P3 & P4 Parents’ Seminar

Springdale Primary School

P3 & P4 Parents’ Seminar

Mathematics

Strategies for Model Drawing

26 Jan 2018

1


Objectives

At the end of this session, parents will be able

to:

• understand the rationale of using the model

approach in solving problem sums

• solve middle primary story sums using the

model approach

• guide their child to solve story sums using the

model approach

2


Outline

• Why? (Introduction to model method)

• What? (Explanation of different types

of model)

• How? (Hands-on practice with drawing

of model)

• How? (Home support for your child)

3


Curriculum Framework

4


Processes – Thinking Skills

• Thinking skills are skills that are used in a

thinking process, such as

– Classifying

– Comparing

– Analysing parts and whole

– Identifying patterns and relationships

– Induction

– Deduction

– Generalising

– Spatial visualisation

5


Processes – Heuristics

• Heuristics are problem-solving strategies

when the solution to the problem is not

obvious. These include

– making a guess (e.g. trial and error/guess

and check, making a supposition)

– walking through the process (e.g. acting it

out, working backwards)

– using a representation (e.g. drawing a

diagram, tabulating)

– changing the problem (e.g. simplifying the

problem, considering special cases)

6


Using a Representation

• Representations allow students to

– reflect on them;

– modify them; and

– link them to suitable problem-solving strategies

• Representations include

– Picture

– Model

– Diagram

– Table/List

7


Mathematics Syllabus

Primary 1 and 2 Primary 3 and 4 Primary 5 and 6

• Whole Number

• Money &

Measurements

• Fractions

• Whole Number

• Money &

Measurements

• Fractions

• Decimals

• Whole Number

• Money &

Measurements

• Fractions

• Decimals

• Ratio & Proportion

• Percentage

• The model method is one of the most frequently

used problem-solving heuristics throughout

primary school.

8


Features of Model Drawing

• Simplify the problem.

• Visualize the problem from abstract to

concrete.

• Make sense of and manipulate the

information pictorially.

• Length of the rectangular bars is drawn

proportionately in relation to one

another.

9


Features of Model Drawing

• The available information is recorded

onto the models.

• Question marks are used to indicate the

unknown information.

• Translate the problem into mathematics

steps.

10


Types of Model

• Part-whole Model

• Comparison Model

• Before-after Model

11


Part-Whole Model

Part Part

Whole

12


Part-Whole Model

Part

Part

Whole

13


Comparison Model

Comparison

X

Y

14


Comparison Model

Comparison

X

Y

15



Examples

1. A container with 2 packets of milk has a mass of 1200 g. The same container with 4 packets of milk has a mass of 2200 g. What is the mass of the empty container?

17

What do I know?

There are containers and

packets of milk

1 container has 2 packets

of milk

Another container has 4

packets of milk

Asked to find?

Mass of container


Examples


18

Have I solved similar

questions before?

What skill should I use?

Guess & Check?

Systematic Listing?

Draw a diagram?

Draw a model?


Examples


1200

2200

milkcontainer milk

milkcontainer milk milk milk

?

19

Difference


Examples


20

2200 – 1200 = 1000

2 units = 1000

Method A:

1200 – 1000 = 200

Method B:

4 units = 1000 × 2

= 2000

2200 – 2000 = 200


Examples


21

200 g makes sense?

Check for:

Number Transfer

Units

Calculations

The mass of the empty

container is 200g.


Examples


22

Refeed the answer back

to the question:

Container A:

200 + 1000 = 1200

Container B:

200 + 2000 = 2200


Examples

2. 3 shirts and 1 dress cost $84. 3 shirts and 3 dresses cost $132.Find the cost of a dress.

23

What do I know?

There are shirts

There are dresses

Cost of 3S and 1D

Cost of 3S and 3D

Asked to find?

Cost of 1D


Examples


24


questions before?


Guess & Check?

Systematic Listing?

Draw a diagram?

Draw a model?

Using equations?


Examples


132

84

?

