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.

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…

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.

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.


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

Further resources include resources from the exam boards; you can also check all these excellent sites for resources.
See, for example:

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 womeninmaths.org 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 womeninmaths.org, 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.

Math Whiteboard

Math Whiteboard

Math Whiteboard is a Collaborative Whiteboard designed specifically for Mathematics Teaching and Learning. With a very attractive interface and underlying powerful Computer Algebra System, this is an outstanding (completely free) resource for Mathematics Educators.

I was very fortunate to have a demonstration of this incredible resource from Donald Carey, one of the creators of this tool; there is no doubt that he and his team are passionate about Mathematics and the learning and teaching of Mathematics.

Select Create or join a whiteboard now.

Next select the new whiteboard button …

You can scroll through the ‘getting started with math whiteboard’ section to get an overview of the wonderful functionality of this resource.

I have created a new series of pages on Math Whiteboard in my Use of Technology series so this will be easy to find from the top menu tabs. I will be adding further pages as I explore this more, but it’s certainly something I will be using! (Not to be confused with Maths White Board, another fantastic resource with an extensive question bank ready for use which has also been added to the series.)

Using a whiteboard you can have everything in one place, multiple representations.
We could insert a text box with instructions for students, you could insert a video in the text box. You can also paste an image as you can see in an example below, where I included an image from Don Steward’s Median. This example uses a feature I really like, Highlight equivalence.

Math Whiteboard – Highlight equivalence feature

You can see many examples of whiteboards, scroll down the homepage to see these.

Maths White Board

From Matt Woodfine comes Maths White Board (scroll down the home page to read more about the projects), a treasure trove of resources including:

  • Examples pairs
  • Mini whiteboard activities
  • Retrieval practice
  • Worksheet generator
  • Class share
  • Class interactive keypads
  • Starter activities

Choose Launch MWB to access activities by topic, from here you can obtain example pairs, mini-whiteboard activities, securing exercises and retrieval practice questions.

Choosing for example Complete the square …

Maths White Board Complete the Square – Model it
Maths White Board – Tools

Note the various tools available at the bottom of the screen so you can write on any of these boards and note that it is possible to save a screenshot of your board.

Maths White Board Complete the Square – Recall WB

It is possible to refresh the questions to create as many examples as you wish.

Looking at the home page we see there is much to explore. Choose from Teaching tools, Printables, Starters/Revision activities, Games and Cass interaction tools,

You can choose to use the question bank or create custom activities. I particularly like Last White Board under Revision Activities. Create a custom board with a question from last lesson, last week, last topic and last term.

Maths White Board – Last White Board

Also under Revision Activities, you could choose a Revision White Board, with this tool you can design your own board or select from pre-made packs; as with other resources, you can select the difficulty level.

Maths White Board – Revision Board

As well as the Maths White Board you can use a Blank White Board which enables you to write over other websites, this is very simple to use, simply type in the web address and you are ready to go. I tried this and loaded this website, you can see my annotation over last week’s post on the highly recommended Challenge Prompts.

This set of projects is a very valuable resource and one which Matt Woodfine is constantly adding to the question bank; you can check his progress; scroll down the screen on the home page to check the current state for each topic area. As you scroll down note too the commentary on the research which shaped three activities.