Here’s the diagram…

What’s the question?

(This post is an update of my post from 2013 and now includes the more recent, excellent resource Goal Free Problems from Peter Mattock.)

Using diagrams as prompts like this is excellent for Retrieval Practice.

triangle diagram

Seeing this well-received resource, GCSE Question Prompts on TES reminded me that I have successfully used this idea myself before. For example for GCSE revision I have given students a selection of various triangle diagrams and asked them what the question might have been. This proved to be a useful way of revising several topics – some of which students sometimes mix up! For several of these triangles there are many possibilities and students can be asked which lengths and / or angles they could work out.

Further excellent examples come from Mark Greenaway – GCSE Visual Prompts for both Higher and Foundation. Mark’s resources (Prompts 1) show the diagram first and then also include the complete question. Prompts 2 provides 25 starter prompts for students to produce their own questions and answers using the given information.

Visual Prompts

Higher – Visual Prompts

Foundation_Tier_Prompts 1 ppt Foundation_Tier_Prompts2 ppt
Higher_Tier_Prompts 1 ppt Higher_Tier_Prompts 2 ppt
Foundation 1 pdf Foundation 2 pdf Higher Prompts 1 pdf Higher Prompts 2 pdf

Algebra Snippets 2

On a similar theme, not a diagram this time but an extract from a question: see Algebra Snippetts.

From Andy Lutwyche comes a very high-quality series of ‘The answer is …What was the question?’ resources. A variety of topics are covered and all answers are provided.
Here's the answer...

Peter Mattock has created Goal Free Problems, a site he set up, in his own words “to allow teachers to access and share goal free problems created by myself and others. Goal free problems have been proven to support pupils in improving their knowledge and understanding by removing the cognitive load of the goal and therefore not prompting means-end analysis of a problem.”

Here you will find hundreds of questions categorised by topic; there are also mixed questions available. A wonderful resource.
Algebra - goal free

Diagram - T SherringtonOn the subject of diagrams, I really like Tom Sherrington’s post “Empowering students to own their learning solves maths problems“; a great idea to start with a diagram with no labels at all as a way into a problem. I tried this with Year 10 (very able students), presenting them with only Tom’s diagram and was very pleased indeed with the outcome. I didn’t even give them the question – just the diagram (a small copy each) and we started by deciding what the question might be. We quickly got onto areas as a possibility so then answered Tom’s original question ‘what fraction of the shape is shaded?’. The class happily discussed how to solve the problem and a student asked ‘can we write on the diagram?’ which of course was perfect – absolutely they could write on it. We solved the problem, revising some basics and had the discussion about what to do when you don’t know what to do! I will certainly use diagrams with no labels again.

This idea could be used for a starter on just about any topic – provide students with an image or perhaps just an expression and ask them to write a question to go with the image.

MEI Starters

MEI have an excellent free collection of GCSE startersDesigned for the start of a GCSE lesson, the diagrams and questions are very clear and will display well on the IWB. There are several starters under the following headings: Mathematical Reasoning, Number, Algebra, Geometry and Measures and Statistics and Probability. Files with the answers and teachers notes are also provided. Many of the diagrams here could be used for students to write their own questions. It is not always possible to have the IWB up and running, particularly if you are coming from a different room and I do like to get students working straight away. Experimenting, I found that I could take a screenshot (I do like the Windows snipping tool) and fit eight to a page! I used a Word document with very small top and bottom margins and a two column layout.

Staying with MEI, Bernard Murphy has some great ideas here on using pictures in A Level trigonometry. Look at this diagram – all the trigonometric ratios!

How creative can you be? I wonder what they would make of something like this…

See also – Numbers – Visualizations

puzzle demo
For a wonderful introduction to equations using diagrams – see Mobile Puzzles – Algebra

Perhaps a photo from the Bad Maths collection on flickr

cc licensed ( BY ND ) flickr photo shared by Danny Nicholson

Click here for the complete collection.

or a holiday snap!


Neuwied – Germany, photo by David Young


Mathematical Miscellany #21

For A Level Teaching, Dr Jamie Frost has as always been very busy! These high-quality resources (slides/worksheets) provide complete coverage for the new specification.

