5 Maths Gems #115

Welcome to my 115^th gems post. This is where I share some of the latest news, ideas and resources for maths teachers.



1. Distributive Law
Miss Konstantine (@GiftedBA) shared a set of questions on the distributive law. Visit her blog to download the PowerPoint.

2. Frayer Models
@MrMattock, @JoLocke1, @MisterRapley, @timdolan and @StudyMaths have created the website frayer-model.co.uk. Here you can download editable Frayer Models for a large number of maths topics (there are currently over sixty on there). 

3. Area Mazes
I know I've featured Area Mazes in my gems posts before but here are a few more...  While on holiday I played some of the Area Quiz app on my phone. Most of the problems in this app require a bit (or a lot) of reasoning. They are suitable for people of any age as long as they know how to work out area. A lot of these problems would work well at school to deepen pupils' understanding of area.

4.  Photo Signs
I'm not crafty enough to make one of these myself! Howie Hua @howie_hua‬⁩ made this sign for his pupils to hold up for photos. If you check out ⁦‪his Twitter timeline you'll see his pupils proudly sharing photos of themselves meeting their new maths teacher and embracing the idea of being a 'math person'.

Inspired by this, other teachers have made signs for their maths classrooms. Here's an example from Alice Aspinall (@aliceaspinall).

5. Starter Questions
Thanks to Mary Atherton from Carmel College in Darlington for emailing me about the pack of Starter Quizzes she intends to use with Year 10 this year. She tried a similar daily activity with other year groups last year and found that pupils responded well to the routine. Ahead of each lesson Mary will put one sheet of questions into each pupil's document wallet (her form will help her do this) and then every pupil will get started as soon as they arrive at the lesson. They get 10 minutes to answer the questions in any order - when the buzzer goes off Mary calls out the answers and the students mark their own work, recording their marks out of 10 on the record sheet that's glued to the front of their document wallet. Thanks to Mary for sharing her quizzes.

Update
I've had very little time for blogging over summer. As well enjoying a lovely (if slightly rainy) family holiday in Jersey, I've been busy trying to write my book 'A Compendium of Mathematical Methods'. It's really hard work! I hope people like it. I've only got one chapter left to write. If all goes well it will be out in December or January. I've learnt a huge amount of new maths through researching interesting, unusual and antiquated methods. I can't wait to share what I've found.

In other news, thanks to Jamie Frost for hosting another lovely evening for maths teachers at his house and local pub.

And thanks to TES for hosting a great TES Maths Panel meeting in Sheffield with a delicious lunch in the sunshine.

Another thing I've done recently that I highly recommend is visit the MA library at the University of Leicester. All MA members are allowed to visit the library. I went with friends Ed Southall and Tom Bennison. MA librarian Mike Price very kindly showed us round and gave us access to the amazing books in the Special Collection. It was incredible!

If you plan to go to any events in the next academic year then do check out my updated conferences page. In September I'll be at #mathsconf20 in Edinburgh where I'm presenting on methods and adfected quadratics, and in October I'll be at #mathsconf21 in Peterborough where I'm presenting on the 'Calculator Crisis'.

If you want to attend the one day training course 'Marvellous Maths Teaching' with Craig Barton and me then it would be a good idea to get your request in at school soon before it sells out. 

I pulled together the MA's August 2019 eNews which came out yesterday, sharing news, puzzles and resources for maths teachers. Do check it out, and subscribe to receive future issues here.

I visited my brand new school building last week with my daughters.

Here's my five year old being a teacher in my new classroom.

I've planned first lessons for a couple of my classes but I am fortunate to have four Inset days before my pupils start so I'm leaving most of my work for then rather than do it in the holidays. I can't wait to get started with lessons! I'm really excited about meeting my new classes. Though of course I'm a bit sad about summer coming to an end.

If you're not back at school yet and you're planning first lessons for September then you might find some of my posts from previous years helpful, such as Year 7 Maths Activities. Also, check out my large collection of maths displays.

If you're back to school soon, good luck with the start of term.

