5 Maths Gems #121

Welcome to my 121^st gems post. This is where I share some of the latest news, ideas and resources for maths teachers.

1. Linear Graphs

It's always good news when Paul Rowlandson publishes a new blog post. His latest post looks at various ways of presenting information that leads to the equation of a straight line. If you're teaching this topic I really recommend reading this and making use of the tasks and ideas featured.

Related to this, check out this awesome task shared by Tim Honeywill (@HoneywillTim).

2. Knowledge Organisers

It looks like some maths departments are being asked to produce knowledge organisers as part of a whole school policy. It's not clear whether they are as useful in maths as they are in other subjects, but if you're in a school where you've been asked to produce them for every maths topic then you're faced with a mammoth task. To avoid duplication of effort across schools, you might find this set of knowledge organisers a useful starting point. Also, check out the set of Year 8 knowledge organisers shared by Delta Trust, and these lovely knowledge organisers shared by Nicola Whiston (@whisto_maths). Nicola will continue to add her knowledge organisers to this folder as she makes them. They are designed around the White Rose scheme of work.

These resources can be used in numerous ways - even if your school doesn't have a policy of self-testing using knowledge organisers then you still might find this resource helpful for revision.

Nicola Whiston (@whisto_maths) has also made a learning journey for the White Rose Scheme of Work - schools following this scheme of work will find this helpful, and even if you're not following this scheme of work then it is interesting to see the way the maths curriculum has been mapped out.

3. A Level Maths Activities 
The Mathematical Association has just published a book, written by David Miles, which contains photocopiable activities for A level lessons. It's selling very well - you can order now from the MA shop for £6.30 (members) or £9.00 (non-members). I will also be selling these books from the MA stand at #mathsconf22 in Manchester.

Although it's already available to buy, this book will be officially launched at the MA Conference in April, along with a couple more exciting new publications - Geometry Juniors by Ed Southall and Hooked on Mathematics by Jenni Black. Do come along to the book launch if you're at the conference.

4. Etymology
Caroline Spalding (@MrsSpalding) shared a picture of a poster she saw in a science lab at her school. It shows etymological roots of science vocabulary.

This prompted Ben Gordon (@mathsmrgordon) to make a maths version of this which he has shared here. Thank you Ben!

Following this, Sudeep (@boss_maths) shared an amazing resource to help relate maths terms with key words from other subjects or words in everyday use.

This large (and growing) collection is outstanding. I love Boss Maths resources - they are always high quality.

I love featuring etymology in my lessons. It's great to see resources like this to help teachers develop their subject knowledge.

5. New Resources
There have been lots of new resources shared on Twitter recently. Here are some examples.

Don Steward has published lots of new tasks, including fraction shading and order of operations with expressions.

Do scroll through his blog to see his new resources - there are many!

Andy Lutwyche (@andylutwyche) regularly shares new resources. Check out his newest Transforming Shapes Codebreaker which is bound to be popular with students (and not just because of the terrible joke!).

As always, all of these resources will be listed in my resource libraries for easy access.

Chris McGrane (@ChrisMcGrane84) shared an interesting task on negatives numbers that really got me thinking.

Emma McCrea (@MccreaEmma) tweeted about a couple of openmiddle.com tasks. I have blogged about this website before but haven't had time to explore all the tasks. I particularly like this one on index laws.

Mr Russell (@mathsDRL) reminded me about Jon Orr's (@MrOrr_geek) Polygon Pile Up activity. I know I saw it a couple of years ago and I was sure that I'd featured it in a Gems post, but I can't find it so maybe I didn't. Here it is - read the accompanying blog post for the resource, discussion, and a more difficult version.

Update
Here are my recent blog posts in case you missed them:

• Rule of Three
• Know Your Workings

On Monday I visited Sheffield to record another two Topics in Depth video CPD podcasts with Craig Barton. You can check out our indices and angles in parallel lines videos if you missed them. Hopefully the next two will be published by TES soon.

My book is still getting good feedback - I am always delighted to receive tweets from teachers who have learnt something new from it.

