5 Maths Gems #113

Welcome to my 113^th gems post. This is where I share some of the latest news, ideas and resources for maths teachers.

1. L Shapes
In MathsPad's July update they shared a free interactive tool for finding missing sides of L-shapes. This can be used to demonstrate how the vertical lines on one side of the shape match the total length on the other side and to help pupils separate horizontal and vertical measures when finding missing lengths. There are accompanying worksheets too.

I know I've said it before, but I love MathsPad! It's well worth the subscription. I'll be teaching Year 7 and 8 next year and have been busy working on resourcing the Scheme of Work over the last two weeks. Many of my recommended resources for each topic come from MathsPad.

2. Maths in Science
Thanks to science teacher Adam Bilton (@heroteach) for sharing a practice booklet for maths in science. Look at this alongside the Maths in Science CPD I shared in Gems 102 to get a better idea of the maths content of the science curriculum. Knowing this stuff helps maths teachers work better with their science department in terms of consistent methods, order of topics and so on.

3. Puzzles for the Classroom
Thank you to Sarah Carter (@mathsequalslove) for sharing Erich Friedman's website. There are loads of great activities to explore here. Sarah has picked out a couple of examples that would work well at school:

Arithmetic Sequence Puzzle
In the Arithmetic Sequence puzzle students fill in some of the blank squares with digits 0-9. Each row and column should contains exactly 3 digits, and these digits should form an increasing arithmetic sequence. Here's an example:

There are 15 increasingly difficult puzzles to complete and solutions are provided.

Number Mazes
Here you start with the number on the left. By moving through the maze and doing any arithmetic operations on the number that you encounter, you exit the maze with the result on the right. You may pass through an operation several times, but you can not make a U-turn. The results of all operations will be positive whole numbers. Each maze has several solutions, but has a unique shortest solution.

There are 16 puzzles and solutions are provided. I've had a go at this - it's fun and the maths is accessible, so I've added it to my 'First Lessons with Year 7' post.

Finally, also check out the 'What's Special About this Number' page. If in your classroom you like to share interesting number facts (eg relating to the date) then this will be very helpful! The list goes all the way up to 9999.

4. Circle Game
This circle sketching game was shared by Graham Walton (@mr_g_walton). Although the scoring seems a bit inconsistent at times, it's fun to try and draw the best possible circle. It works on a phone and a computer. Have a go!

5. Topic Grids
Both BossMaths and MathsBot recently launched new features which can be used to instantly generate mixed topic retrieval activities.

A couple of months ago BossMaths added 'Top Topics' to their site. Using this tool teachers can create grids of up to six questions on topics of their choice. There are over 120 question variants and it's very easy to adjust the difficulty level. I like it that once you've shown the answer you can instantly generate an almost identical question, so if pupils struggled with it then they can have another go after you've gone through it.

On MathsBot there's now a 'last lesson, week, month, year' retrieval starter tool. A lot of teachers have been using this approach to lesson starters since the idea was first popularised last year - this MathsBot tool will save a lot of time in making those starters.

Thank you to @boss_maths and @StudyMaths for their generosity in sharing these free tools for teachers.

Updates
June is always my busiest month in terms of conferences and events - it's been awesome but I'm pleased to have my weekends back now! I also started my new job last Monday. It's wonderful to be properly back in a school after a year on the central team. My school currently just has Year 7s and they are absolutely lovely. I can't wait to teach them maths next year. Along with my new team I'm planning to try some exciting stuff - like ringbinders instead of exercise books in maths for the whole of Year 7 and 8. No doubt I will blog about how it all goes.

In case you missed them, my two recent blog posts were:

• Looking at Year 6 SATs
• MEI Conference 2019

And here are two of my older posts that are always popular at this time of year:

• End of Term Resources
• Activities for your First Maths Lesson with Year 7.

Even though it was a few weeks ago, I haven't yet had the chance to report back on La Salle's #mathsconf19 which took place in South Yorkshire. It was a great conference in a particularly lovely venue. I enjoyed pre-conference drinks, ran the MA bookstand, presented on the evolution of maths vocabulary, caught up with my Twitter pals Ed and Tom, and went to some great workshops.

