# Bar modelling- a powerful visual approach for introducing number topics

Building on my recent post about a taxonomy for deep learning in maths, I have been trying to think a bit deeper myself about what each type of ‘deep learning link’ might look like. In particular, I have been researching and putting a lot of thought into what effective ‘visual models’ look like for the ‘key nodes’ I have previously identified as the most important foundation maths knowledge for students to master before starting their GCSE maths course. These are principally number topics.

Last year I became aware of the ** Singapore Maths Bar Modelling** approached have recently found the time to research it further. I bought some Singapore textbooks and read about the work of Dr Yeap Ban Har. This video, featuring Dr Ban Har shows an exemplification of the approach for a typical functional maths problem:

In short, I really like the approach and am convinced it could enhance my own practice significantly by giving students powerful, but simple visual models they can draw upon and use to solve problems. I have been experimenting with some of the models in my lessons this year and have seen the positive effect they have had on student understanding of topics. What these visual models give you is an entry point when teaching a topic that all students seem able to grasp. It presents the concept in its rawest, simplest form without the distraction of lots of words or mathematical notation. The diagrams don’t replace the eventual algorithmic methods, but they provide an entry point where students seem to understand what it is they are trying to solve; something that often gets clouded when algorithms are presented to early on.

In primary education in Singapore, maths teachers follow a Concrete-Pictorial-Abstract (CPA) sequence when teaching maths topics. They start with real world, tangible representations, move onto showing the problem using a pictorial diagram before then introducing the abstract algorithms and notation.

The particular power of the bar modelling pictorial approach is that it is applicable across a large number of topics. Once students have the basics of the approach secured, they can easily extend it across many topics.

I have spent some time putting together some pictures showing how the approach can be used for different topics. They are not teaching slides, but rather ‘notes for teachers’ to demonstrate how this single model can be adapted to be the diagrammatic entry point for many topics.

To start with students are given blank (bar) rectangles (on plain paper) and then get used to dividing the bars into halves, thirds, quarters etc:

They can then calculate a fraction of a quantity by first drawing the fraction in the bar, showing the length of the bar to be the quantity and then calculating the length of the shaded part:

Again, you’ll end up at ‘divide by denominator, multiply by numerator’ eventually, but this does show the concept of what’s going on very nicely and is a good route into showing where the algorithm comes from.

Next up, equivalent fractions:

Then simplifying fractions:

A ‘fraction wall’ (as many teachers use traditionally in England) can be used for ordering fractions:

Adding fractions with the same denominators:

Adding fractions with different denominators:

Multiplying fractions:

Dividing by fractions (works ok so long as you have integer answers). That’s enough to get across what is going on… Then you can lead into the method…

Converting mixed numbers and improper fractions:

Next up, understanding place value in decimal numbers. This approach lets you deal with lots of misconceptions like 0.62 not being larger than 0.7 etc. * Importantly, this is now taking the bar model and putting a decimal number line onto it.* This forms the basis for many topic models that follow:

Now they have an understanding of how the decimal number line works, and they can draw bar models for fractions, they can combine the two on one diagram to convert between fractions and decimals:

Next they can learn that percentages are hundredths, and in doing so can put a * percentage number line* under the bar model:

They can now combine the fractional bar model with both the decimal and percentage number lines directly underneath it to convert between fractions, decimals and percentages. They draw the fraction bar first, then put on the decimal increments (by dividing 1 by the denominator) and finally put on the percentages (by dividing 100 by the denominator):

Similarly to fraction of an amount earlier, you can use this approach to introduce percentage of an amount. Starting with showing how to find 10% of a number (and thus why you divide by 10), it serves as a nice way into multiplicative reasoning approaches to ‘build off the 10%’ to find other percentages:

Other percentage topics then follow, such as percentage increase and decrease:

By putting both the decimal and percentage number lines on the bar model for this, you can clearly show where percentage multipliers come from, including the ones less than unity for percentage decreases:

You can introduce calculating a percentage change using the bar model approach:

It’s a particularly nice diagrammatic way in to teaching reverse percentages:

Once the above techniques have been mastered, it can be used for showing how compound interest works as follows. Particularly nice is if you turn the bars so they are vertical, it shows the exponentially increasing relationship:

It works for ratio too:

I’ve used fraction walls for years, but it is the inclusion of decimal and percentage number lines built onto the fraction bars that is new for me. It opens up diagrammatic routes into so many topics and in such a coherent, simplistic way. The universality of the approach is what particularly impresses me; from humble beginnings of shading rectangles, the same model leads all the way up to reverse percentages and compound interest. If done in the correct order, there is a beautiful journey of progression all using one simple model. Each topic builds off the last logically as the model is manipulated in different ways.

