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...!

10 June 2018

5 Maths Gems #90

Welcome to my 90th gems post. This is where I share some of the latest news, ideas and resources for maths teachers.

1. Probability Task
Thanks to John Rowe (@MrJohnRowe) for sharing this challenging probability puzzle. If you like this then you'll find similar tasks for a wide range of topics on the fantastic website openmiddle.com which I've blogged about before.

2. Exam Wrapper
Thanks to Alice Leung (@aliceleung) for sharing an exam wrapper for students to complete after an assessment. I like it that students are asked to reflect on whether they did sufficient preparation for the exam.
 3. Number Properties Puzzle
Here's a great number puzzle from @OCR_Maths.
4. A Level
If your Year 12s have internal exams coming up, you might find my revision quizzes for pure and statistics helpful. I've told my students to print off a load of these and test themselves at home until they get everything right. The statistics quiz has a huge number of definitions! 
There was a helpful discussion about A level taster lessons on Twitter this week, initiated by Adam Creen (@adamcreen). You can read the full thread here. I have found in the past that Pascal and Binomial works well. I've also used Susan Wall's lovely 'Find the coordinates' task successfully as a starter activity in A level taster lessons.
5. Euclid
Thanks to @MrsMathematica for sharing this video about Euclid - I enjoyed watching this.

Also check out The History of Non-Euclidian Geometry - Squaring the Circle and The History of Non-Euclidian Geometry - The Great Quest.

In case you missed it, I recently wrote a subject knowledge post about algebraic highest common factor and lowest common multiple. Writing this made me rethink my approach to teaching this topic. 

I also published three sets of breakfast warm up resources for both Higher and Foundation GCSE. It's been lovely to see so many schools using these to calm nerves and warm up brains on the morning of exams.

I also published a set of Year 3 topics in depth packs created by Nikki Martin - please share these with primary colleagues.

There have been a number of good maths education blog posts lately that are worth reading, including:

I'm looking forward to presenting at #mathsconf15 in Manchester in two weeks - my workshop is not just an opportunity to share loads of cool indices stuff, but also to explain the principle of planning and teaching topics in depth. In a world of quick fixes, I'm going in the opposite direction... I wrote a post about this last year.

If you're coming to the conference, print a copy of my #mathsconf15 bingo in advance and play along on the day.

Finally, did you see these awesome biscuits made by Ella Dickson (@elladickson) for her Year 13 students? Amazing!

7 June 2018

GCSE Breakfast Warm Ups: The Final Set

Following on from my previous two posts about breakfast warm up activities, I have now published the final set of resources here (Set B):

Breakfast GCSE Warm Ups - Calculator

As with the last set, there are two levels for Foundation and two levels for Higher. Each sheet contains 20 quick questions which I've not numbered so that students can work on them in any order. I hope these help students feel confident going into the exam next Tuesday.

These resources are suitable for all awarding bodies. I have designed them to be used every year for the foreseeable future - they are definitely not tailored for this year's exams. If you want to edit these resources to remove topics that have already come up in Papers 1 and 2, feel free - I have provided Word versions so you can do so.

Do keep an eye on Adam Creen's (@adamcreen) very helpful annual blog post for updates on resources specifically designed for next week's Paper 3.

Good luck! We're on the home straight now.

30 May 2018

Algebraic HCF and LCM

The topic 'Highest Common Factor and Lowest Common Multiple' is one where there's a lot of scope for subject knowledge development. There is so much for both teachers and students to explore in this topic. Ask a group of maths teachers how they find the highest common factor of two numbers and I expect a number of different methods will be mentioned. There is no universally accepted 'best way'. I wrote about a number of alternative approaches back in 2015.

Identifying Highest Common Factors
When I think of highest common factor questions, I think of questions involving integers. For some reason I've never taught a lesson on 'Highest Common Factor of Algebraic Terms'. On reflection, I think it might really help develop student understanding of the underlying concepts.

Look at these two examples:

Find the Highest Common Factor of 420 and 96
Find the Highest Common Factor of 10ab and 5a2b2

In the first case we want to find the greatest integer that is a factor of both 420 and 96. The most conceptually clear approach is to list all the factors of 420 and all the factors of 96 and then find the biggest factor that's on both lists. But it can take a while to list all the factors, so we often use primes to arrive at the answer more quickly. We write

420 = 2 x 2 x 3 x 5 x 7
96 = 2 x 2 x 2 x 2 x 2 x 3

Many students then use a Venn method to identify the highest common factor. I suspect that although students can follow the Venn procedure fairly easily, they may not really know what's going on. When teaching this from first principles, it's helpful to rewrite the factors like this

420 = 2 x 2 x 3 x 5 x 7 = 12 x 35
96 = 2 x 2 x 3 x 2 x 2 x 2 = 12 x 8

And then it's so much clearer why 12 is the highest common factor.

