Mathematical Miscellany #2

(Note this will not be the format each week, it just happens to be appropriate this week!)

Click on the image and move that slider!

Last week in Mathematical Miscellany #1 I included a reminder that Valentines Day is approaching – save your money and send a Desmos math-o-gram!

This morning I was reminded once more of the value of a quick peek at Twitter (it’s a bit like a lucky dip!) when I came across Chris Smiths’ tweet on his lovely Valentine relay race which you can download with his other relays from TES Resources.

Valentine Relay - Chris Smith

Valentine Relay – Chris Smith

Dot to dot
I read an article from Science Alert stating that Australian researchers have discovered that school children fare better at solving maths problems when they trace their fingers over practice examples, outperforming students who simply read the questions without touching them.

Polar dot to dot

Polar curves – join up the dots (in the correct order!)

Well that’s certainly easy to try and in fact reminded me of my love of dot to dot as a child. Some years ago I created some polar curves for my students. So Year 13 can try these this week, they can work out the correct order to join the dots themselves and trace out those curves! (Join the dots 4 curves for the file in case anyone wants it). These were created with Colderado University’s Mathematical Visualization Toolkit note you can download the application if Java causes you problems trying to use the application online).

Dotty Desmos

Desmos – join the dots

You can easily create some graphs in dots on Desmos – see Desmos dot to dot. Here’s a dotty version of f(x) =x3.

On the subject of polar curves, with Year 13 last week we strayed from the simpler curves to rather more sophisticated ideas; they suggested all sorts of curves we could try – great fun and way off the specification; but then I think they understand the specification content better barbecue we go beyond it where appropriate. Thinking about their suggestions I have something I think they will like this coming week, look at r=acos(kcosθ)!


Note the use of the slider which enables students to see how the curve is traced out as the angle increases. For more such polar resources – see Polar Graphs on Mathematics for Students).
(Polar curves – always remind me of a childhood favourite toy – Spirograph).

Thinking about sophisticated curves lead me to this rather nice publication : Fifty Famous Curves from the Department of Mathematics, Computer Science, and Statistics Bloomsburg University.

Problems & Activities
The Problems and Activities page has been updated with several new entries – so many wonderful free high quality resources – a massive thank you to the teachers who willingly share their work.

Rich Questionsquestions, a popular page has been updated with the addition of the Multiple Choice Questions collection.



Mathematical Miscellany #1

Just over three years ago I started writing compilation posts, a kind of newsletter style – a mix of mathematical goodies. I was inspired by Doug Belshaw’s Thoughts This Week – now Thought Shrapnel. It’s been on my blog to do list to revisit this idea again; how appropriate that Doug’s Thought Shrapnel today includes newsletters he enjoys. I was very happily distracted by some of these great compilation posts this morning and particularly struck by a post from Doug’s first suggestion Austin Kleon – important for any blogger, What to leave out and what to leave in. Some great quotes there including Elmore Leonard’s:

“Try to leave out all the parts readers skip.”

(Wondering which bits you are all skipping here!)

So I decided it was time for a new title for posts of this kind – hence ‘Mathematical Miscellany’. Previous compilation posts can be found under the category (note the drop down menu on the right for categories) Mathematical Miscellany.

In class
With my Year 10 GCSE class I used some ideas from a site I often use: Mudd Math Fun Facts. Squares Ending in 5 and Multiplication by 11 both made excellent starters, we looked at proofs as well as enjoying the mental Maths tricks! You will find more lightening arithmetic suggestions on the site.

OCR Check In TestsOCR have published some very useful new resources; I have written before on their Check in tests. New tests for Foundation and Higher have been added. The Higher resources include Triangle mensuration, Language of functions and Algebraic expressions; these really are excellent resources and I will definitely be using all of them.

The GCSE New Content pages are updated regularly, note all the further resources at the end of that page, including from the wonderful Just Maths, GCSE questions by topic.

Published this month we have the KS2 Mathematics 2016 teacher assessment exemplification, and also the KS1 version. Clearly teachers of secondary age children need to know what students are covering in Primary school. These documents have been added to the GCSE New Content page.

If you read just one blog post this week – from 2014, but Tom Sherrington’s 10 Silver Arrows: Ideas to penetrate the armour of ingrained practice is as valid now as it was then. I think these ideas are important for any teacher to reflect on – I’m pleased to see 6, a subject I have often written on (see Mini Tests).

Tests are good.

…and finally
It’s coming up to that time of year again, I see Valentines cards in the shops, save your money and send your loved ones (or anybody!) a math-o-gram!

Click on the image and move that slider!