I was also chuffed to see my book shortlisted for the Chalkdust Book of the Year 2019. You can read the Chalkdust review of my book here and vote for it here (though to be fair, all these books are awesome and deserve more votes than mine!).

I'm looking forward to two big events that are coming up soon. On Pi Day I will be in Manchester for #mathsconf22, presenting a new talk from my Topics in Depth project: Surds in Depth.

In the Easter holidays I will be presenting at the MA Conference. I can't wait for this: two nights in a spa hotel with loads of amazing workshops and plenty of entertainment.

At the MA we're really pleased that our new conference format has been so well received. Over one hundred delegates have already booked their place. There aren't many full residential places left so book now!

I recently passed a milestone of 30,000 followers on Twitter. I'm not sure there are many other females on EduTwitter who have done this and are still teaching. I'm pleased people find my account useful, and I really appreciate the support.

Finally, did you see the Joanne dress from Popsy? If, like me, you enjoy going to work (and maths conferences) in mathematical attire, then you'll like this one...

Know Your Workings

I have been fortunate that in the last ten years I haven't worked in a school that requires me to mark classwork. I always get students to mark their own answers in lessons, and I give them time to ask me for help if they find they have incorrect answers. I don't believe it is necessary for teachers to mark the practice that students do in class every lesson - and I can't imagine the horrific workload involved in such a thing. For years my students have received feedback in five main ways:

1. Verbal feedback during lessons, where I look over their shoulder and comment on their work - correcting misconceptions, suggesting alternative approaches, advising on layout etc. I'm not great at using mini-whiteboards, but that's a brilliant way to give feedback and is done very effectively by my colleagues.

2. Feedback through Hegarty Maths - instant marking of homework is very helpful for students and a huge time-saver for me.

3. Feedback from regular low stakes quizzes - I mark these myself, but it only takes about 10 minutes to mark a whole class set and gives me a good idea of common misconceptions, so I see this as time well spent.

4. Feedback from 'formal' assessments. When I taught Year 11 this was two sets of mock exams a year - at 240 marks per student per set, this was a large chunk of marking and feedback. For big classes it was a gargantuan workload.

5. Feedback from written homeworks. This is something I do very little of at Key Stage 3 and 4, but did a lot of when I taught A level. By giving big questions to complete on lined paper at the end of a topic, it gave me the opportunity to give feedback not just on students' understanding of the maths, but also on how they lay out their answers. In long A level questions it's important to be able to follow workings in order to spot where they've gone wrong. This is something that most Year 12s need to be trained in because they probably haven't answered many long questions before. This can be done by sharing examples of good work under a visualiser (I wrote about this here).

In my new school we only have Year 7 and 8 and we do two big MAT-wide assessments per year. I find marking these very informative. However, the nature of Key Stage 3 maths assessments means that the questions are typically one or two marks, and an allocated space is provided for workings. What this means is - if I never mark classwork and all homework is done on Hegarty Maths - I never really get an idea how my students set their work out, and I never get the chance to start preparing them for setting out those big A level questions that they will meet in the future.

Just before half-term I decided to take ten minutes of my Year 7 lesson to have my students complete a task on lined paper. I told them that I would mark these in detail and provide feedback. We'd just finished a unit on percentages and I took the questions from an OCR topic check-in. It was intentionally a relatively short and straightforward assessment, and I emphasised the importance of showing clear workings.

Marking these has been very revealing. I'm a bit embarrassed that it has taken me until halfway through the school year to become aware that some of my students don't really know how to set their workings out. Next year I will do a 'blank page assessment' in the first half term so I can address these things much earlier.

I have categorised the workings I saw when I marked these tests into three main types of student: the separators, the minimum possibles, and the one-liners. I have known all three types of student in the past so I guess they must be fairly common. I wonder whether you recognise them too.

1. The Separator
This student has done all of her workings down the side of the page and has written her answers seperately. Note that the workings aren't numbered, though I am able to work out which set of workings goes with each question.

