The Singapore Bar Fiona Allan July 2015

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

The Singapore Bar http://www.youtube.com/watch?v=E5Bte7K_Mxg

Is this a ‘hard’ question?

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?

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 http://math.coe.uga.edu/tme/issues/v14n1/v14n1.Beckmann.pdf 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.

Using the Singapore Bar for Addition and Subtraction

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

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

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

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

“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 http://schoolsofthought.blogs.cnn.com/2012/10/10/my-view-americas-students-can-benefit-from-singapore-math/

Generalisation

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?

Multiplication and Division

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

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?

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?

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”

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?

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

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

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

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

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

Draw 5 bars Mark on 50% and the remaining proportion

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.

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?

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?

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

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?

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?

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

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?

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?

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

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?

The solution

And another solution

And the solution using algebra http://www.singaporemathplus.com/2010/07/singapore-grade-four-question-without.html

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, 374-406

References Aki Murata (2008): Mathematics Teaching and Learning as a Mediating Process: The Case of Tape Diagrams, Mathematical Thinking and Learning,10:4, 374-406 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.

Useful Resources This Maths No Problem video https://www.youtube.com/watch?v=Em2yERb3Kfs 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 http://www.mathplayground.com/thinkingblocks.html

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 http://wn.com/singapore_math_a_spool_of_ribbon_fractions_bar_model Model Drawing in Singapore Schools Introduction to Singapore Maths Part 2 http://www.youtube.com/watch?v=U2bPwW4wc2E .