1. Solving word problems using the bar model
Materials developed by Paul Dickinson, Steve Gough & Sue Hough at MMU

2. Thank you
Sue, Steve and Paul would like to thank all the teachers and students who have been involved in the trials of these materials
Some of the materials are closely linked to the ‘Making Sense of Maths’ series of books and are reproduced by the kind permission of Hodder Education

3. Note to teacher
In this Algebra PowerPoint, students consider ways of representing problems presented in words through the use of two or more bars.
In the Number module students worked with a variety of contextual situations which when drawn were very ‘bar like’ in representation. The focus in Number was to use one bar as a means of representing, to scale, quantities such as percentage, fractions, money, lengths, ratios and so on. Sometimes the single bar was thought of as a ‘double number’ line where the top line of the bar represented say, a percentage scale and the bottom line represented say price of a car.
The use of the bar (or more than one bar) in this session is as a ‘model for’ or a ‘method for’ solving lots of different problems. Although the problems refer to imaginable real world problems, they do not necessarily look ‘bar like’ when you draw them.
Students will have their own ideas about how to represent these problems on a bar and part of the sessions will be about developing a common understanding of how the word information transfers onto the bar. So, for example strategies such as :
Precisely labelling lengths across the top of a bar to represent a specific amount
Using comparison and shading to identify blocks within a bar which represent the same amount
Making effective choices about what is the ‘unit block’ within a particular problem
……………will help students to become more effective in using the bars to solve a variety of problems.

4. Note to teacher
. There are three worksheets linked to this PowerPoint (A5, A6 & A7). Worksheet A7 gives students an opportunity to revisit the sorts of problem they tackled within the Number module. This session also develops ideas around comparison of quantities, which are inherent in the solving of equations. In other words this session subtly shows links between topics in Number and Algebra. As a result of working, on these problems, it is intended that students and their teachers will start to see how drawing bars is a way of approaching and thinking about many areas of GCSE mathematics

5. Ages problem Dad is 37 years older than his son Joel.
Dad is 4 years younger than Mum.
The total of their ages added together is 99.
How old is Mum?
Spend two minutes trying to solve this problem on your own.
Share your ideas
Get the students to try this problem on their own…….it is unlikely that they will be able to solve it. There is no need to go into detail looking at their solutions at this stage. You can talk about where you normally see these sorts of problems, as they often appear in puzzle books or newspapers. Students in countries like Singapore and China, Finland and Holland tend to be more successful at solving problems of this kind. In these countries, they are encouraged to develop ways of representing the information on a bar. Remind your students that they developed ways to solve problems with fractions, ratios, percentage and proportion by drawing a bar, when working on the Number module. In this session they will be looking at how drawing a bar or even two or more bars to represent problems like the one above can really help them to see the problem and so lead them to solve the problem.

6. Word problems in general
Word problems like the age problem from the previous slide are known to be challenging to solve- for people of any age.
We will work on a method which involves drawing bars which will make the problems much easier to see.
See notes relating to the previous slide

7. Comparing quantities – Crisp packets
Aksa eats 6 packets of crisps a week
Latifa eats 10 packets a week
Which bar refers to Aksa ?
Which bar refers to Latifa?
Say how you know
The next few slides can be looked at with the class as a whole in order to establish some common ways of labelling the bars to represent the information.
Working across the representations is an important part of this session, so here we are working between the problem represented in words and then represented on the bars. Follow up questions such as ‘Where can you see the ‘6’ in this picture / ‘Where else can you see the ‘6’?’ / ‘where can you see the crisps in this picture?’ / ‘Where can you see Aksa in this picture?’ can be extremely useful as a way of revealing how the student interpretations of the words on the pictures may differ from each other, and from those of the teacher. For example some students describe the 6 as the length across the top of the first bar, some talk about the number of boxes in the first bar and so on.
Initially students may resist answering these questions as they may perceive them to be trivial, but they soon start to see the subtle differences in each other's interpretation and therefore the need to be more precise in describing their interpretations. Getting students to come up to the board and mark their ideas on the slide is helpful. (Make sure you are on ‘Notes page’ view in order to view this image.)
It is important that the length lines do cover the whole lengths of the bar, teachers should be particular about this if it does not naturally happen.