Here's another one where the answers are separate from the questions, but this one is worse because the workings are in a big unnumbered jumble at the bottom of the page. This reminds me of one of those GCSE questions that are impossible to mark because there are numbers and random calculations dotted all over the page (there's an example of a question like this in AQA's recent publication).

Here's another one like that. It took me by surprise to see this separation of workings so many times.

Here's an even more extreme example. I thought this boy had done no workings at all...

... but then I found them on the back. I never told him to do this, promise!

2.  The Minimum Possible
This student has done the minimum possible amount of workings. He finished the test in about three minutes and loudly put his pencil down and turned his paper over so that everyone knew it. This student prizes speed over all else. If it wasn't for the fact I told him he had to write some workings, he wouldn't have written any at all. He doesn't (yet) see the value in them.

3. The One Liner
I know we're trying to save trees, but there's no need to limit every question to a single line. The maths here is mostly good but I will encourage this student to use a new line for each step of working. I once saw a Year 12 try to answer a six mark quadratics questions in one line. It wasn't pretty.

Thankfully this is all easily fixed. For a start I can show them some good examples using my visualiser. There are some good examples below, and each has a positive feature that I want to highlight to the class. These aren't perfect, but for the most part the workings in these examples are clear and well laid out. The maths isn't all correct, but it's easy for me to follow their thinking, which means I can give them helpful feedback and I can understand misconceptions better.

The last one here made me happy because I am constantly nagging this boy about how messy his classwork is - in fact he's probably the only student I regularly talk to about workings - and this piece of work shows a huge improvement. He has even underlined his answers like I do. It's nice when you realise that someone has been paying attention...

I know my readers probably do this kind of activity all the time, but if you haven't looked at your students' written work in a while, why not ask them to answer some questions on paper like I've done here. It was a very short test so it wasn't a huge workload for me. And given how surprised I was by what I saw, I think it was worthwhile...

The Rule of Three

Multiplication is vexation;

Division is as bad;

The Rule of Three doth puzzle me,

And Practice drives me mad.

You may know this classic nursery rhyme, which was apparently written by John Napier in 1570.

Most modern English maths teachers don't know much about the Rule of Three. I hadn't come across it myself until I started reading old textbooks, where it is ubiquitous. In Hodder's Artihmetick (1702) the Rule of Three is described as the 'Golden Rule' (for as Gold transcends all other metals, so doth this Rule all others in Arithmetick').

Some of you will not know what the Rule of Three is, so allow me to explain. The dictionary definition is as follows:

---

noun

DATED•MATHEMATICS

noun: rule of three; plural noun: rules of three

1. a method of finding a number in the same ratio to a given number as exists between two other given numbers.

---

When I explain it in conference presentations, I usually use this amusing example which I found on a maths website explaining how to solve proportion problems. The problem given is:

"Today we are going to go on a school excursion and we need to make sandwiches for the whole class. If we need 2 loaves of bread to make sandwiches for my 4 siblings, how many loaves of bread will we need in order to make sandwiches for all 24 students in the class?"

Now I think we can all agree that this question is silly. Two loaves of bread to make sandwiches for four siblings? Those sandwich are immense.

Anyway, the website determines that this is a direct proportion problem, so tells us to use the Direct Rule of Three (as opposed to the Inverse Rule of Three, which is different). It provides the following method:

Now I know that many of us will be confused as why they felt the need for a formula here. What they've essentially done is cross-multiply: 4 times x is equal to 2 times 24, then we solve for x. But this isn't how I would approach this question. I'd just say something like this:

"He needed 2 loaves for 4 sandwiches, and now he needs six times the number of sandwiches so he needs six times the number of loaves".

I don't use an equation to solve problems like this. I use multiplicative reasoning. It's logic. 

In England, teachers tend to solve proportion problems using a unitary method and/or scale factors. A popular layout is seen in this lovely resource from Don Steward. 

The Rule of Three is explained very clearly on its Wikipedia page. Here it says that:

"The Rule of Three was an historical shorthand version for a particular form of cross-multiplication that could be taught to students by rote. It was considered the height of Colonial math education and still figures in the French national curriculum for secondary education.".

