1. Solving Problems using the Bar Model
Material 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
This unit is designed to provide a visualisation for percentages and allow students gain both an understanding of, and a strategy to solve, percentage problems. Contexts are used throughout, carefully selected to allow the students to appreciate how the bar can be a representation of the whole. Students are encouraged to engage in and embrace the context, and return to the context when they are struggling to deal with abstract problems. The bar model is used as a visualisation and is a strategy that allows students that find percentages challenging an opportunity to make sense of them. For students with some prior understanding of the topic, it is important that they engage with the bar in more simple problems to allow them to use it later for more challenging problems (such as reverse percentage questions). These students are encouraged to compare and explain their methods with the solution through bar model .

4. Buying Ribbon
When you buy ribbon it is possible to purchase any amount. It costs £1.80 for 80cm of this spotty ribbon
Draw a picture to represent this information and try to make it look realistic.
In trials, the context of buying ribbon has proved to be a successful context for introducing the percentage bar. The shape of the ribbon leads the students towards a rectangular representation and students appreciate that the more ribbon one buys, the more it will cost. However, occasionally students may introduce the idea of some sort of ‘bulk buy’ discount and this should be praised for getting into the context but ruled out – perhaps mention that this is a hard up market trader who cannot afford to offer discounts. With some students/classes, it might be appropriate to bring in a length of ribbon and indicate it’s cost and then fold it in half and ask the cost of half etc.
Ideally the pictures would be initially drawn on a mini whiteboard or plain A4 paper, and students should work on this for a couple of minutes before comparing with the others on their table and then the rest of the class. There are no right or wrong responses here and it will be productive to hear their perception of the similarities and differences of the drawings. If the students do work independently, there will usually be a range of drawings from a life like spool of ribbon to a mathematical looking bar with amounts for cost and length clearly marked. Again, the merits of each picture could be discussed.
The rectangular bar with amounts labelled will allow them to make progress mathematically but if this does not appear, then it is introduced in the next slide.
Compare your picture with others in your class. What is the same and what is different about your pictures?

5. Buying Ribbon
The pictures above were drawn by two students from another class.
Are both pictures realistic?
If these two drawings are already around in the classroom you may decide to use them rather than the two on the slide. Again, both could be considered ‘realistic’ in their own way but the bar is clearly easier to label.
If you wanted to work out the costs of other amounts of the ribbon, which picture would be easier to label?

6. Buying Ribbon
Task 1: Draw a picture to show that it costs £1.80 for 80cm of the spotty ribbon. Now mark on three more lengths that you know the cost of.
For these two tasks, the students may use either the picture they originally drew or a new picture of the ribbon if they think it would make it easier. Students often opt to use the rectangular bar for this activity.
In Task 1, halving and quartering are very common giving 90p for 40cm, 45p for 20cm and £1.35 for 60cm. Some students may divide by 10 to get 18p for 8cm, then 36p for 16cm etc. It is important to encourage students to label 0p for 0cm on their bars.
In Task 2, it requires the students to lengthen their bars. Some will add 20cm (45p) and 80cm (£1.80) to get 100cm £2.25. Others will notice that counting up in 20cm means adding an extra 45p. When they describe their methods it is important to encourage them to stay within the context of buying ribbon (i.e. they should be justifying their ideas by talking about the price of lengths of ribbon)
Task 2: Have you worked out the cost of 100cm (1m) of ribbon? If not, try to work it out and mark it on your picture.

7. Buying Ribbon
The picture above has been used to work out the cost of 140cm of ribbon.
What is the cost of 140cm of ribbon?
This slide shows a more abstract representation of the ribbon and models the type of bar that will be used by students in future problems. Along with some pairs of lengths and costs it has zero/zero marked and the scales labelled ‘cm’ and ‘£’.
The question about the cost of 140cm of ribbon (£3.15) is designed to ensure students are re-engaged with the bar.
The answer for 140cm has been arrived at by initially finding the cost for 20cm by halving 80cm and then halving again. The cost for 20cm (£0.45) and the cost for 80cm (£1.80) have then been added to get the cost for 100cm. Finally, the cost for 40cm has been added onto the cost for 100cm to arrive at £3.15 for 140cm. However, it will be interesting to discuss any other possible routes to the answer.
Although a unitary approach is not the objective here, if it is introduced by a student allow the student to explain their route to the answer in the context of length and cost of ribbon.
In what order do you think the numbers in the drawing have been filled in?