If you missed it on Twitter at the time, do check out this thread from Sam Blatherwick which features exercises from his workshop on inequalities. There were some questions in this set of exercises that I've never asked before and this made me think about the gaps in my explanations.

Last Monday I delivered the closing plenary at the Kent and Medway Maths Hub Conference. It was lovely that I got to have dinner with some of the speakers and organisers the night before, including James and James from my PGCE course.

I really enjoyed the workshops I attended. In Peter Hubble's session on algebraic proof he shared a funny Abbott and Costello video that I'd never seen before.



In Pietro Tozzi's workshop on Edexcel A level, he told us that some schools have started using the large data set to teach statistics at Key Stage 3 and 4 to build early familiarity. For example schools who do Edexcel A level use weather data for GCSE box plots and so on. He also shared some large data set resources.

Another thing that Pietro shared that I hadn't seen before was a set of really good videos from Pearson explaining how to use the Classwiz at A level. Teachers who are teaching A level for the first time next year would benefit from watching these videos to ensure they are familiar with all the new calculator functions.

And finally from this conference, here's a great problem shared by Charlie Stripp.

In other news:

• I wrote June's edition of the MA eNews. We have recently relaunched this newsletter - not only does it contain news from the MA and the wider world of maths education, but it also now contains puzzles and resources for maths teachers. Subscribe here to receive future editions.
• This year's Big Math Off is underway. I competed last year and that meant I had the honour of picking an entrant for this year's competition. I nominated Vincent Pantaloni - you can read his entries, and all the other wonderful maths being shared, on aperiodical.com.
• Don't forget to book your ticket for Marvellous Maths Teaching which is being run by me and Craig Barton in October 2019. We've sold around 90 tickets already, which is exciting given that we know that most teachers won't be in a position to book until September. 

I'll leave you this video, shared on Twitter by @DrFrostMaths. It's an American Youtuber completing an Edexcel Maths GCSE paper. I found it surprisingly enjoyable! It's weird, and quite insightful, to hear someone's entire thought processes as they do maths.

Looking at Year 6 SATs

I think it's a good idea for secondary school maths departments to spend some time looking at Year 6 SATs papers. Ideally this should be done at the start of September when our new teams are in place and we're planning lessons for the first few weeks of term.

When we look at papers it's important to bear in mind that just because a topic was on their end of year assessment, it doesn’t necessarily mean that an incoming Year 7 knows that topic well. Just like at GCSE, grade boundaries are relatively low. This year a Year 6 child needed to get 53% of the answers right in their maths SATs to achieve 'Expected Standard' (ie a scaled score of 100). It's a very similar situation to the Year 11 to Year 12 transition - when a student starts A level with a Grade 7 it's important for us to be aware that they only knew how to do around half the questions on their GCSE exam. Plus we all know that a lot of forgetting happens over the long summer break. So both at A level and in Year 7 (and in fact at the start of every school year) we should never just assume that topics that have been previously taught and assessed are totally secure.

Saying that, we know that many pupils joining us in Year 7 are brilliant little mathematicians. Often they have excellent fluency in arithmetic and times tables and well developed reasoning and problem solving skills. But we won't know that just from looking at their SATs score. A good SATs score might only tell us that back in May the child had a good level of fluency in written methods of arithmetic. A short, focused baseline test with incoming Year 7s might help us work out if that's still the case and give us an idea of a sensible starting point for each pupil.

When my previous school had incoming Year 7s sit baseline tests in September, we were surprised by the variation in results. The graph below shows pupils' Key Stage 2 SATs result against the score they got on their Year 7 baseline assessment. Although we see a general correlation as expected, the middle chunk of the data (expected scores between 100 and 110) was all over the place. For example notice that pupils who got 104 on their SATs scored everything from 35% to 80% on their September baseline test. If your school sets in Year 7 based solely on SATs results you need to be aware of this potential variation.

Schools are able to access data for their incoming Year 7 pupils that shows their SATs marks by question. Like the QLAs we use for mock exams at Key Stage 4, this data is of limited use. Getting a single question right or wrong on a particular topic at a particular time doesn't really tell us very much about a child's understanding of that topic. We have no idea how much a child really understands something until we've spent time questioning them, working with them, assessing them, drawing out misconceptions. Clearly we need to get to know our students and we can't make any assumptions based on data in a spreadsheet.

