8 December 2018

5 Maths Gems #100

Welcome to my 100th gems post!

This is a big milestone for me. I've published 327 posts since I started writing my blog four and a half years ago. One hundred of those posts have been part of my 'Maths Gems' series - each one has featured a selection of news, ideas and resources for maths teachers.

When I joined Twitter I couldn't believe how many ideas and resources were being shared by maths teachers everyday that weren't being seen by the hundreds of thousands of maths teachers who aren't on Twitter. I remember trying to convince a colleague to join Twitter and he said 'I just don't have time for it. I wish someone could just summarise the ideas for me'. So that's what I try to do.

I used to write one gems post a week (I was on maternity leave!) but now I work full-time I only manage one or two gems posts a month. Twitter still provides a constant stream of material to pick from, so I continue to summarise and share some of the best ideas, in the hope that I can get these ideas into classrooms all over the world.

To celebrate the fact that this is my 100th gems post, I'll be recording a special podcast with Craig Barton next weekend. In preparation for this podcast Craig wants you to choose your favourite gem (there are 500 to choose from, all indexed here). Tweet, DM or email Craig (or comment below) to explain why you chose it, and it could end up on the show.

On with the gems...

1. Mastery Learning Cycle
Over the course of the last two years Mark McCourt (@EmathsUK) has published a series of posts on mastery. You must read these excellent posts if you haven't already! Oliver Caviglioli (@olicav) and Mark have worked together to create a poster which visualises Mark's model of the Mastery Learning Cycle.
This is just an extract - download the poster to understand what it's all about. This is definitely something to share with all trainee maths teachers (and probably all experienced maths teachers too).

No doubt Mark's book 'Teaching for Mastery', due to be published in Spring 2019, will be a must-read too.

2. Manipulatives
If you want to use manipulatives in your teaching but don't know where to start, Craig Barton and Bernie Westacott have recorded a video podcast that you will find very useful. You can access the videos through Craig's Youtube playlist. For example here Bernie explains how to use manipulatives to teach negative numbers:

I've blogged before about Jonathan Hall's (@StudyMaths) amazing library of virtual manipulatives. It has continued to grow.
One of the new additions is a bar modelling tool. It's very easy to use and I'm sure lots of teachers will find it helpful.
3. Indices Tasks
Miss Konstantine‏ (@GiftedBA) shared a task for exploring indices. Students sort the cards and find the odd one out.
Peter Drysdale‏ (@pwdrysdale) made a Desmos version of this card sort.

I can't keep up with all the ideas and resources that Miss Konstantine‏ (@GiftedBA) has been sharing lately! For example I enjoyed her recent perimeter problems. Check out her Twitter feed and blog for lots more great stuff.
Speaking of indices, I made an indices task for a recent Teach Secondary article. This resource is designed to be used with a Year 7 class after spending some time on index notation, but it would work with other year groups too. The full resource is here - it contains four introductory indices activities.
4. Area
Here's a nice idea to help students develop an understanding of what area is. Ilona Vashchyshyn (@vaslona) challenged her students to write their name so that it covers an area of exactly 100cm2. Read the thread for ideas on how to extend this activity.

5. Assessment and Questioning
Mrs Budak (@mrsbudak) tweeted an interesting idea from ⁦‪@teacher2teacher‬⁩ that could work in every subject. At the end of an assessment students are given the opportunity to write down everything else they know about the topics on the test. It stops students being frustrated when a test doesn't cover the things they revised, and it's probably a good use of time to retrieve stuff from one's memory and write it down (it definitely beats sitting there waiting for the test to end!).

Finally, for some reason this reminded me of another gem that I've been meaning to share for ages...

It such a simple idea little change, and so easy to do! It's worth reading the thread for discussion and ideas. Credit to Howie Hua (@howie_hua) for first tweeting about this back in May. Here's another of Howie's ideas:

I've had a busy few weeks visiting a number of different schools, including one where I saw silent teacher in action for the first time.

I helped to run the first day of a new London-wide Maths Hub Work Group on developing A level pedagogy, which is led by Carlos Karingal. We were fortunate to have a fantastic group of teachers attend and I look forward to seeing how our A level teaching develops throughout the year. In my session I talked through some of the ideas in this excellent piece written by Chris McGrane about approaches to teaching calculus.

Next week I'll be supporting Chris Reilly in running another Maths Hub Work Group - Challenging Topics at GCSE. I'll blog about this soon.