8. Buying Ribbon Some ribbon is more expensive than other ribbon.
“Love Heart” ribbon costs £1.80 for 60cm. Draw a bar to represent this. Now use the bar to work out the price of 75cm.
“Animal Print” ribbon costs £1.50 for 90cm. Draw a bar to represent this. Use the bar to work out the price of 120cm.
All students should be encouraged to use a bar here, even if they believe that they can answer the questions without the bar. If they do complete the question without a bar, ask them to show where their calculations would be on a bar.
The ‘Love heart’ ribbon question can be solved by halving twice to find the cost of 15cm, then adding this on to 60cm to find the cost of 75cm (Answer: £2.25).
The ‘Animal Print’ ribbon cannot be solved by just halving and adding and requires the students to come up with an alternative strategy (Answer: £2.00).

9. Buying Skirting Board
DIY stores sell skirting board in the same way dress making shops sell ribbon.
The skirting board above costs £12.80 for 2400mm. Draw a bar to show this information.
This question is designed to encourage students to recognise that the bar can be used in a number of different contexts. In this case, the bar is representing a length of skirting and again the skirting, like the ribbon, looks like a bar.
Again, it is important that a bar is used to answer this question. Students should be encouraged to use the context when justifying their method and answer (Answer: 1500mm would cost £8.00).
Use your bar to work out the cost of 1500mm of skirting board.

10. The Download Bar This is an image of a download bar on a computer.
1) When did you last see something like this?
2) What information is there in the window?
This slide introduces a new context where the real life leads to the mathematical model of a bar. Students are initially encouraged to discuss and make sense of the context before engaging numerically with the problem. Some students may have noticed that downloads do not always happen at a constant rate, however we will assume that they do for this and subsequent questions.
3) How long has the program been loading so far?
4) How many megabytes (MB) are there left to load?

11. The Download Bar
10%
This download bar only says what percentage has already been installed. After 4 minutes the window looks like this
1. How long does this program take to install?
The picture here has become slightly more mathematical although still resembling the download bar from the previous slide.
The picture is approximately to scale and it is important that the labelling is kept to scale throughout this work.
2. How much longer will it be before it is 100% installed?

12. The Download Bar
On Worksheet N5, work out the total installation time for each program using the 10 bars.
Work out the total installation time for the program below: 6 ?
minutes 0%
25%
100%
2. Now work out the total installation time for this program:
The slide displays the first two questions from the worksheet. You may wish to model these before the class complete the worksheet, go through them after giving the class a few minutes to have a go or allow students to work on these at the board themselves.
It is important that the students use the bars to reach their solutions. If any solve them using a remembered method ask them to match their method with the labelling on the bar. 3 ?
minutes 0%
10%
100%

13. Percentage Increase
When you put your money into a savings account a bank will pay you money. This is called interest. The savings account at FirstWest pays 4% interest.
Work out how much interest you will receive at the end of the year if you start with £600. Use a Percentage Bar like this to help you.
It would be beneficial to have a short discussion about saving accounts prior to these questions.
Again, it is vital that the students use the bar (Answer: 4% of £600 is £24). Question 2 can be omitted if no one remembers another method.
2. You may remember another way of working out a percentage increase. If you do, use it to increase £600 by 4%. Where on the bar can you see your calculations?

14. Percentage Increase
At another bank, 3% interest is paid on savings. At the start of the year there is £1300 in the account.
Use a Percentage Bar to work out how much interest will be paid at the end of the year.
With some students, you may want to scaffold the solution by drawing a bar labelled with £1300 at 100%, £0 at 0% and a “?” at 3%.
2. If this bank increased the interest rate to 4%, what would happen to the amount of interest paid?

