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.

27 May 2018

More Breakfast Warm Ups

I was really chuffed when loads of people told me that they used my non calculator breakfast revision resources last week. Although I collate and share a lot of resources, I don't often make my own (because it's ridiculously time consuming!) so I'm glad it was worth the effort. We used them with our 240 Year 11s at my school and they worked well.

A few people asked me to make another set of breakfast revision resources for the next paper, so I have done so. You can download them from TES here:

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've provided editable versions in case you want to tailor these for your own students. Answers are provided too.

I've named these 'Calculator A' because I hope I might have time to make a third set ('Calculator B') to use on the day of Paper 3. 

Paper 2 and 3 Preparation Resources
We are very fortunate in maths that lots of helpful teachers provide mid-GCSE resources to help students prepare for Paper 2 and 3. These are created after Paper 1 so any topics that have already come up are removed. Of course, there's no guarantee that these topics won't come up again in Paper 2 and 3, but I think it's still worth doing. On Friday I emailed all of our Year 11s with a link to John Corbett's Paper 2 Preparation Materials along with Pixl's Practice Papers. 

There are loads more Paper 2 and 3 preparation resources - the best place to see the full collection is in Adam Creen's (@adamcreen) very helpful annual blog post where he lists everything as it's published (this now covers Edexcel, AQA and OCR).

And don't forget that if you're seeing Year 11 for a revision lesson during or after half term then my GCSE revision resources post has dozens of excellent resources. 

23 May 2018

5 Maths Gems #89

Welcome to my 89th gems post. This is where I share some of the latest news, ideas and resources for maths teachers.

1. Boss Maths
BossMaths.com is a great maths website that I haven't seen before. It has over 200 lessons matched to GCSE specification references, each with examples, escalating practice exercises, misconceptions, and exam-style problems. Some topics also have worked example videos. There's loads of great stuff here.

All of this content is free. Teachers can easily use the exercises and examples in their lessons. I can imagine this website might also be helpful if your students have tablets or laptops in their lessons as they can access the content directly without a login.

Here's an example of an SSDD style task for percentages.
And here's an example of a challenge question from the topic 'arcs and sectors of circles'.
And here's an example of an exercise from 'Algebraic manipulation involving surds and fractions'.
For some of the new GCSE topics including Systematic ListingGrowth and Decay, Equation of a Circle, Iteration and Inverse Functions, it's really helpful to have a set of examples, exercises and challenge problems all in one place. I'll add these to my resource libraries.

The creators of BossMaths.com are going to make extra premium features available on subscription. This includes worksheets, activities and answers.

2. Year 12 Revision
My Year 12 mathematicians have internal end of year exams in June. To help them prepare I'll be using the 'Question Countdown' revision resources from CrashMaths. A set of four revision questions was published each day in the 10 days preceding this year's AS exams, so that's 40 revision questions ready to use.
Also, creator of new website Naikermaths.com has taken legacy specification exam questions and edited them to make them suitable for the new A level specification. For example when teaching Hypothesis Testing it was useful to have a set of Binomial Hypothesis Testing questions from S2 that weren't mixed in with Poisson questions. It was also useful to have a set of Binomial Distribution questions with references to expectation and variance removed. This website is under development so it's worth following @naikermaths for updates.

3. Probability
I really enjoyed this excellent 10 minute video, shared by @natkin. It's about the probability of winning fairground games and it's really worth a watch. Fascinating stuff - I learnt a lot from this!

4. Apps
Thanks to @mhorley and @DrPMaths who discovered a fantastic number line tool. Have a play with this - it's great.
The other apps on this site are also worth exploring - they're very user friendly and no download is required. It looks like the fractions tool might also be useful for demonstrating bar modelling as you can instantly create and shade bars split into parts.

5. Circle Puzzles
Thanks to Catriona Shearer (@Cshearer41) for sharing a challenging set of puzzles. The aim is to work out what fraction of each circle is shaded. The 12 dots are equally spaced, and the only point used inside the circle is the centre.
The original puzzle (bottom centre) is by @HenkReuling - check out Gems 86 for more of his great puzzles.

