5 Maths Gems #112

Welcome to my 112^th gems post. This is where I share some of the latest news, ideas and resources for maths teachers.

1. Classic Errors
I like this activity suggested by James Tanton (@jamestanton). James asks 'Would taking classic and tempting errors "head-on" be a worthwhile classroom activity and discussion?'.  It's worth reading the responses in the thread. In it Cathy Yenca‏ (@mathycathy) recommends the classic resource 'Algebra Atrocities' which is worth a look.
2. Geogebra Percentages
Thanks to Mark Horley (@mhorley) for sharing some helpful Geogebra applets created by ⁦‪@orchiming‬⁩ .

These interactive tools help with explanations of percentages. For example the profit and loss applet shows really clearly how the calculation works.

3. Area Mazes
I know I've featured Area Mazes in my gems posts an number of times before, but I think they're really fun so there's no harm in mentioning them again. Have a go at this one -  it was tweeted by @Errs5 and is taken from the Area Quiz app which looks excellent.

4. Calculator Crunch
In my post about Year 11s lacking basic calculator skills I mentioned that MEI are running Calculator Crunch to encourage schools to get younger pupils using calculators. Aimed at Year 6 and 7, they are sharing a daily calculator challenge between 10th and 20th June. You can download all the daily challenges from the MEI website and use them at any time. They have also provided a Year 6 lesson plan and a Year 7 lesson plan that you can download here.

Check out the hashtag #CalculatorCrunch to see examples of pupils' work.

5. Place Value
Another great idea for using calculators is in a lesson on place value. Tom Francome shared this 'Place Invaders' task in a conference workshop - it's taken from the ATM book Practising Mathematics which he wrote with Dave Hewitt.

Grab a calculator and have a go and you'll see why this is a great task to use when teaching place value. I think that place value is one of the hardest things to teach in Year 7 so it's helpful to see a good quality activity for this. Tom suggested it would also work well with standard form.

Update
Exam season is nearly over, and it's been an unusually turbulent one this year. We've had questions that appeared to be copied from textbooks, a mistake in an A level, controversial contexts, a leaked exam and some surprisingly challenging A level papers. Never a quiet day in maths education!

I recently made a final set of GCSE revision resources which you can find on my GCSE revision resources page. You might find these resources helpful if you have Year 10s preparing for end of year exams.

I presented at ResearchED Rugby yesterday. My session focused on multiplication - first we looked at a range of multiplication methods and then I presented a critique of each of the reasons the Government gave for prescribing 'formal' methods at Key Stage 2. For the remainder of the day I attended excellent sessions in the maths strand. Rugby School has an incredible history so was a cool venue for a conference.

Next weekend I'll be presenting on the Evolution of Maths Vocabulary at La Salle's conference in Yorkshire. Read my blog post "Whatever happened to vulgar fractions?" to get an idea of the sort of thing I'll be talking about, and if you find it interesting then do come along to my workshop. I'll also be manning the MA bookstand for most of the day so do come and say hello if you get the chance.

Bookings are going well for the training course I'm running with Craig Barton in October. This course is open to all maths teachers and we have two dates available - a northern leg and a southern leg. Ask your school or college to book your place now if you want to come along.

At the end of May Ed Southall and I relaunched The Mathematical Association's eNews. This monthly newsletter provides an update of what's going on with the MA and also in the wider world of maths education. We now feature exclusive monthly puzzles for your students (written by Ed), and there are more new features coming soon. You can subscribe here to receive eNews from the end of June.

I've been kindly sent a couple of books by their authors recently. Kyle D Evans sent me a copy of 'Here Come the Numbers' which my daughters (aged five and seven) love as a bedtime story. It's both mathematical and lovely.

Mark Ritchings sent me a copy of 'GCSE Maths Challenge' which contains 40 GCSE style questions that are aimed at pupils who are working at a Grade 8 or 9. This short book might be helpful for a GCSE tutor, or possibly a top set teacher who needs a few extra ideas for extension questions to write on the board.

Maths teacher Richard Atkin (@RichardAtkin1) - whose uncle was a Trustee of the MA - very kindly delivered an enormous number of old maths books to my house. This will keep me busy for a while! I've already found lots of treasures that have taught me new things. I estimate that I now own at least 300 different maths textbooks dating from 1800 to 2000. I love this!

