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.

23 October 2018

5 Maths Gems #97

Welcome to my 97th gems post. This is where I share some of the latest news, ideas and resources for maths teachers.

1. Loci Match Up

Miss Konstantine (@GiftedBA) has shared a new loci activity. The square has side length 18cm, E is the midpoint of the square and the radius of the circle is 5cm. Students need to work out what colours match each statement, and find the area of each of the coloured sections. Alternatively you can give them the diagram and ask them and write their own statements.
I do love a loci activity that doesn't involve pairs of compasses...

It's worth following @GiftedBA on Twitter because she often shares ideas, resources and examples of student work. Check out her most recent post 'Squares/Roots using Area', and her post 'Circumference and Area of a Circle' in which she suggests an activity where students identify errors from a list. 
Inspired by this, Sarah Evans (@maths_missevans) has been trying out 'Find and Fix with Feedback' tasks. Having a list of feedback statements to choose from helps students to identify and understand mistakes and misconceptions.
2. Calculator Posters
Casio (@CasioMaths) has shared a set of high resolution posters of the Classwiz calculator for A level classrooms.
3. Pythagoras
James and Nicola from MathsPad have shared another lovely set of resources. I love their new worksheet and interactive tool for 3D Pythagoras.
MathsPad's October update was packed full of new resources for trigonometry and Pythagoras. One example is the set of Pythagoras puzzles where students are given one side and a set of possible answers for the other two. As James says, this might nudge thinking towards the theorem's converse.

Subscribe to MathsPad to access the full collection.

4. Equipment Check
A level teacher Stuart Price @sxpmaths shared some things he's trying this year when marking assessments. His tests now include an equipment check for four marks - this a fun way of getting students to bring the right equipment to lessons!
Stuart uses a 'mistake' stamp when he is sure his A level students can self-correct and a 'misconception' stamp when something is fundamentally wrong.
5. Algebraic Area
Thanks to Catriona Shearer (@Cshearer41) for sharing a great set of resources on TES that she created as part of the Mathematical Reasoning at GCSE project run by Cambridge Maths Hub. I've recently discovered that lots of excellent resources are made by Maths Hub workgroups that don't end up being widely shared and I'm determined to change that!

Catriona's Algebraic Area Reasoning Tasks are based on a WJEC GCSE question about forming and solving a quadratic equation for the area of two rectangles. The tasks vary in difficulty.
For a similar set of area tasks with a more numerical approach see Catriona's Compound Area Reasoning Tasks. Also check out her set of Probability Reasoning Tasks (based on the infamous Hannah's Sweets question).

It's great that Catriona is involved in developing resources for Maths Hubs given the quality of her puzzles, which I wrote about in Gems 94. You can now download three pages of Catriona's beautiful puzzles here.
I was excited to have an article about indices published in Teach Secondary magazine. It includes ten ingredients to teach indices in depth, but those ten ingredients could apply to any topic so do have a read. The full article is online here. You can also download a free resource containing four activities which you might find useful when teaching index notation.
Also look out for my article in next month's Teach Secondary magazine about algebraic order of operations.

On Friday I met up with Megan Guinan after school and we went along to the launch of Chalkdust Magazine Issue 8 at my old university UCL. We had a great time and really enjoyed the quiz. Chalkdust magazine is brilliant - you can read Issue 8 online, and if you're coming to my LateMaths event on Saturday then you'll be able to pick up a copy.
All 100 tickets have been sold for LateMaths and I've been busy making all the final arrangements over the last few days. I can't wait!

If you're a teacher in London then do join one of my Maths Hub workgroups! I'm supporting the workgroup led by Chris Reilly on the Challenging Topics in the New GCSE. In this workgroup we'll look closely at the teaching of ratio and hopefully develop and trial some new resources for this topic. I'm also supporting Carlos Karingal on the London-wide Developing A Level Pedagogy workgroup which will meet at Chestnut Grove in Balham. I'm so excited about this workgroup - I think it will have a big impact on my A level teaching. Get in touch if you want more information. Everyone is welcome to get involved.