I produced a new set of resources for breakfast revision for the non calculator GCSE exam - if you missed them, you can download them here. I hope to make calculator versions over half term. I have continued to add new resources to my GCSE revision post so do check it out if you're seeing Year 11 again this year.

I'll be doing a keynote presentation on Thursday 5th July at the BBO Maths Hub Secondary Conference in High Wycombe. Bookings are now open. I will also be speaking at #mathsconf15 in Manchester and the JustMaths Conference at Alton Towers. I hope to see lots of you at these events - do say hello! For a full listing of summer term events, see my conferences page.

I recently bought an 'absolutely' awesome new maths t shirt. Loads of people asked me where I got it so I thought I should share that here: check out spreadshirt.co.uk for a great range of good quality maths stuff.

If you haven't seen it yet, do read Ben Orlin's post 'The State of Being Stuck'. It's fantastic.

Good luck with the rest of exam season! And happy half term.

17 May 2018

GCSE Breakfast Warm-Ups

Year 11s have already gone off on study leave at many schools, but I still have my lovely Year 11 class in lessons (when they're not in exams) and revision sessions. At this time of year I stop my structured topic-based revision lessons and instead use mixed topic resources like these Higher GCSE workouts. These workouts take a full lesson and consolidate a wide range of key skills, with the added bonus of not requiring much printing.

At my school we have breakfast revision timetabled before every GCSE and A level exam. It's not compulsory but quite a few students like to attend. I see it as a good opportunity to warm up their brains, answer any last minute questions, and wish them luck.

Last year I published some GCSE Breakfast Workouts for teachers to use in these short revision sessions. I've now created another set of Breakfast Warm-Ups - including two levels for Foundation and two levels for Higher.

You can download these resources from TES here.

Each sheet contains 20 quick questions designed to get students' brains into maths mode - hopefully this will minimise the risk of them making silly mistakes in the first few questions when they start their exam an hour later. I've intentionally not numbered the questions so that students can work on them in any order. The last thing we want to do is to cause a panic just before the exam by giving unfamiliar questions.

Thanks to Mark Moody at Caldew School for creating and sharing a helpful PowerPoint for the first Foundation warm up, and to Japleen Kaur (@japleen_kaur1) for making a PowerPoint for the Higher warm ups.

I hope these resources are helpful. I've provided editable versions in case you want to tailor these for your own students. I've also added these resources to my GCSE revision page. Good luck everyone! Nearly there...

5 May 2018

5 Maths Gems #88

Welcome to my 88th gems post. This is where I share some of the latest news, ideas and resources for maths teachers.

1. Show That
These mixed topic 'show that' questions created by Josh Evans (@JevansMaths) and adapted by Catriona Shearer (@Cshearer41) are useful for revision. GCSE students often struggle to understand what is required in a 'show that' question. See Josh's tweet and Catriona's tweet for more like this.
2. Plans and Elevations
I always enjoy receiving my monthly email from MathsPad about their new resources. This month they featured a lovely free interactive Plans and Elevations tool, and (for subscribers only) a number of accompanying resources.
3. Foundation Key Skills Booklets
Mr Knowles (@SK18Maths) shared a set of three booklets focusing on key Foundation GCSE skills (algebra, number and data). I've already found the algebra booklet helpful - I used it with an intervention group I work with at another school. Each page has a 'personalised questions' section so I think these would be useful for one-on-one tuition sessions too.
4. A Level
A few A level resources have been shared on Twitter over the last couple of weeks. First, this integration challenge shared by Luciano Rila (@DrTrapezio):
And similarly, a great collection of 'Integration Lookey-Likeys' from Jonathan Dunning (@WaysWithMaths). Check out his blog post for more like this. I'll use these in C4 revision lessons.
If you subscribe to MEI's Integral, check out the new Rich Tasks for Further Maths, written by Jonny Griffiths (@maxhikorski). MEI will be releasing the full set of RISPs for Further Maths over the coming weeks.