15. Percentage Increase
In most savings accounts, the interest is added on to the savings at the end of the year.
£400 is invested at 5% interest.
Challenge
You want to invest some money for 3 years. Explain in detail which account is better?
An account that pays 5% each year
An account that pays 15.5% at the end of the third year
Use a Percentage Bar to work out how much money will be in the account at the end of the year.
Most students will attempt to solve question 2 by calculating the interest at the end of the second year. Some will come to the correct answer of £441 whilst others may arrive at £440. This would provide an opportunity to discuss the importance of deciding what it is that we are finding the percentage of and the importance of using the correct “whole”.
2. Explain why the amount of money in the account at the end of two years is not £440

16. Percentage Reduction
A computer is in a sale with 20% off. It’s normal price was £540
Explain in detail the method above for calculating the sale price.
In what order would the numbers on the bar have been filled in?
On the worksheet, the bars have already been drawn to encourage the students to continue to use them. Although labelling to scale continues to be important, it is not necessary for the students to measure and divide up the bars.
3. Work out the sale prices on Worksheet N6

17. Percentage Reduction
A different computer normally costs £520 but has 20% off in the sale. A student has attempted to find the sale price using this bar
1. Describe in detail what this student has done
2. Explain why she will not be able to find 10% using this method. Why is 10% important?
In this example, the student has started with 100% and halved repeatedly. Some students will double and halve in the hope of reaching the correct percentage but in this case it will not be productive.
During the discussion of question 2, it will be useful for the students to communicate why dividing by 10 gives 100% and what this division looks like on a percentage bar. Also, does dividing by 20 give 20% etc.
3. Use the method of finding 10% on the bar to work out the cost of this computer in the sale

18. Percentage Reduction using a Calculator
At another shop there is a sale with everything 7% off. The normal price is £630
If the order of the labelling of the bar is not obvious, it would be worth labelling a bar with all the information from the question prior to answering question 1
Explain in detail this method for calculating the sale price. In what order did the student fill in the numbers on the bar?
2. Work out the sale prices on Worksheet N7

19. Percentage Reduction using a Calculator
This time, £700 is reduced by 9%. Use a percentage bar to find a 9% reduction of £700
Do you remember another method for percentage reductions? If you do, use your method to reduce £700 by 9%. Compare your method with the bar method. Do the numbers you calculate also appear on the bar?
Again, students with a remembered strategy are encouraged to compare their method to the bar method to develop their mathematical understanding (Answers: Q1. £637, Q2. £78.96)
2. Use a percentage bar to reduce £84 by 6%

20. Depreciation
All the cars in the pictures are 10 years old and cost £2000.
Ford Mondeo Honda Jazz
Which car would you buy? Give reasons for your answer.
Jaguar S Type Peugeot sport
2. Sarah is 20 years old and is buying her first car. Which car would you advise her to buy?
Question 1 is designed to encourage an open discussion to allow students to engage in the context. In trials, students have given a variety of answers for question 1 including decisions based on colour, potential speed of the car, reliability of the car, “coolness” of the cars etc.
Some students will be aware of the cost of insurance for young people when answering question 2.
Any comments about the resale value of the car can lead to an introduction to the idea of depreciation.

21. Depreciation
Sarah thinks she will have more money in 3 years’ time and plans to buy a different car then. She wants to know what her car will be worth when she replaces it.
Ford Mondeo: This car will go down in value by £250 per year.
Honda Jazz: This car will go down by 25% over the 3 years
Jaguar S-type: This car will be worth half as much in 3 years’ time.
Peugeot 206 Sport: After 3 years, this car will be worth 70% of what it was worth at the start
1. On the right is information about the value of the cars in 3 years’ time. Work out the value of each of the 4 cars in 3 years’ time
Ask the students why cars do not depreciate in a linear way. Do any of the class have examples of cars depreciating either quickly or slowly?
In answering question 1, a bar should be drawn for each car.
2. Which would be the best car for Sarah to buy?

