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.

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!

18 April 2018

5 Maths Gems #87

Welcome to my 87th gems post. This is where I share some of the latest news, ideas and resources for maths teachers.

1. Regular Recall
Starting a lesson with mixed topic questions seems to be fairly common practice these days. Many teachers use resources like Corbett Maths 5 a Day. Taking this idea a step further, some teachers on Twitter have recently shared examples of tasks where students are asked a question on what they did last lesson, last week, last term and last year. Here's a great example from @MissBanksMaths.
And here's an earlier example from @JonONeillMaths, who was inspired by the original idea in @87History's blog post about retrieval challenge practice grids.
This is one of those rare ideas that works well in most subjects. Here's an example for physics by @alexpboulton.
2. GCSE Papers
I've blogged about CrashMaths (@crashMATHS_CM) before. Their website is a good source of practice exam papers for both GCSE and the new A level. They've recently added a new set of Edexcel-style Higher GCSE papers (Set B), which includes this nice non-calculator question:
They've also recently added a couple of Higher AQA-style papers to Set A and some helpful GCSE worksheets.

Another useful set of resources on CrashMaths is for the Edexcel A level large data set. This includes an information and guidance document for students and six practice questions.

3. Goal Free Problems
If you've read Craig Barton's book 'How I Wish I'd Taught Maths', you'll already be familiar with Goal Free Problems. In Chapter 4, which is all about focusing thinkingCraig explains that while most exam questions are goal-specific, he now makes use of goal-free problems in the early knowledge acquisition phase. He also uses goal-free exam papers to kick-start the revision process. If you haven't already done so, read Craig's book to see some examples of these problems and to fully understand the goal-free effect and how it relates to Cognitive Load Theory.

Thanks to @MrMattock who has now created a free website - goalfreeproblems.blogspot.co.uk - which shares a large number of these problems for both Higher and Foundation tier. In each of these exam questions, the actual question has been removed and replaced with the words 'Work out what you can from this information'.
4. What Went Wrong
Thanks to Year 6 teacher @MrBoothY6 who has shared a large collection of common maths misconceptions on TES:

See my misconceptions page for more resources relating to common misconceptions.

5.  GCSE Revision
@AccessMaths has been busy making resources - his latest revision resource 'Progressive Overload' covers a number of key algebra skills and works well printed on A3. @podroberts helpfully worked out the answers too!
Also check out his new 'Fill in the Blanks' graph revision resources.
For more GCSE revision resources check out my GCSE 9-1 Revision post. I also have an A level revision post for the legacy specification.

My 300th blog post was 'New GCSE: Bounds' - in this post I took a close look at the GCSE specification and resources for this topic. Before that I wrote 'BCME 9 Reflections' which included slides from my recent workshop 'Ideas that transformed my teaching'. Next week is the fourth anniversary of my blog, which means it's time for my annual gem awards!

Here's some other news that you might have missed:
  • MEI have archived all of their Monthly Maths and M4 magazines and categorised their classroom resources by GCSE topic here.
  • Tickets for the JustMaths conference are on sale. It takes place at Alton Towers on Monday 25th June (I'm looking forward to presenting at this one!).
  • I've added a few new events to my conferences page - including Don Steward presenting at an MA/ATM event in London in June.
  • I love MathsPad resources and this excellent new similarity proof resource is no exception! I've added it to my resource library.
  • For legacy further maths revision materials, check out drolivermathematics.com. Thanks to @gismaths for sharing this website with me.
  • If you teach A level maths in the London area, do take your students along to the IMA 16+ Lectures on 28th June at UCL - the programme is fantastic. 
  • If you're an MA member and willing to help out on the MA bookstand at #mathsconf15 in Manchester on 23rd June, please get in touch.
  • Thank you to Hannah Fry for sharing @oliviawalch's wonderfully illustrated "Some Myths about Math". 
I'll leave you with this excellent maths joke, created by @treemaiden and illustrated by @aap03102. Have a great week!

13 April 2018

New GCSE: Bounds

I like the bounds content in the new GCSE. I think that the introduction of error intervals has provided some clarity. Previously some students just couldn't get their head around why we use 3.65 as the upper bound for 3.6 ("but Miss, 3.65 rounds to 3.7. This doesn't make sense!"). The use of inequality symbols (in conjunction with instruction using number lines) really helps with conceptual understanding here.
3.55 ≤  x < 3.65

Of course error intervals aren't new - they just weren't previously assessed at GCSE. Upper bounds and error intervals are clearly explained in this extract from the CIMT MEP textbook chapter on Estimation and Approximation from the late 1990s:
I also like the inclusion of truncation in the new GCSE specification, because it means that students need to think more carefully before answering a bounds question. Previously, bounds questions at GCSE were predictable, and students only really needed a superficial, procedural understanding of the topic. Things have changed.