A group of us are going to Matt Parker's Humble Pi comedy show on 16th July in London (it's a preview of his Edinburgh Fringe show). If you want to join us, book a ticket here and let me know you're coming. There are similar comedy nights in London listed here, including one that features the amazing FoxDog Studios. FoxDog Studios will also be at Edinburgh Fringe and are doing gigs in various locations before then - dates are listed here. They are amazing.

I'll leave you with this great activity shared by Sarah Carter (@mathequalslove). You can download the full resource from her blog. This puzzle might work well in a first Year 7 lesson in September, along with the other tasks I listed in my post Year 7 Maths Activities.

5 Maths Gems #111

Welcome to my 111^th gems post. This is where I share some of the latest news, ideas and resources for maths teachers.

1. Multiplication Tasks
In a Twitter chat on multiplication I saw two nice resources shared. First, Sharon Malley (@mathsmumof2) mentioned these 7 Times Table Reasoning Activities by krisgreg30 on TES. These tasks require children to use known facts to reason how to solve other calculations.

Second, Jonathan Hall (@StudyMaths) shared a lovely task he designed for his Year 7s to get them thinking a bit deeper.

2. MathByExample
Three years ago I wrote about AlgebraByExample in Gems 54. This set of tasks prompts students to analyse and explain misconceptions in algebra problems.

The team at SERP Institute (@SERPInstitute) have now launched their MathByExample website. The tasks are similar to AlgebraByExample but they are for topics that children meet at Key Stage 2.

There are loads of great tasks to explore on this website. In each case children are given a correct answer and an incorrect answer with questions about each one, and then they are asked to solve similar problems themselves.

The question prompts help children develop a better understanding of each concept.

3. Compound Area
Amie Albrecht (@nomad_penguin) shared a smart way to take a standard textbook-style exercise and add a higher level of thinking. Instead of just completing the exercise, pupils are asked to consider the features and difficultly level of each problem before deciding which problems to solve.

4. Linear Sequences
Thanks to Dan Lewis (@4301maths) for sharing a series of tasks on linear sequences.

Follow Dan on Twitter for more like this, including examples of his pupils' work.

5. Question Generators
Thanks to Jonathan Payne (@DrPMaths) who has built a collection of helpful question generators.

For example if you are creating some angle questions for your explanations or for your pupils to practise, then you can use his missing angles generator to create a set of customised questions.

And here's one that creates arithmagons.

Updates
I've been busy making more GCSE revision resources. This is the last time I'll do this for a while because next year I'll only be teaching Key Stage 3.

Because the non-calculator revision mats and the calculator revision mats I recently made went down well with pupils, I was asked by a colleague to make another set. So now I have a second set of calculator revision mats. Again, they have four levels of difficulty so you can pick the right level for your pupils.

I also made a Higher and Foundation 'Spot the Mistake' revision activity for something a bit different.

I also made a couple of revision mats with topics that might come up on AQA Paper 2. These are just a collation of questions taken from Maths Genie. Because these are 'temporary' resources (ie designed specifically to prepare for AQA Paper 2 June 2019), these are not on TES but are linked through Adam Creen's blog. Every year Adam pulls together all the 'between-paper' resources on his blog for easy access.

Don't forget you can use my breakfast resources as pre-exam warm-ups before Paper 2 and Paper 3. And my GCSE revision post continues to be the place where I collate all free GCSE revision resources (with the exception of the 'between-exam' resources that have a limited shelf-life).

***

I was delighted to announce this week that I will be teaming up with Craig Barton to offer two full day training courses in late October. Visit mathscpd.weebly.com for all the information. Bookings are already going well.

My recent post 'Calculator Woes' rung true with many teachers. It was selected as a Schools Week 'Top Blog of the Week' by Amir Arezoo and featured in Ollie Lovell‏'s weekly Twitter takeaways. I think there's quite a serious problem with calculator skills across the country and I really hope to see teachers try to remedy this by getting lots of Year 6s and Year 7s involved in MEI's Calculator Crunch next month.

Thanks to Teresa Robinson at The Russell Education Trust who used my post to create a lesson on calculator skills.

Thanks also to @pgonlinepub for sharing a free worksheet on the 'Top 5 calculator hacks for GCSE Maths'.