I'll leave you with the New York Regents archive website which was shared by Benjamin Dickman (@benjamindickman). It's beautifully organised by topic and year, containing exam questions going back to 1866. I love stuff like this!

14 October 2018

5 Maths Gems #96

Welcome to my 96th gems post. This is where I share some of the latest news, ideas and resources for maths teachers.

1. Starting Points
Chris McGrane (@ChrisMcGrane84) has started a new website startingpointsmaths.blogspot.com where he shares a collection of tasks and starting points to help teachers plan for richer and more effective learning experiences. The collection includes rich tasks and intelligent practice for topics from Key Stage 3 to Key Stage 5. There are loads of lovely activities to explore so do visit the website and have a look around. Tasks include suggested teaching points and questions for discussion with learners.
2. Mysteries
Thanks to Richard Perring (@LearningMaths) for sharing a couple of great maths mystery activities - one for completing the square and one for the nth term of a linear sequence. The idea is that learners collate the information from the cards in order to solve the overall problem.
If you like these then buy Richard's ebook Talking Maths, published by the ATM, which contains lots of similar activities.

These tasks remind me of the Durham Maths Mysteries which many teachers have recommended to me over the years.

3. Maths Club Activities
Thanks to Emily Fleming who helps to run a charity called SAMI that supports maths clubs in Africa for emailing me a free maths club pack that teachers and/or students can use to run a maths club. This is a lovely resource full of great activities. Also check out Emily's own maths club website.
4. Formulae
I was intrigued by this 'Spot the Mistake' activity from Stephen Bodman‏ (@stephenbodman). I have no idea how best to get students to memorise formulae, but I think there might be some value in short tasks like this. Here, there is something wrong in each formula and students have to spot what it is.
At GCSE the hardest formulae to learn off by heart are probably the quadratic formula and the Cosine Rule. I ask my students to learn these formulae at home and test themselves until they get it right every time - it sounds so straightforward! Lately my six year old daughter has been trying to learn tricky spellings for her weekly spelling test (words like beautiful) and I'm seeing how much she struggles. I've started to appreciate that memorising some formulae and spellings might be more challenging than it sounds. 

5.  Big Mistake
Speaking of errors, thanks to Sandra at MathsBox (@mathsbox1) for sharing a free Foundation GCSE revision resource. In this activity students spot the mistakes in a collection of algebra questions. I have used the fantastic Higher version of this for years in the run up to GCSE exams. I've added both resources to my 9 - 1 GCSE Revision page.

Lots of great stuff has been shared on Twitter lately so I'll have another gems post out soon. Isn't it lovely that half term is just around the corner? Over half term I hope to finish off a number of posts that I've been working on for a while.

It was shame I couldn't sit down for my usual post-conference chat with Craig yesterday! Alas, he was on holiday. Hopefully we will be back with more mathsconf podcasts next year.

Rather than share a detailed write up of the day, I will just finish my gems post with a few thank yous. Thank you to La Salle Education for again running a huge Saturday event at a very low price for maths teachers, allowing us to attend high quality CPD that we otherwise would not have access to. Thank you to Rob Smith for doing a million things at once including the raffle, tuck shop, MA stand and ATM stand. Thank you to everyone who presented sessions, and everyone who said hello and shared their thoughts with me, and everyone who said encouraging things, and everyone who complimented me on my maths dress! I'm sorry I had to rush around everywhere all day - I had a lot to do! I so love chatting to maths teachers. Special shout out to Jo Locke and David Faram.

I didn't do a topics in depth session this time because I wanted to do something related to old textbooks. I focused on quadratics. I think there's a real lack of this kind of subject knowledge CPD for teachers, so I wanted that to be my focus this time rather than something pedagogical. I think it was well received. My slides are here (they make more sense with my commentary!) - download them to see the references in the notes.

I now have 90 guests coming to #latemaths. I'm excited to announce that I will now be selling mathematical earrings there too (great stocking fillers!), on top of all the other entertainment. There are 10 tickets left and you have one week before sales close! See latemaths.weebly.com for details. 

I'll leave you with this lovely puzzle, shared by Rachael Horsman (@Rach_Read)