Dr Frost KS5

Dr Frost Maths – A Level Resources

Whilst Dr Frost mentions Edexcel, the content for A Level Maths is the same for all examination boards; Further mathematics has some common Pure content.

Dr Frost has also provided very clear instructions for using the Casio FX99-1EX Classwiz calculator which you will find on his site. To really challenge your students Dr Frost has created such a useful resource with his STEP, MAT and AEA questions all aligned to new A Level chapters. This document is 156 pages of categorised questions (brief answers are given). Also available is a pdf file of just the STEP questions.

There is a page on Dr Frost’s resources as part of the A Level (16+) Resources series of pages.

From the older students, let us turn our attention to the younger students.

Corbett 5-a-day Primary

Corbett Maths – 5-a-day Primary

The excellent Corbett Maths Primary includes 5-a-day Primary illustrated above. The page of resources for younger students has been updated with this brilliant addition.


5x8 Card

SERP – 5×8 Card

Emma McCrea alerted us to The 5×8 Card generated by a SERP Team. You can read more information on the background to this resource here.

Not having come across the SERP website before I investigated further and found more delights. I particularly like MathByExample and AlgebraByExample which is a set of Algebra 1 assignments that incorporate worked examples and prompt students to analyze and explain. These look like good resources for discussing common misconceptions.

SERP - Algebra by Example

SERP – Algebra by Example

Note the tabs at the top of the home page, from the Materials tab you can access all the resources. Note the free downloads at the SERP download centre where you can download individual resources, or very simply, use the menu on the left to download the complete workbook; answers are also available as are introductory guides for teachers and students.

With UK A Level results having recently been published, the Results 2018 page has been updated and checked. Links to grade boundaries and results statistics are provided and note the useful resources from Ofqual. GCSE Links will be checked on Thursday 23rd August.

It is still Summer holiday time for some of us, so I’ll end with this lovely puzzle I had not come across before. See plus magazine – Finding the nine. (There is a link to a very clear solution in the video). The

Polya – Let us teach guessing

This week I finally listened to a podcast I have been meaning to for a considerable time – Craig Barton’s podcast of his discussion with Anne Watson and John Mason.

As you can see from Craig’s notes the discussion is wide-ranging; so well worth a listen, this will leave you with plenty to think about.

I was particularly struck by Anne and John’s Big 3 (or Big 5!) (scroll down Craig’s notes). The only website they mentioned is Underground Maths; a personal favourite – Underground Maths pages here). John Holt’s How Children Learn was my first reading on my teacher training course decades ago – still a classic.

This morning, I have been distracted from writing watching Anne and John’s recommendation – Polya’s video “Let us teach guessing”. (Also on YouTube). To hear the man himself talking to his class is a joy. Not a traditional lecture – join in with his class as they guess!

Much food for thought – you can hear Polya’s summary of what is important in reasonable guessing at 54:16. Including the all-important reminder that we must test our guesses.

Polya Guessing from ihor charischak on Vimeo.

From the University of California, Berkeley see this summary of Polya’s problem-solving techniques – including a summary, in the Polya’s own words, on strategies for attacking problems in mathematics class from the book, How To Solve It, by George Polya, 2nd ed., Princeton University Press, 1957, ISBN 0-691-08097-6.

For a useful document on cutting space with planes, the problem discussed in Polya’s class, see this University of Toronto resource.

Seneca Learning

Seneca Learning

I included Seneca Learning in my latest sessions on Retrieval Practice (Retrieval Practice – it’s not all about quizzes); a free revision and homework platform, Seneca has applied research in cognitive neuroscience and psychology to provide an engaging environment for students. To answer the question What is Seneca? we can turn to this blog post from Stephen Wilks. The blog is helpful for learning more about Seneca, also note the Frequently Asked Questions which includes guides for teachers, students and parents. Further features are planned for teachers.

Exam Board specific GCSE content is available, my Year 10 students have tried this and reacted very favourably; this is certainly something I will be using with my students in the coming academic year; the step by step explanations and examples look very helpful indeed. My students certainly liked the AQA GCSE Maths sections we looked at – they also strayed into the many other subjects available, liking the content they saw.

Happily, Seneca has just announced that KS3 and A Level courses will also be free – such excellent news.
Free KS3 & A Level

Currently, several GCSE courses are available for UK Exam Boards.
GCSE Courses