Saying that, let's have a quick look at a bit of national data just because it's interesting. The following two questions were the worst answered questions in last year's maths SATs.

Paper 3 2018. Answered correctly by 24% of pupils. Content first taught in Year 4.

Paper 2 2018. Answered correctly by 31% of pupils. Content first taught in Year 6.

And here's the worst answered question on the 2018 arithmetic paper:

Paper 1 2018. Answered correctly by 45% of pupils. Content first taught in Year 6.

I think this is a pretty grim question. I'm surprised it's only one mark. I'm also surprised how many pupils got it right given the unnecessarily fiddly numbers. It would be interesting to see how many Year 7s can remember how to do this in September. And it would be interesting to see how many Year 11s can get this right.

I think it would be good maths department CPD to discuss all these questions in detail and think about what pupils would have found difficult.

The breakdown of topics in the 2019 Year 6 SATs was similar to previous years. A large chunk of the questions were calculations and fractions, decimals and percentages. This is unsurprising. There weren't many questions on shape and statistics.

I find that Year 7 teachers are sometimes unsure what algebra their pupils have seen before. Here are two algebra questions from this year's Paper 3.

Here's the primary curriculum content relating to algebra:

It's mainly 'missing number' stuff. So when we teach collecting like terms etc in Year 7, it's their first time manipulating algebra in this way.

I've been trying to find time to write this post for about 18 months so I'm glad I finally finished it! I hope it's a helpful starting point. The main point I'm making is that SATs data is of limited use - teacher judgement is way more important - but I strongly recommend looking through all three 2019 maths SATs papers in a department meeting in September to help teachers get to know the primary curriculum better. If you don't have time to look through all three papers then perhaps just have a look at the Third Space Learning blog which has an analysis of this year's maths SATs papers including some examples of questions to discuss.

By the way, if you're teaching Year 7 in September (I am!) then you might like my post about activities for a first lesson with Year 7.

MEI Conference 2019

I was fortunate to have the opportunity to attend the MEI Conference from Thursday to Saturday last week. I normally only attend conferences at weekends and in the holidays so getting permission from work to attend a term-time conference was a rare treat. The last time I attended the MEI Conference was in 2015 - I blogged about it here.

Residential conferences are a very different experience to one day events. There’s more time in sessions to discuss ideas and do maths, and there’s more time outside of sessions to socialise, meet new people and chat about teaching. At lunch on Friday I sat with Ben Sparks plus Christopher Rath (@Rathematician) and his colleagues - they showed me cool maths stuff that I'd never seen before. So much fun. Since I got back I've been showing everyone my new tricks! Literally every time I see Ben Sparks he blows my mind with some clever maths magic. He did the same in his workshop on Saturday morning where we played a poker game in which I genuinely thought I had a full house – until I realised he’d rigged it. How does he do that…?

It was wonderful to stay for all three days of the conference - I loved every minute and felt incredibly relaxed hanging out on a university campus in glorious sunshine. The evening entertainment was top quality - the after-dinner speak Harry Baker was absolutely wonderful. Everyone was blown away by how lovely he was. Watch this video to get an idea, but go see him perform if you can (I first shared this video back in 2015 in Gems 26 so it was cool to actually meet him in person!).

I’d always assumed that the MEI Conference is aimed mainly at A level teachers but I found that there was a wide range of sessions to appeal to all secondary maths teachers. It had a bit of an academic feel to it in parts - particularly the first two plenaries. The first keynote was by Dr Eugenia Cheng (she’s my new hero) and second keynote was delivered by Professor Alison Etheridge – she delivered a fascinating lecture on genetics which made me feel like I was back at university! I love the high expectations that the organisers have of their audience - it’s wonderful to be immersed in intellectual challenge every now and then. Craig Barton’s keynote on the final day was brilliant, as always. He also had the coolest clicker I've ever seen. Did I mention that I’m running a course with Craig in October? You can book your place here.

