6 August 2018

5 Maths Gems #93

Welcome to my 93rd gems post. This is where I share some of the latest news, ideas and resources for maths teachers. It's the summer holidays! Many of you will be on a Twitter break right now, so this post will fill you in on some things you might have missed in the last couple of weeks.

It was in the summer holidays - four years ago this week- that I first started writing my gems posts. You can see the full collection here.

1. Triangles
Thanks to John Rowe (@MrJohnRowe) for sharing this right-angled trigonometry activity. Students have the find the length x.
Benjamin Leis‏ (@benjamin_leis) has blogged about his approach to this problem here.

If you like this then you might also enjoy this problem from UEA which requires knowledge of the sine and cosine rules - I featured it in Gems 54. It takes a while to solve

You might also like the angle chase problems I shared in Gems 35. I love angle chases!

Speaking of triangles, thanks to Mark Horley (@mhorley) for sharing @DrPMaths' triangle generator. This helpful tool creates possible triangles with integer sides, area and height. If you make your own resources then this will be helpful to ensure you don't include impossible triangles.
2. Sums and Products
Thanks to Shaun Carter (@theshauncarter) for sharing a new resource which he calls 'Diamond Problems'. He blogged about it here.
The questions are similar to those on this shorter 'Sum Products' worksheet which I always use before my students start factorising quadratics. Of course, as Shaun says in his post, these puzzles are suitable for students of any age even if they're not studying quadratics - they are good practice for working with negative numbers and decimals.

Thanks also to Meredith Purser (@MeredithPurser) for suggesting that students first work out the rule themselves before completing the blank grids.

3. #midweekmaths
The White Rose Maths Secondary Twitter account (@WRMathsSec) is sharing a weekly problem throughout the holidays using the hashtag #midweekmaths
It's worth following @WRMathsSec because they regularly share lovely tasks for students. 
Also check out their new secondary five year plan.

4. Interleaved Homework
I have always set homeworks that directly relate to the topic I'm teaching, but for the last couple of years I have been meaning to change that. Continually revisiting past topics through homework is a great way to help students remember things.

Thanks to David Wees (@davidwees) for sharing 125 interleaved practice assignments - although these are aligned to a specific curriculum, it's so helpful to see the format and approach. This is definitely on my list of things to start doing!
5. Geogebra Whiteboard
I know many of you already use Geogebra, but you might not have seen this beta version of Geogebra Whiteboard. Thanks to Pip (@AccomplishEdu) for sharing this. The interface is brilliant - it's so easy to construct and annotate diagrams. Have a play with it and you'll see what I mean.

I've updated my conference listings for 2018/19. There are lots of great events coming up. Next term I intend to go to #mathsconf17 and MathsJam. And I hope to announce my own event soon...

I've been working on my A level resource libraries - they need a total rewrite because of the new A level specifications. It's a huge, time consuming job!

If you're a member of The Mathematical Association and interested in volunteering, please get in touch. I chair the Publicity and Media Committee and am looking for a couple of new members of my committee. I'm also looking for people to help man the MA bookstand at conferences. Please let me know if you can help. A small honorarium will be paid to conference volunteers.

In other news:

I'll leave you with this arithmetic maze from @MathforLove, shared by @MathsEdIdeas. Without passing through the same cell twice, what’s the largest total you can make? This might be a nice activity for Year 7 to have a go at in September.

22 July 2018

5 Maths Gems #92

Welcome to my 92nd gems post. This is where I share some of the latest news, ideas and resources for maths teachers.

The summer holidays are here - hurrah! Time to relax, have fun and let our mental and physical health recover before we go back for fresh starts with new classes in September.

1. Equivalent Ratios
I like this equivalent ratios activity from Richard Tock (@ticktockmaths). Below is an extract showing some of the ratios that must be simplified and grouped. These have been carefully chosen to reveal misconceptions.
Henk Reuling‏ (@HenkReuling) shared an interactive version of a similar activity here.
2. Probability Problems
Thanks to Jason Steele‏ (@steelemaths) for sharing a nice probability task. Here students have to write numbers on the spinners to meet the stated conditions.
Do check out Jason's other resources too - he recently shared a nice set of questions on angles and ratio.
3. Real-Life Graphs
MathsPad has a brilliant set of interactive resources which are great for classroom demonstrations. Their latest interactive tool for subscribers demonstrates real-life graphs based on filling containers with liquid at a constant rate. There's an accompanying worksheet too.
4. Open Middle Angles
Here's another fantastic Open Middle problem from John Rowe (@MrJohnRowe):
5. Symmetry Display
Thanks to David Morse (@Maths4Everyone) for sharing a new alphabet symmetry display. This might be useful for making 'Welcome to the Maths Department' signs.
If you want to brighten up a maths corridor or classroom, you will find a large collection of maths displays here.