25

S SS

SSS

D

D D D


Examples


26

132 – 84 = 48

2 units = 48

1 unit = 48 ÷ 2 = 24


Examples


27

Better alternative?

3S + 1D = $84

3S + 3D = $132

2D = 132 – 84 = 48

1D = 48 ÷ 2 = 24

Better alternative?

S S S D = $84

S S S D D D = $132

$84 D D = $132


Examples


28

24 makes sense?

Check for:

Number Transfer

Units

Calculations

The cost of a dress is $24.


Examples


29

24 makes sense?

If dress is $24, 3 shirts will

be 84 – 24 = 60

1 shirt = 60 ÷3 = 20

Refeed into the question:

3S= 60

3D= 24 × 3 = 72

Total = 60 + 72 = 132


Examples

3. Aaron is 28 years older than Ben. Ben is 4 years older than Carl.If their total age is 84 years, what is Aaron’s age?

30

What do I know?

A older than B

Comparison

B older than C

Comparison

Total of A + B + C = 84

Asked to find?

A’s age?


Examples


31


questions before?


Guess & Check?

Systematic Listing?

Draw a diagram?

Draw a model?


Examples


28Aaron

Ben

Carl

84

?

4

4

32


Examples


33

28 + 4 + 4 = 36

84 – 36 = 48

3 units = 48

1 unit = 48 ÷ 3 = 16

16 + 4 + 28 = 48


Examples


34

Check by refeeding

A = 48

B = 48 – 28 = 20

C = 20 – 4 = 16

Total = 48 + 20 + 16 = 84


Examples

4. The sum of 2 numbers is 1568.The difference between them is 580.What is the greater number?

35

What do I know?

There are 2 numbers

Sum = 1568

Difference = 580

One number is greater than

the other

Comparative

Asked to find?

The greater number?


Examples


36


questions before?


Guess & Check?

Systematic Listing?

Draw a diagram?

Draw a model?


Examples


580Big

Small1568

?

37


Examples


38

1568 – 580 = 988

2 units = 988

1 unit = 988 ÷ 2 = 494

494 + 580 = 1074

The greater number is

1074.


Examples


39

Check by refeeding

Greater number = 1074

Smaller number:

1074 – 580 = 494

Sum:

1074 + 494 = 1568


Examples

2

3of the sales from potatoes is as much as

2

5of the sales

from tomatoes. The sales from the potatoes is $500 less than the sales from tomatoes. Find the total sales from the potatoes and tomatoes.

40

5.

What do I know?

2

3of the sales from potatoes

as much as

2

5of the sales from tomatoes

$500 less from sales of potatoes

Asked to find?

Total sales?


Examples

2


2

5of the sales


41

5.

Making sense of ….

2

3of the sales from potatoes

as much as

2

5of the sales from

tomatoes.


Examples

2


2

5of the sales


42

5.


questions before?


Guess & Check?

Systematic Listing?

Draw a diagram?

Draw a model?


Examples

2


2

5of the sales


Potatoes

Tomatoes

?

500

43

5.


Examples

2


2

5of the sales


44

5.

2 units = 500

1 unit = 500 ÷ 2 = 250

8 units = 250 ×8 =2000

The total sales from the

potatoes and tomatoes is

$2000.


Examples

2


2

5of the sales


45

5.

Check by refeeding

Sales from potatoes:

3 units = 250 × 3 = 750 2

3of it =

2

3×750 = 500

Sales from tomatoes:

5 units = 250 × 5 = 12502

5of it =

2

5× 1250 = 500


Examples

6. At a concert, there were twice as many girls as boys. The number of girls is 4 times as many as the adults. There were 360 fewer adults than girls. How many people were at the concert?

46

What do I know?

Concert

Girls = 2 units; Boys = 1 unit

Adults = I unit; Girls = 4 units

Fewer adults than girls

Comparative

Asked to find?

Total no of people?


Examples


47

Making sense of ….

The number of adults is 1

4of

the number of girls


Examples


48


questions before?


Guess & Check?

Systematic Listing?

Draw a diagram?

Draw a model?


Examples


Girls

Boys

Adults

360

?