Click on the image and move that slider!

In what happily seems to have become an annual tradition Desmos have provided you with the means to send a math-o-gram to the mathematicians in your life!
Desmos Valentine instructions

Geeky people you could even use the Desmos API …

Valentine’s Day seems an appropriate time to express love for Desmos!

Elsewhere – express your feelings for WolframAlpha!

I Love YOU

Transum Valentine Puzzle
and here’s a logic starter from Transum for Valentine’s Day!

Mathematics Stories

MEI M2 question

MEI M2 question

Working with my Further Mathematics students on a work / energy question based on the Jack and Jill nursery rhyme started a lively discussion on what other stories and rhymes we could turn into Mechanics questions! We all immediately thought of Humpty Dumpty! I think the examiner was having fun with this one – the question carries on with poor Jack falling over and hurting himself. Jill seems to get off lightly unlike the original! If anyone is interested this is question 4 on the June 2007 MEI M2 paper.

This theme also reminded me of a long forgotten resource I used to use with Year 7 – The Three Little Pigs as a mental test – this is definitely coming out again!
I found this many years ago on as a free sample.

Three Little Pigs

I cannot mention Mathematics stories without reminding everyone of  ‘John and Betty’s Journey into Complex Numbers‘ which has always made me smile.

John & Betty’s Journey into Complex Numbers

Staying with the subject of Complex Numbers, with the same students, another discussion we had recently when looking at Complex Numbers was the need to write clearly, a z can look like a 2 for example. See Mathematical Handwriting on Mathematics for Students. Also on Mathematics for Students I have updated the page giving links to A Level Exam papers to include some really useful solutions resources such as David Smith’s worked solutions.




Multiple Choice Questions

Daisy Christodoulou on Life After Levels - conclusion

Daisy Christodoulou on Life After Levels – conclusion

Daisy C Multiple Choice

Daisy Christodoulou on Multiple Choice Questions

From researchEd 2015 and highly recommended, Daisy Christodoulou discussed the removal of National Curriculum Levels. Read Daisy’s post and see her slides on her blog you can also watch the presentation.
Note – researchEd Maths and Science 2016 – I already have my ticket!)

In that presentation Daisy discussed the use of Multiple Choice Questions, something I have always liked using in my teaching. Daisy’s discussion  of making questions harder by changing the number of correct answers reminded me of the, in my opinion excellent, Multiple Choice A Level papers which the then London board included in their A Level Mathematics examinations (London Syllabus B). I can see see some old papers in the ‘Very Past Papers’ section of the Edexcel emporium but sadly not Paper 1, the Multiple Choice paper. Fortunately I have print copies which I can use and I think a worthwhile project would be to use some of the question types from those papers in lessons, something I will work on.

In the meantime, happily we do have access to some Multiple Choice questions online and I thought it would be useful to bring them together in one place.

Diagnostic Questions

Diagnostic Questions

I have written several posts on the excellent Diagnostic Questions site which hosts thousands of multiple choice questions written by teachers designed to address student misconceptions. I have found it worthwhile to discuss the wrong answers with students so we are all aware of the kinds of mistakes it is easy to make. It is a site I use regularly, teaching Year 10 about the equation of a tangent to a circle at a given point for example, I created a Desmos page and a Diagnostic Questions Quiz using questions on circles (centre the origin) from Diagnostic Questions.
(pdf: quation of a Circle & Gradient of Tangent).

Note the various collections on the site including those from AQA and OCR to support the teaching of GCSE Mathematics.

A Level topics are also covered on Diagnostic Questions; see for example this quiz I created on  Logarithms and Exponentials.

Ben Cooper Resources

Mega Maths Quiz from Ben Cooper

For a superb collection of Multiple Choice questions, great for starters, plenaries or any time, head for Ben Cooper’s resources; see for example Mega Revision from Ben Cooper and 30 Number StartersBen’s website is coops-online where he shares very high quality resources. Resources (all free) by Ben can also be found on TES.

Don Steward Multiple Choice

Don Steward – Rearrangements Multiple Choice Quiz

Don Steward has so many outstanding resources on his Median blog, including multiple choice quizzes, see this on rearrangement for example. Note that he also has a separate blog for GCSE practice resources. many of these quizzes are multiple choice.