From my conversations with teachers I get the impression it is not just taught in France, but is taught all over the world. In fact, I get the impression they think it's weird that we don't use it.

Yesterday I attended the Harris Federation Maths Conference and, coincidentally, saw teachers from other parts of Europe use the Rule of Three on two separate occasions. They described it as cross-multiplication rather than the Rule of Three.

The first was a teacher from Poland who spoke to me after I delivered my session on Unit Conversions (I didn't catch his name - but if he's reading this then thank you for speaking to me - I love talking about methods!).

Converting metric units is done shockingly badly at GCSE in this country so I ran a workshop on how to teach it in more depth. I talked about various methods for doing unit conversions and showed them (for interest, not because I recommend teaching it) the zany factor-label method. At the end he asked me whether I'd considered using cross-multiplication to convert metric units.

Say we want to convert 300 milliliters to litres - this is the sort of question that is often answered incorrectly at GCSE. He said that he'd lay out his workings like this:

Then he'd cross-multiply.

Cross-multiply is a term I tend to avoid using but I know it is widely used around the world. I once worked with an American teacher who used it all the time, for lots of different problems, assuming that our students knew what she meant (I don't think they did). The dictionary definition of cross-multiply is "to clear an equation of fractions when each side consists of a fraction with a single denominator by multiplying the numerator of each side by the denominator of the other side and equating the two products obtained". Apparently the term has only been used since the 1950s (if that's true then it's a relatively new term in mathematics). 

In this case we have no fractions, but we have quantities that are in proportion which means they can be treated (or indeed, written) as fractions. We cross multiply like this:

This gives the equation 1000? = 300, so to find the value of ? (or whatever symbol we use here), we solve the equation by dividing 300 by 1000.

For students who know how to solve simple equations, it is a very reliable, efficient and straightforward approach for unit conversions. And indeed, some teachers may use it in this country, but I believe it is far more popular in other parts of the world. 

I think that the Rule of Three fell out of common use in England quite a long time ago. Perhaps this is because it was considered mechanical and done without understanding...?  I don't know for sure but I'd be interested to hear from teachers who were educated in England in the 1950s and 60s who can tell me whether they were taught the Rule of Three.

I have a little book called 'How to Teach the Method of Unity' from 1883 which appears to argue against using The Rule of Three for solving proportion problems, in favour of using what we now call the unitary method.

If solving a problem such as "5 pencils cost £1.25, how much do 3 pencils cost?", it tells us to first work out how much one pencil costs, instead of using a Rule of Three formula or 'statement'. 

In its preface we are told that the New Code of 1882 says that teaching of the Rule of Three is to be by the Method of Unity, and then goes on to explain the justification for this. It tells us that the great aim of teaching arithmetic is to improve reasoning powers.

The second occasion in which I saw a cross-multiplication method yesterday was in a workshop about ICCAMS. 

At the start of the workshop we were asked a question: 

Three teachers went to the board to show their methods. The first use a scale factor method, noting that we could multiply by 1.5. He set his workings out in what I'd call the box style, similar to the Don Steward resource shown above. The second teacher use a visual approach (essentially a bar model).  The third teacher (I didn't catch where she was from unfortunately - somewhere in Europe) showed a cross multiplication method:

It was interesting to see this approach used twice in one day. I was already aware that these methods are widely used around the world, but what I am interested in is when and why we (for the most part) moved away from them in this country. As I've said above, I assume there was a move towards approaches that involved reasoning as opposed to procedure, and I am interested in how this move came about.

The approach of using cross-multiplication to solve proportion problems is one that all maths teachers in this country should know, even though most of us don't teach it. We need to know it because:
a) some of our students will learn it from elsewhere (e.g. from tutors)
b) some of our colleagues may teach it
c) some of our students will have learnt it in other countries before moving to our schools.

If we have students who use it, we just need to check that they understand why it works and are able to use it correctly when solving different types of problems (e.g. those involving inverse proportion).