Yesterday I had coffee with Simon Singh, the author of my favourite maths book. We discussed what parents can do to encourage and support their mini-mathematicians at home. Simon mentioned the coding app Box Island. When I got home I downloaded for my daughters. It's awesome!

By the way, if you don't currently receive the MA's eNewsletter then you can sign up here. I've been working with Ed Southall to relaunch it. From now on it will include an exclusive monthly puzzle for your pupils to try - one for primary and one for secondary. Sign up now!

I'll leave you with this graph, shared on Twitter by @lizardbill, which is probably the best example I have ever seen of a really really bad graph. There are more amusingly terrible graphs in the thread.

Enjoy half term!

Marvellous Maths Teaching

I have exciting news! I'm teaming up with Craig Barton to deliver a brand new training course for maths teachers this October. 'Marvellous Maths Teaching' offers a full day of high quality maths CPD.

Your school or college will only have to pay £90 + booking fee to send you on this course, which makes it an absolute bargain - many other one day training courses are considerably more expensive. Even better, we're running both a Northern Leg (at the home of White Rose Maths in Halifax) and a Southern Leg (at my awesome new school in Sutton). So basically we're going on tour, and we're very excited about it.

Check out the full programme, and if this looks like something you might benefit from then get your absence/training request in quickly - there are only 100 places available on each course.

We can't wait to get started on this! It's going to be great. For more information and to book a place, visit mathscpd.weebly.com. Hope to see you there!

Calculator Woes

One of the Year 11 classes I teach is what we call a withdrawal group. They all got a Grade 0 in their Higher GCSE mock back in November so we moved them to Foundation tier. This was definitely the right decision. I teach them for two double periods a week. The pupils in this particular group are very complex. Trying to teach them has been the most challenging thing I have ever done. I'm crossing my fingers that they all get a Grade 4 this summer.

Last week I attempted to engage this class in a revision lesson with one of the revision mats I recently made. This required calculators, which I lent them. This particular revision mat was deliberately designed to identify gaps in their knowledge of how to use basic calculator functions. I very quickly discovered that there were indeed many gaps. In this post I hope to help teachers of similar GCSE classes be aware of things to look out for and address.

The first question to cause me concern was this one on multiplying fractions:

The answer they wrote down was 4.2. There are two worrying things about this: 

1. that they don't recognise that the answer should be greater than 21 (an area model would be a good way to demonstrate this)
2. after almost five years of using calculators at secondary school, they don't know where the mixed number key is. 

Their answer of 4.2 came from them inputting the fraction ½ and then using the cursor to move back and put a three in front of it. The calculator sees this as 3 x ½ rather than the intended 3 + ½. 

In this country (though apparently not all others), writing 3½ means 'three and a half' and we call it a 'mixed number' - I'm pretty sure my pupils know this much. But when I went round and corrected each of them individually they all swore blind that they had never before been shown the mixed number button.

Now, I have been teaching teenagers long enough to know that the old 'we've never been taught this' line is normally nonsense. However in this case it may well be true. When we teach fraction arithmetic we do so in lessons that deliberately don't involve calculators. So unless these pupils have had a 'calculator skills' lesson then perhaps they haven't been shown it.

The second question to cause me concern was this one:

Thankfully all but one pupil wrote down the correct answer without using a calculator because they have some understanding of place value (phew!). The pupil who got it wrong wrote one seventh. 

He could have typed 7% in his calculator and then pressed the S-D button and it would have done it for him. The percentage button simply divides by 100. It's nothing special but it's there if they need it. For example, think about the various ways to do this question: 

We know that the most efficient thing to type into your calculator is 43 x 1.16. If pupils are not au fait with multipliers then perhaps they can type 16% x 43 instead, making use of the percentage button, and then add their answer to 43.

Do they do this? Hell no. Interestingly this is a topic where most students working at Grades 1 - 3 are very good at non-calculator methods. They will have done a lot of work on using multiplicative reasoning to find 10%, 5% and 1%. I'm super pleased that they can do this, but it's frustrating to see them use this method when they have a calculator at their disposal.

