Mudd Math Fun Facts

Mudd Maths Fun Facts

This searchable collection, Mudd Math Fun Facts from Harvey Mudd College Math Department has resources that can make great starter activities, perhaps try Squares Ending in 5 and Multiplication by 11 both made excellent starters. I have looked at proofs for these with students as well as enjoying the mental Maths tricks!

It is possible to search by topic, difficulty level and keywords.

Examples to try:

Bell Work

At the beginning of a lesson, I like to get everybody busy straight away, making a calm start to the lesson and very much like the idea of so-called ‘bell’ work. Give students a task that is simple to understand and requires no more than a simple instruction, question/s and/or diagram on the board (no technology required – unless you are in the room ahead of your students which offers more possibilities). This is a particularly useful idea if students arrive at different times. Students are expected to get to work as soon as they enter the room.

In current times, the students may well be in the room before their teacher, so could be given instructions at the end of a lesson on what they are expected to be working on at the beginning of the next lesson.

Some ideas

  • A short question or questions on a topic studied recently.
  • Ask students to write down all they can remember on any topic. They could perhaps draw diagrams or just jot down examples or vocabulary – anything at all – a ‘Brain Dump’, see ‘Brain Dumps: A small strategy with a big impact’ on Retrieval Practice.
  • Ask for some specific facts, eg write down the names of all the quadrilaterals they can with a quick sketch for each.
  • Students make up some short questions to review a topic – they could then put their questions to the class.
  • Provide students with a diagram, they write a question, (See ‘Here’s the diagram ….’). Particularly useful for providing a diagram or a question to write up quickly is Peter Mattock’ wonderful 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.
How Many Factors – nzmath
  • Prime numbers can be used for an exploring numbers type starter. Find numbers with exactly two factors. Three factors? (A square of a prime number). Four? Five? Or generalise (perhaps rather too long for a starter!) This investigation, How Many Factors on nzmaths requires students to  find ways to group numbers, which numbers have only two factors and which have only three factors? For further ideas see these possibilities from Nrich. Two Primes Make One Square or Penta Primes for example could make suitable starters.
  • Prime Number ideas and numerous other ideas can be found in Colin Foster’s KS3 Instant Maths Ideas (3 books) which are freely available online; these contain a wealth of activities to try in the classroom. Colin Foster is a Reader in  Mathematics Education in the Mathematics Education Centre at Loughborough University.
Colin Foster – KS3 Instant Maths Ideas

  • Also from Colin Foster on Nrich we have Mathematical Etudes where he discusses lovely rich tasks and tedious exercises! Note his Mathematical Etudes Project; scroll down the page for examples of Mathematical Etudes on Different Topics, there are many activities here for which instructions can be given to students easily.
  • Problems from Open Middle can be very simply explained, but can really get your students thinking.
  • Countdown type problems or Make 24, see 4 Numbers Game. If you just want to make a note of a few problems and solutions then you could use this page from Print out some Make a number puzzles with solutions from Brain Food. Note the other problems available on Brain Food, a Logi-Number puzzle could be written up quickly for example. Many such problems are available, see for example As Easy As 1234 from
  • UK Maths Challenge questions can make excellent starters and you don’t even have to provide the multiple choice answers!
  • Some of the problems on Transum Software – Maths Starter of the Day are simple enough to easily write up on the board. Note that there is a complete index of starters including the topic of the starter. Many of the Shine and Write activities would also make good lesson starters.
    A+ Click Maths provides another source of possible starters.
  • A book I like very much is ‘Thinkers’ from ATM, many questions here would be very simple to put to students at the beginning of a lesson.
  • At the end of a lesson – tell them what you expect them to do the minute they walk into the next lesson, so they know what there bell work is before thy even get to the lesson!

Mental Tests
Many schools are providing students with booklets for use in lessons. Alternatively or in addition to, why not give a mental test where the teacher simply reads a short question which the students can write down and then answer can make an excellent start to a lesson, or in fact can be used at any point in a lesson. These should be very much low stakes activities. There are many sources of questions you can use, for example, see

Corbett Maths Class Quizzes

Class Quizzes from Corbettmaths for a collection of questions designed to help students remember key facts. Looking at these will probably give you ideas for writing your own quizzes too.

A really useful source of questions which can be used this way are the mental tests from CIMT; these are included with their resources for Years 7, 8 and 9 and also for GCSE. For Key Stage 3 (ages 11-14) scroll down this page for the Year 7, Year 8 and Year 9 course material, the resources include mental tests as part of the teacher support material. On the GCSE page scroll down to the teacher support material and note the mental tests available for most units, see this on Formulae for example.

CIMT Resources – Mental Tests
CIMT – Mental Tests
CIMT Mental Test – Formulae

Talking of mental tests reminds me of a long forgotten resource I used to use with Year 7 – The Three Little Pigs as a mental test. I found this many years ago on as a free sample.

Why not start a collection of such ideas? Many of these ideas could also be used for those odd moments in a lesson when you find you have some extra time.

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Knowledge Organisers – Mathematics

In a post on Cognitive Science in the Classroom, I mentioned Knowledge Organisers, or to be more precise I mentioned Kris Boulton’s  “When shouldn’t I use knowledge organisers?”. Kris has written on why they are less applicable to maths. Certainly, I had not used knowledge organisers for Mathematics myself with one exception, I have used  William Emeny’s Angle Facts; as Kris Boulton says in his article, “Maths is super-dense with concepts, and processes, but really only very few facts.” Noting topics where students do need to know more facts, he includes angle facts.