8. Comparing quantities - Pets
Zahra has 6 pets
Libby has 4 pets
Which is Zahra’s bar?
Which is Libby’s bar?
The problem is deliberately similar to the previous situation, but now the bars are blank. This means that labelling the length of the bar as the quantity makes more sense now there are no obvious divisions to the bar.
It is worth asking whether the students think the bars are drawn to scale or not? In trials, this prompted students to come to the board and start comparing the relative lengths of the bars by matching the end of the first bar down onto the second bar. An example of this is shown on the next slide.

9. Comparing quantities - Pets
Zahra has 6 pets
Libby has 4 pets
Which is Zahra’s bar?
Which is Libby’s bar?
A possible labelling of the bars, showing that the bars are roughly drawn to scale, given that the extra piece on Zahra’s bar is roughly half the length of Libby’s bar.
Similar ‘where can you see’ type questions like those used with slide 5 can be helpful here. i.e. ‘Where can you see the 6 pets?’, ‘where else can you see the 6 pets?’ and so on

10. Comparing quantities - Wages
Niyal earns £8 more than Neha a week
Which is Niyal’s bar?
Which is Neha’s bar?
How much does Niyal earn in a week?
Students can find the ‘more than’ description hard to interpret, so it is worth asking how they know which is Niyal’s / Neha’s bar.
Also ask, where to label the £8. The gesture of using your hands to represent the start and the end of the first bar, then transferring this gap onto the second bar to indicate the matching part of each bar is helpful. A possible labelling is shown below: (make sure that you are in ‘Notes page’ view in order to view this image.)
Some student’s say you can not tell how much Niyal earns in a week.
Some student’s say roughly £24. This estimate comes from assuming that the bar is drawn roughly to scale.

11. Comparing quantities - Coats
Jake has 3 fewer coats than Daniel.
Which is Jake’s bar? Which is Daniel’s bar?
Where could you show the ‘3’ coats?
Students may find it hard to interpret ‘fewer than’ in terms of the bar representation; how do they know which is Jake’s bar?
In the trials students gave various suggestions as to how to show the ‘3’ coats on their pictures. Mini white boards or a visualiser are useful here, so that the subtle variety of drawings can be quickly compared.
Although student’s natural interpretations are worth exploring, it can be helpful to suggest some guidelines. For example labelling the ‘3’ as a length rather than inside the bar may allow for more precision. Using the labels ‘Jake’ and ‘Daniel’ at the side of the bars are useful conventions to point out now, if student’s are still missing these. Even better might be to use the labels ‘Jake coats’ and ‘Daniel coats’ as this gives even more of the information given in the words.
There are various ways of indicating the ‘3’ on the bars, but these tend to fall into two categories.
Either the ‘3’ is indicated within Daniel’s bar as the extra part compared with Jake: (make sure that you are in ‘Notes page’ view, in order to view these images.)
Or the ‘3’ is indicated as an extension to Jake’s bar:
Notice the use of dotted lines in this case, to indicate that this ‘3’ is not actually part of Jake’s quantity of coats.

12. Word problems – the swimming party
There are 25 children at Lola’s swimming party, including Lola.
There are 13 more girls than boys at the party.
How many girls and how many boys are at the party?
Look carefully at the bars drawn and say how they represent the information in the question
Where are the 25?
Where are the 13?
Where is Lola?
Where are the boys?
It is helpful to bring students to the front so that they can gesture the parts they are referring to on the actual bars. They may gesture lengths to mean particular amounts, or the whole bars themselves (i.e. areas). These two ways can be represented using a pen, either by marking across the top of the bar as a length, or by shading inside the bars (i.e. using an area representation.), or both. Introduce the convention of drawing a bracket at the right hand side of the two brackets labelled as 25 in this case, to indicate this is the total amount represented by both bars when counted together. If they haven’t naturally labelled the bars with the name ‘Boys’ and ‘Girls’ in the case of this question, then remind them to do this or something similar. These labellings can be seen on the next slide.
Problems may arise in worksheet 1 and it will probably be necessary to look some of the questions with the group as a whole.
Copy the bars and use them to find how many girls and how many boys are at the party.

