Maths to share  CPD for your school
In this issue of Maths to share we look at a technique that is used in Singapore to help children bridge the gap between concrete mathematical experiences and abstract representations. In Singapore, children are encouraged to use visual models such as ‘bar models’, ‘ten frames’, arrays and place value charts. We are going to focus on the ‘bar model’ which is specifically used to help children make sense of word problems. The approach is meant to reveal the structure of the mathematics in the problem. It is not a tool for performing a calculation. In Singapore, children begin to solve multistep word problems from the equivalent of our Year 4 using this technique. This is applied to solve increasingly more difficult problems. Some schools in England are now beginning to apply this approach and are finding it really helpful. If you have experience of using the ‘bar model’, we would love to know if it has made a difference to your children’s ability to make sense of and solve multistep word problems.
It would be helpful if you had colour rods or small cubes available during your meeting so that colleagues can fully explore this method both practically and visually.
Begin your meeting by setting this problem:
‘Ben spent ^{2}⁄_{5} of his money on a CD. The CD cost £10. How much money did he have at first?’

would this problem cause difficulties for the children in your school? Why/why not?

if this appeared on a KS2 SATs paper how many Year 6 children would confidently attempt it?
Next ask colleagues to solve the problem for themselves. Invite volunteers to share how they found their solution. Discuss the mathematics needed to achieve a solution.
Ask colleagues to copy you as you demonstrate how to find the solution using the ‘bar model’ practically with colour rods…
Line up five identical colour rods or cubes to represent five fifths
Move the rods or cubes that represent ^{2}⁄_{5} to one side.
If these represent £10, how much does one cube represent?
Agree £5.
There are five cubes altogether, so the total amount of money Ben started with was £25.
Next demonstrate this on a whiteboard or flipchart:
Total amount of money Ben had before he bought the CD is £25.

what do colleagues think of this practical and then visual approach?

do they think that it helps to ‘open up’ a problem?
Set one or two problems for them to solve, first practically using rods or cubes and then by drawing the ’bar model’ as you did.
You could use these examples:
1. Peter has four books. Harry has five times as many books as Peter. How many more books does Harry have?
Peter’s books
Harry's books
Harry has 16 more books.
2. There are 32 children in a class.
There are 3 times as many boys as girls. How many girls?
Each square is 8, so there are 8 girls and 24 boys.
3. Sam had 5 times as many marbles as Tom. If Sam gives 26 marbles to Tom, the two friends will have exactly the same amount. How many marbles do they have altogether?
Tom’s marbles
Sam’s marbles
Each part is 13, so 78 marbles altogether
4. A computer game was reduced in a sale by 20% and it now costs £48. What was the original price?
Each part is £12, so the original price was £60.
Hopefully, your colleagues will be quite excited about how this technique helps to make sense of problems!
It may be worth spending some time discussing how this technique can be introduced in Reception and KS1 so that the children will become familiar with it and competent in using it when they are in KS2.
Here are some suggestions to get you started:
In Reception and KS1, simple calculations can be explored practically and when the children are ready they could also be represented pictorially. For example:
Using sweets, set this problem: Sally had 3 sweets. Armani gave her 2 more. How many does she have now?
Using red and blue cars set this problem: Rasheed had 5 red cars and 3 blue. How many more red cars does he have?
This becomes a generalisation where a whole will represent 5 and not distinct squares:
Rosie had 4 pencils. Samir had twice as many. How many pencils did Samir have?
Make a list of the ideas you and your colleagues think of to distribute after the meeting.
Finish the meeting with this problem:
‘Sophie made some cakes for the school fair. She sold ^{3}⁄_{5} of them in the morning and ^{1}⁄_{4} of what was left in the afternoon. If she sold 200 more cakes in the morning than in the afternoon, how many cakes did she make?’
Clue:
There are five more morning parts than afternoon parts, so each part is 40 (200 ÷ 5). She made 400 cakes.
And finally…
Ask colleagues to try this technique out with their classes. They could discuss how their children responded at a future meeting.
We hope that you have found this article helpful. If you decide to use it for staff professional development, please let us know (either by posting a comment below or emailing us at info@ncetm.org.uk)  we'd love to hear what you did.
Explore further!
If you've enjoyed this article, don't forget you can find all previous Maths to share features in our archive, sorted into categories, including Calculation, Exploring reports and research, and Pedagogy.
Image credit
Page header by Roland O'Daniel, some rights reserved