My second Teach Secondary article is now available to read online. It's about order of operations, and opportunities to interleave this topic with fractions, decimals and algebra.
Did you see that Pearson have a new series of maths textbooks coming out? It's called Purposeful Practice and you can view sample pages here.

Finally, Christmas is fast approaching so you might find a use for my collections of seasonal resources (here and here) - some are topic based and some are for enrichment. I've written various posts about Christmas presents for maths teachers over the years (here is last year's post on TeachWire if you're looking for inspiration). If you want to treat yourself during December, the MA has a Christmas advent calendar where you can get daily discounts on MA books.

I'll leave you with this lovely problem shared by James from MathsPad (@MathsPadJames). There are a range of solutions in the comments (spoiler alert!), but it can be solved in a matter of seconds without any workings using GCSE level maths.

24 November 2018

5 Maths Gems #99

Welcome to my 99th gems post. This is where I share some of the latest news, ideas and resources for maths teachers.

1. Non-Examples Quizzes
I blogged about the new website nonexamples.com from Jonathan Hall (@StudyMaths) back in Gems 95. There are now some great new features on the website including Frayer Model templates and Multiple Choice Quizzes.
The multiple choice quizzes come with a QLA allowing instant identification of misconceptions.

On Twitter Bernie Westacott (@berniewestacott) described how he used the multiple choice quizzes with an intervention group:

2. Sum Fun
I had an email from a maths teacher looking for books by the author of this A level resource:
It turned out the original books are out of print but Twitter came to the rescue and thanks to Hans Stroeve (@stroevey) we now have the full collection of books scanned in and available to download here. They remind me of Maths with Pizzazz resources and I know they won't appeal to everyone! They include resources for topics from Key Stage 3 right up to matrices and polar coordinates! Warning: always check that the joke is appropriate before using these resources with students.

3. A Level Questions by Topic
Thank you to Chris Ansette (@mransette) who has collated old Exexcel exam questions for Pure and Mechanics and organised them for the new A level. You can download them here. These well formatted collections of exam questions with answers are really helpful for A level maths teachers.
4. Calculator Poster
In Gems 97 I featured a link to a poster of an A level calculator. Casio Maths (@CasioMaths) have also shared a set of high resolution posters of the fx-83GT Plus, which is commonly used at GCSE. You can download the posters from the Casio website.
On the subject of calculators, read this thread from @literallyjustq for some calculator tips.

5. Times Table of the Week
I see a lot of Year 7s really struggling in lessons (on topics like multiplication, division and fractions) because they don't know their times tables. I have long been a big fan of Times Tables Rock Stars to help fix this. I like @DynamicDeps's idea for a times table fact of the week. The suggestion is to use the Times Table Rock Stars heat maps to identify a multiplication fact that students struggle with, and to put up a poster of that fact everywhere in the maths department, and in fact all over the school.
Bruno Reddy has now made a set of 'Times Tables of the Week' posters for all multiplication facts which you can download here.
Perhaps maths teachers and form tutors could regularly quiz individual students on the weekly multiplication fact. Given that there are only 21 facts to memorise (assuming students already know their 1, 2 and 5 times tables), you can easily get through the whole lot in a school year if you do one a week.
The 21 Facts from Kangaroo Maths
I had an absolutely wonderful time at the MathsJam Annual Gathering last weekend. Thank you to the organiser Colin Wright and to everyone else involved. I absolutely love everything about the weekend and would really like to run a maths education event with the exact same format. Maybe next year!
Me, Mariana, Ed, Tim and Joe at the MathsJam Annual Gathering 2018

If you like the idea of social puzzling, do check out the monthly MathsJam events, and also Puzzled Pint's monthly social puzzle event in pubs all over the world. You can download Puzzled Pint's awesome puzzles for school maths clubs too.

I had another article published in Teach Secondary this month. It's about order of operations, and opportunities to interleave this topic with fractions, decimals and algebra. It comes with a free algebraic order of operations resource!
I also presented on order of operations to Harris Heads of Maths this week. The idea was to show that 30 minute CPD sessions on specific topics that are coming up on the scheme of work are a good use of maths department meeting time.
Hannah Fry has agreed to become the 2020 President of The Mathematical Association, which is very exciting news for all MA members.

Next month I will publish my 100th gems post (I have some cracking gems lined up!) and will record a celebratory podcast with Craig Barton. Craig will be asking listeners to get in touch with him in advance to share their favourite gems from over the years. So if there's something you use in your teaching that you found out about through a gems post then do let Craig know! Check out my gems index to see the whole collection!