49


Examples


50

3 units = 360

1 unit = 360 ÷ 3 = 120

7 units = 120 ×7 = 840

There were 840 people at

the concert.


Examples


51

Check by refeeding

1 unit = 120 (A)

4 units = 120 × 4 = 480 (G)

2 units = 120 ×2 = 240 (B)

twice as many girls as boys

number of adults is 1

4of the

number of girls

Total: 120 + 480 + 240 = 840


Before-After Model

• A basic change situation involves

3 elements

– the initial value of a quantity

– the change, which can be an increase or

decrease, and

– The final value of the quantity

• 2 models are drawn for comparison.

• It is not always the case when the

before model is drawn first.

52


Examples

7. Aaron had 5 times as many marbles as Ben. After Aaron gave 72 marbles to Ben, they each had the same number of marbles. How many marbles did Aaron have at first?

53

What do I know?

2 people: A and B

A has more than B

A has 5 times as many as B

A gave 72 to B

After: they had the same

number of marbles

Asked to find?

A at first?


Examples


54


questions before?


Guess & Check?

Systematic Listing?

Draw a diagram?

Draw a model?


Examples


Aaron

Ben

?

72

Aaron

Ben

BEFORE

AFTER

55


Examples


56

2 units = 72

1 unit = 72 ÷ 2 = 36

5 units = 36 ×5 = 180

Aaron had 180 marbles

at first.


Examples

7. Aaron had 5 times as many marbles as Ben. After Aaron gave 72 marbles to Ben, they each had the same number of marbles. How many marbles did Aaron had at first?

57

Check by refeeding

Aaron = 180

Ben = 180 ÷5 = 36

Aaron gave 72:

180 – 72 = 108

Ben received 72:

36 + 72 = 108


Examples

8. Aaron had 6 times as much money as Ben at first. When their mother gave them $300 each, Aaron had 3 times as much money as Ben in the end. How much money did Aaron have at first?

58

What do I know?

2 people: A and B

A has more than B

A has 6 times as much

money as B

Mother gave them $300

each

After: A has 3 times as

much money as B

Asked to find?

Amount Aaron had at first?


Examples


59


questions before?


Guess & Check?

Systematic Listing?

Draw a diagram?

Draw a model?


Examples


Aaron

Ben

?

?

Aaron

Ben

BEFORE

AFTER 300

60

300

300 300

300

300


Examples


61

6 units + $300 = 3 units +

$900

3 units = 600

1 unit = 600 ÷ 3 = 200

6 units = 200 ×6 = 1200

Aaron had $1200 at first.


Examples

8. Aaron had 6 times as much money as Ben at first. When their mother gave them $300 each, Aaron had 3 times as much money as Ben in the end. How much money did they have at first?

62

Check by refeeding:

Aaron = $1200Ben = $200

Mother gave them $300 each: Aaron = 1200 + 300 = 1500Ben = 200 + 300 = 500 Aaron had 3 times as much

money as Ben in the end.


A Final Word

What Makes Model-drawing Difficult?

• Knowledge Factors

– Linguistic knowledge

– Algorithmic knowledge

– Conceptual knowledge

– Schematic knowledge

http://repository.nie.edu.sg/jspui/bitstream/10497/132/1/ME-2-1-93.pdf

63


A Final Word

What Makes Model-drawing Difficult?

• Affective factors

– Interest and motivation

– Confidence

– Perseverance

http://repository.nie.edu.sg/jspui/bitstream/10497/132/1/ME-2-1-93.pdf

64


A Final Word

Common Model-drawing Pitfalls

• Incorrect representation of the story sum

• Incomplete representation of the story

sum

• Transfer error

65


A Final Word

Resources for Model-drawing

• Facebook group – “Maths Model Method

– Singapore”

• http://www.teach-kids-math-by-model-

method.com/

• http://citeseerx.ist.psu.edu/viewdoc/dow

nload?doi=10.1.1.555.5563&rep=rep1&ty

pe=pdf

66


Q & A

THANK YOU

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Documents

P3 & P4 Parents’ Seminar