Mathematical Miscellany #75

A collection of impressive resources…

Jake Gordon – Worked Examples

Jake Gordon has been rather busy! Have a look at his “monster of a PowerPoint”. These detailed worked examples are based on the book Teaching math with examples by Michael Pershan.

Also inspired by Michael Pershan’s book and her research on self-explanation, have a look at Karen Hancock‘s journey into worked examples.

From Nathan Day – a brilliant collection of resources on Distributivity: Partitioning, Grid Method, and Expanding Brackets The 15 tasks increase in demand. Note the complete thread.

Nathan Day

Also from Nathan Day – a wonderful collection of Starters – again, note the complete thread.
I have added this collection to my Starters page, a collection that includes Advanced Level Starters.

Andy Lutwyche – Erica’s Errors series

Andy Lutwyche’s collection of Erica’s Errors where students must identify errors in solutions can be an ideal starter for either retrieval practice for an earlier topic or to consolidate learning for a current topic.

Added to my Primary page – curriculum mappings from NCETM

Also added to the Primary page:

On DrFrostMaths there is a growing collection of Key Skills for Primary students.

I have mentioned DrFrostMaths more than once recently, a new instruction manual for teachers is now available.

DrFrostMaths Key Skills – Algebra

From the GCSE/IGCSE/L2 Further Maths page, the pages for individual topic areas all include Sites with clear resources by topic, see the Algebra page for example. These resources by topic include links to the relevant DrFrostMaths key skills.

Custom Maths Revision Starters

Thinking about practising skills for GCSE, why not create some custom starters?

Jake Gordon – Skills Grid Creator
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Jake Gordon – Skills Grid Creator

From Jake Gordon, try his Skills grid creator – customise exactly as you want from 90 skills, and answers are included.

On Jonathan Hall’s Test Maker on MathsBot we could create a test to test many of these skills. Choose the skills you want from the menu, the number of questions, and the initial and final difficulty of the questions on a scale of 1 to 10, also note the Paired Solutions option to show one example with an answer then one to try. An Answer Key is provided for any test you create.

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MathsBot – Test Maker

We could also use the many brilliant GCSE resources from MathsBot.

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Transum Mathematics – Refreshing Revision

Alternatively, using Transum’s Refreshing Revision, we could create a custom resource to check some of these skills. The resource allows teachers to select the number of questions and the topics to include; scroll down the page and choose the topics you want from the Concept Selection, it is also possible to drag the panels so your questions are displayed in the desired order. A very nice feature is the fact that you can save a particular selection of topics as the URL for your selection will be generated. Every time you refresh the page you get different revision questions. Note also the Advanced Level version. This resource like all the Transum resources is free to use, an extensive collection of resources such as the various KS3 and KS4 self-checking exercises mentioned below provide answers for any user. Answers for some resources such as examination questions and this Refreshing Revision resource are provided for subscribers; extremely good value for money in my opinion.

Transum has an extensive library of self-checking exercises, so we could easily provide examples on inequalities for example. On Transum, there are several ways to search for resources, for example, try:

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Maths White Board – Maths Starters based on advance information

See on Maths White Board: Revision Board – generate a starter based on the Advance Information

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Interactive Maths Generators – Dan Rodriguez-Clark

On Interactive Maths Generators from Dan Rodriguez-Clark (@InteractMaths), design your own sets of questions on more than 50 maths topics for your students to practice a variety of skills. There are many customisation options, generally as well as for individual topics. Full instructions are on Dan’s site.

On DrFrostMaths try the Question Explorer. A brilliant feature (well one of the many brilliant features) of Dr Frost Maths is the extensive collection of key skills for UK KS2 to KS5 (ages 8-18), practice as many examples as you want, and perhaps watch a video on this key skill.

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DrFrostMaths Key Skills

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Unlimited Edexcel GCSE 1 mark questions

On TES, from salimnore, this Excel spreadsheet generates an endless supply of 1 mark questions with the option to display the answers, making an ideal starter. Two versions are available, you can choose 5 or 10 questions to display.

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:

You will find more lightning arithmetic suggestions on the site.

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.

For a collection of Cornell notes linked to topics found in the Maths curriculum see Mr Hawes Maths.

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

Mrs D Maths – Knowledge Organisers

Mrs D Maths has been writing Knowledge Organisers for A Level Maths, ( free on TES Resources). She has also started a Year 2 set of organisers, see Year 2 Functions and Graphs.

These organisers are included in the Knowledge Organisers collection which includes resources for KS3, KS4 and KS5 (ages 11 – 18).

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.)

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. From the Subjects menu, choose a subject and you will find this includes Knowledge organisers for Year 7 through to Year 11, see Maths for example.

CIMT’s GCSE course includes Facts to Remember for each unit.