One thing I like about the MEI conference is that there's a very different range of speakers from the events I usually attend so I get to hear lots of new perspectives. I recorded a conference podcast with Craig Barton on the Friday night (after a few glasses of wine…) so have a listen to that to hear some reflections on a few of the sessions I attended. I didn’t get a chance to speak about every workshop. In this blog post I’m just sharing a couple of things that I didn’t mention in the podcast that I thought teachers might be interested in.

Football Loci
Rob Eastaway (@robeastaway) ran a session called 'What's the difference between a puzzle and a maths question?' which was partly about how to turn a dull question into an interesting one. Something that really jumped out at me related to loci. We often see really contrived scenarios in loci questions but he shared a fascinating scenario that I loved.

Rob showed us a set of football rules from 1891, when the rules of the game were evolving. It said:

"The goals shall be upright posts, eight yards apart, with a bar across them eight feet from the ground.

When the ball is kicked behind the goal-line by one of the opposite side, it shall be kicked off by any one of the players behind whose goal-line it went within 6 yards of the nearest goal-post..."

Show pupils this and ask them what it would look like. Where can the player stand to take a goal kick according to these rules? Perhaps ask pupils to try to draw a scale drawing of it.

They should end up with a couple of arcs rather than the six yard box we are now so familiar with.

Once they've done this you can show them a picture of the 1901 FA Cup Final when Tottenham Hotspur played Sheffield United. There are arcs! It's very cool. The Wikipedia page on goal kicks says "In 1891, pitch markings were added to define the six-yard radius from each goal-post".

In 1902, the term 'goal area' was introduced for the place from which the goal kick was taken - it assumed its modern dimensions as a rectangle extending six yards from each goal post.

I loved this workshop and I think that this lesson on loci and scale drawing would go down really well.

Fascinating stuff from Rob Eastaway, as always.

Circle Theorems Chase
Tim Honeywill (@honeywilltim) led a session called ‘Expecting the Unexpected’ where he shared a pack of problems which are suitable for IGCSE and A level pupils. I love angle problems and there were a few in the pack, including this beauty called 'Crazy Circle Theorems'.

Tim's pack of questions contained loads of nice problems. Here are a few more examples:

Very kindly, Tim has let me share the whole set of resources from his workshop here. Thank you Tim!

Sectors of Circles
I ran a workshop on resources in which I talked through the process of planning a sequence of lessons on sector area. I showed a large number of places we can go to get resources including some classics such as SMILE. I talked about different types of task and discussed how they meet different purposes.

For this workshop I made a new Fill in the Gaps task on sector area (extract pictured below) which is intended to be used as part of the initial instruction phase. The idea is that pupils develop an understanding of how to identify what fraction of the circle is required, and make connections between sectors of different sizes. I also decided that I wanted to get pupils more comfortable with sketching diagrams, so I included that in the task. If you use this resource, let me know how you get on.

There's lots more I could say about the MEI Conference but to keep this post to a manageable length I will leave it there. Thank you to everyone - speakers, delegates and organisers - who made it so fantastic. Well done MEI! Great work.

Hanging out with Chris Shore and Tom Bennison

The next residential maths education conference I attend will be the MA Conference 2020 which involves two nights away at a spa hotel in the Easter holidays – I’m looking forward to it already!

5 Maths Gems #112

Welcome to my 112^th gems post. This is where I share some of the latest news, ideas and resources for maths teachers.

1. Classic Errors
I like this activity suggested by James Tanton (@jamestanton). James asks 'Would taking classic and tempting errors "head-on" be a worthwhile classroom activity and discussion?'.  It's worth reading the responses in the thread. In it Cathy Yenca‏ (@mathycathy) recommends the classic resource 'Algebra Atrocities' which is worth a look.
2. Geogebra Percentages
Thanks to Mark Horley (@mhorley) for sharing some helpful Geogebra applets created by ⁦‪@orchiming‬⁩ .

These interactive tools help with explanations of percentages. For example the profit and loss applet shows really clearly how the calculation works.

3. Area Mazes
I know I've featured Area Mazes in my gems posts an number of times before, but I think they're really fun so there's no harm in mentioning them again. Have a go at this one -  it was tweeted by @Errs5 and is taken from the Area Quiz app which looks excellent.

