Factorising by Inspection

In 2018 I decided to write more short posts about approaches or methods that teachers might not have seen before. When I share these posts, I am well aware that there will be many people who already know the thing I'm blogging about, but I figured that it's still worth sharing even if it's only new to a handful of people.

My first post in this series was about using vectors for enlargements. Today's post is about a way to save time when factorising a non-monic quadratic expression (ie a quadratic where the coefficient of x² is greater than 1) by inspection.

Back in September I had a post published a blog post for La Salle about methods for factorising non-monics. I showed that by looking at old textbooks we can see how methods have changed over time.

I explained that I prefer to factorise by inspection, but most teachers these days factorise by grouping (ie 'splitting the middle term'). In response to that post I had lots of teachers either (a) try to convince me that grouping is better than inspection (b) try to convince me that another method is better than inspection. Often teachers argue that their method is great because their students really liked it in the lesson, but I'm more interested in the extent to which methods 'stick' in the long term. When I used to teach factorising by grouping my students were always happy with it at the time, but a few months later they would struggle to remember the full procedure.

Factorising by inspection is intuitive and logical, so there's no procedure to memorise. I appreciate that most teachers still prefer the grouping method, and I do not intend to try to convince anyone to change their mind. But for those of you who like factorising by inspection, here's a tip from Susan Russo (@Dsrussosusan):

Let's say you have to factorise 6x² + 17x + 12.

Factorising by inspection is super-quick once you get the hang of it, but here both 6 and 12 have multiple factors so this one might take a bit longer than others.

If you list all the possibilities and check each one, there are twelve cases to check.
(6x + 1)(x + 12)
(6x + 12)(x + 1)
(6x + 2)(x + 6)
(6x + 6)(x + 2)
(6x + 3)(x + 4)
(6x + 4)(x + 3)
(2x + 1)(3x + 12)
(2x + 12)(3x + 1)
(2x + 2)(3x + 6)
(2x + 6)(3x + 2)
(2x + 3)(3x + 4)
(2x + 4)(3x + 3)

Yawn!

An expert would probably work out the correct combination fairly quickly without writing down all the options. For a novice it's a pain that there are twelve options to think about here. At first it seems like it might take a while to select the correct combination.

Susan pointed out something which should be totally obvious but hadn't occurred to me before. We can teach our students to refine their guesses in order to make this method more efficient. Here's the key: each bracketed expression shouldn't contain any common factors. For example if you have a 2x then you can't put an even number in with it.

Let's look at that list again and immediately disregard any option where there's a common factor in one or both brackets.
(6x + 1)(x + 12)
(6x + 12)(x + 1)
(6x + 2)(x + 6)
(6x + 6)(x + 2)
(6x + 3)(x + 4)
(6x + 4)(x + 3)
(2x + 1)(3x + 12)
(2x + 12)(3x + 1)
(2x + 2)(3x + 6)
(2x + 6)(3x + 2)
(2x + 3)(3x + 4)
(2x + 4)(3x + 3)

It turns out there are actually only two cases to check by inspection. Students fluent in expanding brackets should be able to do it in seconds. You can immediately see that the first option will give a large coefficient of x, so we check (2x + 3)(3x + 4) and find that it works.

For some reason I've never shared this time-saving tip with my students. I'm very grateful to Susan Russo for bringing it to my attention. Let's try it again with one more example: factorise 12x² + 11x - 15. Here's the massive list of 24 options to consider:

(12x + 15)(x - 1)
(12x + 1)(x - 15)
(12x + 5)(x - 3)
(12x + 3)(x - 5)
(6x + 15)(2x - 1)
(6x + 1)(2x - 15)
(6x + 5)(2x - 3)
(6x + 3)(2x - 5)
(3x + 15)(4x - 1)
(3x + 1)(4x - 15)
(3x + 5)(4x - 3)
(3x + 3)(4x - 5)
(12x - 15)(x + 1)
(12x - 1)(x + 15)
(12x - 5)(x + 3)
(12x - 3)(x + 5)
(6x - 15)(2x + 1)
(6x - 1)(2x + 15)
(6x - 5)(2x + 3)
(6x - 3)(2x + 5)
(3x - 15)(4x + 1)
(3x - 1)(4x + 15)
(3x - 5)(4x + 3)
(3x - 3)(4x + 5)

Removing those with a common factor in one or both brackets gives us this:
(12x + 15)(x - 1)
(12x + 1)(x - 15)
(12x + 5)(x - 3)
(12x + 3)(x - 5)
(6x + 15)(2x - 1)
(6x + 1)(2x - 15)
(6x + 5)(2x - 3)
(6x + 3)(2x - 5)
(3x + 15)(4x - 1)
(3x + 1)(4x - 15)
(3x + 5)(4x - 3)
(3x + 3)(4x - 5)
(12x - 15)(x - 1)
(12x - 1)(x + 15)
(12x - 5)(x + 3)
(12x - 3)(x + 5)
(6x - 15)(2x + 1)
(6x - 1)(2x + 15)
(6x - 5)(2x + 3)
(6x - 3)(2x + 5)
(3x - 15)(4x + 1)
(3x - 1)(4x + 15)
(3x - 5)(4x + 3)
(3x - 3)(4x + 5)

So this time we eliminated half the possibilities. It's not as time-saving as in the first example, but still helpful. My next step would be to try the less extreme numbers (ie not those involving a 12x or a 15) so that gives me only four options to test initially. For an experienced factoriser it's fairly quick to see that (3x + 5)(4x - 3) works.