Here are a few things you might have missed:
  • My most recent post was about surds and how the teaching of the topic has changed over the years.
  • If you're launching Hegarty Maths in September, you might find Ben Gordon's (@mathsmrgordon) launch PowerPoint helpful.
  • If you're setting up a group or individual intervention with Key Stage 3 students next year, or if you're a private tutor taking on a new tutee, you might find this assessment helpful for identifying topics to focus on.
  • If you're teaching Year 12 in September, check out my entry assessment and first lesson slides.
  • In case you missed it, do check out the lovely blog post "Not As Complicated As It Looks…" by Mr Rowlandson (@Mr_Rowlandson).
  • I had a fantastic time with three of my Year 7 students at the London Rock Wrangle a couple of weeks ago. We didn't win the helicopter ride, but we came sixth out of 43 schools overall which was awesome. If your school doesn't already use Times Tables Rock Stars, check it out.
  • If you have time over summer, do listen to Craig Barton's latest podcast which features 55 individuals answering the question “what have you learned this year?”.
  • I really enjoyed taking part in The Aperiodical's The Big Internet Math Off over the last few weeks. I was honoured to be invited to take part. I wrote two pieces for the competition - one on hexaflexagons and one on Klein bottles. I was knocked out in the second round by Matt Parker! Do look out for the final on 24th July.
  • Thanks to David Faram (@dagsmaths) for showing me the maths education research pages from the University of East Anglia which are full of interesting reads for maths teachers. Have a look at the tasks here and here and the student responses.
  • Congratulations to Jemma Sherwood on the publication of her excellent new book "How to Enhance Your Mathematics Subject Knowledge". 

Finally, a quick shout out to the wonderful Glyn maths teachers who joined me for the best leaving do ever! We did bottomless brunch, an escape room, pedalos in Battersea Park and burgers in Clapham. What a great day! It was my first ever escape room and I loved it so much I now intend to try every escape room in London! Thank you to everyone on Twitter who recommended it.

I have a few things to work on over summer, but not a huge amount as I really need a rest! I will continue my old textbook research and my topics in depth project. I plan to update my resource libraries - in particular, I need to totally reorganise my A level libraries. I also need to update my conference listings. But before I do any of that, I'm off on holiday with my daughters... first stop, Cadbury World.

I'll leave you with this nice little puzzle from MathsPad's latest standard form resource.
Have a great summer!

15 July 2018

A Look Back at Surds

This year I've written a number of posts about maths textbooks from the last 300 years. Buying and reading historical maths textbooks has fast become my favourite hobby - it's fascinating, and it has done a lot for developing my subject knowledge this year.

Subject knowledge for maths teachers isn't just about being able to do the maths - that's the easy bit - it's also about knowing how to explain concepts clearly, knowing multiple methods and approaches, knowing common misconceptions, knowing history, etymology, narrative and so on. So, in the interest of subject knowledge development, today I bring you some maths from the past. I'm focusing on surds here (because everyone loves surds!), and I hope to bring you similar posts about other topics over the coming months.

In Elementary Algebra for Schools (Hall & Knight, 1885), the chapter entitled 'Elementary Surds' starts with some definitions:

Whether we'd still classify those algebraic terms as surds is debatable, but otherwise the wording of the definition ('when a root cannot be exactly obtained') has been pretty consistent over the years. Here it is again in 'A Shorter Algebra' (Baker & Bourne, 1927):
The book later goes on to refer to 'surdic expressions' - I've never used the word surdic and I'd like to see it back in common usage!
In 'The Essentials of School Algebra' (Mayne, 1961) we have the following definition of surd, which offers more clarity by providing a few examples and non-examples:
All three books explain what is meant by the 'order' of surds. There are exercises on transforming surds of different orders into surds of the same order - something that we don't do in secondary mathematics anymore.

Note also that surds of order two were sometimes referred to as 'quadratic surds'. This is another expression that we seem to have lost from our secondary school vocabulary. We now deal almost exclusively with quadratic surds, in fact I have seen a number of websites and resources claim that only square roots can be called surds, which is just plain wrong.
Contrast the precise vocabulary and detailed definitions in old textbooks with the disappointing one line description we have in a modern day textbook (Edexcel GCSE Maths, OUP, 2016).
I wonder why textbook authors gave up on thorough definitions.

