Mathematical Miscellany #51

MathsPad – Equations

My recent post on Bar Modelling has had some updates, note Pearson Edexcel GCSE Maths Networks on YouTube; you can watch Naomi Bartholomew-Millar’s excellent session on Bar Modelling.

Additionally, two slideshows have been added to illustrate The Mathenæum from Ken Wessen one on Modelling Word Problems and the other on Fractions, Decimals and Percentages. Also added is a worksheet from MathsPad; you can download the resource free from the Mathspad site here; it is the worksheet, Using Bar Pictures to Solve Equations. This provides a great introductory exercise for students getting used to solving equations with bar models. I like the way the questions progress; questions include giving the students the equation only, they must provide the diagram and the solution; questions are also provided where only the diagram is given. Note the other free resources to try in this set of resources; MathsPad is a subscription site (excellent value in my opinion) but there are numerous free resources you can try.

I wrote about Maths White Board recently (now included in the Use of Technology series available from the top menu). This excellent site continues to go from strength to strength, Matt Woodfine is adding tutorial videos, you can see the first on the Question Level Analysis tool now.

We also have interactive class games – brilliant!
Note the video on how to use the keypad tool on the Maths White Board site.

Transum Mathematics – Simultaneous Shopping

For a great introduction to simultaneous equations, try Simultaneous Shopping from Transum, a reminder of just how helpful diagrams can be. Level 1 is illustrated.

Simultaneous Shopping – Level 5

Level 5 uses coupons to introduce the idea of negative coefficients.

AMSP – Motivating students at KS4

From AMSP, see Motivating students at KS4, this is a package of support designed to support Key Stage 4 students who have had their maths learning disrupted due to the pandemic. The resources aim to help maintain student interest, motivate their learning and encourage them to consider studying maths post-16. Several resources are already available and more will be added this summer term.

I do like ‘A formula for prime numbers’ which could be used for a lesson on proof. The examples usefully illustrate that it is possible for lots of examples to work for a given formula, but this is not evidence that proves the result.

blutick – Conditional Probability

I included information on blutick in my post on Transformations; a recent tweet from blutick reminded me to revisit these resources; each blutick topic includes clearly explained videos, worked examples and interactive questions. 

You can explore topics that provide comprehensive coverage for UK KS3 and KS4 and several other curricula. See blutick’s resources centre for Curriculum Mapping Guides which include Common Core, Singapore, IB, England KS3,4, England KS2, Cambridge International and Australia. We see from the pricing information that this library of videos and interactive questions will always be free.

Amanda Austin has produced some more of her excellent revision grids, it’s good to see some resources for IGCSE as well as GCSE. See the Revision section (under Algebra and Probability on Dr Austin Maths.

From Ofsted see their publication on Mathematics (25th May 2021) in its research review series; Schools Week has published a speed read summary. You can read more on the principles behind the research reviews here. An important read alongside the review is Anne Watson’s response on

A reminder of the approaching Core Maths Festival, the full programme is here. I added a Core Maths page to the 16+ series recently. (Included in Updates)

On the subject of festivals, we also have the Pearson Maths Festival 2021, a two week celebration of Maths.

Bar Modeling

For an Introduction to Bar Modelling have a look at the NCETM Secondary Professional Development Materials  Choose the section on Using mathematical representations at KS3 for 10 different guidance documents including Bar Models. This 17-page document for professional development provides a very comprehensive introduction to Bar Modelling and includes example problems.

Barvember – White Rose Maths

Barvember, from White Rose Maths, was created to encourage the use of the bar model. Bar models can really help children visualise and then solve maths problems. You could solve the problems using this great digital tool from White Rose Maths, Bar Model

On the subject of White Rose Maths have a look at this brilliant resource from Ben Gordon: Representations through the Curriculum.

Thinking Blocks – Math Playground

The section on Further Resources in the NCETM guidance document includes Thinking Blocks, an outstanding resource that I included in a recent post on Ratio and Proportion. The relevant information on using the resource is included for completeness in this post. Further resources for Bar Modelling are also given.

Thinking Blocks from Colleen King’s site Math PlayGround – is a wonderful way to visualize Ratio and Proportion problems. This works on all devices.

Models like this really help students to visualise problems. Note the numerous videos showing how to model problems (scroll down the page). You can use the Thinking Blocks Tool to model your own problems; watching the videos will help you learn to use the tool. There is a very clear help section also.

Thinking Blocks Modeling Tool

I have used the tool here to model a problem from Andy Lutwyche’s Clumsy Clive on Ratio resource.

Thinking Blocks Modeling Tool – use for your own problems

Also featured in NCETM’s resource section is The Mathenæum from Ken Wessen which includes Modelling Word Problems and Fractions, Decimals and Percentages.