I've noticed a lot of my kids (with tutors) use 'the rule of three' "formula" when dealing with ratios. Fine so long as you get the numbers in the right place. Hard to encourage them to do otherwise.

— Maths Teacher (@MathsTeacherKYP) October 24, 2019

Can’t wait to read it! And love the design! Do you mention “the rule of three”? It is widely used in Spain for pretty much anything but not popular in England... Thanks for your work on this

— Sandra Escribano 🎗 (@madonna4alex) September 18, 2019

Thank you to all teachers who shared their approaches with me yesterday. It's so important to talk about methods.

5 Maths Gems #120

Welcome to my 120^th gems post. This is where I share some of the latest news, ideas and resources for maths teachers.

1. GCSE Tips
TeachIt Maths and AQA published a free guide 'GCSE mathematics - small things make a big difference'. It's designed to help Year 11 students prepare for their GCSEs. Teachers might go through it in class before mocks or final exams.

It's packed full of example responses from real GCSE papers and contains a lot of helpful information for teachers and students.

2. Tutor Time Activities
Thank you to @AccessMaths for sharing a large collection of form time numeracy activities.

3. Number Bonds Chart
Thank you to Jonathan Hall (@StudyMaths) for sharing a new number bonds chart on his awesome website MathsBot. This is an interactive version based of the chart shared by the NCETM in 2017.

4.  Vocabulary
Thanks to Ben Gordon (@mathsmrgordon) for sharing a visual summary of techniques to improve students' maths vocabulary. This might be a helpful thing to discuss at a maths department meeting.

5. Goal-Free Paper

The 'goal-free effect' is discussed in Craig  Barton's book How I Wish I Taught Maths. There are numerous goal-free resources available to download, or you could make your own. Ed Southall adapted a whole GCSE paper to make it goal-free: you can download it here. Some of the questions in this paper work really well with the goal removed. The idea is that students work out what they can from the information given. This often encourages less confident students to have a go at questions that they previously might have skipped.

Update
Another one of my topics in depth video podcasts with Craig Barton was recently published by TES. This one is on angles in parallel lines. I hope people are finding these useful. We're recording a couple more in February half-term.

Last weekend I attended BrewEd Maths in Croydon. It was a brilliant event and I really enjoyed it.

I've had to cut down on conferences this year because I'm really busy at work, but I'm looking forward to #mathsconf22 in Manchester in March - I've booked my hotel and train. See my conferences page for a list of upcoming maths events.

This big event of the year for me is the new-look MA and NANAMIC Easter conference which I blogged about here. Ticket sales have been excellent so far, which means that the MA has been able to extend the early-bird discount to the end of January. With that discount it only costs £299 to attend the whole conference which includes food and accommodation at a lovely spa hotel, and a fantastic programme of speakers and activities. Rates are cheaper for trainees and NQTs, and there's the option of attending just part of the conference for those who can't make all three days.

I've been really pleased to see people learning new methods from my book A Compendium of Mathematical Methods which was published in December.

Using @mathsjem's fabulous book to discuss methods with trainees. Really useful to be thinking about how we teach things, why we use the method we do & how we can develop understanding. My campaign for subtraction using partial differences is gaining support! 😀 @RedKiteTT pic.twitter.com/F0OaF9hwkc

— MrsD (@MrsDMaths) January 9, 2020

It's wonderful to see maths teachers discussing methods, which is exactly what I hoped would happen. I love this blog post from Jack Nicol (@geomathsblog) where he summarises a discussion with his maths department on methods for factorising non-monic quadratics.

Did you hear that Craig Barton has a new book coming out? I'm lucky enough to have read it already. It's excellent. I know it is going to be very popular. It's out next month but you can pre-order it here.

Another new book out is Clarissa Grandi's Artful Maths. There is a teacher book and an activity book - you can buy them from Tarquin. I've been using Clarissa's beautiful website artfulmaths.com this week to help me set up my new origami club.

I'll leave you with this interesting bit of etymology which was shared by @solvemymaths on Twitter last week: the word average comes from the apportionment of financial liability from goods lost or damaged at sea.