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!

Easy Multiples

In 2018 I decided to write a series of short posts about approaches or methods that teachers might not have seen before. When I share these posts, I am well aware that there will be many people who already know the thing I'm blogging about, but I figured that it's still worth sharing even if it's only new to a handful of people. My first post in this series was about using vectors for enlargements and my second post was about factorising by inspection. I then got really busy writing my book, and didn't add to this blog post series for four years! Oops. So today I'm relaunching the series with a very simple little 'trick' (not a trick at all, just maths).

My Year 6 daughter has recently learnt long division. To be clear on what I'm referring to, long division looks like this:

Whereas 'short division' looks like this (this is sometimes colloquially referred to as a 'bus stop method'):

The only difference between the two methods is that in short division we work out the remainders in our head and jot them down in the dividend, but in long division we work out the remainders on paper in a more structured format. If your divisor is greater than twelve (for example if you're dividing by 28) then it might be tricky to work out remainders in your head, so that's typically when the long division format might be preferred. But they're essentially the same method, just with a slightly different structure for processing the calculations.

It was funny to see my daughter learning long division as it's something that I literally never teach in secondary school. I was pleased with myself for remembering how it works. For many students it exists in Year 6 alone, never to be seen again. A typical Key Stage 2 SATs question might look like this:

But something like this is highly unlikely to come up at GCSE. Students do sometimes have to do divisions by hand in their non-calculator GCSE exam (an example is shown below, from the Foundation tier), but I think most students would choose to use short division.

Some people argue that the long division algorithm is used again when students learn algebraic division in Year 12. This may have been the case ten years ago, but I think that most(?) A level teachers now prefer more intuitive methods of polynomial division, like the factor method shown below for example. 

So for the most part, long division resides solely in Year 6. And my daughter, who is in the 'middle' group for maths, was coping fine with it, but she told me that she finds it tricky to write out the multiples at the start. For example when she's dividing by 28, she's been told to begin by writing out some multiples of 28. She finds this time-consuming, a bit tricky, and rather dull.

But don't worry, because there's a really simple way to write out the multiples of 28. My colleague Sian showed me this - she picked it up a few years ago from her daughter's Year 6 teacher. I showed my daughter, who loved it - she was then able to master long division as she'd found a way round the tricky bit.

To quickly and easily write out the multiples of 28, just write the multiples of 20 and the multiples of 8 and add them together:

As long as the child knows their standard times tables fairly well, listing the two sets of multiples is straightforward. And the addition is pretty straightforward too, as they are always adding to a multiple of ten.

Here's another example: multiples of 17.

This may already be really widely used by Year 6 teachers. But in case anyone hadn't thought about this super simple way of listing multiples, I thought it worth sharing here. As I've always said, even if it just helps one person then it's worth taking the time to write about it.

5 Maths Gems #154

Welcome to my 154^th gems post. This is where I share some of the latest news, ideas and resources for maths teachers.

1. Mixed Six

In Gems 153 I shared a resource from @SegarRogers which utilises Pointon and Sangwin’s task design taxonomy. There are more tasks like this on Chris McGrane's website. @mathsmrgordon has now created a large collection of tasks with a similar structure. These tasks focus on the key skills and topics from Edexcel's advance GCSE information. The full thread is here, and a couple of examples are shown below. Ben has also invited volunteers to make and share their own tasks using his template. Thanks Ben!

2. Maths Genie Mini Tests

Thank you to @mrnp4tel for sharing a great GCSE resource from Maths Genie that I've not seen before. I use Maths Genie all the time - it's a very user-friendly source of exam questions and past papers - but I'd never explored the resources section. He tweeted about the Target Tests which feature 20 question tests at four different difficulty levels: Foundation, Foundation (Grade 5), Higher (Grades 4 - 6) and Higher (Grades 7 - 9).

3. A Level Resources
Thanks to @pmyygrm for sharing the London Academy of Excellence's A level schemes of work and resources on TES. The resources contain numerous examples for each topic, with lots of space for students to take notes in lessons. Some questions have been adapted from exam papers and from Madas resources.

4. Ratio Booklet
Thank you to @ChrisMcGrane84 for sharing another excellent curriculum booklet. This one is on ratio - it was created by @ShivMcKenna55 and is hosted on Chris's website Starting Points.

5. KS2 Spot the Mistake

Thank you to @SarahFarrellKS2 for sharing copies of the full 2018 and 2019 Key Stage 2 reasoning papers that she has completed badly to show lots of misconceptions and common mistakes. Importantly, not all the answers are wrong. I find this is a really helpful strategy in 'spot the mistake' resources.

