classes):

I can’t do any work on curriculum timings until my leadership team confirm the dates of next year’s assessment windows (they in turn have to wait for the MAT to confirm the dates – I never understand why the MAT waits until so late in the school year to decide the following year's assessment calendar). It means I can’t do all my curriculum work until the last couple of weeks of the summer term. And it’s not just rewriting the scheme of work – website documents have to be updated too, and I need to load the curriculum for each year group onto Sparx which is a big job.

Sparx will look a bit different for our Year 11 next year - previously we’ve set normal Sparx homeworks right up until Easter but we’re going to change that. We like the new GCSE revision tool on Sparx so we're going to set that from February half term of Year 11, instead of topic-based homeworks.

Assessment Writing
We write new mid and end year assessments for Years 7 to 10 every year. This is partly to ensure students don’t get hold of the previous year’s assessments and memorise the answers. An advantage of writing new assessments every year is that we can use previous years' papers as revision material. I enjoy writing assessments, particularly searching for (or creating) novel questions to challenge students. But it’s such a huge job that it’s hard to get in done during the busier periods of the school year, so I try to make a start in the summer term where possible (I try to – but it requires a lot of focus which is hard to achieve at school, so I normally end up doing most of it in the summer holidays or October half term).

This year we wrote Year 13 mocks from scratch for the first time. Last year we used Edexcel’s secure mock papers (which are excellent) but students had clearly managed to find these online, or perhaps through tutors, so we ended up with lots of inflated predictions. It was so frustrating to waste time marking mock A level papers for students who had clearly seen the papers in advance. One boy got 100% in his mechanics mock (!) – his solutions were flawless, seemingly written by a mechanics genius – but he went on to score 62% in the mechanics paper of his actual A level exam in June... We’d had enough of all this cheating so this year we gave up on the secure mocks and instead made our own A level mock papers using questions from a variety of sources. It worked very well, so we’re making one more set. My colleague Linda will do this during her Year 13 gained time – this is an excellent use of time in the summer term.

Resource Improvements
We use warm booklets for retrieval practice at the start of our lessons, and at the end of each topic we give a test (we call these ‘snapshots’). All of these are teacher-made - sometimes they were made in a rush six years ago but get re-used every year. The summer term is a good opportunity to review and improve these resources. For example our snapshots are sometimes a bit short of space so some re-formatting is needed. Here's an example taken from one of my own Year 11 snapshots - in the top image there wasn't enough space for students to present their work clearly. The bottom image is an improvement. 

Improving the quality of shared resources is a good task to direct my team to complete after Year 11 have left. 

Subject Knowledge Development

The summer term is a good time for teachers to prepare for any new courses they're teaching next year and to develop their subject knowledge in the courses they already teach. 

The day after GCSE and A level exams have taken place I hand the spare papers out to teachers of those qualifications and expect everyone to complete the papers themselves. We become experts in the qualifications we teach when we regularly complete exam papers.

As well as completing exam papers I also hope to spend some time this term reading Colin Foster's new website bigmathematicalideas.org. 

Timetabling, Groupings and New Staff Induction
I’ve already done the class allocations for next year (i.e. deciding who will teach each year/set) and now we’re waiting for the timetabler to run it all through the system and we’ll see what comes out the other side. My hope, as always, is for minimal split classes. I have a rule that no classes of low attainers should be split between two teachers, so when the timetable comes through I’ll probably have to make some tweaks to ensure this happens (I don’t like any class being split but it's less of a concern for classes of high attainers).

We’re lucky that every year our timetable is ready for new staff induction day, which is on 29th June this year. On this day I run a department meeting after school (joined by new members of staff) where I hand out provisional timetables and set out the big picture and expectations for the next academic year. It's important that new staff go home from their induction day knowing exactly what they're teaching in September.

Another one of my big jobs in the summer term is allocating students to maths sets for next year. It’s important to get this right (particularly when it impacts GCSE tiering) so it takes time – it’s mainly spreadsheet work but I also need to get input from teachers. Our setting is based on performance in formal assessments (mid and end year) but we also take into account classwork throughout the year. I have to wait until Key Stage 3 assessment results are ready before I can get started on this job, and I have to wait for SATs results before I can start grouping the new Year 7s (these are released on 7th July but there’s always a delay getting all the UPNs we need - so this is another job that ends up being completed right at the end of term). I also get heavily involved in groupings in other subjects (careful allocation of students to bands is a key strategy for behaviour management) - all of this happens right at the end of term when there's lots of school events going on (I'm leading two Year 7 trips in the last week of term!) so it gets a bit crazy. Don't ever let anyone tell you that the summer term is quiet!