Age is a good way to teach truncation, given that students will already be very used to truncating their own age, rather than rounding it to the nearest integer.
Ed is 36 years old. His age is represented by x. Give the error interval for x.
 36 ≤  x < 37
Jemma Sherwood has written a helpful post about truncation resources.

Discrete Bounds
My Year 11s really struggled with a bounds question in their mock exams this year. It's from a recent AQA paper so I can't share the actual question here, but this question is similar:
Two integers are rounded to the nearest 10. The rounded numbers are added to give to 40. What's the maximum total value of the orginal numbers?
Let's say that both numbers were 20 when rounded. The maximum each number could be is 24, so the highest possible total is 48. The common mistake here is to take the upper bound to be 25, giving an incorrect final answer of 50. So why was this such a common mistake? I expect it's because most of us teach bounds in a context of measurement, not counting. We spend a lot of time on continuous bounds, and very little (if any) time on discrete bounds.

Interestingly, my GCSE textbook doesn't contain a single question on discrete bounds. A bit of internet searching shows that BBC Bitesize has one question on discrete bounds:
The number of people on a bus is given as 50, correct to the nearest 10. What is the lowest and highest possible number of people on the bus?
The OCR Check-In Test on Approximation and Estimation has this question:
 Explain why the error interval of 400 cars to the nearest 50 cars could be written as  375 ≤  c  424 or 375 ≤  c < 425.
and the AQA Rounding Topic Test has a couple of discrete bounds questions, including this one:
Two performances of a show are each attended by 175 people, to the nearest 5. Work out the maximum possible difference between the numbers of people attending.

It's worth noting that the Government's GCSE subject content doesn't specifically refer to discrete bounds. This is the official content for bounds at GCSE:

"15. round numbers and measures to an appropriate degree of accuracy (e.g. to a specified number of decimal places or significant figures); use inequality notation to specify simple error intervals due to truncation or rounding

16. apply and interpret limits of accuracy, including upper and lower bounds"

OCR's specification does mention discrete bounds though, saying that students must "Understand the difference between bounds of discrete and continuous quantities".

AQA's excellent Teaching Guidance has this:
"Upper bounds do not necessarily require use of recurring decimals. For example, if the answer to the nearest integer is 7, the maximum could be given as 7.5, 7.49.... , or 7.49.
If this value of 7 represented £7, £7.49 would be expected for the maximum.
For continuous variables, students may be asked for the lower and upper limits rather than the minimum and maximum values."
The money example here is interesting - it's not something I have explicitly covered with my students. Don Steward has a good bounds exercise including money questions - I'll  make sure I use this next time I teach this topic.

Edexcel doesn't specifically refer to discrete bounds in either their specification or supporting material, but of course that doesn't necessarily mean they won't come up in an Edexcel exam.

Exam Questions
Bounds questions involving calculations can be fairly challenging for students, particularly as sometimes it's not immediately obvious that the question involves bounds. Here's an example from an old Edexcel Linked Pair paper:
"Sian is driving on a motorway.
Sian drives for 2.8 miles, correct to the nearest tenth of a mile.
It takes her 200 seconds, correct to the nearest 5 seconds.
The average speed limit on this part of the motorway is 50 miles per hour.
Did Sian drive at a speed within the average speed limit? You must explain your answer."

Another type of challenging GCSE question is one that says "by considering bounds, work out the value of x to a suitable degree of accuracy, justifying your answer". For examples of questions like this, see Dr Frost's 'Full Coverage: Bounds' resource which has an example of every different question type from past Edexcel papers.

I've already mentioned Don's excellent resource, the CIMT resources and Dr Frost's full coverage GCSE questions. For a comprehensive list of bounds resources, see my resources library.

It's worth noting that John Corbett has helpful videos, textbook exercises and practice questions on limits of accuracy and limits of accuracy: applying. And Edexcel's new content resources include a helpful worksheet on bounds.

I'm doing a revision lesson on bounds with my Year 11s on Monday and will be using this excellent bounds GCSE revision resource from Maths4Everyone. 

For Interest
Finally, in my research into how mathematics was taught in the 18th, 19th and 20th centuries, I've found few references to rounding. However, in 'Practical Mathematics for All' (McKay, 1942) I found this nice explanation of 'limits of error':

If anyone knows of any earlier references to upper and lower bounds, I'd love to see them.

Thanks for reading! This was my 300th blog post, so a bit of a milestone for me. To read other posts about teaching specific GCSE topics, you can view my topic collection here.