I'll leave you with this puzzle that your pupils might enjoy - it was shared on Twitter by @theexperttutor after she spotted it in a magazine. There are lots more like this here - thanks to @Caminomig for the link.

Two Maths Apps for Children

A little while ago I had a comment on my blog from a parent who asked if I could recommend some apps for their children. I couldn't at the time, but now I can. I recently did some work researching some of the best apps related to maths, logic and problem solving for all Key Stages (from age four through to eighteen). There are many! In the process of doing this research I downloaded some brilliant apps for my own children to try out. My daughters Maddie (7) and Hettie (5) got to be my guinea pigs, and thankfully they really loved the apps I chose for them. I've decided to write about two of my favourite apps here. This is the first time I've written a post aimed at parents rather than teachers. Note that I bought these apps myself - no one has paid me to promote their app (I don't do that!).

Thinkrolls
As I write this post my five year old is sitting next to me playing Thinkrolls. She loves it! Every now and then she asks me to give her a hint when she gets stuck. I try not to give her a hint - I know she can work every level out herself, once she's had a think about it. Thinkrolls is a beautiful little game and absolutely perfect for her age. Easy mode is aimed at three to five year olds and hard mode is for ages five to eight. The idea is to move cute little characters around mazes. It takes logic and creative thinking. It's super fun, and well worth the £3.99.

I've always been really hesitant about paying for apps but now I realise that's a bit silly - I buy books and board games, I rent films and subscribe to Netflix, I pay for swimming lessons and Kung Fu and so on - so I'm not sure why I thought that I shouldn't spend £3.99 on a clever little educational app.

I'm very pleased to see that the developer of Thinkrolls has a number of similar apps so that when we finish the first Thinkrolls we can try another one in the series. 

Slice Fractions
My seven year old has been playing Slice Fractions. This is directly related to the maths curriculum but I don't think she thinks of it as a maths game. To her it's just a game involving woolly mammoths in funny hats.

She's done a little bit of work on fractions at school but she's very much a novice. As soon as the app introduced fraction notation she started to find it a bit tricky because she doesn't know the basic concepts (eg what a quarter is), but she's not been put off. She loves it and she seems to be picking up the concepts pretty quickly. The makers of this app say that research shows that "Slice Fractions significantly improved students’ performance in a very short amount of time". So perhaps this game will help to develop my daughter's understanding of fractions, as well as giving her opportunities to solve problems and think logically. And even if it doesn't help her understand fractions better, then there's no downside because to her it's just a fun game.

Slice Fractions is £3.99, and there's a sequel (Slice Fractions 2).

If you have children a similar age to mine, I recommend both Thinkrolls and Slice Fractions. In my opinion they're worth buying.

The other maths apps that are really well respected are those from Dragonbox. They have apps for kids from age four up to teenagers (including algebra for both five and twelve year olds). I haven't tried them yet but I've heard they are awesome.

By the way - I haven't featured the brilliant apps Box Island and Sumaze Primary here because I've blogged about them before. I should mention Numberblocks though - they have two lovely apps, one of which is free.

Enjoy!

My guinea pigs!

5 Maths Gems #114

Welcome to my 114^th gems post. This is where I share some of the latest news, ideas and resources for maths teachers.

1. Polypad
Philipp Legner, writer of the beautiful website Mathigon, has been working on some new resources for schools and teachers. The first new tool - published today - is a library of virtual manipulatives called Polypad.

There are polygons, fraction bars, number and algebra tiles, Cuisenaire rods, pentominoes, and many other features. Like all of Mathigon, it's completely free to use and works on all mobile and desktop devices.

2. Don Steward
Don has been publishing loads of new resources lately. I'd love to share all of them here but it makes more sense for you to visit his blog and scroll through them yourself! I've started to add them to my resource libraries.

One example is his recent set of exercises titled 'Linear equations extras'. Next year I'll be teaching a couple of Year 8 classes. The scheme of work includes a unit on linear equations. They first met this topic in Year 7. Once I've assessed how good my pupils are at solving linear equations, I might dip into Don's questions if appropriate - they present a good opportunity for stretch and challenge as well as interleaving. Take for example these linear equations that involve decimals.  Here pupils can develop their fluency in solving equations and at the same time practise working with decimal operations.

