Mathematical Miscellany #63

On Transum Mathematics, we now have Equatero which John Tranter created having seen the TV game show, Lingo.

Transum Mathematics

The rules are simple – find the calculation – any numbers or symbols in the correct place are shown as green, any numbers or symbols that are correct but in the wrong place will be yellow and any numbers or symbols that are not included in the calculation will be  red. Rather an addictive game – but plenty of thinking! This has been added to the Number collection in the Puzzles & Games series.

We also have a Maths words version – Vocabero.

MathsBot – Bar Modelling Tutorial

It’s great to see Jonathan Hall adding manuals for his great resources on MathsBot. Here is Bar Modelling for example.

See post on Bar Modelling for problem examples and resources.

Thinking Blocks – Math Playground

It’s really helpful when sites have a clear ‘What’s new?’ type section or you can easily search by recent additions and/or subscribe to a newsletter. Just a few examples from the many excellent sites for Mathematics:

On Transum see Breaking News where you will find information on new and updated resources. Current news includes the resources mentioned above, Equatero and Vocabero.

From NCETM, see News and Features.

Nrich – sign up for their Newsletter.

What’s new at Desmos? This is really comprehensive – a great way to keep up to date.

On Dr Austin Maths select New from the Menu, I see some more of her always high quality resources – I’m a fan of Fill in the Blanks (as well as all her other resources!) For a whole collection of Fill in the Blanks type resources from various sources, see this post.

MathsBot was mentioned earlier, keep up with Jonathan Hall’s latest MathsBot resources.

Jonathan Hall – MathsBot

An absolute favourite resource Author – Andy Lutwyche, search his resources by newest on TES. I see some brilliant new Erica’s Errors on Mechanics.

Andy Lutwyche – Erica’s Errors series

On Purposeful Maths from Phil Bruce and Martin Green, see What’s New; we see for example added in December, Adding Negative Numbers.

Purposeful Maths

Starting Points Maths is arranged by date, most recent first, you can also look at Latest Tasks by topic area.

Starting Points Maths

The task continues with further questions and students draw a graph; I really like the way students see the multiple representations here.

From MEI – on Integrating technology into your scheme of work, we see published in December, GeoGebra resources for Further Maths A level.

MEI GeoGebra Tasks

Misconceptions in Mathematics

I must start this post with something Dan Meyer said at the MEI Conference 2021 that really struck a chord with me, he said “There are no mistakes or misconceptions, just takes and conceptions.” Dan Meyer mentioned WW Sayer who said:

Most remarks made by children consist of correct ideas very badly expressed. A good teacher will be very wary of saying ‘No, that’s wrong’. Rather he will try to discover the correct idea behind the inadequate expression. This is one of the most important principles in the whole art of teaching.

WW Sayer

classic mistakes 2
Nevil Hopley’s excellent Classic Mistakes site.

For a starter addressing common misconceptions try the excellent Classic Mistakes resources by Nigel Hopley.

A superb resource to use in class (or for students to use at home) to address misconceptions is of course Craig Barton’s and Simon Woodhead’s Diagnostic Questions site. The site has many thousands of questions with carefully designed multiple-choice responses to address common misconceptions.

The Insights feature is so helpful for learning about misconceptions, suppose we look at a White Rose Quiz on Algebraic Notation, for example, looking at the Insights we can see for any question the number of responses for each option from the many students who answered this question.


NCETM – resources for supporting KS3 Maths Topics

From NCETM, these videos and resources for teaching Key Stage 3 maths topics include common misconceptions and pitfalls; looking at Directed Numbers for example we find slides and a pdf support document including as illustrated here, “What things typically go wrong?”

Some years ago a website, which is now no longer available published a very useful document on misconceptions. In 22 sections, in each section misconceptions are given along with the correct version. Further explanations are also provided and also follow up exercises with answers.

The above pdf document includes all 22 sections. The first 8 of these documents, by Ilan Samson & David Burghes, are on the CIMT website.

Malcolm Swan’s excellent ‘Improving Learning in Mathematics‘, includes a section (5.3) on exposing errors and misconceptions. An activity suggested there is to let your students become examiners and mark the work of others, this works very well, I have highlighted some excellent resources for this on the ‘Spot the mistake!‘ page.

