STU Seminar on The Model Method in Mathematical Problem Solving at NTUC Centre, Singapore 10 April 2010 by Yeap Ban Har

word problemsmodel method

of solving mathematical

Yeap Ban HarNational Institute of Education

Nanyang Technological UniversitySingapore

banhar.yeap@nie.edu.sg

Slides are available for download fromwww.mathz4kidz.com

SEMINAR

intellectual competenceintroduction

singapore curriculum

curriculum framework

Wellington Primary School, Singapore

Ministry of Education Singapore 2006

an excellent vehicle for the development and improvement of

a person’s intellectual competence

“

”

educationmathematics

Move 3 sticks to get two squares.

Move 3 sticks to get two squares.

Wellington Primary School, Singapore

Wellington Primary School, Singapore

Wellington Primary School, Singapore

Wellington Primary School, Singapore

Wellington Primary School, Singapore

Wellington Primary School, Singapore

visualization

“…development and improvement of a person’s

intellectual competencies...” Singapore Ministry of Education 2006

John had 1.5 m of copper wire. He cut some of the wire to bend into the shape shown in the figure below. In the figure, there are 6 equilateral triangles and the length of XY is 19 cm. How much of the copper wire was left? 

19 cm x 5 = 95 cm

150 cm – 95 cm = 105 cm

Primary Mathematics Standards Edition

coaching techniquesmodel method

pre-algebra

Siti

Rahim

29 kg

11 kg

Siti packs her clothes into a suitcase and it weighs 29 kg. Rahim packs his clothes into an identical suitcase and it weighs 11 kg. Siti’s clothes are three times as heavy as Rahim’s clothes.

What is the mass of Rahim’s clothes?What is the mass of the suitcase?

Siti packs her clothes into a suitcase and it weighs 29 kg. Rahim packs his clothes into an identical suitcase and it weighs 11 kg. Siti’s clothes are three times as heavy as Rahim’s clothes.

What is the mass of Rahim’s clothes?What is the mass of the suitcase?

Siti

Rahim

29 kg

11 kg

11 kg 18 kg

2 units = 18 kg1 unit = 9 kg

Rahim’s clothes is 9 kg.The suitcase is 2 kg.

We can also find the mass of Siti’s clothes (27 kg) if required.

Siti

Rahim x y

x y y yx + y = 11

x + 3y = 29

2y = 29 – 11 = 18

y = 18 ÷ 2 = 9

Siti packs her clothes into a suitcase and it weighs 29 kg. Rahim packs his clothes into an identical suitcase and it weighs 11 kg. Siti’s clothes are three times as heavy as Rahim’s clothes.

What is the mass of Rahim’s clothes?What is the mass of the suitcase?

and primary onemodel method

in kindergarten

Ali has 3 sweets. Billy has 5 sweets.How many sweets do they have altogether?

mental computationsmodel method

alternate methods

Cheryl has $20 less than David.Cheryl and David have $148 altogether,Find the amount of money Cheryl has.

Cheryl

David 20

$148

Cheryl has $20 less than David.Cheryl and David have $148 altogether,Find the amount of money Cheryl has.

Cheryl

David 20

$148 - $20 = $128

$128 ÷ 2 = $64

Cheryl has $64.

How about David? $84

Cheryl has $20 less than David.Cheryl and David have $148 altogether,Find the amount of money Cheryl has.

Cheryl

David 20

$148

Cheryl

David 20

$148 + $20 = $168

20

$168 ÷ 2 = $84

David has $84.

Cheryl has $64.

Cheryl has $20 less than David.Cheryl and David have $148 altogether,Find the amount of money Cheryl has.

see abstract ideasmodel method

helps average learners

Emil spent 2/5 of his savings to buy a gift and 1/6 of the remainder to buy a snack.Emil then has $7.50 left. Find the amount Emil spent on the gift.   

5 units = $7.50

1 unit = $1.50

4 units = $1.50 x 4 = $6

Emil spent $6 on the gift. How about he snack? $1.50How much is his savings? $7.50

There were three times as many students in soccer as there were in basketball. After 12 students moved from soccer to basketball, there number of students in both sports became equal. Find the number of students in soccer at first.  

soccer

basketball 12

Soccer 12 x 3 = 36

How about basketball?

There were four times as many students in soccer as there were in basketball. After 12 students moved from soccer to basketball, there number of students in both sports became equal. Find the number of students in soccer at first.  

soccer

basketball

There were four times as many students in soccer as there were in basketball. After 12 students moved from soccer to basketball, there number of students in both sports became equal. Find the number of students in soccer at first.  

soccer

basketball

3 units = 12

1 unit = 4

8 units = 32

There were 32 students in soccer at first

problem solvingmodel method

new situations

88 children took part in a swimming competition. 1/3 of the boys and 3/7 of the girls wore swimming goggles. Altogether 34 children wore swimming goggles. How many girls wore swimming goggles on that day?

boys

girls

34 34

88 children took part in a swimming competition. 1/3 of the boys and 3/7 of the girls wore swimming goggles. Altogether 34 children wore swimming goggles. How many girls wore swimming goggles on that day?

boys

girls

34 34

88 – 34 – 34 = 20

34

2 units = 34 – 20 = 14

1 unit = 7

7 x 3 = 21 21 girls wore goggles

leaving examinationmodel method

primary school

Machine A

Machine B 12

Every minute Machine A prints 12 pages more than Machine B. Machine A and Machine B together print a total of 528 pages in 3 minutes. At this rate, how many pages does Machine B print in 1 minute? 