I'm not normally one for motivational posters, but here's a quote that I'd have up by my desk if I had a desk. Often attributed to CS Lewis, this is a great message for a Year 11 class getting mock papers back. Thanks to Jen McMillan at Harris Greenwich for this!

10 November 2018

5 Maths Gems #98

Welcome to my 98th gems post. This is where I share some of the latest news, ideas and resources for maths teachers.

1. Area Activity
Thanks to @MrNiksMathClass for sharing a nice activity in which students apply their knowledge of area formulae.

2. Triangle Puzzles
John Rowe (@MrJohnRowe) shared a new set of triangle puzzles which are inspired by area mazes. They can all be solved without using fractions or decimals.
3. Standard Form
Thank you to Gillian Mathewson (@gmathewson1) for sharing a standard form variation grid. I like the way students convert to expanded form before converting to standard form - it's a good check that they really understand what's going on. I've added this to my number resources library.
4. Multiplying Fractions
Mary-Kate Connolly (@MKConnolly1991) shared a picture of her board which shows how she explains fraction multiplication. Note the use of colour to support the explanation.
Here's the same approach explained in 'Polish Up Your Mathematics’ by Fawdry which was published in 1945.
5. Mock Season
If you have Year 11 mocks coming up (I'm marking them at the moment!) then don't forget that I published a mock revision blog post last year. My GCSE 9- 1 revision post might also be helpful - it contains a long list of excellent GCSE revision resources.

New to my revision page is a unit conversions quiz - this goes alongside the formula quiz I made a couple of years ago.

If you want to print personalised booklets for your Year 11s based on their individual performance in mock exams then you'll be pleased to hear that PinPoint Learning has a free trial until 29th January 2019. The June 2018 QLAs are available so if you used these papers for your mocks then PinPoint is all ready to go.
In case you missed it, my latest post was on map scale. I wrote about approaches and resources for this undertaught and underesourced GCSE topic.

I also wrote 'Turing for the Fifty' which tells the story of the time I worked in banknotes at the Bank of England, and why I am particularly excited that there's an opportunity to nominate a mathematician for the new fifty pound note.

I've set up a new Instagram account to share maths teaching ideas and inspiration so do follow me if you're an Instagram user. I will still be sharing updates on Twitter and Facebook as usual. And if you want to get my blog posts by email then you can subscribe here.

I'm very excited to be approaching a milestone - I hope to publish my 100th gems post in December, and will celebrate by recording a special maths gems podcast with Craig Barton.

My resource libraries have been very busy lately - I'm glad teachers find them useful when planning lessons. My Topics in Depth page has also had a lot of visits as a large number of primary teachers have discovered that I have a really useful set of primary packs there. I have been updating my conferences page with new events and my pret homework site with new contributions.

If you're based in London, don't forget to join to one of my Maths Hub workgroups. We already have a large number of A level teachers signed up up for the A level workgroup which is really exciting. In the GCSE workgroup we will be working on ratio and unit conversions. I have done lots of research into these topics lately so I am excited to share what I've found.

Next weekend I'm going to the Maths Jam annual gathering with friends - I can't wait! What a great way to spend a weekend.

I'll leave you with this fun fractions activity that I found in ‘Fundamental Arithmetic’ by P B Ballard which was published late 1920s.

30 October 2018

Map Scale

This question on map scale in AQA's 2018 Foundation tier GCSE paper was answered really badly:
Most students had no idea. In fact, they would have done better if they'd just guessed. The most common answer given was 1:1000.

The question requires students to understand two things:
  1. how to convert from cm to km. 
  2. how to interpret a map scale given as a ratio with no units (eg knowing that 1:100000 means that 1cm on the map represents 100,000cm in real life). 
I think that metric conversions are relatively easy to memorise (most students know that there are a hundred centimetres in a metre from their familiarity with a metre ruler and the prefix cent, and they should be aware that kilo means thousand). So this leads me to believe that the main difficulty with this question was a lack of familiarity with map scales given as a ratio. 

I have a feeling that this topic is sometimes skipped over by maths teachers. Some students may only see one maths lesson on map scale in their entire time at secondary school. It gets buried in amongst other topics on schemes of work (it's normally in with either ratio or with bearings and scale drawings, though I have seen it in with similarity too). I think it sometimes goes unnoticed and doesn't get the time it deserves. Perhaps this is because it rarely comes up in GCSE exams. 

