Mathematical Miscellany #55


In Mathematical Miscellany #54 I featured two excellent resources from Curriculum for Wales; a third is now available, “The Foundations of Algebra” is suitable for progression step 3 of the new #CurriculumForWales (age 11). The workbook contains chapters on patterns, commutativity, distributivity & associativity.

Foundations of Algebra – Curriculum for Wales

I do like the above exercise which as the Teacher’s Guide acknowledges is based on Don Steward’s work, directed number arithmetic speed up and Chris McGrane’s Alternative representation of integers. A further useful resource for such an exercise is Jonathan Hall’s Directed Number MCQ Generator on MathsBot with which you can generate all the addition and subtraction multiple-choice questions you want; choose between Counters on or off.

Foundations of Algebra – Teacher’s Guide

As with the other two resources, a very comprehensive teacher’s guide is also available. You can see the contents here, this resource with its carefully chosen and varied activities and exercises will help students with the foundations of Algebra.

PhET Sims

On the subject of negative numbers, from PhET simulations we have another excellent resource in their latest addition to the Mathematics collection. To use the number line as a model for ordering real numbers and also to illustrate operations with negative numbers we can use the excellent, Number Line: Distance. Also available are Number Line: Integers, and Number Line: Operations. All are excellent for students to explore.

This resource has been added to my post on Negative Numbers which looks at some resources to develop understanding of operations with positive and negative integers and exercises for practice.

A popular post on this blog is on Venn Diagrams, first written in 2016 this has recently been checked and updated with some new resources including always excellent resources from Amanda Austin on Dr Austin Maths. Included in her Probability resources you will find an excellent section on Set Notation and Venns.

Dr Austin Maths

GeoGebra retweeted this from Javier Cayetano Rod

…the translation:

“Adding some leaves to the stem of a flower can be the perfect excuse to talk about translations and turns in space @debora_pereiro . Added to the flower generator in @geogebra

Which in turn led me to this amazing GeoGebra collection, Flowers 3D, Author: Deborah Pereiro Carbajo.

Have look at Flowers from Curves, simply brilliant!

Flowers from curves – Deborah Pereiro Carbajo

Explore the collection – you could be a while!

Complete Maths has made Robert Smith’s session “Web Autograph, a First Look” freely available

Highlight equivalence – Math Whiteboard

I wrote earlier this year on the excellent Math Whiteboard. This is completely free to use; If you create a whiteboard you can then get a link for that whiteboard which you can share. When I have created a whiteboard I then save a second copy so I can always return to the original.

With an individual subscription ( currently $15 for a year) it is possible to access all the features of Fluid Math including as an authoring tool for creating Math Whiteboard activities. It is also possible to save your activities.

A new feature is available – the ability to create answer boxes to check for correct answers. You can see examples using this feature here.

Venn Diagrams


The DfE document describing the GCSE Mathematics subject content is an excellent starting point for checking new content, all exam boards must include this content.

Note that only the more highly attaining students will be assessed on the content identified by bold type. The highest attaining students will develop confidence and competence with the bold content. See page 4 of the DfE document.

There are many excellent resources for teaching Venn Diagrams; investigate this collection.

Dr Austin Maths

From Amanda Austin, on Dr Austin Maths, included in her Probability resources you will find an excellent section on Set Notation and Venns.

Diagnostic Questions

On Diagnostic Questions – Probability with Venn Diagrams.

Diagnostic questions now has over 21000 questions on Mathematics including wonderful collections of examination questions. The site is completely free (and promises to remain so). Plenty of help is available to help you learn how to use the site.

CIMT Venn Diagrams

CIMT is one of my favourite sites for a very good reason – when I want additional examples for any topic at any level I can always find them on CIMT! Venn Diagrams are no exception to this, you can find Sets and Venn DiagramsSet Notation and Logic and Venn Diagrams in the student interactive resources and the text chapter on Logic from the Year 7 text here; in sections 1.3, 1.4 and 1.5 of the text you will find examples and exercises on Set Notation and Venn diagrams. See also the additional Teacher resources for this unit (Unit 1, Logic) such as Additional exercises are also available as are Aural Tests. Other teacher resources include slides and Revision Tests (you will need the CIMT password for the Revision Tests).

Problem from section 1.4 CIMT Venn Diagrams
AQA – Bridging the Gap

From AQA’s excellent Bridging the Gap resources, Sets and Venn Diagrams is superb, also see the OCR resources, from OCR’s Check In tests, see Combined events and probability diagrams. For further information on the changes which were made to Probability at GCSE see this post.

Transum – Venn Totals

From Transum try Venn Totals 4 different levels of exercises that can be checked are available. Level 1 – Reading information from a Venn diagram containing two intersecting sets. Level 2 – Reading information from a Venn diagram containing three intersecting sets. Level 3 – Adding information to a Venn diagram containing three intersecting sets. Level 4 – Adding information to a Venn diagram containing three intersecting sets with some problem solving required. There are also exam-style questions, to see the worked solutions a subscription is required.

Also from Transum, try Venn Paint.

Level 2 has 3 set Venn Diagrams. Exam Questions are included also (solutions are provided for subscribers).

From David Morse on maths4everyone, choose sets from the drop-down menu on his GCSE Practice Questions by topic, and also from Worksheets and Review Sheets, note that you can choose KS3 or KS4.