Andy Lutwyche

See for example from Andy Lutwyche, his excellent Erica’s Errors series for Spot the Mistake activities or, also, on TES, Andy’s Clumsy Clive series. Andy’s many Spot the Mistakes resources.

Many of the outstanding resources from the Standards Unit address common misconceptions; see PD2, learning from mistakes and misconceptions.

On the SERP website before see MathByExample and AlgebraByExample which is a set of Algebra 1 assignments that incorporate worked examples and prompt students to analyze and explain. These resources can provide prompts for discussing common misconceptions.

SERP – Algebra by Example

From this page you can access all the resources.

From Michael Pershan, see his Math Mistakes site, to quote the Author:

The purpose of this site is to collect, organize and make sense of the mistakes that students make while doing math. I’m also increasingly interested in using mistakes to help us create worked examples that students can learn from.

Michael Pershan


Steve Blades’ site has many excellent resources; on the GCSE page we see under ‘Miscellaneous Worksheets‘, several documents including 18 Common Misconceptions.

All the examination boards publish helpful material which addresses common misconceptions, such resources can promote very useful class discussion as can examiners’ reports.

From Cambridge University, see Common Errors in Mathematics.

Happy New Year 2022

It’s that time of year again and we can play the 2022 NCTM Year Game in our January lessons. Full rules are here; that’s a lot of 2s, multi-digit numbers such as 20, 220, or 0.22 are accepted for 2022 though note that students are encouraged to find solutions using only the single-digit numbers 2, 0, 2, and 2 rather than double digits like 0.2 or 22.. , students are also encouraged to challenge themselves to use the digits in the order 2, 0, 2, 2.

Playing this with younger students has been an opportunity to introduce the factorial function, and we tend to stray into double factorials as students are curious. A good exercise in algebra for your older students – can they find a relationship between the single and double factorial functions?

Excel has a function for computing double factorials, illustrated here. I like to show my students a few examples and see if they can work out what is going on.

Have a look at this article from Wolfram Math World and check this journal article – Henry Gould, and Jocelyn Quaintance. “Double Fun with Double Factorials.” Mathematics Magazine 85, no. 3 (2012): 177–92.

And so to number properties of 2022…

Number Gossip

We can always check Tanya Khovanova’s Number Gossip site for properties of 2022, the common properties of 2022 are shown here. All Number Gossip properties are detailed here. I have come across these before, but we have not had an untouchable number since 2010; untouchable numbers are those that are not the sum of the proper divisors of any number. A chart of the first 600 untouchable numbers is available from Here’s a great post from Iva Sallay on Find the Factors – 516 is an Untouchable Number.

2022 is also an iban number – this has amused me for a long time – get your students thinking outside the box with the iban sequence: 1, 2, 3, 4, 7, 10, 11, 12, 14, 17, 20, 21, 22, 23, 24, 27, 40, 41, 42, 43, 44, 47, 70, 71, 72, 73, 74, 77, 100, 101…

How many ways can you write 2022 as a sum of squares? And can 2022 be expressed as a difference of two squares? I came across this idea from a new year (2009) maths item of the month from MEI, we could be more general – are there any years which cannot be written as the difference of two squares? Here’s a superb resource for this problem on Nrich, What’s Possible? Possible questions and approached in class are included in the article with a printable worksheet for students.

WolframAlpha – 2022hsitorical forms

We can also look at WolframAlpha for further information on the number properties of 2022 including what 2022 looks like in historical numeral forms. We could use the various historical numerals examples to learn how Babylonian, for example, numerals work. I have successfully used this as an interesting starter for January lessons.

The Babylonian system was a positional base 60 system, though interestingly uses ‘units’ and ‘tens’ symbols to create the 59 symbols needed.


For more on the Babylonian system including how fractions were represented see History of Fractions from Nrich.

We could look back and use the excellent MacTutor History of Mathematics Archive from the University of St Andrews, Scotland. We could check today or any day for Mathematicians who were born or died on that day.

The site is searchable in several ways, including the comprehensive index of History Topics.