22. Depreciation When things go down in value they are said to Depreciate.
Why do you think cars depreciate?
This is a slide that focuses on providing purpose to the mathematics to be learnt. In trials we have found that this not only engages the students but gives them insights into the mathematical concepts.
Can you think of…
Two other things that depreciate in value?
Two things that do not depreciate?

23. Depreciation Computers depreciate in value very quickly
Computer 1: Cost £600. The value of this computer will go down by 88% over 4 years.
Computer 2: Cost £750. This computer will only be worth 10% of its original value after 4 years.
Computer 3: Cost £570. This computer will be worth £125 less every year for 4 years.
Work out the value of each of the three computers four years after being purchased.
2. Which computer would you buy? State your reasons.
Students might be able to share examples of the value of computers falling in value.
The value of each computer should be calculated using a percentage bar. These need to be compared with care as although the length of each bar is 100% and so could be drawn of equal length, the amount of money represented by each bar is different and it is this that needs to be compared.
Answers to question 2 will vary depending on the opinion of the students

24. Repeated Percentage change
Sometimes depreciation over a number of years is calculated by reducing the value of something by the same percentage each year.
The value of a car goes down by 20% each year. Explain why the amount the value goes down each year gets less.
Use the percentage bars on the right to explain your answer to question 1?
This is a relatively complicated diagram and it will be worth allowing the students a little time to make sense of it. Ask them where is the initial value of the car?, Where is the loss in value after the first year?, Why are the bars getting shorter?, Why is the rate of reduction of the bars reducing? Why is the 20% at the end of the bar getting smaller? etc
Using the bars for Question 3 is time consuming but will allow them to gain insights rather than just learning a rule that they may later forget.
3. If the car was originally worth £12000, how much will it be worth after 5 years?

25. Repeated Percentage change
Cars usually go down in value each year. The value of a second hand Audi A3 goes down by 10% per year.
If the car is currently worth £8000, how much will it be worth next year?
Again, using the bars is time consuming but will allow them to gain insights into the concept of depreciation. (Answer: In 4 years, the car will be worth £ )
2. If the car is currently worth £8000, how much will it be worth in 4 years?

26. Test Results – Percentage Bar
Test and exam results are often given as percentages
2. Why do you think teachers sometimes use percentages instead of marks when they give you your results?
What was the best percentage test result you have ever had? What is the worst?
In trials, question 1 proved to be an excellent way to engage the students in the context. Some students were particularly enthusiastic to recall their worst results!

27. Test Results – Percentage Bar
Demi is revising for her exams and is convinced that going on revision websites will improve her marks.
Name some websites that she could use to help her revise.
Before she started using websites, she got 24 out of 40 in a science test. Her most recent mark in science is 30 out of 48. Has she improved? Explain your answer.
Question 1 should get them engaged with the context and also allow them to share useful revision websites.
Question 2 could be answered informally at this stage. There are a number of informal strategies that could be used to reach an answer. For example, some students notice that 24/40 is 6/10 and in the second test she got 6 more marks from 8 more question (6/8) so she has improved!

28. Test Results – Percentage Bar
Demi has drawn a percentage bar to help her change her mark of 24 out of 40 into a percentage.
The top bar shows how she started. Explain how she has filled in the top bar.
2. The bottom bar shows how she went on to find the answer. Explain what she did.
The transferring of the information in the question onto a bar is an important skill. It is very beneficial to include 0%/0 marks on the bar, allowing them to see each side of the bar as a linear number line and thus maintaining scale.
Demi has then found 10%(4), 20%(8), 40%(16) before adding 8 and 16 to get 24 (20+40=60%).
Some students may suggest that once she has found that 10% is 4, we could multiply 4x6=24 so 10x6=60%. They could consider which is easier and what other strategies are available.

29. Test Results – Percentage Bar
This is how Demi worked out her percentage mark for the second test (30 marks out of 48):
Explain how she has used the bars. Is there a quicker way of finding her percentage mark?
Has she improved after using the revision websites?
2. Use the bars on Worksheet N8 to work out if Demi has improved in all her lessons.