Preparation for September
There’s a lot of preparation for September that needs to be done in advance. In August I have to spend a lot of time checking papers for A level and GCSE candidates who were close to a grade boundary so I've learnt from past experience not to leave my September planning until August.

I encourage my team to start preparing their resources for September in July (as soon as they know their timetable then they can get started). This includes hole punching exercise books, printing warm up booklets and preparing first lessons. I do this for my own classes too – September Inset Day is always very busy so I know I won’t have time to plan lessons then.

There’s also stuff to prepare for September as Head of Maths – the big one is the maths department meeting on September Inset Day. This is the most important meeting of the year so I always spend a lot of time preparing it. I also need to produce and update posters and slides for my team to share with students in September – for example slides on how to use Sparx (there are new features we need to tell them about) – see my post September Routines for more on this. And of course there’s loads of housekeeping to do in the final weeks of the summer term – updating corridor displays, tidying classrooms and offices, and placing the annual ‘big stationery order’.

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This isn't meant to be an exhaustive list of ‘things to do in the summer term’. In all schools there's a lot going on at this time of year (for us it's things like Year 6 transition interviews, Year 10 work experience calls, vast quantities of UCAS references, Open Evening, trips, induction days, assessment marking and plenty more). This is just a post about ‘things Heads of Maths should be thinking about at this time of year’, in case you’re new to the role and not sure where to start. I hope it helps!

5 Maths Gems #198

Welcome to my 198^th gems post. This is where I share some of the latest news, ideas and resources for maths teachers.

1. GCSE Question Distribution

Thanks to Jamie Frost for producing a detailed breakdown of skills that have appeared on past GCSE papers. Other websites have analyses by topic, but Jamie has such an extensive database of past exam questions that he has been able to produce a much more granular analysis, for all exam boards and including IGCSE. 

I teach AQA at GCSE and when I teach circle theorems I always say 'alternate segment theorem seems to come up a lot in exams' - this was just a hunch, but now I know for sure - it does indeed come up a lot! Of course just because something has appeared in exams a lot in the past, it doesn't mean it will continue to do so (in fact, the opposite could be true - perhaps this year's GCSE exams will contain a lot of topics that haven't been tested in a while) so this data can't be used to make predictions. It's still interesting though. Thank you to Jamie for sharing it. 

Each skill includes a link to a randomly generated example question. Schools subscribed to Dr Frost can generate worksheets from these which can be printed or set as online tasks.

2. E2E Revision Sheets

@ThtPedagogyGeek has created a nice set of resources called E2E (Example to Exam) Revision Sheets. The idea is that you model the worked example - students copy it down and take notes, then complete the three questions. These are designed for GCSE revision.

3. Diagram Creator 

There are numerous online tools to create diagrams but I often find them unintuitive or overly complex, or they require a login. Thank you to @TickTockMaths for sharing a Geometry Tool designed for creating simple diagrams quickly. He made it so he could insert diagrams into lessons created with his Slide Creator (see Gems 197).

4. Corbett Maths Key Stage 1
Key Stage 1 teachers will be pleased to see that John Corbett has extended his popular 5-a-day range to Key Stage 1. He has made the first level (Ruby) with Emerald and Diamond to follow. 

I know that secondary teachers sometimes find these primary resources helpful too (I'm thinking of our Entry Level classes). 

5. A Level Resources

We had a difficult logs and integration question in our Year 13 mock this year that even our strongest mathematicians struggled with, so I'm pleased to see this lovely little task from ‪@mrlevmaths.bsky.social‬ for practising this specific skill.

Also thanks to ‪@karenshancock.bsky.social‬ for a resource on 'Power Equations' which is useful for Year 12 and also Year 11 L2 Further Maths students. 

Other resources on Karen's website that might be helpful for both Year 12 and L2 Certificate in Further Maths include Factorising with Surds and Graphical Solutions. Thank you Karen!