I'm sure you're already familiar with RISPs for A level as they've been around for a long time, but if not then do check out the collection at risps.co.uk.

5. UKMT Wallpaper
Thank you to Jon O'Neill (@JonONeillMaths) for a lovely idea. He suggested that we hold onto the workings from last week's UKMT Junior Maths Challenge and use it as backing for corridor displays. After my students completed the challenge I collected their workings and it was fascinating to look through them! It's lovely to see real mathematical thinking.

In case you missed it, to mark the fourth anniversary of my blog I recently published my annual gem awards. This post featured a couple of new resources, including Jonathan Hall's great new Recall and Recap Quiz.
I've been so busy at school and home lately, I haven't had time to finish off a number of blog posts that I've had in draft for a while. But hopefully things will calm down soon. As I mainly teach exam classes, the second half of the summer term will be quieter for me in terms of teaching, but like everyone else I will have lots to do both inside and outside of school - I'll be attending a number of events in June and July and also working on my big summer project (watch this space!).

In other news from the last couple of weeks:
  • One of Ed Southall's puzzles randomly went viral, appearing in hundreds (perhaps thousands) of newspapers, magazines and websites all over the world. 
  • Don Steward published loads of great resources including a set of GCSE ratio questions that will be really helpful for revision over the next few weeks.
  • AQA revised their Level 2 Certificate in Further Maths qualification. First exams are in 2020. This is a popular choice for schools looking to extend their strongest Year 11s.
  • Tom at PinPoint Learning launched a new free tool to create personalised revision packs.
  • The Primary Maths Challenge 2018 is now open for entries. This lovely competition for Year 5 and 6 takes place in November. 
  • Rachel Mahoney (@RachelMahoney14) shared an adapted version of an idea I shared in Gems 87. This is a retrieval practice starter with the addition of a Corbettmaths Conundrum. I like this tweak. Read more about this on her new blog.
Finally, I'll leave you something mathematically delightful from Christian Lawson-Perfect (@christianp). He's the one who made the awesome and addictive Is This Prime? game. Christian has now made a fun online tool which performs a cool trick: writing any number as the sum of three palindromic numbers. Have a go now! You can read more about it in The Aperiodical.

Have a lovely sunny long weekend!

27 April 2018

Gem Awards 2018

This week it's resourceaholic.com's fourth birthday! It's become a tradition for me to mark the anniversary of my blog by publishing an annual 'Gem Awards' post. Here I look back at all the ideas I've shared in my gems posts over the last year and choose some of my favourites.


1. Best New GCSE Resources
The award for best new GCSE resources goes to David Morse for his collection of 'Maths4Everyone' resources for GCSE topics. I've been using these a lot in my Year 11 revision lessons over the last couple of weeks.
I like the clear format and well written questions. Check out his set of exam-style quadratic 'show that' questions to see the kind of excellent resource David shares.

2. Best New A Level Support
I'm very grateful to everyone who has shared resources to support the teaching of the new A level. The award in this category goes to MarkIt, a website that I first featured in my gems posts and later wrote about in my BCME 9 Reflections. I love this website for its well written A level questions (which are freely available), its high quality functionality for setting A level homeworks, and its helpful blog about teaching A level. I look forward to seeing how this website develops.
Honorable mentions in this category go to Dr Frost (who has produced slides for every topic in A level maths, plus useful resources for the ClassWiz calculator and large data set), Dr Madas (who has created new IYGB papers), and to CrashMaths (who have shared excellent new A level practice papers).

3. Best New Primary Support 
This award goes to John Corbett for his new primary 5-a-day tasks. 5-a-day is already widely used in secondary schools and it's great to see the collection expanding further. John has also started to share new primary worksheets.
Honorable mention to Nikki Martin, who attended one of my workshops last year and was inspired to make a set of 'Topics in Depth' packs to support primary school teachers. These packs cover approaches, methods, tools and resources for all the topics taught in Year 6 and Year 2. Nikki has kindly shared the whole collection for free here.