The next question that alerted me to a knowledge gap was this one:

There were two issues here: 1. not knowing where the cube root button is (I've known pupils to do 3√64 in the past) and 2. not following the order of operations. Some of them wrote 15 x 4, having worked out the cube root of 64 on their calculator after asking me the whereabouts of the button. I reminded them that their calculator has 'been taught the order of operations and understands it well', so we can trust it to correctly evaluate the multiplication first if we type in the whole calculation in one go. The problem of course comes with something like this:

In this case the calculator does indeed follow the order of operations, so if you type -3² then it will correctly square the three before multiplying it by negative one. But of course that's not what we wanted it to do. So if we insist on squaring negative three using a calculator, then we must put it in brackets. In fact if using a calculator I'd suggest putting all substituted values in brackets ie typing in 5(-3)² -2(-3). I'd prefer to see this one done without a calculator really though. 

Evaluating the square of a negative incorrectly using a calculator is a problem that persists. We know that GCSE students make this mistake all the time when asked to plot quadratic graphs. They end up with weird looking 'parabolas'. I remind them that they know they've gone wrong if their graph isn't symmetrical. Pupils must use brackets if they use a calculator to square a negative! I feel like I've said this a million times and I will keep saying it until I'm blue in the face. 

The next question is one that I deliberately included in the resource to draw my pupils' attention to a useful calculator button. In my question about prime numbers, most of them assumed that 51 is prime. I pointed out that using the FACT button will show the prime factorisation of that number, which is rather exciting.  If they're asked to write a number as a product of primes it will probably be in the non-calculation paper, but it's still helpful to know this function. It gave me the opportunity to have a helpful conversation with my pupils about what prime numbers are. 

I think they were impressed by this button. 'This button is peng, fam'.

I had shown them it before, back in January. I'd also shown them the button they can use for time conversions. They were impressed then too, but had clearly forgotten all about these buttons which is unfortunate.

The final question of interest is this one on standard form.

Like fractions, standard form is often taught in a non-calculator context. And even if we have a calculator, it doesn't help us convert between standard form and ordinary numbers. However, if we're asked to use standard form in a calculation and we have a calculator to hand, it might be helpful to know how to use the x10^x button. 

On reflection, this resource ended up being more than just a revision mat. It helped me identify and address significant gaps in calculator knowledge, albeit much more last minute than I would have liked. They should have been using these buttons for years. 

I should mention that when I took on this class at Christmas there were even more gaps in their calculator knowledge. At first they weren't making use of the Ans button. Plus, to do say 5.7² they would type 5.7 x 5.7 because they didn't know about the x² button. So we have made some progress. 

Ironically, as they left the classroom I reminded them that they should bring compasses and a protractor to their maths exam and they told me that they don't need to because it is all provided for them in a pencil case. 'Oh. In that case just make sure you bring your calculator. It has to be the one you've been using for the last five years - the one you know how to use'. 'Nah miss - they provide calculators too, we'll just use whatever they give us'. 

*******

This lesson got me thinking. I have a feeling that many schools don't do enough with calculators. In every school where I've taught Year 7 we haven't used calculators at all for the entire year - this is such a shame when there are many opportunities to explore and make sense of numbers in Year 7 using calculators. Also, although we need to build fluency with arithmetic in Year 7, I am convinced we should allow calculators for topics such as angles. It makes sense, and pupils like it.

Pupils need to be explicitly shown how to use their calculators throughout secondary school - I think that teachers often assume pupils already know the main keys. We need to bear in mind that much of it may not be obvious to a novice.

If you currently teach Year 6 or Year 7 then you might want your class to take part in MEI's Calculator Crunch in June. In this challenge there will be nine daily problems to solve using a calculator. MEI will be providing two lesson plans too.

There are lots of fun 'know your calculator' activities in my resource library. These are great, but we must make sure that calculator use is embedded throughout the topics we teach too, not just covered separately. I'm sure that some schools already do this really well.

*******

Note that the calculator pictures in this post are from my 'peng' Casio fx-83GTX which was kindly given to me by Science Studio (and engraved with Mrs Morgan, which I love!). The buttons I've featured are in the same position on the Casio fx-83GT and fx-85GT (the ubiquitous calculators in the current Year 11 cohort).

5 Maths Gems #110

Welcome to my 110^th gems post. This is where I share some of the latest news, ideas and resources for maths teachers.

1. TES Author
When we did the latest round of TES Maths Panel reviews, my fellow panelist Damian Watson discovered the wonderful free resources of TES author cparkinson3. This author's PowerPoints are really well designed - they are slick and professional with neatly animated worked examples plus exercises with solutions.

For example check out the two lessons on volume of prisms - one for Foundation tier and one for Higher tier.