You may be wondering if the ability to use a map scale of the form 1:50000 comes up in geography. In the geography GCSE syllabus it says that students must "use and interpret OS maps at a range of scales, including 1:50 000 and 1:25 000 and other maps appropriate to the topic". It doesn't specifically say that students have to measure and convert distances, though this is implied. I found this on a geography revision website:
"The scale number on an OS map indicates how many centimetres on the ground are represented by a centimetre on the map. On a 1:100,000 scale map, one centimetre on the map represents 100,000 cm on the ground, in other words, one centimetre on the map represents one kilometre in reality. A scale of 1:5,000 therefore means that a centimetre on the map represents a distance in real life of 5,000 centimetres (50 metres). This method of representing the scale of a map is called the fractional method, but you will also see graphical representations or written representations like 2 cm = 1 km."
I wasn't aware that the method of representing map scale in the form 1:50000 is called the fractional method. I like knowing proper names for things. Though I'm not sure whether this term is used consistently - I've seen other sites refer to it as a ratio scale or a fractional scale or a representative fraction.

Wikipedia lists different types of map scale including lexical (ie expressed in words - also known as verbal or stated scales), linear or graphical scales (represented as a bar), ratio scales, and fractional scales. It points out that a lexical scale in a language known to the user may be easier to visualise than a ratio, but lexical scales may cause problems if expressed in a language that the user does not understand or in obsolete or ill-defined units (eg one inch to a furlong or one pouce to one league). So ratio scales have pros and cons. When I read this I straight away thought that there could be some really nice opportunities for enrichment in this topic - I'd love to talk to my students about antiquated units of measurement!

Anyway, it looks like 'map skills' in GCSE geography focuses mainly on recognising symbols and using grid references. I haven't found many geography resources on ratio scale, other than a couple of PowerPoints on TES that run through it very quickly. So it seems that measuring lengths on maps and performing unit conversions using ratio scales isn't something they spend much time on in the GCSE geography course.
We definitely do need to spend some time on it in maths lessons, and it fits well at both Key Stage 3 and 4. The maths GCSE specification says (in both the ratio and geometry sections) that students should know how to use scale diagrams and maps. AQA's Teaching Guidance helpfully provides additional clarification: "Scale could be given as a ratio (for example 1:500 000) or as a key (for example 1cm represents 5 km)."  

Approach and Resources
Without sound knowledge of both place value and metric unit conversions we can't even get started on this topic. So that's the first thing we need to check. I remember once giving my top set Year 10 a simple starter asking them to put these lengths in ascending order:
It took way longer than expected.

If metric conversions need teaching, there are loads of good resources for this, including:
Students who struggle with simple unit conversions might find it helpful to draw out a ratio table each time, writing their known fact at the top and using multiplicative reasoning to fill in the gap:
Once students are fluent in unit conversions, it would be sensible to remind them of how ratios work before moving onto map scale. There are lots of great resources available for this wide ranging topic (see my post on ratio), but the focus here is simplifying ratios with mixed units (ie converting the antecedent and consequent to the same units), and on expressing ratios in the format 1:n. Useful resources include:

Now we just need to combine these ideas to understand map scale.

If a map has scale 1:50000, how do we work out what 6cm on the map represents in km on the ground? The two steps involved (the unit conversion and the measurement conversion) can be done in either order. I'd suggest something like this:
Here I've start by writing the ratio scale with units - any units work but cm is usually preferable. Instead of writing 1cm = 50,000cm (which is a horrible use of the equals sign!) I've used a table.

I did my unit conversion in two steps, going via metres as the base unit.

Using a similar approach to answer the question: "If 4cm on a map represents 100km on the ground, what's the ratio scale?", we have the following process:
So the answer is 1:2500000.

Of course there are lots of different ways to set the workings out here - the table is optional.

Most resources I've seen online for ratio scales skip through it very quickly - it's often covered in a couple of slides at the end of a related lesson. Teaching it properly - in depth - should probably take two or three dedicated lessons. The CIMT material 'KS3 Scale Drawing' is very useful, as are the Boss Maths lessons 'Using scale diagrams and maps' and 'Scale drawings'. Corbett Maths has some exam style questions on this topic.

I am eagerly awaiting something on variationtheory.com on all this!

Once students are fluent in metric unit conversions and working with ratio scales, they might enjoy a bit of map work to consolidate their learning. MathsPad have a free online map scale tool which is helpful for demonstrating map skills on the board. The Mapzone website shows what different OS map scales look like - this is not on the maths curriculum but might be of interest to students. This Reading Map Scales Activity from emtay on TES gives students the chance to practise using a ratio scale on a map of Europe.