4. Best New Book
This award goes to Craig Barton for his book 'How I Wish I'd Taught Maths'. A thoroughly enjoyable read, it brings us honest reflections and research-based recommendations on how to teach maths more effectively. The book been incredibly popular and seems to have encouraged a large number of teachers to try new things. Do read it if you haven't already done so. It's fantastic.
5. Best New Puzzles
This award goes to 'consecutive number puzzles'. The first example I saw was shared by Chris Smith in his weekly maths newsletter.
Similar activities include Jonathan Hall's interactive activity 'Consecutive Number Types' and MathsPad's free worksheet 'Consecutive Chains'. These are lovely, accessible activities for Key Stage 2, 3 or 4 - they're great for exploring number properties.
6. Best New Revision Resources
This award goes to the website Access Maths. Grant has shared a large collection of GCSE revision activities including an A3 grid of 100 questions which worked perfectly as a cover lesson for my Year 11s in the run up to their December mocks.
Honorable mention to Mel at JustMaths for her excellent 'Preparing for your exam' slides.

During exam season, don't forget to check out my collections of GCSE revision resources and A level revision resources.

7. Best New Online Tool
This award goes to MathsPad for their interactive tool for transformations. The tracing paper overlays work really well here, enabling the teacher to demonstrate transformations while pupils complete the matching questions.
MathsPad's free constructions tool and graphing tool are also worth exploring.

8. Best Display
Recent research suggests that displays in the classroom may lead to cognitive overload. Whilst minimising distractions makes total sense, I've taught in dingy, display-free classrooms before and I think we have to get the right balance here... Dozens of facts and formulae stuck around the whiteboard at the front of a classroom may well be problematic, but I certainly don't think that we should totally strip our departments of the wonderful displays that get students talking about mathematics. Displays in the maths corridor and at the back of the classroom can really enrich students' maths education. I often see students gathered around posters outside my classroom, deep in conversation about the maths on display.

So, though some of you may have become a bit anti-display in recent months, I'm still a fan. The winner of this year's best display award is the lovely Maths Word of the Week posters by @missradders.
And honorable mention to @jaegetsreal for her First 1000 Digits of Pi Posters and her post about the wonderful conversations this display has sparked between students.
I have a large collection of lovely display resources on my displays page. If only I had time to put them up!

9. Generosity
The generosity award goes to Rob Smith, a maths teacher from Leicester who works tirelessly to support the maths education community. Amongst other things, Rob runs branch events for the ATM and MA, masterclasses for students, and a charity tuck shop at conferences (which I love!). Whenever I see him at MA Council meetings he is full of kind words and encouragement. Rob is incredibly thoughtful and is always going above and beyond to make people happy. Thank you Rob!
10. Lifetime Achievement
This year my lifetime achievement award goes to Jonathan Hall, creator of awesome websites including MathsBot.com and formtimeideas.com. I've blogged about MathsBot extensively before, but since then tonnes of new features have been added including a library of virtual manipulatives, an interactive number line and a place value tool. His most recent addition is a great Recall and Recap Quiz which might work well in form time in the run up to GCSE exams.
Considering the rate at which Jonathan produces new material, and how incredibly responsive he is to teacher requests, he must work around the clock. Barely a week goes by where Jonathan doesn't share something new. He is creative, smart and generous, and on behalf of all maths teachers I'd like to thank Jonathan for everything he does.


That's it for the 2018 Gem Awards! Thank you to everyone who tweets about what they've tried in their classroom. It's so inspirational. Congratulations to all winners. Apologies that I can't afford to send you actual prizes. I'd really love to hold a glitzy award ceremony one day!
I know I've failed to mention a great many awesome people and resources in this post. If you visit my Gems Archive you'll find an index of 87 gems posts - they are all full of great ideas and resources. Also bear in mind that I've been careful not to give an award for the same thing twice so it's worth checking out the Gem Awards 2017, Gem Awards 2016 and Gem Awards 2015 to see who has won awards previously.
Happy 4th birthday resourceaholic.com. Thank you to my readers for all your support!