This resource on curve sketching is excellent.

I really like the bar modelling lesson too.

Check out the full collection. I've added them all to my resource libraries.

2. Vectors in Transformations
There's been some updates to BossMaths including changes to their Geogebra applets. The rotations applet now lets you show and hide tracing paper. You can also use vectors as an alternative to using tracing paper. I like this idea - I find using tracing paper a bit unsatisfactory so I will try vectors next time I teach this.

The enlargements applet now lets you show vectors from the centre to a vertex simply by clicking the vertex. This is useful if you use vectors for enlargements (which is particularly helpful for negative enlargements) - I blogged about this here.

3. SATs Support
I know this is a bit late for Year 6 teachers because SATs are next week, but these resources might be helpful to teachers of other year groups, and to Year 6 teachers next year.

Mr Morgs (@_mrmorgs) shared a PowerPoint explaining the language used in the KS2 Maths SATs. This follows on from his detailed analysis of exam language. This is really interesting and comes with lots of advice for teachers. I hope that one day someone does a similarly detailed analysis of the language used in GCSE maths exams.

Greg Chantler (@gregchantler) put all Key Stage 2 SATs arithmetic questions together in a PowerPoint and split them by curriculum year. Each question comes with discussion prompts.

4. Defend Yourself
Amie Albrecht (@nomad_penguin) shared a new idea for an activity. Students are assigned to a representation or method and have to try to explain why it's best. I think this would prompt rich discussion and thinking.

5. Shifting Times Tables
This activity from Miss Konstantine (@GiftedBA) provides a really clear representation of linear sequences as 'shifting times tables'.

Here we can see the four times table (4n) highlighted in the first row, then we see that the times table has been shifted down by one in the second row, meaning the nth term of the sequence is 4n - 1. I love the visualisation here. Check out Miss Konstantine's regularly updated blog for lots of great maths activities and resources.

Update
My blog has been receiving record number of visitors in the last couple of weeks. I guess a lot of people are using my GCSE revision resources page and my recently published post on Foundation revision resources.

I've recently uploaded two sets of revision mats to TES. I made these at work when I realised it's hard to find suitable resources for students who are working at a Grade 1 or 2. Most GCSE resources are inaccessible to them, and this can result in disengagement and sometimes bad behaviour. So I made A3 revision mats that are more accessible, and while I was doing that I made similar resources for other classes. So there are four levels of difficulty. You can download these resources here:

• GCSE Calculator Revision Mats: Higher and Foundation
• GCSE Non-Calculator Revision Mats: Higher and Foundation

I also made some Foundation workouts which are particularly 'print budget friendly'.

GCSE exams are just round the corner now. Don't forget that last year I made breakfast revision resources for all three exams. These can be used either on the morning of the exams or in the lessons leading up to them. 

Last week I published by 5th Annual Gem Awards. Do check it out if you missed it so you can see all the best gems of the year.

Congratulations to Emma McCrea on the publication of her new book 'Making Every Maths Lesson Count'. This is an excellent book for maths teachers.

Here are some other things you might have missed:

• In 2017 I published a post on approaches to answering AQA GCSE ratio questions. It's one of my most popular posts with over 20,000 reads. I've now made a lesson to accompany this post.
• Adam Boxer (@adamboxer1) shared a post on observing expert teaching which includes an 'expert teaching observation form'. I contributed to this post by testing the form out in a maths lesson.
• Richard Tock (@TickTockMaths) published a number of new lessons including some on indices which draw a lot of ideas from my Topics in Depth CPD. Check them out on his blog. 
• One of my Year 11s was flummoxed by rotational symmetry last week so I used the FlashMaths rotational symmetry tool to show him how it works. This has been around for a very long time so I was surprised that a lot of teachers on Twitter hadn't seen it before.

Don't forget that there are a number of events in the summer term that are now open for booking (see my events listings here). I will be presenting at researchED Rugby 2019, the Complete Mathematics Conference, the MEI Conference and the Kent and Medway Secondary Maths Conference. Speaking of events, I also have something a bit different to announce soon - watch this space.

I'll leave you with this video on why zero is the naughtiest and most important number from BBC Ideas.

Why zero is the naughtiest and most important number

via @bbcideas pic.twitter.com/pRJ0M6uEdC

— BBC Teach (@BBC_Teach) 2 May 2019