Mathematical Miscellany #61


From Matt Woodfine on Maths White Board revision cards and online tests are now available.

Looking at the latest newsletter from Chris Smith, (highly recommended), I see he mentions Stella’s Stunners, this is an excellent site, I featured it in my post on Proof, it has so many excellent resources.

Stella’s Stunners

Stella’s Stunners is a library of more than 700 Non-Routine Mathematics Problems for Middle and High School Students; these problems certainly encourage mathematical thinking. You can filter the problems to narrow your search and usefully create a pdf document of your chosen problems. Perhaps try Symbol Pushing or Logic. Try the Introductory Problem Set which consists of twenty problems designed to show you how to use the problem library. A comprehensive user guide is available.

I’ll be keeping my Advent and Christmas pages updated – here’s another to add from STEM Learning.

If you have just a few minutes, then why not try a Maths in a minute article from Plus Magazine. A recent article by Kate Stansfield is on Maths and music. If this is a subject that interests you and you have more minutes, you can find a whole collection of articles exploring Maths and Music. The article in the Maths in a minute series includes a discussion on why some pairs of notes sound “nice” to us.

We could hear some different intervals on WolframAlpha which will also give you the pitch difference between the two notes.

Additional information includes the frequencies of each note, you can also find related musical queries. I spent rather longer than a minute on this, particularly having discovered other Music examples on WolframAlpha! You can get information on songs and artists, audio waveforms, and musical instruments as well as Music Theory.

Maths in a minute reminded me of The Mathematical Moments site which features many downloadable pdfs, posters that show the role that Mathematics plays in Science, Technology and Human Culture. A short or more detailed version of each poster is available and a search is provided. This is included in my post on Posters.

Mathematical Moments - Sticking the Landing

Continuing with the theme of Maths in minutes, From Pearson Qualifications, try Tea Break Training, these short videos which offer targeted professional development training on teaching strategies, content areas and support for GCSE Maths.

If you are preparing students for STEP examinations, have a look at this webinar from Nrich which In this webinar Claire explains how STEP questions can be used to help students prepare for a University Maths degree and demonstrates the resources available on the STEP Support Programme. ( This has been added to the University Admissions Tests page which includes many resources for students aiming at high A level grades.

Advent Calendars and Christmas Resources

December is fast approaching, so an annual job is a check and update on Mathematical Advent Calendars and Christmas Resources.

Two new pages have been created, currently visible in the Featured Posts menu on the right.

Both pages will be kept up to date and any new resources added through December in the run-up to Christmas.

Both pages have had all links checked and new resources have been added.

E Weston – GCSE Higher Revision Quiz
Transum Mathematics

Mathematical Miscellany #60


Andy Lutwyche – Lazy Lionel series

In my Fill in the blanks post, I featured many excellent resources where students must completely partly worked examples including several from a favourite resource author, Andy Lutwyche. I mentioned some of Andy’s latest resources include his Lazy Lionel resources, Lionel does not show his working, so loses marks! Andy has added further to this collection. We also have Hasty Hazel and Methodical Mabel, these are so good and I’m sure can promote excellent conversations in the classroom on misconceptions and showing sufficient working.

See also Compare & Discuss/Multiple Strategies

White Rose Year 11 end of term assessments

From White Rose Maths, Year 11 end of term assessments for autumn are now available free from their secondary assessments page, scroll down for Autumn Foundation and Higher resources. Mark schemes are also provided.

Staying with White Rose Maths …

White Rose Maths – Barvember 2021

It’s still November, so remember Barvember, from White Rose Maths, which was created to encourage the use of the bar model. Now you can solve the problems using another great digital tool from White Rose Maths, Bar Model. White Rose Maths completed the solution to their day 10 challenge on the Bar Model tool:

Bar models can really help children visualise and then solve maths problems. See Bar Modelling.

NCETM Secondary Mastery Professional Development Materials – Operating on number

NCETM have published KS3 subject knowledge audits; there are 17 audit documents with each one based on one of the core concepts from the NCETM Secondary Mastery Professional Development Materials. For each document, following audit questions with check boxes you can select to show how confident you are, you will find exemplifications and explanations, and further support links. There is space at the end of each document where you can reflect on the material. Further KS3 teaching resources can be found on the 17 core concept webpages.


Discussing increasing and decreasing functions with my A level students recently I made the happy discovery that WolframAlpha provides a clear visual representation for such a query, as well as returning the values of x for which the function is increasing and decreasing; we also have the first derivative and critical points.

Since last week’s post on Literacy Skills in Mathematics, I have added resources to that collection. In my post on Proof, I began with the importance of spoken language. Spoken language is so important in students’ development, the KS3 Programme of Study quite rightly stresses the importance of spoken and written language. Both spoken and written language and notation are key. This applies to all key stages from definitions and examples and non-examples at key stage 3 to proof by contradiction – writing negations of statements at KS5. For many students, if their ability to write mathematics clearly and succinctly is poor this may well be a result of their not speaking about mathematics clearly. Conversations in class where students explain their thinking are so valuable.

Resources, where students practice writing expressions from algebraic statements and vice versa, are now included in the literacy post.

Also added are further Transum Mathematics resources as you can find a whole collection of Vocabulary activities on Transum.