4. Calculator Crunch
In my post about Year 11s lacking basic calculator skills I mentioned that MEI are running Calculator Crunch to encourage schools to get younger pupils using calculators. Aimed at Year 6 and 7, they are sharing a daily calculator challenge between 10th and 20th June. You can download all the daily challenges from the MEI website and use them at any time. They have also provided a Year 6 lesson plan and a Year 7 lesson plan that you can download here.

Check out the hashtag #CalculatorCrunch to see examples of pupils' work.

5. Place Value
Another great idea for using calculators is in a lesson on place value. Tom Francome shared this 'Place Invaders' task in a conference workshop - it's taken from the ATM book Practising Mathematics which he wrote with Dave Hewitt.

Grab a calculator and have a go and you'll see why this is a great task to use when teaching place value. I think that place value is one of the hardest things to teach in Year 7 so it's helpful to see a good quality activity for this. Tom suggested it would also work well with standard form.

Update
Exam season is nearly over, and it's been an unusually turbulent one this year. We've had questions that appeared to be copied from textbooks, a mistake in an A level, controversial contexts, a leaked exam and some surprisingly challenging A level papers. Never a quiet day in maths education!

I recently made a final set of GCSE revision resources which you can find on my GCSE revision resources page. You might find these resources helpful if you have Year 10s preparing for end of year exams.

I presented at ResearchED Rugby yesterday. My session focused on multiplication - first we looked at a range of multiplication methods and then I presented a critique of each of the reasons the Government gave for prescribing 'formal' methods at Key Stage 2. For the remainder of the day I attended excellent sessions in the maths strand. Rugby School has an incredible history so was a cool venue for a conference.

Next weekend I'll be presenting on the Evolution of Maths Vocabulary at La Salle's conference in Yorkshire. Read my blog post "Whatever happened to vulgar fractions?" to get an idea of the sort of thing I'll be talking about, and if you find it interesting then do come along to my workshop. I'll also be manning the MA bookstand for most of the day so do come and say hello if you get the chance.

Bookings are going well for the training course I'm running with Craig Barton in October. This course is open to all maths teachers and we have two dates available - a northern leg and a southern leg. Ask your school or college to book your place now if you want to come along.

At the end of May Ed Southall and I relaunched The Mathematical Association's eNews. This monthly newsletter provides an update of what's going on with the MA and also in the wider world of maths education. We now feature exclusive monthly puzzles for your students (written by Ed), and there are more new features coming soon. You can subscribe here to receive eNews from the end of June.

I've been kindly sent a couple of books by their authors recently. Kyle D Evans sent me a copy of 'Here Come the Numbers' which my daughters (aged five and seven) love as a bedtime story. It's both mathematical and lovely.

Mark Ritchings sent me a copy of 'GCSE Maths Challenge' which contains 40 GCSE style questions that are aimed at pupils who are working at a Grade 8 or 9. This short book might be helpful for a GCSE tutor, or possibly a top set teacher who needs a few extra ideas for extension questions to write on the board.

Maths teacher Richard Atkin (@RichardAtkin1) - whose uncle was a Trustee of the MA - very kindly delivered an enormous number of old maths books to my house. This will keep me busy for a while! I've already found lots of treasures that have taught me new things. I estimate that I now own at least 300 different maths textbooks dating from 1800 to 2000. I love this!

A group of us are going to Matt Parker's Humble Pi comedy show on 16th July in London (it's a preview of his Edinburgh Fringe show). If you want to join us, book a ticket here and let me know you're coming. There are similar comedy nights in London listed here, including one that features the amazing FoxDog Studios. FoxDog Studios will also be at Edinburgh Fringe and are doing gigs in various locations before then - dates are listed here. They are amazing.

I'll leave you with this great activity shared by Sarah Carter (@mathequalslove). You can download the full resource from her blog. This puzzle might work well in a first Year 7 lesson in September, along with the other tasks I listed in my post Year 7 Maths Activities.

5 Maths Gems #111

Welcome to my 111^th gems post. This is where I share some of the latest news, ideas and resources for maths teachers.