The extract below is taken from 'New General Mathematics' (first published in 1956 and still in use in the 1980s). We can see the conceptual understanding being embedded much more carefully than it is now. Whereas modern textbooks often jump straight to questions like 'find the HCF and LCM of 45 and 60', which requires a number of separate skills, here we can see isolation of the specific skill 'identify the common factors'. This seems a far more sensible starting point from a cognitive load perspective.

Students are not required to do the prime factorisation first, nor are they being asked to find both a HCF and LCM. They are simply selecting the common prime factors. In the first question, all three numbers contain 3 x 3, meaning they are all multiples of 9. There is no other factor that appears in all three numbers, so 9 is the highest common factor. Later in the exercise students will need to do the prime factorisation themselves, but not yet.
Of course, the idea is the same for the algebraic terms. But it's easier, because it's immediately clear what the algebraic factors are:

10ab3 = 2 x 5 x a x b x b x b = 5ab2 x 2b
5a2b= 5 x a x a x b x b 5ab2 x a

For numbers, finding a highest common factor can take a bit of work - it's harder to identify the factors of a number than the factors of an algebraic term. But the underlying mathematical concept is identical. Perhaps it might even be better to start with algebra before moving onto numbers.

Factorising with Highest Common Factors
Factorising an algebraic expression by identifying the highest common factor of two or more terms is relatively easy once you understand how index notation works.

10ab+ 5a2b2  = 5ab2(2b + a)

We don't often factorise in the same way with numbers, but it's fun to play with:

 Factorise 56 + 42

Here we take out the highest common factor, giving us 14(4 + 3). And this is nice, because we can now see that 56 + 42 is the same as 14 x 7. That's because it's four lots of 14 plus three lots of 14.

Here's another one:

Factorise 24 + 36

Here we take out the highest common factor of 12, giving us 12(2 + 3). So we can now see that 24 + 36 is the same as 12 x 5. Which is obvious when you think about it, because it's (12 + 12) + (12 + 12 + 12). It's fun to play with numbers like this.

There are plenty of fun numerical factorisation questions in textbooks, old and new.
Identifying Lowest Common Multiples
Identifying lowest common multiples using primes can be quite conceptually challenging. Students can be successful when using Venn methods and similar, but I bet few would be able to explain how their method works.

I like this introduction from a 1950s textbook for starting to build an understanding of what's going on:

What factors must necessarily be in a number so that 2, 5 and 7 will divide into it exactly?

What factors must necessarily be in a number so that 3, 2 x 3 and 3 x 5 will divide into it exactly?

What factors must necessarily be in a number so that 4x, 2ax and 6a will divide into it exactly?

I think it might be worth developing this as a starting point when introducing this topic.

At primary school, students learn to add and subtract fractions using lowest common denominators, but they may not realise that in doing so they are actually finding LCMs. I wonder if we could make better links to this prior knowledge.

Questions for Developing Fluency
This question came up in an AQA mock last year:
AQA GCSE Practice Set 4 (Paper 1H) 

When marking this I noticed that although I hadn't covered it directly with my students, most were able to work it out using their understanding of HCF and LCM.

Later I stumbled upon a huge set of questions just like this! The extract below is from Elementary Algebra for Schools, a textbook from the late 1800s that has been fully digitised.

It makes a lot of sense to develop fluency here, before moving onto related skills such as factorisation into single brackets and simplifying algebraic fractions.

The book takes this concept a lot further, looking at LCMs and HCFs of compound expressions including quadratics and cubics.
This chapter on algebraic lowest common multiples comes before the chapter on 'Adding and Subtracting Algebraic Fractions'. This makes a lot of sense!

When I teach adding algebraic fractions I don't directly teach the separate skill of finding the LCM of algebraic expressions beforehand. Perhaps I should. To add or subtract algebraic fractions like those shown below, there are a number of key skills:
1. identifying the lowest common multiple of the denominators
2. converting fractions so they share a denominator
3. adding the numerators once the fractions have the same denominator
4. simplifying
There's a lot going on here. We need to take these skills one at a time.
There are dozens more pages on algebraic highest common factor and lowest common multiple in this Victorian textbook. It was clearly a big part of secondary school mathematics in the 1800s.

There are similar activities in textbooks from the 1950s. They contain loads of practice on HCF and LCM of algebraic terms and expressions.
In stark contrast, there are only a handful of relatively simple questions in a modern GCSE textbook on the same skill:

It's funny how things change.

I hope you've found this useful, or if not useful then at least interesting! If you have anything to share regarding algebraic highest common factor and lowest common multiple, please tweet me or comment below. Thanks for reading.