UKMT Extended Solutions

From UKMT, the UK Maths Challenges provide a wonderful library of multiple choice problems; note the these past papers where you will find questions and full solutions. Check also the Individual Challenges pages where you will find details of the challenge and see sample materials. Check the Junior Challenge for example. Note the addition for 2015 of extended solutions.

mathisfunMathisfun has an extensive library of very clear diagrams and explanations and also multiple choice questions for most topics. Use the Index to find the topic you want and note that for most topics you will see some questions at the end.

mathisfun multiple choice

mathisfun – Simplifying Square Roots – select image

As you can see from the Index all ages are catered for including older students; I have used the clear explanations and questions on finding the inverse of a 3×3 matrix with Further Mathematicians for example. Once you have selected an answer a complete solution is provided. Note the Question Database – some exploring to do I think!

PowerPoint Millionaire

The Who Wants To Be A Millionaire format can provide a fun way to present Multiple Choice quizzes. A Google search returns various resources and of course provides you with PowerPoint Millionnaire templates you can adapt for your own use. Some highlights from that search, the Primary Resources files use a simple and clear format and William Enemy has described a resource on Great Maths Teaching Ideas which also uses that template; such a good idea to have all students answer all questions and add up their winnings! Another example – Algebra on TES.

For a alternative and slightly more sophisticated template see PowerPoint Games which includes templates for several games including Who Wants To Be A Millionaire?

Oxford Maths Admissions Test

Oxford Maths Admissions Test

FDesmosor an extensive collection of Challenging questions for Advanced Level students we can turn to the Oxford Maths Admissions Test, note the menu on the right hand side of the page, many papers and also solutions are available. I can never resist a quick graph on Desmos but note the very comprehensive solutions provided.


That QuizThere are various options for creating your own multiple choice questions.That Quiz is simple but effective – all free and no adverts. There are many quizzes already available on a variety of topics, it is also possible to create your own quizzes. Teachers can register and add classes if they wish. You can search the many quizzes available, searching for Fractions for example led me to this quiz.

Testmoz Directed Numbers


For an alternative way to set up a simple quiz try Testmoz. No registration is required. This has been written by Matt Johnson, an undergraduate student – the instructions are all very clear and you can check out the FAQ!  (I love those FAQ! For example: I lost my quiz URL can you retrieve it for me? Answer: No). Try this test on Directed Numbers – log in as a student, the passcode is cy090610

More to Explore!
British Columbia exam question
In Daisy’s talk mentioned at the beginning of this post she mentioned British Columbia questions in her discussion of multiple choice questions, a quick search led me to this Pre-Calculus paper for example; I also found a site I’ll return to have another look at: Mathematics 30-1.

Finally, from Australia, New South Wales Government, we have Multiple Choice Questions for the School Certificate and Higher School Certificate.
Updates: Thank you Mark Greenawaymore resources! 
MG Twitter

Algebra – expand three brackets

Following a recent test for Year 10 (UK age 14-15) I wanted to use colour to show clearly a way to make sure all terms have been included when three brackets have to be expanded (new to the UK GCSE qualification). (Colour can be so useful in Mathematics – something I have written on before).

Expand Brackets Colleen Young

I have created this example using Excel. This is just a simple example with positive coefficients only; I wanted to start by being clear on making sure that all combinations are included. There are various FOIL and quick methods available if you search but my own preference is a more systematic approach. An advantage of using Excel is the ability to trace precedents (Formulas menu).
Excel trace precedents

The numbers could be changed and of course any examples could be made up and checked on WolframAlpha.
WolframAlpha brackets

I have also collated some Algebra resources (many topics) – examples and exercises on Mathematics for Students. Note the very clear workbooks from Plymouth University

Functions - Plymouth University

Functions – Plymouth University

New Year Resolutions

Resolutions for Maths Teachers PowerPoint file
or as a pdf Resolutions for Maths Teachers 

Rather than follow the now usual tradition of commenting on popular posts of the year I thought I would simply check on my Resolutions for (Mathematics) Teachers – many popular posts are included there anyway. This was updated at the beginning of the academic year and I am happy to report that I am indeed conscious and act on the resolutions shown in these slides in my planning.

In particular:

I try and use a good variety of resources including those that students can use at home and use technology (I would not be without Desmos or WolframAlpha!) where it enhances learning but always strive to put the learning firstwhy this or that resource? How will it help them learn?

A calm and prompt start to a lesson is so important, using some sort of activity for everyone including the older students sets the expectations for the lesson immediately. Something like Corbettmaths 5-a-day for Core 1 for example would get your Sixth Form students busy and provide some useful revision too.

I have used RAG123 regularly for marking and found it particularly successful with Year 7 (age 11-12). I use it very simply – it’s a way for them to let me know if they are not too sure on something (I hope they would ask in a lesson anyway – but it can be useful to note in their books, knowing that I will see their comment).

