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, in this month’s newsletter, we see 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.

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

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

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

Mathematical Miscellany #59


Today is not only Halloween, it is Magic Day!

MathsPad Number Tricks

My post on Proof includes Magic or Number tricks which can be an ideal way into algebraic proof, try MathsPad – Number Tricks, one of the very generous collection of free resources on MathsPad (search on free), or be dramatic with Number Jumbler from Nrich! I have used this successfully with students from Year 7 to Year 12. Also try the suggested task, Double-Digit.

For a collection of mathemagical activities, try Transum Mathematics.

Transum Perfect Magic Square

You can find many activities on Transum, including Perfect Magic Square, which has 9 such puzzles, by the time we get to Puzzle 9 only two clues are given.

Wayne Chadburn – A little bit of maths each day

Tomorrow sees the start of November, so a good time to remind readers of Wayne Chadburn’s monthly calendars. He writes these calendars to provide regular, varied practice – a little bit of maths each day. In this academic year, he is producing calendars for year 11 students each month from September to April; the calendars for November 2021 are available. Note the three versions including crossover questions, Foundation, Foundation plus and Higher.

For November we have Barvember, from White Rose Maths, which was created to encourage the use of the bar model. Bar models can really help children visualise and then solve maths problems. See Bar Modelling.

Barvember – White Rose Maths

Nathan Day – factor Tree Puzzles

Back in July, Nathan Day published his very valuable CIMT Complete PDF. He now has a new website: Mr Day Maths where he is adding all the brilliant tasks, displays, resources, and thoughts he has shared on Twitter over the past few years to one place – a new treasure trove to explore! Try Factor Trees, for example, a set of (rather tricky) factor tree completion problems, that’s another lovely resource added to this Fill in the blanks collection.

For some quadratic questions to really get your students thinking, try these non-routine questions from @boss-maths. Follow the thread to the end for the answers!

Recovery Resources

Matt Woodfine continues to add to his brilliant site, a new addition is Level Up. a programme that targets basic numeracy.

From MEI, for KS3 have a look at these free GeoGebra interactive online classroom activities to support maths recovery at Key Stage 3. You can try the first three of five Algebra resources. Each lesson not only has a GeoGebra activity for use in the lesson, but also one for student independent learning. Full lesson plans and comprehensive notes for teachers are provided.

The collection will be expanded over the coming months to include similar sets of lessons for Number, Geometry and Measure, Ratio, and Statistics and Probability.

On the subject of GeoGebra, for the other end of the school, you will find on the Maths Emporium, a GeoGebra guide for A level Further Maths available to download or it is available with no login needed from the Edexcel website here. These activities have been produced to support teaching or could be used by students for independent study.

I have included many GeoGebra Resources on the GeoGebra series of pages. An extensive library of GeoGebra resources is available from Edexcel for GCSE (with many of these helpful for younger students also) and A Level.

Edexcel GeoGebra Resources

Edexcel GCSE GeoGebra Resources


Perton Maths Department is so good at providing us with problems for various events, check @PertonMaths for daily half-term challenges.

For more Halloween problems try the windows on Transum Mathematics Halloween.

Transum – Halloween starters

A selection of Mathematical Halloween themed resources:

Perton Pumpkin 1
I do appreciate the dynamic Perton Maths Department, I have mentioned their various puzzle collections before – they are busy again, this time with Halloween puzzles!

Halloween Relay
Halloween relay – Chris Smith

From Chris Smith, try his great Halloween relay and note the whole set. I have used many of these very successfully – have fun whilst doing plenty of Maths!

From OCR Maths who are rather good at providing us with puzzles, try their fiendish number grid puzzle. OCR Maths regularly publish excellent puzzles many of which I have successfully used in class.

OCR Maths Puzzles
OCR Maths Puzzles

Nrich Halloween

From Nrich, thinking about systematic listing strategies with younger students, check this Halloween Investigation, just how many possible Halloween costumes are there?

Or a good starter for Thursday perhaps, try Halloween Day on Nrich, a problem taken from the UKMT Maths Challenges.

I like the way Nrich suggest related problems and strayed into Helen’s Conjecture here, one of many Nrich problems on Factors and Multiples similar to Helen’s Conjecture.

ghost curve

And to finish, a ghost from WolframAlpha!

Checking popular curve examples on WolframAlha, a discovery – I had not realised that it was possible to, for example, create a randomly colored ghost curveRefresh the page for different color ghosts!

Mathematical Miscellany #58


Following last week’s post on Mathematics for Psychology, note also these Desmos activities for Biology by Sarah Denison from AMSP/ASE Biology. My Mathematics for Science page has been updated with these Biology Resources.

Desmos Activities – Surface area to volume ratio, Sarah Desnison

As with the Psychology resources the Advanced Mathematics Support Programme [AMSP] are supporting Biology teachers of Psychology in their teaching of the maths and statistics requirements of the psychology specifications and provide free CPD and resources. Maths for Teachers of A Level Biology is an event starting on 30th November. The course is free to attend, priority is given to teachers from state-funded schools and colleges in England and trainee biology teachers. Full eligibility details as well as a detailed study schedule can be found on the AMSP site.

Seeing Jonathan Hall’s latest Mathsbot resource, Place Value and counting to 100, reminded me of a task that has been on my list for some time – an update to a resource list for Primary Mathematics, which now has its own page on the top level menu. This is still work in progress; more resources will be added in the coming weeks.

In case you missed this post – Compare & Discuss/Multiple Strategies

A new feature on Desmos – lists.

Seeing OCR’s latest puzzle, it struck me how ideal Autograph is for exploring this as plotting points and joining them with line segments is very straightforward. See the Autograph pages for more on Autograph.

On UK Exams and Assessments 2022, you can find updates from the exam boards including the formula sheets for GCSE Maths. For example, from AQA

Formula Sheets

Formula Sheets: Students will be given a formulae sheet in GCSE Maths and a revised equations sheet in GCSE Physics and GCSE Combined Science covering all the equations required in the subject content.
We will make these sheets available to schools for use in teaching and so students can familiarise themselves with them. They will also be provided in the exams. AQA

AQA have stated that everything in sections 1 and 2 of the appendix on pages 43-44 of the specification will be provided for the examination in 2022; anything required from section 3 will be given in the question as normal. You can view sections 1, 2 and 3 here.


For GCSE Maths, the exam support will be a separate formula sheet which gives students the formulae they may need to refer to in their assessment, appropriate to their tier of entry. This will be available as an additional insert on the day of the examination. The formulae sheet will shortly be accessible via the website and learners should then familiarise themselves with the formulae sheet ahead of the summer 2022 examinations. Edexcel

OCR Update on the formulae sheet for GCSE (9-1) Maths assessments in 2022.

What impact will Ofqual’s chosen grading system in 2022 have

fft education datalab – 30 September 2021:
What impact will Ofqual’s chosen grading system in 2022 have?