Teacher Resources on Line
trol calendars

On the subject of dates and the new year, from trol, Teacher Resources on Line, we can make a calendar for 2022. I do like the fold and tuck models – no glue required.

Wishing educators and students everywhere a very Happy New Year.


Puzzles & Games

I do enjoy puzzles and the Christmas break offers time for more! During the Summer of 2021, I incorporated a series of pages, Puzzles & Games on this blog, available from the top menu. Since then new additions can be found in all categories; you can see from the index that you can try puzzles and games by category as well as access collections.

To highlight just a few of this large collection…

Nrich features on many pages including reading; Nrich has many excellent articles on the use of games in the classroom; hence a Reading page.

On Geometry of course follow Catriona Agg who regularly posts new puzzles and has made her collection of screenshots freely available.

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Catriona Agg
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Following each puzzle, you can find a discussion on the puzzle. Have a look at this puzzle and discussion for example.

In response to this puzzle, you can see various solutions including an Autograph file created by Rob Smith. Rob has this up to change both squares and you can move also move a point on the smaller square.

Here’s a recent puzzle you can see with many replies:

Another brilliant Geometry puzzle author is Ed Southall who made 40 puzzles available to celebrate his 40th birthday earlier this year; he has also provided slides for teachers. Check his Geometry Cheat Sheet.

The collections page includes sites many sites featuring numerous resources.

Mathigon puzzles and games

For example for some very attractively presented Puzzles and Games – try Mathigon.

In the category Probability and Statistics games…

Guess the Correlation
Guess the Correlation – Omar Wagih

The collection includes from Cambridge PhD student, Omar Wagih ‘Guess the Correlation‘, a rather addictive game with a purpose – Omar Wagih is collecting the data on the guesses collected and using it to analyse how we perceive correlations in scatter plots. Select About to read the rules and further details.

Algebra includes the lovely SolveMe mobiles puzzles.

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Or for a rather different approach, try Shuttle Mission Workshop from Math Playground where students build and solve their own visual math puzzles. Play Shuttle Mission Pro first to practise.

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Math Playground – PEMDAS Exhibit

On Number, Math Playground, PEMDAS Exhibit provides great practice for order of operations. Touch an operation to form that part of the operation.

The Underground Maths page includes several resources on puzzles and games.

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Underground Maths – Equation Sudoku

From Underground Maths try Equation Sudoku for example.

We can write down equations:

f+g+a =19

m+k+c =16

and so on.

As always on Underground Maths we have suggestions and a complete solution as well as printable/supporting materials.

Christmas Cards

Dr Tom Bennison

I have used many of Tom Bennison’s resources, including his Christmas colouring for A Level students – these make perfect Christmas activities for older students. For more cards have a look at Tom’s Christmas themed resources.

Chalkdust 2021

For the last few years, I have made Christmas Cards for students using Chalkdust resources, Matthew Scroggs has designed the 2021 Chalkdust card. You can use an interactive or pdf version. The 2021 card contains 14 puzzles.

See also the cards from 202020192018, and 2017.


GCHQ Christmas 2021

Recently published, is this year’s official Christmas card from Director, Sir Jeremy Fleming. Each year the card contains a festive brainteaser, this year’s card has seven puzzles that have been specially designed for 11 to 18-year-olds. GCHQ is encouraging everyone to test their skills and see if they can complete the challenge.

For more puzzles, a search on the GCHQ site returns many puzzles to keep you busy! We can also search on Christmas, this search returns cards and Christmas puzzles from earlier years.

For arts and craft lovers – construct a Christmas card from Clarissa Grandi on Artful Maths. I see for 2017, Clarissa used Spirograph, a great idea for Christmas cards. As a child, my Spirograph was definitely a favourite toy, if you don’t have a Spirograph set you could use this brilliant digital version, Inspirograph by Nathan Friend. Try altering the gears so that the fixed and rotating gear are the same size, or make one size a factor of the other, make the two sizes have a common factor, or not! Investigate. You can change the colours too and create a work of Art! Now there’s an app too – Spirograph on your phone!

Inspirograph by Nathan Friend

Or we could create cards with Desmos designs!