Updates

Here are a few things you might have missed:

• A number of teachers on Twitter have shared useful resources relating to the 2022 GCSE advance information. I'm not sharing these resources on my blog because I don't want to duplicate work being done elsewhere. Colleen Young has written a really helpful blog post collating the information and resources. Thanks Colleen!
• Did you see @MrDraperMaths' brilliant blog post about Cognitive Payload? It's a must-read.
• Puzzle of the Week is back! The competition started on Monday.
• I've been busy writing two presentations this week: one for the MA Conference which is in the Easter holidays, and one for my #mathsted22 event which takes place in two weeks. I'm delighted that I'll be welcoming 130 maths teachers to my school for an evening of CPD with myself and Steve Wren from Ofsted.
• My school is recruiting a maths teacher! It's a brilliant place to work - we're a science-specialist comprehensive school with excellent behaviour and attitudes. We will have Year 11 for the first time next year, and we'll be setting up our Sixth Form too, so there will be opportunities to teach A level the following year. It's an exciting time to join. We welcome applications from both new and experienced teachers. The closing date is 7th March. Apply here, or email me to discuss.
• Some of the resources in this post have a GCSE revision focus, so it might be a good time to remind you that I have an extensive collection of GCSE revision resources here. This includes a number of resources I have made myself, such as my revision mats which are suitable for students working at four different grade levels (Grades 1 - 3, Grades 3 - 5, Grades 4 - 6 and Grades 6 - 9).

I'll leave you with these puzzles for your students, shared by OCR on their hashtag #OCRmathsPuzzle.

What Secondary Teachers Should Know About The Key Stage 2 Maths Curriculum

How confident are you that you know what your students were taught at primary school?

Look at the questions below. How many of these skills are covered by the Key Stage 2 curriculum?

I'll give you a clue: three of them are covered at primary school, the rest aren't. Do you know which ones?

For an upcoming talk I'm planning, I've been busy researching the feedback that maths departments have received from Ofsted (and 'mocksted') deep dives. One thing that I've noticed come up a few times is:

"Teachers of mathematics do not know enough about what students have learned in the past"

Secondary teachers should definitely know what's on the primary curriculum. And it's not good enough to have a 'rough idea'. Knowing the primary curriculum well helps us build a strong Key Stage 3 curriculum, and improves the effectiveness of our teaching, particularly in Year 7. 

This is easy to fix. The main problem is time. I think most secondary teachers agree that it would be good to know what's on the primary maths curriculum. But we struggle to find the time to find out. 

So I have tried to be helpful! I researched the primary curriculum from a secondary teacher's perspective (I had a pretty good knowledge of the primary curriculum already, but there were a few things I didn't know, so it was a very worthwhile activity for me). I then made a presentation summarising the key points. And then I recorded it and put it on YouTube:

 

This is only 43 minutes long, so if you want your whole department to watch it together then it should be short enough to fit in a department meeting.

The majority of the presentation is me describing the key features of the primary curriculum. In the last ten minutes I offer some advice for Key Stage 3 teaching.

The slides are here.

I hope this is helpful! Basically from reading Ofsted reports I identified a gap in maths teacher knowledge and decided to try and fix it.

This took quite a chunk of time out of my Christmas holidays, so if you find it helpful and want to say thank you, you can buy me a drink here. Cheers!

5 Maths Gems #138

Welcome to my 138^th gems post. This is where I share some of the latest news, ideas and resources for maths teachers. I've got a bit behind on my blogging lately because school has been overwhelmingly busy, so here I'm trying to catch up by featuring ten gems instead of my usual five. 

1. Prime Factors Tool

Jonathan Hall (@StudyMaths) has published another new tool on the brilliant mathsbot.com. This lovely prime factors tool uses virtual prime factor tiles to help students make sense of prime factorisation, highest common factor and lowest common multiple. Click on the cells to toggle through blanks, products, and prime factors.

2. Certificate in Further Maths Resources

Teachers delivering the AQA Level 2 Certificate in Further Maths should check out Amanda Austin's (@draustinmaths) website which contains practice resources for this qualification. I like the format of her practice strips which are very helpful for sticking into books.

Do check out the rest of Amanda's website, which features a collection of KS3 and KS4 IGCSE and GCSE Maths resources ordered by topic. 

3. Tasks

Since Chris McGranes's (@ChrisMcGrane84) book on mathematical tasks was published and he started running courses in task design, there has been a flurry of excellent tasks shared on Twitter and Chris's blog. Here I share a couple of examples - check out his blog for more like this.