176

Jim bought some chocolates and gave half of them to Ken.  Ken bought some sweets and gave half of them to Jim. 

Jim ate 12 sweets and Ken ate 18 chocolates. After that, the number of sweets and choco;ates Jim had were in the ratio 1 : 7 and the number of sweets and chocolates Ken had were in the ratio 1 : 4. 

How many sweets did Ken buy?PSLE 2009

chocolates

Jim

Ken

sweets

18

3 parts 12 + 12 + 12 + 12 + 18 = 661 part 22

Half of the sweets Ken bought = 22 + 12 = 34So Ken bought 68 sweets.`

12

12

12

12

12

12

Monday

Tuesday

Wednesday

Thursday

Friday

20

20 20

20 20 20

2020 20 20

Siti started saving some money on Monday. On each day from Tuesday to Friday, she saved 20 cents more than the amount she saved the day before. She saved a total of $6 from Monday to Friday. How much money did she save on Monday? 

At first Shop A had 156 kg of rice and Shop B had 72 kg of rice. After each shop sold the same quantity of rice, the amount of rice that Shop A had was 4 times that of Shop B. How many kilograms of rice did Shop A sell? 

A

B

156 kg

72 kg

At first Shop A had 156 kg of rice and Shop B had 72 kg of rice. After each shop sold the same quantity of rice, the amount of rice that Shop A had was 4 times that of Shop B. How many kilograms of rice did Shop A sell? 

A

B 28

156 kg

72 kg

3 units = 156 kg – 72 kg = 84 kg

1 unit = 28 kg

Each shop sold 64 kg of rice.

preliminary examination

model method primary school

Mrs Liu spent some of her monthly salary on a handbag, some of the remainder on a vacuum cleaner and saved the rest of her monthly salary. She saved $1890. Find her monthly salary.

Mrs Liu spent 1/5 of her monthly salary on a handbag, some of the remainder on a vacuum cleaner and saved the rest of her monthly salary. She saved $1890. Find her monthly salary.

Mrs Liu spent some of her monthly salary on a handbag, some of the remainder on a vacuum cleaner and saved the rest of her monthly salary. She saved $1890. Find her monthly salary.

Mrs Liu spent 1/5 of her monthly salary on a handbag, 1/4 of the remainder on a vacuum cleaner and saved the rest of her monthly salary. She saved $1890. Find her monthly salary.

3 units = $1890

Mrs Liu spent some of her monthly salary on a handbag, some of the remainder on a vacuum cleaner and saved the rest of her monthly salary. She saved $1890. Find her monthly salary.

Mrs Liu spent 1/5 of her monthly salary on a handbag, 3/8 of the remainder on a vacuum cleaner and saved the rest of her monthly salary. She saved $1890. Find her monthly salary.

5 units = $1890

Mrs Liu spent some of her monthly salary on a handbag, some of the remainder on a vacuum cleaner and saved the rest of her monthly salary. She saved $1890. Find her monthly salary.

Mrs Liu spent 1/5 of her monthly salary on a handbag, 4/7 of the remainder on a vacuum cleaner and saved the rest of her monthly salary. She saved $1890. Find her monthly salary.

12 units = $1890

There were 192 apples and pears in a box. John removed 2/5 of the apples from the box and he added 24 pears into the box. As a result, there was an equal number of apples and pears left in the box. How many more apples than pears were there in the box at first?

apples

pears 24

192

There were 192 apples and pears in a box. John removed 2/5 of the apples from the box and he added 24 pears into the box. As a result, there was an equal number of apples and pears left in the box. How many more apples than pears were there in the box at first?

apples

pears 24

192 + 24

8 units = 216

8 units = 160 + 56

1 unit = 27

apples

pears

2 27 2727 27

24

192

27

?

Apples = 27 x 5 = 135

Pears = 27 x 3 – 24 = 81 – 24 = 57

There were 135 – 55 – 2 = 78 more apples than pears at first.

A librarian counted the number of adults in the library and found that 2/5 of the number of women was equal to 2 times the number of men. When another 12 men entered the library and 45 women left the library, the ratio of the number of women to the number of men became 5 : 2.

men

women

12

45

30 5 units = 30 + 45 = 751 unit = 15

Men = 2 x 15 = 30

Women = 10 x 15 = 150

leaving examinationother heuristics

primary school

Draw a Diagram

David and Michael drove from Town A to Town B at different speeds. Both did not change their speeds throughout their journeys. David started his journey 30 minutes earlier than Michael. However, Michael reached Town B 50 minutes earlier than David. When Michael reached Town B, David had travelled 4/5 of the journey and was 75 km away from Town B.

Michael

David

75 km4/5

4 x 75 km = 300 km

(a) Town A to Town B is 375 km.

(b) 50 min 75 km

10 min 15 km

1 h 6 x 15 km = 90 km

(c) 50 min 1/5 of the journey250 min whole journey

Michael took 80 min less.He took 170 min.