Transum Mathematics – Circle Pairs

Literacy Skills in Mathematics

Part of the Lesson Planning series includes Vocabulary, which includes a variety of resources including vocabulary activities, clear definitions of mathematics vocabulary and also command words used in examinations.

Edexcel – Teacher’s Guide to Command Words

See for example, from Edexcel – this Teacher’s Guide to Command Words, not only do we have commentary on what is expected from students when a particular commonly used command word is used but examples of questions to exemplify the use of the word.

Pearson – Comprehension Tasks

From Pearson, see Maths – Diversity and Inclusion, a set of resources including comprehension tasks. This set has been designed for each year group to engage in one themed literacy task per term. Look at the Term 1 resource on Famous Mathematicians, this four-page document has an activity for each of Years 7, 8, 9 and 10. A short paragraph on a famous mathematician is followed by 10 questions to answer.

The theme for Term 2 is Engineering.

These resources have been written by Jenny Hill Parker, you can find all these Comprehension resources here where you will also find Term 3 on Astronomy and Term 4 on Sport.

Jenny hosted a Twitter mathscpdchat where she asked “what does literacy in mathematics mean to you, and how do you address it in your teaching?” A summary of this session hosted by NCETM can be found here. The summary includes all the resources mentioned in the session as well as the conversation. The discussion included Freyer Models.

Freyer Model from

From Teachit Maths, there are some excellent resources to develop literacy skills in Maths. You can see these and other literacy resources here.

Ben Gordon has updated Nicola Whiston’s guided reading activities.

I do like The language of ratio which is a collection of 21 ratio problems for students to sort into categories and then complete. The problems are designed to help students decide what a ratio question is asking them to do. There are three categories of problems: create/simplify a ratio, use a ratio to find one quantity when others are known, divide quantities in a given ratio.

Or perhaps try a Mathematics comprehension exercise; this resource includes an excerpt from the prospectus for the fake BODMAS international school for students to read, a set of 12 questions about what they have read and full solutions. See also the free resource, a whole school approach, Closing the word gap: activities for the classroom – secondary which includes a PowerPoint and pdf file, the subjects covered are english, maths, science, geography, and history. Resources for Primary are also available.

Teachit Maths Comprehension exercise

From Sarah Farrell, Maths comprehensions on 12 different topics.

On the language of ratio, remember AQA’s modular Teacher Training packs, including a pack on ratio that includes activities and discussion prompts on switching representations, combining ratios, problem-solving with ratio and dividing quantities in a given ratio. This and many more resources for learning and teaching about ratio can be found in my post, Ratio and Proportion.

AQA Teacher Training pack – Ratio

Explaining in Maths is a set of 4 posters that show model answers and common misconceptions. Another poster on the mode is also available.

Teachit Maths – Explaining in maths posters

Mathematics Key Stage 3 Programme of Study

In my post on Proof, I began with the importance of spoken language. Spoken language is so important in students’ development, the KS3 Programme of Study quite rightly stresses the importance of spoken and written language. Both spoken and written language and notation are key. This applies to all key stages from definitions and examples and non-examples at key stage 3 to proof by contradiction – writing negations of statements at KS5. For many students, if their ability to write mathematics clearly and succinctly is poor this may well be a result of their not speaking about mathematics clearly. Conversations in class where students explain their thinking are so valuable.

“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 of the art of teaching.”

W. W. Sawyer (2012). “Vision in Elementary Mathematics”, Courier Corporation

Dan Meyer really struck a chord with me at the MEI Conference in July 2021 – showing this quote from WW Sayer he said “There are no mistakes or misconceptions, just takes and conceptions.” That approach in our classes should certainly encourage an atmosphere where they are willing to share ideas and talk about mathematics.

In the slideshow you can see Jonathan Hall’s Worded Expressions, as always with MathsBot resources we have lots of choices – for example, hide either the sentences or expressions. With the ability to generate new expressions we have an endless supply. This is ideal for self-study as well as for use in class. From Don Steward, we have translating English to algebra, expressions, see also translating English to algebra, relationships. Also included here is an activity, A1 from the Standards Unit on Interpreting algebraic expressions. This includes 4 card sets to match, ideal for looking at multiple representations, students match algebraic expressions, explanations in words, tables of numbers and areas of shapes. One of the goals of the activity is to help learners to translate between words, symbols, tables, and area representations of algebraic shapes. The Standards Unit resources can all be accessed without a login from the very clear to navigate University of Nottingham site linked to in the Standards Unit post.

Thinking about language and notation, and writing expressions from algebraic statements and vice versa there are some ideal activities for this, some suggestions….

One of Chris McGrane’s Starting Points MathsCurriculum Booklets – Algebra 1 from Phase 3 features some great activities for writing algebraic statements, featured on the slides you can see a Smile activity, and Jo Morgan’s lovely Introduction to Writing Algebraically – this is such a good idea, as Jo says in the resource description if they know how to do it with numbers, then they just do the same thing with the algebra.

Further excellent resources on this skill are available on Maths4Everyone.

OCR’s A Level Mathematics B (MEI) H640 includes a comprehension paper. Legacy papers include C4 with its Section B Comprehension Section. Steven Walker’s post on preparation for this paper includes links to the 2016 and 2018 papers which could be used as extension tasks for GCSE or early in AS classes. 2016, Photomontages applies trigonometry in a practical situation and 2018 ‘Rain stopped play’ looks at the various models used to estimate fair results in cricket.

Further Reading