Fraction Stories (thanks to @limeswright for tweeting about this one - I think I will use it next week to assess prior understanding in my first fractions lesson with Year 7):

Simultaneous Equations (I'll be teaching this soon to a class who will really benefit from the  scaffolding in this activity):

4. GCSE Statistics
Thank you to Helen Scott (@HelenScott88) for sharing her GCSE Statistics Retrieval Roulette. Aimed mostly at Higher Tier candidates, the chapter references in this resource relate to the Pearson GCSE Statistics textbook. For guidance on how to use Retrieval Roulette, read Adam Boxer's blog post.

5. Volume

Thank you to @SegarRogers for sharing this cuboid volume task. He designed it for a class in which some students have trouble seeing individual cubes, requiring a gradual removal of the isometric grid, and other students are ready to try working backwards.

6. Recall

All the way back in Gems 87 (April 2018) I shared an example of a regular recall starter from @JaggersMaths (formerly @MissBanksMaths). Mrs Jagger has now updated her R^Infinity resource, which teachers can use to quickly create personalised four-question starters. Visit her website to download the resource and to find out how it works.

7. Sequences and Fractions Task

Richard Perring (@LearningMaths) recently shared two interesting tasks. The first is a sequences task in which students generate their own sequences.

The second task is based on a similar idea, but relates to fractions. I really like these tasks.

8. Number Sense Maths
Clare Christie (@Ms_Mathsteacher) got in touch about the website numbersensemaths.com. There are tonnes of free resources to get schools going on structured number fact teaching, and a full scheme of work for those who want to commit to it. 

This website will be of particular interest to primary schools, and I think that SEND departments in secondaries will find useful strategies and resources here too.

9. Place Value
@taylorda01 had a nice idea for a place value task. Present students with the following numbers and ask them how many times bigger the value of the red digit is than the value of the blue digit. Then ask how much bigger the value of the red digit is than the value of the blue digit.

10. Pictograms
Thanks to @giftedHKO for a new pictograms resource. I have linked to this in my data resource library.

Update

Last weekend I enjoyed spending the morning in Scotland (virtually!) for the Northern Alliance N5 Maths Conference. The workshops were excellent. I presented on teaching quadratics. I will present a similar workshop at La Salle's '#MathsConfMini' which is on a Friday evening in January.

Another event I'm presenting at in January is White Rose Maths Secondary Maths Brunch.

In case you missed them, I published two new blog posts in the last couple of weeks. They were:

• Lines and Angles on Square Grids (a post written by Anne Watson about a collection of Don Steward tasks)
• Zooming In (a post about a great teaching tool I recently used for the first time in a Year 7 lesson).

Another thing I was involved in last week was a discussion about Don Steward's tasks on Tom Manners' ResourceFULL channel. This is worth a watch if you want to see some outstanding examples of rich tasks.

Other things to check out if you haven't already:

• Marvellous Maths 2, which is a CPD course for maths teachers from me and Craig Barton. It's been great to hear of teachers using ideas from this course in their teaching this week.
• Ed Southall's new book, Geometry Juniors. This is aimed at 8 to 12 year olds. It looks great!
• The review of my book A Compendium of Mathematical Methods published in the MA's journal Mathematics in Schools. I loved this review and was very pleased to read an account written from the perspective of the book's main target audience (maths teachers!).
• I've had a few questions about intervention resources for Year 11s recently. Do check out my GCSE Revision Resources page if you're looking for this kind of thing.

I'll leave you with this wonderful pentagon tessellation tool from Mathigon.

Regular pentagons don't tessellate – but there are 15 other types of pentagons that do. We’ve collected all of them at https://t.co/PDYbPhRhMa. Can you fit them together? pic.twitter.com/a7I5Z32des

— Mathigon (@MathigonOrg) November 6, 2020

 

5 Maths Gems #134

Welcome to my 134^th gems post. This is where I share some of the latest news, ideas and resources for maths teachers.

This is a double gems post! Lots of ideas and resources have been shared on Twitter in the last few weeks and - given it's the calm before the storm (i.e. the weekend before the start of what might be the craziest school year yet!) - I better post them all now, before things get too busy.

1. Games
Thank you to the wonderful @Ayliean for sharing her favourite short classroom games. They’re suitable for use with a whole class whilst the students sit down and the teacher runs the game on the board from the computer - this is good when social distancing measures are in place. I can imagine that some of these games might work well when getting to know a new tutor group. Open the file here for clickable links, and also check out the teacher guide with tips and curriculum links. 

I've just played some of these games - lots of fun!

2. New Resources
Thank you to @Miss_R_Em for sharing some some great angle reasoning worksheets on Twitter.

            

3. Quotes Display

I love this 50 Mathematical Quotations display from @nathanday314. I have added it to my Displays Page.

