1. The Singapore Bar
Fiona Allan
July 2015

2. Aims
To consider representations to link the conceptual understanding of the structure of the mathematics with selecting appropriate mathematical operations in order to solve problems
To think about how using the ‘Singapore Bar’ might help our learners

3. The Singapore Bar

4. Is this a ‘hard’ question?

5. Try this question
A farmer has twice as many ducks as chickens. After the farmer has sold 413 ducks and 19 chickens died, he has half as many ducks as chickens. How many ducks does he have now?

6. Why was Singapore successful?
Laura had $240. She spent 5/8 of it. How
much money did she have left?
Overall percent correct, Singapore: 78%,
(United States: 25%).
Why were Singapore so successful?
They used a particular representation which
enabled pupils to access the structure of the mathematics
This is taken from a research article, which can be accessed on the web
Reference: Beckman, S. (2004). Solving algebra and other story problems with simple diagrams: A method demonstrated in grade 4–6 texts used in Singapore. The Mathematics Educator, 14(1), 42–46.

7. Using the Singapore Bar for Addition and Subtraction

8. Addition - Aggregation
There are 3 footballs in the red basket and 2 footballs in the blue basket. How many
footballs are there altogether?

9. Addition - Augmentation
Peter has 3 marbles. Harry gives Peter 1 more marble. How many marbles does Peter have now?
Concrete Abstract

10. Subtraction - Comparison Model
Peter has 5 pencils and 3 erasers. How
many more pencils than erasers does he
have?

11. Moving to the abstract
Peter has 5 pencils and 3 erasers. How many more pencils than erasers does he have?

12. “The bar model is really a form of pictorial algebra
“The bar model is really a form of pictorial algebra. The bar model converts the abstract words of the problem into an easy to understand pictorial diagram that helps students solve algebraic problems without using formal algebra. In later years the question mark can be replaced by a variable, making the transition to formal algebra easy.” Bill Jackson, Special to CNN

13. Generalisation

14. Problems to Solve
Tom has a bag of 64 marbles, his friend gives him 28 more,
how many does he have now?
Kelsey was running a 26 mile marathon, after 18
miles she felt very tired. How many more miles did
she have to run?
Carly bought an apple for 17p and a banana for 26p,
how much has she spent?
Ali had £10, he bought a DVD for £6.70 and a CD for £2. 90,
how much money did he have left?

15. Multiplication and Division

16. Harry has five times as many books as Peter. How many books has Harry?
Peter has 4 books
Harry has five times as many books as
Peter. How many books has Harry? 4 4

17. Multiplication
Henry ate 10 meatballs at the Christmas party. Shane ate 3
times as many meatballs as Harry . How many meatballs did they eat altogether?
Helen has 9 times as many football cards as Sam. Together they have 150 cards. How many more cards does Helen have than Sam?
The sum of 2 numbers is 60. One number is 9 times as big as the other. What is the bigger number?
The sum of 2 numbers is 64. One number is 7 times as big as the other. What is the smaller number?

18. Division
108 children are going on a field trip to the art museum. Each bus must carry 12
children. How many buses are needed?
Mr Smith had a piece of wood that measured 36 cm. He cut it into 6 equal pieces. How long was each piece?

19. According to Singapore's Handbook for Mathematics Teachers in Primary Schools, this model drawing approach is helpful for several reasons: (1) It "helps pupils visualize situations" (2) It "creates concrete pictures from abstract situations." (3) It "satisfies the pupils' learning through seeing and doing." (4) It "transforms words into recognizable pictures”

20. Try this!
Sam had 5 times as many marbles as Tom. If Sam gives 26 marbles to Tim, the two friends will have exactly the same amount. How many marbles do they have altogether?