30. Reverse Percentage Calculations
At Yellow Two (a mobile phone provider), there is a 20% reduction in the bill for the first 6 months.
At this point we move onto the challenging topic of finding the original amount after a percentage change. In trials, some students have been able to solve this problem without any teacher input. Their approach was to label the bar with the information from the question as they had been doing in previous questions. The process of creating a visualisation of the information helped them recognise that £28.80 was 80% and that the answer that they were looking for was 100%. From there the process of calculation 100% was similar to previous questions.
If the bill, including the 20% reduction, for the first month is £28.80, what would the bill have been without the reduction?
HINT: Start by putting the information onto a bar.

31. Reverse Percentage Calculations
Grace and Nicole try to work out what the phone bill would be. Grace has worked it out to be £34.56.
Grace says: “I found 20% of £28.80 which was £5.76 and added it back on to get £34.56
Nicole say: ”but if you take 20% off £34.56 you do not get £28.80”
Grace’s method is a very common incorrect approach to answering these questions and one which some in class may have used for the question on the previous slide. Nicole is offering an argument that questions this approach.
One possibility would be to set up a mini debate within the class with some students arguing for Grace and others for Nicole. The teacher could suggest that a bar could make their justifications stronger. The class could then make a judgement about who is correct (hopefully the correct one!)
What do you think about Grace and Nicole’s ideas?

32. Reverse Percentage Calculations
Nicole starts her calculation by putting the information on a bar (see above)
Explain why Nicole has filled in £28.80 at 80% on the bar. What could she do next?
Nicole now offers a different approach and the students are encouraged to explain this approach in as much detail as possible.
Nicole finds 20% and then adds it on to 80%. How much would the bill be before the 20% reduction?

33. Reverse Percentage Calculations
At Shoo, there is a “25% off everything” sale. A pair of boots cost £63.75 in the sale. What would the original price have been?
It is beneficial to ask members of the class to explain the labelling of the bar before they begin to solve the problem (Answer: £85.00)
Look at the bar above. Find 25%, and then find 100% (100% will be the original price)

34. Reverse Percentage Calculations
If you are lucky, you will get a pay rise at the end of each year. Kevin’s monthly salary after a 10% pay rise is £2255 but he cannot remember how much he got before the pay rise.
This is the first problem where a bar is not offered to the students and it is vital that they learn to label the bar correctly and continue to use it. It is also the first problem involving a percentage increase – you may like to point this out, or you may wish to see if the students notice this and can adapt their bars accordingly.
(Answer: £2050)
Draw a bar and label it with his new salary under the appropriate percentage.
Use your bar to work out his original salary.

35. Reverse Percentage Calculations
There are a number of valid ways of reaching the answer using a bar and if students have solved the question correctly they should compare their bar with the one on this slide.
Students that did not answer the question correctly should compare their method with the bar above and then explain in detail how the bar has been used
Kevin used this bar to work out his old (original) salary.
Explain what he has done.
What is his old (original) salary?

36. Reverse Percentage Calculations
In each of the following problems, work out the original amount:
1. After a 5% pay rise, Tom’s new salary is £27, What was his pay before the pay rise?
2. Stacey’s new car has gone down in value by 30% in the first year. It is now worth £6860. What was it worth a year ago?
3. Olly’s ticket to see 1D cost £ This included a 7% handling fee.
What was the cost of the ticket without the handling fee?
In order to consolidate the use of the bar, you should insist that the students draw a bar for each of these questions. You may wish to set more questions of this type for homework.
(Answers: Q1. £26250, Q2. £9800, Q3. £85, Q4. £180)
4. The cost of Lauren’s new coat was £108 in the Next sale. 40% had been taken off the price.
What was the original price?

37. Summary (Don’t stop drawing)
When you are answering questions on percentages it is often useful to draw a Percentage Bar.
1. The cost of a train ticket from Slough to London Paddington is £7.20. The cost is increased by 4%. What is the cost of the new train fare?
For most students, the bar is a useful visualisation that can be used to not only to introduce percentages but can also be used to solve problems. Students need to be reminded that the bars can be used in exams and that they are not in any way “babyish” or non-mathematical.
2. Explain how this drawing has been used to work out the cost of the new train fare.