4. Videos

In Gems 133 I shared a video made by @McGuirea499. Mr McGuire has now made his own Youtube Channel. He's shared a number of new videos recently including a series on ratio tables. Here's an example:

@McGuirea499 has also launched the website makeemthinkmaths.com which features resources, ideas and videos.

5. Further Maths Resource

@MrsMathematica spotted a helpful resource on TES for anyone who teaches A Level Further Maths. Revision 4 Questions for AS Further Maths by Mel Murray is a set of tasks that could be used as starters for Year 2 students to revise AS content. 

A Level teachers might also be interested in @mathsaurus's new pages of A level exam questions by topic for each chapter of the Year 1 and Year 2 textbooks. All questions are from sample and 2018 papers for Edexcel, OCR A and B and AQA for the new syllabus. 

And whilst on the subject of A Level, I love this task on related binomial expressions from @360maths. 

6. White Rose

Those of you who follow the White Rose Scheme of Work will love @mrshawthorne7's spreadsheet for Years 7 to 9. It contains hundreds of links corresponding to the White Rose small steps, including tasks from Don Steward, ⁦‪@giftedHKO‬⁩ and ⁦‪@ChrisMcGrane84‬⁩. The link is on Charlotte's website. 

White Rose users will also like @BenBent05826467's work on linking UKMT questions to the White Rose scheme of work. 

7. Primary Resources
In Gems 132 I wrote about the publication of the Department for Education's of Mathematics Guidance documents for the Key Stage 1 and 2 national curriculum, and the accompanying CPD videos. 

The NCETM (@NCETM) has now shared a set of resources to accompany the ready-to-progress criteria. Written in collaboration with Nrich, the first of 79 PowerPoints can be downloaded from the NCETM's newly revamped website.

8. Vocabulary Map

@MissACMaths created a lovely map of maths vocabulary showing how it all links together.

9. Mr Carter Maths A Level

@MrCarterMaths has launched an exciting new A Level website which will be free for at least the first year.  

You can watch a video here to see how a pupil uses the site, and one here on how a teacher uses the site.

A similar site for GCSE is planned for next September.

10. Timers

I love a slick, professional PowerPoint. Thank you to @DrStoneMaths for sharing some transparent timer gifs for use in PowerPoint, based on a design by @nathanday314. They range from one minute up to ten minutes. 

On the subject of good resource layout, do have a look at this thread where @nathanday314 makes some great suggestions:

Update

For those of you who took the summer off and have missed some of my posts, here's what I've written since July:

• Maths Teacher FAQs
• 5 Maths Gems #133
• Warm Up Booklets
• 5 Maths Gems #132
• Catching Up?

I also updated my popular Year 7 Maths Activities post with ideas for first lessons with Year 7.

Another change to my blog was the addition of a 'buy me a drink' button! Thank you so much to the lovely people who have already used this - I am really touched by the kind messages.

Earlier this week I ran two webinars for The Mathematical Association called 'Preparing for September'. A global Zoom outage meant that the Monday afternoon session couldn't go ahead, but I was able to rearrange it for Wednesday. Thank you to everyone who joined me for one of the two webinars. The recordings can be viewed here and the slides are here.

 

Another thing I did over summer was made a resource based on the old textbook exercise below. I like the way it presents the problems in pairs: one with numbers, one with algebra. I've done this in class discussion before but not as an exercise. 

I've started planning my workshop for #mathsconf24 which will be run online on Saturday 3rd October. I'll be sharing ideas for Key Stage 3. Tickets are still available!

A few weeks ago I tweeted an open invitation to Friday night's 'socially distanced maths teacher drinks' in London to celebrate the start of a new school year (and commiserate the end of the holidays!). Thank you to the ten tweeters who joined me - it's always great to meet new people, and to catch up with old friends after so long. I really missed the socialising of the summer conference season this year. 

Thank you to @MathsTeacherKYP⁩ for the giant calculator. I will lend this to students!

Here are a few things you might have missed if you've been staying away over summer:

• @DrStoneMaths ran a survey on maths timetabled hours - you can read the results here.
@jamesgrime organised a group of Maths YouTubers to release videos on their favourite numbers over one million. The full playlist is here.

• Chika - the London schoolboy I mentioned in Gems 116 - has won an award for coming up with a test for divisibility by seven. This is such a great story to tell your students!
• Maths teacher @timdolan shared a really helpful video '20 Ideas for How To Use a Visualiser in the Classroom'.
• The NCETM published Covid recovery guidance for secondary maths teachers.

I'll leave you with this lovely activity 'Number Pyramids' from @MathforLove, along with helpful slides made by @fawnpnguyen, which include extension tasks.

 

Those of you going back to school next week - enjoy! And good luck. See you on the other side.