21. Take a strip and a paperclip
Explain that the strip of card is the bar

22. Your Strip Represents 10p
Show me 5p
Show me 2p
Show me 8
Show me 7p

23. Your Strip Represents £1
Show me 50p
Show me 20p
Show me 80p
Show me 70p
Link to previous slide

24. Your Strip Represents 1 metre
Show me 50cm
Show me half a metre
Show me 20cm
Show me 80cm
Link to previous slide

25. Your Strip Represents £5
Show me £3
Show me £4
Show me £3.50
Show me £3.59
What would the half way mark represent?
Link to previous slide

26. Draw 5 bars Mark on 50% and the remaining proportion

27. Dividing by a fraction https://www.ncetm.org.uk/resources/43609
Springfield Y6 - bar model dividing by fractions) where Year children are first use the bar model to divide a whole number by a fraction, divide by a unit fraction and divide by non-unit fractions. The children then use the same model to solve problems involving division by a fraction.

28. Fraction
George had 355 marbles. He lost 1/5 of the marbles and gave ¼ of the remaining marbles to his brother. How many marbles did he have left?

29. A ‘difficult’ fractions question
Mary spent 1/3 of her money on a book. She spent 3/4 of the remainder on a pen. If the pen cost £6 more than the book, how much money did she spend altogether?

30. Ratio Tim and Sally share marbles in the ratio of 2:3
If Sally has 36 marbles, how many are there altogether?

31. Ratio question
Aaron buys a tie and a belt. The prices of the tie and belt are in the ratio 2 : 5. If both items cost £53.90, (a) what is the price of the tie? (b) what is the price of the belt?

32. Solving ratio problems
A herbal skin remedy uses honey and yoghurt in the ratio 3:4. How much honey is needed to mix with 120 g of yoghurt? A health bar sells desserts with chopped apricot and yoghurt in the ratio 2 : 5. How much chopped apricot will be mixed with 150 g of yoghurt?

33. Money problems
A dress costs £32, it is reduced in price by 20%, how much does it cost now?

34. Money Problems
A Super Mario Game costs £45, it is reduced in price by 25%, how much does it cost now?
A computer game was reduced in a sale by 20%, it now costs £40, what was the original price?
A computer game was reduced in a sale by 40%, it now costs £60, what was the original cost?
Laura had £240. She spent 5/8 of it. How much money did she have left?

35. Basic algebra
Peter thinks of a number. He multiplies the number by 3. He then adds 2. The answer is 20. What was the number?

36. Algebra
Could you use a bar model to help learners solve 4(y + 5) = 28 OR 5x + 7 = x + 9

38. Try this again!
A farmer has twice as many ducks as chickens. After the farmer has sold 413 ducks and 19 chickens died, he has half as many ducks as chickens. How many ducks does he have now?

39. The solution

40. And another solution

41. And the solution using algebra

42. Reflection
As learners come to use particular representations in learning activities, the representations help guide the learning process and become a part of the learners’ cognition.
Murata (2008 p376)
With the aid of these simple strip diagrams, children can use straightforward reasoning to solve many challenging story problems conceptually.
(Beckmann 2004 p46)
Note the Murata article is from Japan, where the diagrams are also used
Aki Murata (2008): Mathematics Teaching and Learning as a
Mediating Process: The Case of Tape Diagrams, Mathematical Thinking and Learning,
10:4,

43. References Aki Murata (2008): Mathematics Teaching and
Learning as a Mediating Process: The Case of
Tape Diagrams, Mathematical Thinking and Learning,10:4,
Beckman, S. (2004). Solving algebra and other story problems with simple diagrams: A method Demonstrated in grade 4–6 texts used in Singapore. The Mathematics Educator, 14(1),
42–46.

44. Useful Resources
This Maths No Problem video may help you gain a deeper understanding of how the bar model can be used to solve problems. You can practise problems online with Thinking Blocks

45. More Useful Resources
Bar diagrams can be used to solve many word problems. To see this in action watch these videos: (Watch numbers 1 and 6 first Model Drawing in Singapore Schools Introduction to Singapore Maths Part 2 .