No doubt someone will tell me that when teaching this topic I *must* give my students full size OS maps and send them outside on some big orienteering project! Hmm. I'm not sure that's practical on a main road in Croydon... I also believe that although they may remember the activity, it probably won't help them either understand or remember the maths. So I'll probably skip that.
Since we've all now got sat nav on our phones this topic isn't as much of a 'life skill' as it once was. That's ok though. Thankfully we don't teach mathematics for its utility.

What's good about this topic is that as well as sewing together two key areas of school maths (ratio and metric units), we get the chance to come back to it when we teach bearings, and again when we do area and volume scale factor with questions like this:
A map has a scale of 1:50 000. A park is shown on the map as a rectangle measuring 6cm by 4.2cm . What is the actual area of the park?
The Boss Maths lesson 'Converting between metric units of measures of area and volume' covers this.

I'm involved in the London Thames Maths Hub workgroup on Challenging GCSE Topics in which I hope to look at unit conversions and ratio. Do get in touch if you want to get involved in developing some resources for this topic.

In the meantime - let's all make sure that map scale gets the time it deserves in maths!

28 October 2018

#LateMaths - thank you!

This is just a quick post to say thank you to everyone involved in my LateMaths event last night. I held a big mathematical party at The Fable in Central London to celebrate the launch of Ed Southall and Vincent Pantaloni's second book 'More Geometry Snacks'. Over 100 mathematicians came out in the freezing cold weather to join me - I'm very grateful to everyone who did so. Over the last four years I have increasingly felt part of a growing community of maths enthusiasts who have come together through social media - it's a wonderfully welcoming community and I am proud to be part of it.

I'd like to thank Ed Southall and Vincent Pantaloni for letting me hijack their book as an excuse to throw a party! It was lovely that their families got to see them signing so many books. Vincent's family had come all the way from France and Ed's family had come down from Yorkshire. Even though they have written two books together, it was the first time Ed and Vincent had met in person. Thank you to both of them for their excellent talks. There's nothing I like more than learning about triangles and Archimedes whilst drinking vodka on a Saturday night! Thank you also to Andrew from Tarquin for all his hard work in getting the books printed in time for the launch.
Waiting for my guests
With Vincent. The venue all ready to go.
With the authors Vincent and Ed
The authors with their new book, hot off the press

Our after dinner speaker was the wonderful Ben Sparks. His talk was absolutely awesome and everyone thoroughly enjoyed it, particularly the part with the human microphone stand!
Ben Sparks (plus Justice, with the microphones)

Thank you to my brilliant helpers Martin Noon, Natalie Palmer and Lizzie Stokes. Special mention to my ex-colleague and good friend Lizzie who ran the registration desk and sold earrings all night, and has helped me out at every event I have ever run. What a star.
Lizzie and Natalie
Me and Lizzie, plus Matt and Ben photobombing

We sold all 40 pairs of maths earrings - huge thanks to maths teacher Cara (who makes these herself) for letting me sell her designs.

Thank you to the people at The Fable. The venue was lovely - a book themed bar with mathematical light shades - and the bar staff were excellent.

Thank you to TD Dang and Matthew Scroggs from Chalkdust who brought along copies of their brilliant maths magazine and sold Chalkdust T shirts.

Thank you to Matt Parker, Katie Steckles and Zoe Griffiths from Think Maths for bringing along all sorts of mathematical goodies including awesome stuff from Maths Gear. Matt gave me a Dodecaplex Puzzle - it's driving me crazy. Somehow he managed to do it in seconds - I've been trying for hours and am totally clueless!
Matt Parker, me and Ben Sparks
Dodecaplex Puzzle from Maths Gear

All of this maths has really got me in the mood for the MathsJam Annual Gathering next month. It was lovely to see the organiser Colin Wright last night, even though he did confuse me with a very simple but brilliant magic trick.

Thank you to those of you who complimented my playlist! It was mainly stuff I liked when I was a grungy teenager in the 1990s - an acquired taste! I had fun making it.

I hope guests enjoyed the quiz - if you weren't there and want to have a go then you can download it here, along with all the quizzes and answers from my previous events. Seven teams submitted answers - all entries were very impressive. I'm pleased to announce that the winners were Martin Holtham (@GHSMaths), Christopher James (@TeacherBowTie) and Cindy Wells (@cindy44uk) with a score of 44 out of 49.

It takes a surprisingly huge amount of work to organise an event so I'm pleased it all came together well and people enjoyed themselves.

Do check out the tweets on the hashtag #latemaths to see more photos of the night. Thank you once again to all my guests for coming to support me at my fourth annual maths party. Let's do it again one day.