On the subject of Feedback, I see that the Verbal Feedback stamp idea thought to be wonderful not that long ago seems to be now on the scrap heap of ideas – why does everything have to be good / bad, black or white – isn’t there some middle ground? Exercise books are for students to learn from, what’s wrong with the stamp as a signal to the student that they should record an aspect of the verbal feedback which will be useful to them in future? So long as the emphasis is on usefulness for their learning and not merely to please somebody else I see no problem at all. I expect students to make a note of useful feedback in their books anyway, but now and again perhaps the stamp as a signal to do so could be helpful, particularly for younger students.

In any planning we should consider the very useful What makes great teaching? Review of the underpinning research. Robert Coe, Cesare Aloisi, Steve Higgins and Lee Elliot Major (October 2014). When I asked students about good teachers, it seemed to me that their comments fitted very well with the categories in the review.

Making things stick is as important as ever and we need to help our students easily recall all the basics if they are to be able to make links between all the material they learn and apply this to more involved tasks. It strikes me that with the greater emphasis now on Problem Solving Simon Singh’s wonderful quote is highly relevant and encouraging students to be great problem solvers is something I want to continue working on this year.

.Somon Singh quote

In fact I would say we want all our students to be happy with being baffled (appropriately for each student) and help them find ways to get unstuck. Isn’t that a good resolution for Mathematics educators everywhere?


GeoGebra Graphing Calculator

GeoGebra AppGeoGebra announced the release of their GeoGebra Graphing Calculator earlier this month; currently available for Android, the app will also be available for iPhone and Windows – watch for announcements.

For all the information you need to learn how to use this outstanding app for Mathematics see “What is the GeoGebra Graphing calculator?” and all the Tutorials available.

Having downloaded the app to my phone I can confirm it works really well and i was easily able to follow the instructions given in the links here.

I very quickly created the diagram in the screenshot from my phone below. Using a finger one can drag point D around and note the angle at the circumference. With an account you can also save your files which are standard GeoGebra files.

GeoGebra App

GeoGebra App

Follow GeoGebra on Twitter or Facebook

Problem Solving

You will find that there are hyperlinks throughout the slideshow; images link to the appropriate web page or document.
Files:  pdf: Problem Solving in Mathematics 

PowerPoint:  Problem Solving in Mathematics

Further Notes (relevant links for each section  are also given here)

What is a problem anyway?
GCSE & A Level Reform in particular: GCSE mathematics subject content and assessment objectives where the assessment object AO3 defines what is regarded as Problem Solving at GCSE. It is helpful to look ahead to A Level – see:  A level mathematics working group report which is very clear on what may be considered problem solving.

Ask the students
For our students to be good problem solvers who have the confidence to start on a problem without necessarily know where they are going with it they need a classroom environment where they are comfortable to take risks and know that it is OK to make mistakes. They need great teachers who not only understand their subject but the importance of the teacher / student relationship. Here’s what some Year 9 (ages 13-14) students said about Good Maths teachers and this from Year 7 (ages 11-12).

Teaching Ideas
You can see some questions to consider on the slides.
There are several slides here on using diagrams; sometimes we run the risk of getting too formal too quickly – some great diagrams can help with understanding and be very useful in our teaching. Giving students just a diagram and asking them what the question might be can be a very useful exercise too and also be a good way to discuss the vocabulary used in exams.

I always tell students that going backwards can be great for understanding and also making connections – consider expanding brackets and factorising for example.

Links to the resources in this section:


Making It Stick

Further Resources






Christmaths 2015

Christmaths 2015

Christmaths 2015

I’ll be posting on Monday this week – to coincide with Christmaths 2015 which I am very much looking forward to.

In the meantime – just in case you are not following the various speakers (you should):

Mel Mulowney JustMaths
ColleenYoung     that’s me    (and not to be confused with the other Colleen Young)
Bodil Isaksen & Dani Quinn  Bodil’s blog 
Kris Boulton Back to the Whiteboard
And of course…. Jo Morgan who is making the whole thing happen. (Resourceaholic)

Circles & Tangents

Circle & tangent

Desmos – Circles & Tangents

Use Desmos to explore tangents to circle through given point

Teaching Year 10 about the equation of a tangent to a circle at a given point I have created a Desmos page. I have also created a Diagnostic Questions Quiz using questions on circles (centre the origin) from Diagnostic Questions.
(pdf: quation of a Circle & Gradient of Tangent)

(Also added to GCSE New Content page)

Further Desmos Updates

An early introduction – plot some points and functions
Desmos – points & functions (for PowerPoint file)

 Graphs-GCSE for PowerPoint file