13. Word problems – the swimming party
There are 25 children at Lola’s swimming party, including Lola.
There are 13 more girls than boys at the party.
How many girls and how many boys are at the party?
Here the ‘standard’ block is the ‘Boys’ block and the ‘Girls’ block is drawn in relation to this. i.e it is the ‘Boy’s block with an extra part added on which is representing ‘13 more’. The two bars together represent 25 people. Transferring the end of the ‘Boy’s’ block to the equivalent place on the ‘Girls’ block is a very important strategy in tackling these problems.
Having labelled the bars, students will think on these bars in a variety of ways. One possible strategy is to reason : 2 BOYS + 13 = 25, SO 2BOYS = BOYS = 6.
Some students will subtract the 13 from the 25 in their head to leave 12 and then halve to give 6 BOYS. Which means = 19 GIRLS
Another alternative is to think of the ‘Girl’s’ block as the standard block and draw the ‘Boy’s’ block as 13 shorter than the girls block. In this case the thinking is along the lines of 2 GIRLS – 13 = 25, so 2 GIRLS = 38, 1 GIRLS = 19. BOYS is 13 less than 19 which makes 6 BOYS.
Although the thinking is algebraic, the pictures encourage this as a natural process, rather than students having to think of using rules like ‘doing the same to both sides’. They will have a variety of ways of thinking and recording their strategies, it is not necessary to force one particular way. However in the trials students did experience difficulties in deciding what was their ‘unit block’. So for example in this question some tried to use ‘Boy’s’ as their unit block, others went for ‘Girls’. It may be slightly easier to choose the smallest block as the unit block (in this case Boy’s).
Worksheet A5 enables students to develop their understanding of how the bars are used to represent information given in words and how the bars enable them to deduce other information about the original problem.
Describe exactly what has been written on these bars and why.
Now try the problems on Worksheet A5

14. Word problems – Pocket money
Jack has £12 more money in his pocket than Bella. Together they have £30.
How much money does Bella have?
The shift now is that students need to draw their own bars.
Having completed Worksheet A5, they will have picked up some general ideas such as starting the bars lined up, attempting to draw bars of the same thickness and the need to indicate the end of shorter bars within longer bars, and so on.
After sharing ideas and strategies for the Swimming party problem, students should be able to tackle the pocket money problem independently.
A possible representation is shown below: (Make sure you are on ‘Notes page’ view in order to view this image.)
Thinking of Bella’s amount as the unit block gives: 2 BELLA’S + £12 = £30. So 2 BELLA’S = £18. BELLA has £9 .
Hint: Start by drawing two bars one to represent Jack’s money and one to represent Bella’s money

15. Word problems – Mum’s age
Dad is 37 years older than his son Joel.
Dad is 4 years younger than Mum.
The total of their ages added together is 99.
How old is Mum?
Hint: Start by drawing three bars one to represent Dad’s age, one to represent Mum’s age and one to represent Joel’s age.
This is a return to the first problem from slide 3. This could be tackled before working on Worksheet A6, or part way through Worksheet A6.
The dilemma here is which to choose as the unit block out of Dad’s age, Mum’s age or Joel’s age. Both Mum and Joel’s ages are given as comparisons with Dad’s age, so this may lead to students choosing Dad’s age as the unit block.
Another preference is to choose the lowest age (i.e. Joel) as the unit, so that the other two bars are drawn as ‘add ons’ to this. This is the representation shown below: (Make sure you are on ‘Notes page’ view in order to view this image.)
This representation leads to thinking along the lines of:
3 JOEL’S + 78 = So, 3 JOEL’S = 21 JOEL is 7 years old. Sue is 48 years old.
It is worth seeing alternative approaches from other students and as a class discussing if any ways of thinking are easier to see than others. Sharing in this way, may lead to some students adopting a different approach.
Worksheet A6 gives students the opportunity to develop and refine their ways of drawing and using bars to solve these types of problem.
See Worksheet A6. Solutions and strategies for possible ways to solve the problems.
Try to make up a similar problem for your own family and give it to someone in your class to solve.
Now try the problems on Worksheet A6

16. Word problems – Summary
You can now tackle a large number of problems by drawing a bar or more than one bar to help you think about the problem.
Worksheet A7 gives a wide selection of problems to try. These relate to topics such as ratio, percentage, fractions and others.
If you become stuck on any problem it is always worth drawing something like a bar, as this can help you to think about a problem in clearer way.
Worksheet A7 includes problems similar to those tackled in the Number module. Working through Worksheet A7, helps the students (and their teachers) to recognise just how powerful the bar model is as a way of approaching so many problems in Mathematics (and in the world in general.)
Thinking of Bella’s amount as the unit block gives: 2 BELLA’S + £12 = £30. So 2 BELLA’S = £18. BELLA has £9 .
Now try the problems on Worksheet A7

17. Blank template slide Information Question 1 Question 2
Available if you wish to add to this presentation for your particular group.
Question 2