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The product of two odd numbers is odd Always, Sometimes, Never 1

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Page 1: The product of two odd numbers is odd Always, Sometimes, Never 1

The product of two odd

numbers is odd

Always, Sometimes, Never

1

Page 2: The product of two odd numbers is odd Always, Sometimes, Never 1

Overview

Consider representations to

link the conceptual understanding of the

structure of the mathematics with selecting

appropriate mathematical operations in

order to solve problems

Page 3: The product of two odd numbers is odd Always, Sometimes, Never 1

KS2 2012

Page 4: The product of two odd numbers is odd Always, Sometimes, Never 1

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

Page 5: The product of two odd numbers is odd Always, Sometimes, Never 1

Using the Singapore Bar for Addition and Subtraction

Page 6: The product of two odd numbers is odd Always, Sometimes, Never 1

Addition - Aggregation

There are 3 footballs in the red basket 2

footballs in the blue basket. How many

footballs are there altogether?

Page 7: The product of two odd numbers is odd Always, Sometimes, Never 1

Addition - Augmentation

Peter has 3 marbles. Harry gives Peter 1 more marble. How many marbles does Peter have now?

Concrete Abstract

Page 8: The product of two odd numbers is odd Always, Sometimes, Never 1

Subtraction - Comparison Model

Peter has 5 pencils and 3 erasers. How

many more pencils than erasers does he

have?

Page 9: The product of two odd numbers is odd Always, Sometimes, Never 1

Moving to the abstract

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

Page 10: The product of two odd numbers is odd Always, Sometimes, Never 1

Generalisation

Page 11: The product of two odd numbers is odd Always, Sometimes, Never 1

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?

Page 12: The product of two odd numbers is odd Always, Sometimes, Never 1

Multiplication and Division

Page 13: The product of two odd numbers is odd Always, Sometimes, Never 1

Peter has 4 books

Harry has five times as many books as

Peter. How many books has Harry?

13

4

4 4 4 4 4

Page 14: The product of two odd numbers is odd Always, Sometimes, Never 1

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?

Page 15: The product of two odd numbers is odd Always, Sometimes, Never 1

Division

108 Year 3 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?

Page 16: The product of two odd numbers is odd Always, Sometimes, Never 1

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?

16

Page 17: The product of two odd numbers is odd Always, Sometimes, Never 1

Problems involving proportion

[email protected]

mailto:[email protected]
Page 18: The product of two odd numbers is odd Always, Sometimes, Never 1

Take a Strip and a paperclip

Page 19: The product of two odd numbers is odd Always, Sometimes, Never 1

Your Strip Represents 10p

• Show me 5p• Show me 2p• Show me 8p• Show me 7p

Page 20: The product of two odd numbers is odd Always, Sometimes, Never 1

Your Strip Represents £1

• Show me 50p• Show me 20p• Show me 80p• Show me 70p

Page 21: The product of two odd numbers is odd Always, Sometimes, Never 1

Your Strip Represents 1 metre

• Show me 50cm• Show me half a metre• Show me 20cm• Show me 80cm• Show me 70cm

Page 22: The product of two odd numbers is odd Always, Sometimes, Never 1

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?

Page 23: The product of two odd numbers is odd Always, Sometimes, Never 1

Draw 5 bars

Mark on 50% and the remaining proportion

Mark on 25% and the remaining proportion

Mark on 75% and the remaining proportion

Mark on 40% and the remaining proportion

Mark on 35% and the remaining proportion

Page 24: The product of two odd numbers is odd Always, Sometimes, Never 1

Solving Proportional Problems

Peter has ten sweets he eats half of them how many does he have left?

Ali has 30 sweets, she eats 1/3 of them, how many does she have left?

Stacey has 30 sweets, she eats 2/3 of them, how many does she have left?

A dress costs £32, it is reduced in price by

50%, how much does it cost know?

Page 25: The product of two odd numbers is odd Always, Sometimes, Never 1

Solving Proportional 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?

Page 26: The product of two odd numbers is odd Always, Sometimes, Never 1
Page 27: The product of two odd numbers is odd Always, Sometimes, Never 1

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)27

Page 28: The product of two odd numbers is odd Always, Sometimes, Never 1

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. 28

Page 29: The product of two odd numbers is odd Always, Sometimes, Never 1

Useful Resources

http://www.mathplayground.com/thinkingblocks.html

29

http://www.mathplayground.com/thinkingblocks.html
Page 30: The product of two odd numbers is odd Always, Sometimes, Never 1

Ratio

30

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

Page 31: The product of two odd numbers is odd Always, Sometimes, Never 1

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?

At peak times my mobile phone costs 1 times as much

as it does off-peak. A peak call costs 90p. What would it have

cost off-peak?31

Page 32: The product of two odd numbers is odd Always, Sometimes, Never 1

How would you solve this problem?

Penny had a bag of marbles. She gave one-third

of them to Rebecca, and then one-fourth of the

remaining marbles to John. Penny then

had 24 marbles left in the bag. How many

marbles were in the bag to start with?

(Problem N16, page 19. Overall percent correct,

Singapore: 81%, United States: 41%)

32


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