Mathematical Miscellany #32

Small SatisfactionA nice little starter from Transum, Small Satisfaction. I do like the related activities, I know some of my students will do this first level quickly, the second and third levels will provide them with something to think about, or perhaps the alternative, Satisfaction.

Remember that Transum provides starters for students all the way through to Avanced level.

A divisibility test for 7…
Staying with Transum, Divisibility Tests 2-12 provides a handy summary of divisibility tests. What really caught my eye here is the divisibility test for 7. Scroll down the page for the comment from Transum which tells the story of the origins of the test – a 12-year-old student. And note the proof – something for our A Level students perhaps!

Note the exercises to practise using the divisibility tests. Delightfully Divisible will keep everybody busy! See also, from Plus magazine – Finding the nine. (There is a link to a very clear solution in the video).



Mathisfun is an attractively presented site, the divisibility rules are clearly explained and if you scroll to the end of the page you will see multiple-choice questions to try.

Alternative tests for 7 exist, but I like Chika’s Test!

Mudd Math Fun FactsThe page on the divisibility tests for 7 is a reminder of the site – Mudd Maths Fun Facts. This searchable collection of Mathematics fun facts from Harvey Mudd College Math Department make ideal lesson starters or perhaps useful for those odd moments. Note the search on the left, it is possible to search by topic, difficulty level and keywords.

For example, try:

ITP - Number LineA blast from the past for many teachers perhaps, from MathsFrame, the ITPs have been made available for modern browsers – no Flash needed. Developed in 2002 to help with the teaching of maths in UK Primary Schools, the National Numeracy Strategy Interactive Teaching Programs cover many areas of numeracy including number, shape, measure and statistics.

To finish this collection, some Number Puzzles from Mathisfun.

Mathisfun puzzle

Mathisfun number puzzle

Note the complete Mathisfun Puzzle Index.


Numbers – Visualizations

I have several references in various places on this blog to some great visualizations.
Time to put them all together!

Jeffrey Ventrella’s Composite Number Tree

Jeffrey Ventrella’s Composite Number Tree

From Jeffrey Ventrella this wonderful Composite Number Tree – I have used this successfully with many students. It makes a great starter. Students can work out themselves how the tree is being formed and comment on any patterns they notice.

Stephen Von Worley

Brent Yorgey

Brent Yorgey








Another excellent visualization, animated factorization diagrams comes from Data Pointed. And here is Stephen Von Worley’s blog post, Dance, Factors, Dance which tells the tale of the animation. Noting his reference to Brent Yorgey’s factorization diagrams led me to Brent’s own later post, More factorisation Diagrams. I love Brent’s use of colour here. If you want even more on these great diagrams he has more information and links on this page on his blog, The Math Less Traveled.

Visual Patterns

Fawn Nguyen – Visual patterns,

On the subject of Diagrams generally I have several posts on the subject. You can see Fawn Nguyen’s lovely Visual Patterns in Diagrams in Mathematics.

Review Questions – Underground Mathematics

Underground Mathematics - QuadraticsI wrote earlier on the wonderful resources on Underground Mathematics from the University of Cambridge. Thinking about the new A level specifications  I believe this site will provide us with rich resources for these new specifications.

Each section includes Review Questions, look at Thinking about Algebra for example; scroll down the different resource types for the Review questions for this station.

Alternatively you can browse all the Review questions.

Review Questions - types
An excellent feature of Underground Mathematics is the excellent search facility; we could look at the Review questions by type. One can also search by line ( Number, Geometry, Algebra, Functions or Calcuus) and by Station.
Review Questions - search

See the example question below, for each review question you will find the question, a suggestion, the solution and sometimes suggestions for taking it further with for example GeoGebra resources.

Note the star by the title – if you choose to log on to the site (you don’t have to but it’s a very good idea!) you can save any favourite resources to your collection.

Review Question - Gemetry

I can never resist a quick illustration on Desmos! I think I’ll start an Underground Maths Desmos collection! Note the use of the hyperlink on Desmos to link to the question.
Select the image for the Desmos page.
Desmos illustration

I think we have a wonderful supply of excellent questions here to challenge our students and help them see connections between the various areas of the subject. These are ideal to use with A Level students; some are also useful for higher level GCSE students aiming at those top grades or Level 2 Further Maths students. Any student who wants to study Mathematics at university should certainly be using this site.

Systematic Listing Strategies

Transum - Combinations resources

Transum – Combinations resources


5. apply systematic listing strategies including use of the product rule for counting

The UK GCSE specifications include systematic listing strategies.

Some resources for this topic:

Transum Combination Starters

Transum Combination Starters

Transum Mathematics has many excellent resources including a section on Combinations, where you will find Combinations starters, activities and investigations. The Transum site includes links to some videos includes The Counting Principle which provides a clear introduction. Note Transum’s clear index for teachers and for students, Maths Map with numerous activities to support their learning.

Nrich - Fraction Combinations

Nrich – Fraction Combinations

Nrich has a sophisticated problem, Fraction Combinations which not only asks students to systematically list possibilities but work with fractions too.

Ben Moss Area Problem

benmossmaths – area problem

Ben Moss has a rather nice area problem which it seems to me will involve students in some systematic listing! I tried this with Year 7 and it worked very well.

AQA have a very clear Teaching Guidance document with many useful examples, I use these a lot in class. If you use or are considering using AQA you can register here (there seems to be a problem with the link on the Teaching Guidance document page).

AQA Teaching Guidance

AQA Teaching Guidance

And finally (for now!) Don Steward has this bracelets or necklaces problem.

Don Steward - combinations

Don Steward – combinations

Have a look at this post on Just Maths: Product Rule for Counting which includes Pearson / Edexcel exam questions and solutions.

Just Maths

Just Maths

By Colleen Young Posted in Number