Interestingly, old textbooks claim that the use of surds isn't absolutely necessary because, with extensive number work (ie using the process of evolution to find roots), we can find the value of any surd to a suitable degree of accuracy.

Back in the days when maths was all done by hand, they were generally happy with 'accurate enough' and didn't insist on exactness. In fact in 'A Shorter Algebra' it says,

"Results in surds are only practically useful when expressed as decimals".

It seems that although surds were often used for efficient workings, numerical answers were rarely given in surd form. Here we see that rationalising the denominator is done to make numerical calculations easier, because multiplying by a decimal is easier than dividing by a decimal:
Though we all know and use the phrase 'simplest form' to describe a surd that has been simplified, I'm not sure that many of us use the term 'entire surd' to describe an unsimplifed surd.
Textbooks used to feature exercises where students had to convert simplified surds into entire surds - it's rare to see exercises on this now.

The method for adding surds hasn't changed over the years. We still add 'like surds' or 'similar surds' (perhaps we more often refer to them 'like terms' though),

Adding unlike surds leaves us with a compound surd ('an expression involving two or more surds'):
And the process for rationalising hasn't changed over the last 130 years. We multiply the denominator by a 'rationalising factor'. Where necessary, the rationalising factor will be the conjugate of the denominator. We still refer to the conjugate now - it's good that at least some of the formal language has not been lost.
Of course, standard problems in the 1960s were far harder than they are now.
I don't need to tell you that exercises used to be a lot longer and more challenging than they are in most modern maths classrooms. Here's an example of an exercise on rationalising - this one is from 1885. Note that the first ten questions are just for practising finding the product of two conjugates:

Notice the tricky typesetting of those pesky vinculums!

For comparison, modern textbooks have only half a dozen questions on the same skill.

The books I have been looking at (one from 1885, one from 1927 and one from 1961) continue with square rooting binomial surds and solving 'irrational equations'. I could include a lot more about surds here, but to keep this post to a reasonable length I will leave it there! You get the idea - a lot has changed, all except the mathematics itself.

I've recently bought loads of old textbooks from the early 1900s and am hoping to do a presentation about them at an upcoming conference (possibly #mathsconf17), so if you find this stuff interesting do come along.

Part of my old textbook collection!

30 June 2018

5 Maths Gems #91

Welcome to my 91st gems post. This is where I share some of the latest news, ideas and resources for maths teachers.

1. World Cup Box Plots
Thanks to Southborough Maths (@Mathsteam1) for sharing these box plots created by @johnwmillr. They show the distributions of height by position for players in the World Cup. They make for great discussions with students, and provide a nice demonstration of how box plots can help us make comparisons.
I've been using a similar set of graphs for years, every time I teach box plots (see my post on teaching box plots for more on this). It always goes down well.
2. Variation Theory
Last week Craig Barton launched a new website packed full of sets of well written questions for intelligent practice. Do check out variationtheory.com if you haven't already seen it.
'Rearranging formulae' by Danielle Moosajee
'Mixed Bases' by Joe Berwick 

Like Craig's other resource websites (SSDDs, Venns and Diagnostic Questions), you can submit your own resources for inclusion on this website.

3. Fractions
Thanks to Berkeley Everett‏ (@BerkeleyEverett) for sharing this animation. This can be found, along with loads of other great animations, on the Math Visuals website. 
4. Compound Shapes
Thanks to Mark Ives (@MarkIvesTeach) for showing us how he used Numicon to support students in identifying the lengths of sides in compound shapes.
5. Coordinates Problems
Thanks to Dave Taylor (@taylorda01) for a sharing a set of challenging coordinates problems (see this tweet and this tweet) . Here are a couple of examples:

Do maths teachers all say things in the same way? At the Tweet Up in Manchester last weekend, I recorded a group of teachers saying words that I've heard pronounced differently by different maths teachers. I've picked three of these words for the first video from my pronunciation project:

Thank you to everyone who took part! It may not be the most exciting video ever but I think it's really interesting that students hear different things from different teachers.

Here are a few other things you might have missed recently:

Ten years after we did our PGCE together, I finally met up with Colin Hegarty! He came to my school to launch Hegarty Maths at our first annual trust maths conference. This is really exciting - Hegarty Maths is awesome. I loved trialling it with my Year 11s this year. Thank you to both Colin and Simon Petri from the Surrey Plus Maths Hub for their excellent presentations.