From piximaths there are several excellent resources on Venn Diagrams.

Andy Lutwyche

On TES from Andy Lutwyche, there are several high-quality resources – including a resource from the Clumsy Clive series.

Nrich too can always be relied on to provide resources – a search on ‘venn’ returns these resources.

From teachitMaths, try Venn diagram dominoes (pdf versions of all the resources on this site are free), there are also other resources available from the site.


However, note that some of these questions refer to ‘difference’, examination specifications should be checked for notation, for example AQA’s helpful teaching guidance includes notation such as this illustration.

AQA Teaching Guidance
AQA Teaching Guidance

For a useful way of displaying these regions on Venn Diagrams you could use the demonstration from the Venn Diagrams tutorial on Vivax Solutions. Geogebra or WolframAlpha can also be very easily used as shown near the end of this post.

Included in Jonny Griffiths wonderful RISPS – see RISP 10.

From AMSP, have a look at Vexing Venn Diagrams these resources and the Maths Feast Set Theory task.

From Craig Barton – see this whole collection of rich tasks with Venn diagrams.


Underground Mathematics

To really challenge your students combine Venn Diagrams and Algebra and try this review question from Underground Mathematics. (From a 1969 MEI O Level Additional Mathematics paper.

search finds more problems – all resources on Underground Mathematics include complete documentation including suggestions, a full solution, printable materials and more.

Perhaps try Can we find how many boys study French, Latin and German? or Quadratic Solving Sorter – not a traditional Venn diagram problem but certainly a diagram as in the possible solution presented is a very good idea.

These three interactives from Shodor are a good introduction to Venn diagrams:
Venn DiagramsShape Sorter and Triple Venn Diagram Shape Sorter 

With the Triple Venn Diagram Shape Sorter you can either set the rules or guess the rules by selecting the appropriate button:

WolframAlpha can be used to illustrate Venn Diagrams.
The slideshow here shows several examples.

Alternatively – try GeoGebra.

On a lighter note – a Twitter conversation on a fun idea!
Twitter Venn

If you want to create your own there are plenty of tools to use – there is a good summary here on Cometdocs.

New Scientist Gallery - Venn Diagrams

…and to take Venn Diagrams to their extremes have a look at these wonderful images from New Scientist!


Revised and updated in the Lesson Planning Series of pages (and watch this space for more on that series) is the Starters page.

Starting with ‘Bell Work‘ for a calm and prompt lesson start, starters are suggested from Year 7 to Advanced Level.

The Starters page includes an index with the following categories

  • Collections
  • Retrieval Practice
  • Challenges
  • Number
  • Algebra
  • Advanced Level Starters

The Starters – Posts page includes resources that can work well at the beginning or end of a lesson.
An appropriate puzzle or game can provide a good starter activity.

Starters – Posts page

From Year 7 …

NCTM Illuminations Pan Balance – Shapes

…to Advanced Level

Transum Mathematics

…and everything in between


Andy Lutwyche – Clumsy Clive series

Retrieval Practice

Maths White Board – Last White Board


Starters – Number

Number Operations – from the Number Starter section


Advanced Level Starters

crashMaths Skills Check
CRASHMaths AS Maths Key Skills Checks

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:

Mathematical Miscellany #54

Mathematical Alphabet – Sarah Carter

If you are looking for posters, there are so many excellent free resources for Maths from puzzles to information, careers and many more.

This post, Posters has been updated and all links checked. Check two of the new additions to the post – a brilliant Mathematical alphabet from Sarah Carter on

Casio font for ClassWiz series

Or if you want any calculator symbols – try Casio’s collection of fonts.

Diving Deeper – Curriculum for Wales

Superb resources from Curriculum for Wales:
Diving Deeper with chapters on place value, negative numbers and fractions. The Learner Guide includes many activities, it’s great to see bases other than 10 as well! Scroll down the page for English versions, there are several resources available including answers, a teacher’s guide and additional tasks.

Calculation Guide-pdf or note the Word version also.

Dr Frost Maths – KS3 White Rose Maths

The KS3 (UK ages 11-14) page has had various updates recently including several new sources of resources. From Dr Frost Maths choose KS2/3/4 from the resources view or select the White Rose Maths course. For further resources linked to the White Rose Schemes of Learning, check this very useful feature on Transum Mathematics, where you can find resources linked to the White Rose Schemes of Learning as well as to areas of the National Curriculum. The page includes many valuable transition resources.

Transum Mathematics

From AMSP for students at the other end of the school, check their very helpful new topic plans for AS/A Level. With common content for all boards these are useful whichever board you are teaching. Have a look at this example on Integration:

AMSP Topic Plans AS A Level Year 1 (AS) Pure

The Mathigon page has been updated, note Mathigon’s professional development section that includes details of upcoming free events which are held regularly and recordings of all the past event recordings.

Follow Mathigon for latest news.

News on Mathigon seems to come thick and fast! Their Mobile Apps have recently been updated and on Monday, 8/9 August check the Charts, Statistics and other Data Science tools coming to Mathigon.

Two new additions for this Core Maths page: from the University of Plymouth, CIMT check Core Maths Subject Support which includes notes, examples and exercises.

From @KerryDunton and published on Dave Gale‘s blog, reflectivemath’s blog check this Core Maths Further Reading List.