The visual models won’t ever replace the algorithmic approaches, but instead will I hope provide my students with a better understanding of ‘what is going on’ when we are at the early stages of learning a topic. I hope their conceptual understanding is improved and this in turn enhances their procedural understanding through it giving it a purpose and something visual to hook onto. If they can ‘see the bar picture’ for problems with simple numbers it is my hope that the algorithmic approaches that follow that will enable them to solve problems with more challenging numbers with stick better. If they can represent a problem by drawing one of these models, they may have a better understanding of what the problem wants them to do.

I plan to develop resources to support teacher explanations and student activities in these topics in the coming months with the support of keen beans in my department and will share them with you when they have been tried and tested. There are no silver bullets in education, but this does represent a decent step forward for my teaching. Much to learn still!

Thank you for posting so much detail about this. I’ve been using bars and boxes for operations with fractions for the last couple of years and this has helped to cement what I do and extend it further. I know this will work for my current Y10 group, who need lots of support. I just need to ensure that the resources are in place for them (and will no doubt be useful for other classes of varying abilities).

I can see this being used well with some ‘interactive notebook math’ that I’ve been thinking about too.

Hi Tim,

Thanks kindly for your comments. I’m delighted to hear that you think your students might find this approach useful. I aim to develop some resources to support the ideas as time goes on and will of course share them when complete.

I hope you had a fab Christmas and all the best for 2015!

Best wishes,

Will

We used this with year 6 in primary last year. Romped through fraction, decimals, ratio and proportion. Entire class now think ratio and proportion really easy. Now introducing it in ks1 for addition and subtraction so children really ‘get’ how they are inverse and understand ‘ find the difference’ problems. It’s not worth trying to retrofit this onto older kids who’ve already got it but there might be some secondary students who are really confused about which operation to use when for whom the bar model for + and – is a light bulb moment.

I use these models a lot with KS1 and KS2 pupils, especially the more visual learners just get it. For those that haven’t come across it there is a great website mathplayground.com which has a neat interactive for these models from the very basic through fractions, ratio & proportion etc. Really worth a look. Thanks so much for all these models here, given me inspiration to keep working with them and develop them further with my pupils.

I have been using bar modelling for some of these topics at KS3 for the past couple of years and they really help students to visualise what is happening.

Thank you for the great diagrams and notes. If you don’t mind I would like to use some of your images to work on these ideas with the other teachers in my department.

Hiya,

Thanks for your kind words. Yes, please do feel free to use any of the pictures with your department. It would be great to hear how you get on with the resources.

Will

What an absorbing and useful post. So many useful ideas – thanks so much for taking the time to write.

I have been using the Australianised version of Singapore Maths, Prime, from Scholastic. There is a handbook with the bar model method.

http://au.scholastic.com/en/scholastic-prime-mathematics/teacher-support

Awsome post! I have created some GeoGebra sketches for understanding fractions and you’ve convinced me to slowly include bar models in each.

Take a look:

http://geogebraintheclassroom.blogspot.co.uk/search/label/fractions

I think these sort of dynamic visualisations go hand-in-hand with the more old school approach of drawing them. They are another great scaffolding tool to use at the board when teaching or for students to use individually if the technology is available in the classroom.

This looks great, very informative thank you! I am a class teacher and as much as I think other staff could use these as clear visuals, I think the children would be able to latch onto these the most. Is it ok if I am able to use these and share them?

Josh

Thanks Josh. Yes, please feel free to share

Wait is this 4 primary ppl or secondary? Cause I need to learn about it in ur 7

This is really good. I love using visual representations, and it’s great that it is something that can be used in so many topics so the students will revisit it lots of times. Thank you!

I always believed in visuals to aid learning especially when teaching EAL children. I really believe that this way of teaching will help to secure learning – ARE!

We have introduced the bar model method across our primary school this year. Complete game changer! Found this resource very useful and will be watching for anything else you put together. Thank you!

I am planning to introduce this to our Primary school so that it is embedded by the time it comes to Secondary school. Please share any worksheets or resources as this will be greatly appreciated. There are some apps on the iPad called ‘Thinking Blocks’ which are great and promote independent learning and also the website: http://www.mathplayground.com/ThinkingBlocks/thinking_blocks_modeling%20_tool.html

I’ll be honest, in the past I have been sceptical about this method. But having seen our maths lead use it and now reading this, I’ve been won over!

What a great way to introduce a range of concepts in maths. Your article has made it really clear to me and I am also planning on sharing it with children in maths lessons.

Thank you!

Thanks for this informative and detailed explanation of the bar model! I had never thought about such a visual explanation and will look to incorporating this more of to my teaching.

Hi I have heard references to the bar method and have seen a few scribbled examples, but to actually get some pictures with the accompanying narrative is so valuable. I am teaching my lowest set year 7 tomorrow, and really wanted to try this method; having read this, I now feel that I have the confidence to take it into the classroom. Thank you.

Hi. I’m a parent who’s daughter in year 7 struggles with maths. Where can I purchase the maths books. I believe this will greatly benefit her.

I’m glad I came across your site.

Thank you

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