The Mathenæum – Ken Wessen
The Mathenæum – Ken Wessen

See also:

Sybilla Beckmann paper

Sybilla Beckmann’s paper, Solving Algebra and Other Story Problems with Simple Diagrams: a Method Demonstrated in Grade 4–6 Texts Used in Singapore is an interesting read on the subject of the use of diagrams.

Note Pearson Edexcel GCSE Maths Networks on YouTube; for an excellent introduction to Bar Modelling watch Naomi Bartholomew-Millar’s session.

MathPad – Using Bar Pictures to Solve Equations

Ratio and Proportion

Some resources for problems on Ratio and Proportion…

From AQA Maths, check their excellent GCSE Maths Focus on success modular Teacher Training packs. Looking at the pack on Ratio for example you will find activities and discussion prompts on switching representations, combining ratios, problem-solving with ratio and dividing quantities in a given ratio.

AQA Teacher Training pack – Ratio
Clumsy Clive on Ratio – Andy Lutwyche

From Andy Lutwyche, many excellent Ratio resources, including these from some of my favourite series:

Transum Mathematics – Ratio Problems

Dr Austin Maths

Thinking Blocks – Math Playground

Thinking Blocks from Colleen King’s site Math PlayGround – a wonderful way to visualize Ratio and Proportion problems. This works on all devices. See this post on bar modelling which includes more details on using bar models to solve problems.

Models like this really help students to visualise problems. Note the numerous videos showing how to model problems (scroll down the page). You can use the Thinking Blocks Tool to model your own problems; watching the videos will help you learn to use the tool. There is a very clear help section also.

Thinking Blocks Modeling Tool

I have used the tool to model a problem from Andy Lutwyche’s Clumsy Clive on Ratio resource – see below for Andy’s resources.

Thinking Blocks Modeling Tool – use for your own problems

AQA Problems - Teacher's Guide
AQA Problems – Teacher’s Guide

Part of the Rich Tasks collection, from AQA

From AQA comes an outstanding resource, GCSE Mathematics: 90 maths problem  solving questions. Em  @EJmaths has a brilliant PowerPoint with all the questions and answers – see it here.

The guide includes indices by topic and also by strategy. I have used many of the problems successfully in the classroom with a variety of ages, they really do make students think. Checking the content index, there are several problems involving ratio:

AQA Problems on Ratio
AQA Problems on Ratio
AQA Problem Solving Guide
AQA Problem Solving Guide – Inside Circle

Nrich of course, a search on Ratio returns this selection of problems. Nrich problems always include teacher’s notes and solutions.

Nrich - Mixing Lemonade
Nrich – Mixing Lemonade

Ratio problem illustration
Ratio problem illustration
WolframAlpha – mix 4 white and 1 black

The GCSE specification includes: “relate ratios to fractions and to linear functions”.

Grey paint is made by mixing white paint with black paint in the ratio 4:1. Represent this information by drawing a straight line graph. (AQA GCSE Exemplification)

We can use the excellent Desmos graphing calculator to demonstrate examples; taking the above exemplification question from AQA I created a Desmos page to illustrate the problem. And did you know that you can mix colours on WolframAlpha?!

CIMT Interactive Materials – Year 8

Just as one can always find something on Nrich, the same is true of CIMT. From the Centre for Innovation in Mathematics Teaching, we have Ratio and Proportion from the Year 8 book (UK age 12-13) and Variation from the GCSE course (UK age 14-16). The Year 8 Interactive material includes lots of practice questions where the answers are checked for students.

Unit 7(Ratio and Proportion)
7.1 Equivalent Ratios
7.2 Direct Proportion
7.3 Proportional Division

As well as the Year 8 text, further resources are available such as Activities and Extra Exercises. Most CIMT resources require no password – a few like Teachers’ notes and answers do. An educational institution can obtain the password.

These sorts of problems must be on everybody’s Mathematics specifications all over the world! In the UK there is an increased emphasis on Ratio and Proportion in the latest GCSE specifications; a link to these resources has been included on the GCSE Content page.

Celebrating Women in Mathematics 2021

Why May 12th? As you can see from the information on it is the birthday of Maryam Mirzakhani; from the excellent St Andrews University MacTutor site we can read a biography of Maryam Mirzakhani.

Check the Twitter posts on this day, #May12WIM.
This site, Women in Maths was created by a student member of The Institute of Mathematics and its Applications (IMA)

Exploring this page on, note the video Journeys of Women in Mathematics a film created by the IMU Committee for Women in Mathematics.

It seems appropriate to link to my post Women in Mathematics written earlier this year to celebrate International Women’s Day.