3. Arithmetic Sequences
Sarah Carter (@mathequalslove) shared a blog post about how she uses Step Puzzles from Naoki Inaba when teaching arithmetic sequences.

These puzzles are accessible to any age group. By looking at the example above you can probably figure out how they work without any instructions. The numbers along each straight line must form a linear sequence.

There are lots of examples on Sarah's blog - they start off easy but get much more challenging. There's also an English language version of this resource on TES. I'm teaching sequences to Year 7 next term - I hope to give my pupils some time to play with these lovely puzzles.

4. Templates
Thanks to Ben Gordon (@mathsmrgordon) for sharing a set of editable worked example templates. There are various layouts of Frayer models, example pairs, guidance fading and incorrect examples.

I'll be using Frayer models next year - I have a plan!

5. Puzzle of the Week
Andy Sharpe (@asharpeducator) has been running Puzzle of the Week for a while now. There are now 100 puzzles on puzzleofthweek.com in the Puzzle Library, all free and sortable by topic and difficulty.

Why not encourage your students to submit solutions to the weekly competition? This would work well as part of a school puzzle club. Information about how to enter is here.

Updates
Twitter is relatively quiet during the summer holidays these days. It's really different to five years ago when the summer holiday period was awash with blog posts and tweets about maths teaching and resources. Conversations this summer range from a fierce debate about how to put cream and jam on a scone (I have no idea, but I do quite fancy a scone...) and news about truly horrific political developments (it's like the Handmaid's Tale is actually happening). If you're taking a summer break from Twitter, here are a few bits and pieces you might have missed.

• MEI's (@MEIMaths) latest app Sumaze! Adventure is now out. I love Sumaze!
• I updated my conferences page. It now lists national maths teacher events in 2019/20. If you've not attended before, why not come along to a conference next year? You might love it.
• If you want to attend the one day training course 'Marvellous Maths Teaching' that I'm running with Craig Barton then it would be a good idea to get your request in at school in early September before it sells out. We've already sold 100 tickets. 
• The MA's July 2019 eNews came out, sharing news, puzzles and resources for maths teachers.
• @MathsEdIdeas shared 42 mathsy activities, one for every day of the school holidays. 
• Craig Barton (@mrbartonmaths) published his latest Slice of Advice podcast "What did you learn this year?". Maths teacher Charlotte (@mrshawthorne7) made this awesome sketchnote while she was listening. 

Here are some ideas and resources I've had through on email over the few weeks:

• Maths teacher Laurie Luscombe (@mrluscombemaths) sent me a fantastic description of his approach to introducing the area of a circle. This is well worth a read when you're planning lessons on this topic.
• Australian maths teacher Wendy Taylor emailed me about her website mathsquad.org which includes free homework booklets for Year 7. If you're thinking of using homeworks for mixed topic retrieval then do have a look at these booklets.
• Martin Lofthouse pointed out that Active Maths is now free to use. I used to use this in my NQT year but it required a subscription back then. For example check out the tree diagrams whiteboard worksheet and match the spinner with the graph (see gif below). Also have a look at fishy equations and the speed distance time tool and the ratio generator.

I've been working on my workshop for #mathsconf21 in Peterborough in October. I've found so many great resources I can't wait to use them with my pupils next year and share them at the conference.

I might present the same workshop at #mathsconf20 in Edinburgh but I can't decide!

I've been working hard on my book 'A Compendium of Mathematical Methods'. I'm over halfway through now and have started talking to publishers which is really exciting.

Because I'm focusing on my book plus a couple of other projects this summer, and spending a lot of time with my family (we're off to Jersey next week!), I won't be blogging much during the holidays. If you're planning lessons for September then you might find some of my posts from previous years helpful, such as Year 7 Maths Activities and Bridging the Gap: Revisited (for Year 12 teachers).

I'll leave you with this picture of a weight machine that appears to be show a real life Normal distribution, shared by @gin_and_tacos.

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.