Of course, in reality we never really list out all the options and then decide what to eliminate. What most people actually do when faced with 12x² + 11x - 15 is write down (3x      )(4x      ) and then (often in their head rather than on paper) try some numbers that multiply to give -15. So it's helpful to remember that there's no point putting and 3 or a 15 in the first bracket.

This isn't a big game-changer and it doesn't help with every quadratic, but I like things that save us time. When solving a long, complicated problem at A level, it's good to able to factorise quadratics efficiently.

If you hadn't realised that you can quickly eliminate options in this way then I hope this was helpful.

If you want to have a play with this, there are lots of non-monic expressions to factorise here.

2018 Highlights

Last weekend I recorded a podcast with Craig Barton to celebrate my 100th gems post. I love the way @boss_maths described it: "So informative and entertaining, listening to this felt uncannily like listening to TMS as a cricket fan"! You can listen here and see what you think. The conversations we had in the podcast made me realise how much happens in a year. In this post I look back at 2018 and list some of my personal highlights, and look ahead to what's coming up in 2019.

The start of 2018 saw the publication of Craig's book How I Wish I'd Taught Maths. I can't believe that was only a year ago! Having read the book in advance I knew it was going to be a huge success, but I couldn't have predicted how much impact it would have on the maths education community. It's not just the teachers of Twitter who have told me how much their teaching has changed since reading Craig's book. I go into lots of different types of schools and see both new and experienced teachers trying example-problem pairs, silent teacher and other techniques that I probably wouldn't have seen at all in 2017. These teachers are keen to tell me how much Craig's book has made them more reflective, and how entire departments are now taking more interest in research and seeking to improve their practice.

Thankfully, opportunities for maths teachers to take control of their own professional development have continued to grow. We still live in times where most schools are unable to support requests for term-time subject specific CPD, but increasingly we are able to attend CPD at the weekends if we choose to. Of course weekend CPD is not suitable for everyone, so thankfully it remains entirely optional. I adore being part of the community of hundreds of maths teachers who participate in these events and am very grateful to the organisers for regularly bringing us all together.

In 2018 I attended three of La Salle's national maths conferences and loved every minute. If you're a maths teacher and you want to try some Saturday CPD for the first time, do check out the 2019 events (visit mathsconf.com for details). The next one takes place in Bristol on Saturday 9th March.

At the Manchester conference I presented on indices, and at the Birmingham conference I presented on quadratics.

If you missed these presentations then you  might be able to catch me presenting them again next year. I'm presenting Indices in Depth at the ATM/MA London Branch event on 19th January and How to Solve an Adfected Quadratic at the #HabsGirlsConf on 23rd March.

I will also be presenting new material at a number of events in 2019, including Educating Northants on 30th March, the ATM/MA conference at Easter, and both ResearchED Rugby and the MEI conference in June. There are loads more maths education events happening in 2019 - check out my conference listings here.

In addition to the La Salle conferences, I was fortunate to attend a number of excellent events in 2018. It was an honour to deliver the keynote presentation at the BBO Maths Hub secondary conference, and I thoroughly enjoyed the second JustMaths conference which took place at Alton Towers in the middle of the summer heat wave.

David Faram, me and Craig Barton on the rapids

I also attended my first ever residential maths education conference at Easter. BCME (the British Congress of Mathematics Education) only takes place every four years so it was quite a big deal. I loved it - not only for the CPD, but also for the social programme (did I mention that we won the quiz?!). I also very much enjoyed recording a daily conference podcast with Craig in which we reflected on the sessions we'd attended that day. I can't wait for the upcoming ATM/MA conference which takes place in the Easter holidays - there's a ceilidh, a quiz and a disco! And the daily podcast will be back. 

2018 was the year that I became a bit braver. I often turn down exciting opportunities, normally because of family commitments but sometimes because I think I won't do a good job. After turning down the opportunity to appear in an advertising campaign for Mazda at Easter, I kicked myself for weeks and resolved that I should stop saying no to things. So when I was invited to take part in the Big Internet Math Off, I said yes. And later in the year I also said yes to presenting at the MathsJam Annual Gathering, and to contributing to Craig's 'Slice of Advice' podcasts. In all three cases women were underrepresented, not through the fault of the organisers but because more women did not take the opportunity to participate. This made realise how important it is to keep saying yes, even to things I find a bit scary. I can't complain about women being underrepresented if I say no to everything!