Also thanks to ‪@adamsfax.bsky.social‬ for his 'Edexcel formula sheet in a page'. We'll print these in September for all A level students to keep in the front of their book.

Big Mathematical Ideas

Colin Foster (one of the wisest men in maths education) has launched a new website: bigmathematicalideas.org. He has written a chapter on each of the five big mathematical ideas.

 

I believe this should be compulsory reading for every trainee teacher. No matter how good you are at maths when you train to be a teacher, the thing that everyone needs to develop is their pedagogical subject knowledge (not just how to do the maths, but how to explain the maths). But it's not just for trainees. I've been teaching for a long time now and I think my subject knowledge is pretty strong, but within a few minutes of reading this website I was thinking deeply about things that I haven't thought deeply about before (the big multipliers table in Section 1.5.2 had me wondering why I'd never shared this with students).

Reading this website will probably be my summer holiday project, or perhaps something I do in my gained time this term. I will use it to write next year's professional development sessions for my department.

 

Update

The first half of the summer term is very intense if you teach a lot of exam classes! I know that in some schools Year 11s have already started study leave, but I still have three weeks of lessons left with Year 11. I'm still doing topic-based revision (for example, next week we're revising surds and indices) and mixed revision (often using my revision mats). I always find that my students make masses of progress in our summer term revision lessons so I'm glad my time with them isn't cut short. If you're looking for good revision resources for Year 11, remember my GCSE Revision page has loads of hand-picked recommendations, and Adam Creen's annual 'Best Guess GCSE Dropbox' has been updated with the latest practice papers.  

I had a fantastic to response my last blog post (#mathsconf40) and to my recent conference presentation about Key Stage 3. I received dozens of emails from teachers asking for support on curriculum and assessments, including lots who said they find my gems posts helpful. Since most social media posts are now met with tumbleweed (a few likes and shares, but rarely a response), I often wonder if there's any point blogging for maths teachers these days - it's hard to tell whether anyone is listening. But after the flurry of lovely emails my enthusiasm for writing has returned, so thank you to everyone who got in touch. 💗

No doubt you saw that new research on setting in maths was published by the EEF. There are Year 7s in my school who struggle to add single digit numbers, or to multiply integers by ten, and others who find it easy to solve complex algebraic and geometric problems. Teaching these children in the same class doesn't make any sense to me, and I feel it would particularly let down the students who struggle the most and would benefit from being in a smaller group. Maths teachers are sometimes made to feel like they're evil or lazy for teaching in sets when they are actually trying to do the right thing for the students who need the most support. The narrative that schools often assign their 'worst' teachers to teach the lowest attainers is insulting nonsense. Most Heads of Maths prioritise the students who find maths difficult, and assign absolute superstars to teach them.

Here's a round up of some recent developments, in case you missed anything:

• Jonny Griffiths has updated his website carom-maths.co.uk which houses a collection of forty activities bridging from A Level Mathematics to University Mathematics. This is a great set of resources for a Sixth Form maths club. 
• Maths Genie has had a big update! Set up a free account to access useful resources, like the GCSE mini tests. 
• A few teachers have been in touch to ask about the RULER resources from Ben Tanner. These popular resources used to be available on Ben's website but it no longer exists. Helpfully, Tim Stirrup shared an archived version of the website, where you can still download the booklets.
• The QAMA is back - but this time, as an online version instead of a physical calculator. QAMA requires an estimation before it will work out an answer. It's great for developing number sense. If students don't have access to devices to use the online calculator in lessons, it could instead be used by the teacher in modelling. 
• Recently I've received a fair few emails from teachers who have either used AI to create tools for teachers or have created a platform where students can use AI to support their learning. For example, on gcsemathsai.co.uk students type in any topic or question and get instant step-by-step worked explanations. It covers the full GCSE syllabus and is free, with no account needed. One thing that stood out for me on this website is the misconceptions for each topic (e.g. look at 'common mistakes' for inequalities and arc length and sector area) which might be helpful for teachers planning lessons.
• Hannah Fry is going to be on the next Celebrity Traitors (our favourite family viewing - my daughter is obsessed with it)! Cue me telling everyone who will listen about the time Hannah bought me a drink and we hung out at BCME...

I'll leave you with these new 'squareas' puzzles from Transum. These are fun and come in multiple levels of difficulty. 

Thanks for reading!