1. Multiplication Tasks
In a Twitter chat on multiplication I saw two nice resources shared. First, Sharon Malley (@mathsmumof2) mentioned these 7 Times Table Reasoning Activities by krisgreg30 on TES. These tasks require children to use known facts to reason how to solve other calculations.

Second, Jonathan Hall (@StudyMaths) shared a lovely task he designed for his Year 7s to get them thinking a bit deeper.

2. MathByExample
Three years ago I wrote about AlgebraByExample in Gems 54. This set of tasks prompts students to analyse and explain misconceptions in algebra problems.

The team at SERP Institute (@SERPInstitute) have now launched their MathByExample website. The tasks are similar to AlgebraByExample but they are for topics that children meet at Key Stage 2.

There are loads of great tasks to explore on this website. In each case children are given a correct answer and an incorrect answer with questions about each one, and then they are asked to solve similar problems themselves.

The question prompts help children develop a better understanding of each concept.

3. Compound Area
Amie Albrecht (@nomad_penguin) shared a smart way to take a standard textbook-style exercise and add a higher level of thinking. Instead of just completing the exercise, pupils are asked to consider the features and difficultly level of each problem before deciding which problems to solve.

4. Linear Sequences
Thanks to Dan Lewis (@4301maths) for sharing a series of tasks on linear sequences.

Follow Dan on Twitter for more like this, including examples of his pupils' work.

5. Question Generators
Thanks to Jonathan Payne (@DrPMaths) who has built a collection of helpful question generators.

For example if you are creating some angle questions for your explanations or for your pupils to practise, then you can use his missing angles generator to create a set of customised questions.

And here's one that creates arithmagons.

Updates
I've been busy making more GCSE revision resources. This is the last time I'll do this for a while because next year I'll only be teaching Key Stage 3.

Because the non-calculator revision mats and the calculator revision mats I recently made went down well with pupils, I was asked by a colleague to make another set. So now I have a second set of calculator revision mats. Again, they have four levels of difficulty so you can pick the right level for your pupils.

I also made a Higher and Foundation 'Spot the Mistake' revision activity for something a bit different.

I also made a couple of revision mats with topics that might come up on AQA Paper 2. These are just a collation of questions taken from Maths Genie. Because these are 'temporary' resources (ie designed specifically to prepare for AQA Paper 2 June 2019), these are not on TES but are linked through Adam Creen's blog. Every year Adam pulls together all the 'between-paper' resources on his blog for easy access.

Don't forget you can use my breakfast resources as pre-exam warm-ups before Paper 2 and Paper 3. And my GCSE revision post continues to be the place where I collate all free GCSE revision resources (with the exception of the 'between-exam' resources that have a limited shelf-life).

***

I was delighted to announce this week that I will be teaming up with Craig Barton to offer two full day training courses in late October. Visit mathscpd.weebly.com for all the information. Bookings are already going well.

My recent post 'Calculator Woes' rung true with many teachers. It was selected as a Schools Week 'Top Blog of the Week' by Amir Arezoo and featured in Ollie Lovell‏'s weekly Twitter takeaways. I think there's quite a serious problem with calculator skills across the country and I really hope to see teachers try to remedy this by getting lots of Year 6s and Year 7s involved in MEI's Calculator Crunch next month.

Thanks to Teresa Robinson at The Russell Education Trust who used my post to create a lesson on calculator skills.

Thanks also to @pgonlinepub for sharing a free worksheet on the 'Top 5 calculator hacks for GCSE Maths'.

Yesterday I had coffee with Simon Singh, the author of my favourite maths book. We discussed what parents can do to encourage and support their mini-mathematicians at home. Simon mentioned the coding app Box Island. When I got home I downloaded for my daughters. It's awesome!

By the way, if you don't currently receive the MA's eNewsletter then you can sign up here. I've been working with Ed Southall to relaunch it. From now on it will include an exclusive monthly puzzle for your pupils to try - one for primary and one for secondary. Sign up now!

I'll leave you with this graph, shared on Twitter by @lizardbill, which is probably the best example I have ever seen of a really really bad graph. There are more amusingly terrible graphs in the thread.

Enjoy half term!