Another first for me in 2018 was having two articles published in print for the first time. My article on indices published in Teach Secondary magazine in October, and a month later my article on the order of operations was published. I'm still chuffed about these articles.

My blog posts have been increasingly influenced by the research I have been doing through old maths textbooks. Blog posts I am particularly proud of from the last year include my posts on ratio, bounds, surds, algebraic LCM and HCF, and Year 7 maths activities. Check out my blog archive to see the whole collection.

Every year I host a social event for maths teachers. This year it was LateMaths - a big party in London to celebrate the launch of More Geometry Snacks by Ed Southall and Vincent Pantaloni. It was a great night and I am very grateful to everyone who helped me make it a success.

I have lots of ideas for events I'd like to run in future. In July I'm hoping that Rob Smith and I will run an event which (fingers crossed!) will involve a trip to the amazing MA archive at the University of Leicester. I'd also love to run a two day residential maths education conference at the same venue as the MathsJam Annual Gathering. It's a shame I don't have the time and money to run events more often because I really enjoy it.

What will 2019 bring? If you haven't already read Craig's book, perhaps give it a go. If you've never been to a La Salle maths conference, maybe 2019 is the time to try one. If you've not done one of the big Easter conferences, this might be the year to come along. If you've never listened to a podcast, have a go and see if you like it. If you're not on Twitter, consider joining. We're a very welcoming maths community, and we're stronger together.

We live in exciting but incredibly challenging times for maths teachers. I can't wait to see what 2019 brings.

I'll leave you with some of my favourite photos from 2018.

With my old PGCE buddy Colin Hegarty
 at GLF's first annual maths conference

With ex-colleagues Lizzie, Amelia and Sarah at prom 

Recording a conference podcast with Craig

At MathsJam with friends Mariana, Ed, Tim and Joe

Summer drinks at Dr Frost's house. 

With Megan Guinan at a Chalkdust Magazine Launch Party

Celebrating three new books and one new job with my Twitter besties
Craig, Ed and Tom at #mathsconf14 in Kettering

On the MA bookstand with Rob Smith
 at #mathsconf15 in Manchester

At BCME with Ed, Craig and Hannah Fry

At LateMaths with Matt Parker and Ben Sparks

Have a wonderful Christmas and New Year everyone!

5 Maths Gems #100

Welcome to my 100^th 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.

The challenge: Write your name so that it covers an area of exactly 100 cm squared.

Love the different strategies students used here. E.g. Leia, who chose to make each letter 25 cm^2. Notice the revisions on the A.#mathchat #MTBoS Cc @PaiMath pic.twitter.com/FowEQoL4I6

— Ilona Vashchyshyn (@vaslona) 16 November 2018

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...

Two years ago, I was saying “do you have any questions?”. Last year I switched to “what questions do you have?” It made a difference. Today I tried “ask me two questions”. And they did! And those ?s led to more ?s. It amazes me that the littlest things have such a big impact!

— Andre Sasser (@MrsSasser) 27 August 2018

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:

For tests, I dedicate the first 5 minutes to have students hold the tests and talk to their group about strategies (w/out pencils). This lowers their test anxiety AND when you walk around, you hear some of the richest math conversations. #iteachmath

— Howie Hua (@howie_hua) 28 September 2018

Update
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.

5 Maths Gems #99

Welcome to my 99^th 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:

enabled ps to reflect on their errors, ask me anything and then requiz themselves. Scores typically went from 5 or 7 to 10/10 with much whooping and high fiving. We were able to cover about 8 topics in a short space of time, Requiz next week to look for retention then go deeper.

— bernie westacott (@berniewestacott) 23 November 2018

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

Update
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!

5 Maths Gems #98

Welcome to my 98^th 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.

Too often measurement problems become just an exercise in evaluating formulas. Make it more interesting by taking away all the numbers. In this example, students must find values (no repeats) for the dimensions so that all the shapes have equal areas. #tvdsbmath #iteachmath pic.twitter.com/7DzsN8912M

— Math with P. Nik (@MrNiksMathClass) 28 October 2018

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.

Updates
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.

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. 

Curriculum
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:

• Why the metric system matters - excellent video by Matt Anticole
• The Chalkface - Metric Units of Measurement and Units of Measurement
• CIMT Unit - KS3 Units of Measurement
• MathsPad - loads of great resources if you subscribe including a converting metric lengths activity and a set of metric units of length exercises.
• BossMaths lesson - Converting between metric units of measures of length and mass

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:

• Maths4Everyone - Simplifying Ratio
• Richard Tock - Pixel Picture: Equivalent Ratios
• Dave Taylor - Write Ratios in the Form 1:n

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!