When I have used Will’s angle facts, I have adapted it so some content is missing, particularly the section on basic angle facts, students can be given just the diagrams for example and asked to recall the basic angle facts. I have also asked students to recall as many basic angle facts before they see the list as in the organiser, so using it following retrieval practice or as a retrieval exercise.

As noted in this post on Knowledge Organisers from Durrington Research School, it is not about the knowledge organisers  themselves but how they are used for planning, teaching and testing. Note the example shared of a Maths Knowledge Organiser, Year 10 Expressions, Foundation. I like the fact that key vocabulary is included here, the few key facts needed and of course by far the largest section – key concepts with examples.

To quote the Durrington blog, “Maths are using their range of knowledge organisers to support homework tasks. Firstly, the students can access their maths knowledge organisers are any time using our online system Connect. This means that students have scaffolding in place for when they are working outside of the classroom. Furthermore, every fortnight the maths team set a homework that is based on retrieval quizzing. The students are required to use the knowledge organisers to find the answers to upcoming quizzes and then actually sit the quiz in class on the due date for the homework. Students who score less than 12 out of 15 are then supported in making flashcards on the questions, again gaining the information from the knowledge organiser, and use these to retest until they are successful. This strategy demonstrates how knowledge organisers can be used to support learning through the testing effect.

Algebra KO extract Nicola Whiston

Algebra Knowledge Organiser extract – Nicola Whiston

Nicola Whiston has a superb collection of Knowledge Organisers which follow the White Rose Schemes of Learning, all are available on TES Resources, on TES editable versions are available as well as the free pdf resources. These are really attractive and I believe appeal to students. I think these are excellent to use in class alongside teaching a topic. They could also be used for retrieval practice.

You can hear Nicola talking about Knowledge Organisers with Tom Manners, In his interview, he was joined by Nathan Burns aka @MrMetacognition who has researched these in great detail, as well as Nicola Whiston (@Whisto_Maths) to discuss what maths knowledge organisers should contain and how we should use them effectively.

B Reed Indices KO

Indices GCSE Knowledge Organiser extract – Becky Reed

From Becky Reed a set of Knowledge Organisers for Edexcel GCSE (UK age 14-16). We have here another set of very clear and also attractive set of resources. Like the other resources here I think these are useful in class and for students to use at home also. There are several  examples given which is really helpful.

Sarah Hall has a GCSE (‘WJEC flavoured’ )collection. Sarah’s Knowledge Organiser resources can all be found (all free) on TES Resources. These have many clear illustrations and like others in the Knowledge Organiser collection, are very attractively presented. Sequences KO Sarah Hall


On MathsQ, some free resources are available – a clearly presented set of knowledge organisers.

For A complete set of A Level Statistics and Mechanics Knowledge Organisers – see these resources from Lucyjc.  These resources are available free on TES Resources: Statistics and Mechanics. All include Key Words and Definitions and What Do I Need to Know sections.
Knowledge Organisers - Lucy

And for Pure Mathematics, we have from @misschakera a set for the first year of A Level.
Pure KO

non examples – Jonathan Hall

If I want definitions, characteristics and examples (clarified with the use of non examples), then I could return to the Frayer model. (See Frayer Models.)

Reading further on Knowledge Organisers, I do recommend this collection of blog posts and articles from St Mary’s Catholic Academy, on Retrieval Practice which includes the use of Knowledge Organisers. Once again, the emphasis is, quite rightly, on how they can be used for retrieval practice.

Searching for Mathematics Knowledge Organisers, I have come across some resources I wish to explore further, such as the Henry Box School on Knowledge Organisers where the school are sharing Knowledge Organisers for each subject, recognising the support parents can offer. On TES, GCSE Maths 1-9 Knowledge Organisers is a (free) set of 50 files, described as “A full set of Knowledge Organisers containing the facts, definitions, formulae etc. that students need to know for the new GCSE Maths specification, broken down into individual units. On each Knowledge Organiser, content shaded in grey is for Higher Tier only.

Can be used for homeworks, revision, starters, plenaries or any other ways you might find useful.”

The files have been created in Word, I like the 3 column format which includes, Topic/Skill, Definitions/Tips and importantly, Examples.

I will return to this topic as I have further resources I wish to explore….

…and I want to ask my students what they think.

Advanced Starters – Transum

Transum Advanced Starters

Chancing upon this tweet from Transum proved fortunate. Hidden gems indeed, I had not noticed the Advanced Starters before, some of which I think could be useful for students aiming at the highest GCSE grades as well as for Advanced Level students.
The problem, Find the Radius, illustrated in the tweet is very neat!

Looking at the Main Transum Starters page I see at the foot of the page we have various categories of starters including the Advanced Starters.
Transum Starters
I recommend exploring!

I wrote the above in February of last year and I see from Transum’s latest newsletter that The Advanced Starters collection has grown. I see many starters here I like, thinking about next year it’s a good time to check these. With linear courses, starters provide an ideal opportunity for review.

Looking at Coordinate Distance, I can never resist a Desmos page to illustrate a problem! This starter could be also be used to review some coordinate geometry – find the midpoint? Find the equation of the line?

Coordinate Distance - Transum Advanced Starter

Transum – Advanced Starters

Quad Midpoints - Advanced Starter

Transum – Advanced Starters

For more on Transum see this post which includes some great Venn Diagrams resources and a reminder that there is a very clear index for teachers and another for students. The extensive library of activities on Transum are all free to use.  Activities such as this Algebra resource on Completeing the Square can be checked as part of the free offering.
Completing the Square

A subscription offers teachers answers to for example the Advanced Starters and Exam Questions.


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