It's all been a bit crazy lately. Next week I have an AQA Expert Panel Meeting, the BBO Maths Hub conference, a TTRS Rock Wrangle trip, and prom. Then I can relax!

I'll leave you with this lovely factor tree puzzle from Sarah Carter (@mathequalslove), inspired by @HaroldReiter.

26 June 2018

Conference Round Trip

I was very fortunate to have the opportunity to present at two excellent maths conferences over the last few days. On Saturday I was at La Salle’s #mathsconf15 in Manchester, and on Monday I was at the JustMaths conference at Alton Towers. I did a three night round trip, travelling by train with a huge backpack.

I used to publish very detailed write ups of every maths conference I attended - in fact I used to be one of the only people who did so. But now there are loads of teachers blogging about conferences. This is fantastic! I love seeing so much enthusiasm for maths teaching. So I won’t go into great detail about the conferences in this post - not because I didn’t have a great time and learn a lot, but just because you can read far better write ups elsewhere.

If you do want to hear my personal reflections in more detail, I recommend you listen to the two conference takeaway podcasts that I've recorded with Craig Barton. The #mathsconf15 podcast is already out, and the JustMaths podcast will be out next week.
At #mathsconf15 I presented on Indices in Depth. This presentation took me months to pull together! I've shared the slides here, but bear in mind they make a lot more sense when I talk through them:

Do have a go at the problems in the handout if you have a few minutes. Delegates showed me multiple approaches to the final question.

The idea behind my slides is that you can present them yourself at school when your department is about to teach indices. Links and sources are in the notes at the bottom of each slide.

I hope to follow this up with a presentation on fractional and negative indices at a future conference. My full collection of 'topics in depth' packs (which is growing very very slowly) is available here.

If you'd like to read about the key points from my workshop then check out the conference write-ups from these lovely bloggers: Jess Prior, Atul Rana, Danielle Moosajee, Ed Watson, Richard Tock and Rachel Mahoney.

In response to a tweet about my workshop, Susan Russo (@Dsrussosusan) replied with a brilliant powers resource from YummyMath that I'd not seen before:
I love this. I've added it to my number resource library.

At the Tweet Up I did some filming for my 'Pronunciation Project'. Thank you to everyone who agreed to be filmed - such helpful people! Sorry about the parallelepiped...! I'll share the final video soon. I just need to work out how to edit it in the way I want it.
What I loved about talking to maths teachers about the way they say things is that I learnt some fascinating stuff. Esther (@MrsMathematica) teaches in Belfast and she told me that they say the word 'upon' to indicate there is a bracket present. So for example they say 4(x+2) as "4 upon x + 2". And they say (x+2)(x+3) as "x + 2 upon x +3" with distinct pauses in certain places to tell you it is double brackets. I had no idea! This is so interesting.

Watch out for my videos, coming soon!

JustMaths Conference
Like last year, the JustMaths Conference was wonderfully quirky and unique. The large audience of over 200 maths teachers stayed together all day instead of going to different workshops. There were presentations from the awarding bodies, Ofqual, me, Craig Barton, and the organisers Chris and Mel. I did a short session on resources, and was rather nervous given the size of the audience.

I absolutely loved the session Graham Cumming from Edexcel did at the end of the day - he talked us through some of the contextual questions from this year's GCSE exams and shared some hilarious student tweets.
After the conference we had some time in Alton Towers. I enjoyed a very pleasant ride on the rapids with David Faram, Craig Barton and other conference delegates. I overcame my fears and queued up for the new Wicker Man ride but it broke down because of the hot weather and we gave up in the end! I'll try that one again next year.
Laden down with my heavy backpack, I finally got home shortly after midnight on Monday night. I am exhausted. Teaching today was particularly hard work!

Thank you to all the conference organisers, and to everyone who presented. I got a lot out of both conferences. Also, thanks to everyone who kept me company over the weekend! It was great to catch up with so many lovely people. Special thanks to:
  • Megan Guinan for helping me out at the TweetUp where I took multitasking a step too far
  • Mariana Don Bosco for being such excellent company on the train to Manchester (Pythagorean triple proofs are my new favourite thing)
  • Tom Bennison for kindly filming my pronunciation videos
  • Rob Smith for running the MA stand and getting me a fabulous free t shirt from the guys at Texthelp
  • Craig Barton and David Faram for being my theme park buddies and forcing me to be brave!

I'm really looking forward doing a keynote at the BBO Maths Hub Secondary Conference next week. But until then, after my long weekend of conferences, I need a rest...!