Bar Models for Elementary Grades

Peggy FooMarshall Cavendish Institute

Learning Outcomes

Participants should be able to

understand the rationale of model method as a heuristic/ problem-solving tool.

Draw different types of models to solve a variety of word problems.

Origin

Developed by a project team in MOE in the 1980s

Objective: Help students who have great difficulty with word problems in the early years of primary school.

Drawing a pictorial model to represent mathematical quantities (known and unknowns) and their relationships given in a problem.

Rationale

The Mathematics Curriculum Framework focuses on mathematical problem solving.

Reasoning, communication

and connections

Under ‘Processes’ component, One of the heuristics is model

method Helps to visualize situations and

Maths problems which are usually meant for secondary pupils

DifferencesModel Method Algebraic Method

_________ representation

Abstract reasoning

More effective for _______ pupils who need to see to understand

More suitable for older pupils

Foundation for algebraic thinking (without the use of abstract symbol)

Use of abstract symbol

DifferencesModel Method Algebraic Method

Pictorial representation Abstract reasoning

More effective for younger pupils who need to see to understand

More suitable for older pupils

Foundation for algebraic thinking (without the use of abstract symbol)

Use of abstract symbol

Guidelines Represent the problem using bar(s) The bar(s) are best drawn

proportionately Fill in the diagram with all the given

information The unknown value/ answer is

represented by question mark Interpret the model and write a simpler

mathematical statement (e.g. 11 units + 40 84)

Different types of models

Part-Whole Model Comparative Model Change/ Transforming Model

Part-Whole Model Shows various parts which make

up a whole Find the whole by addition Find the other part by subtraction

Part-Whole Model (using concrete materials)

Ann had 5 books.Bill gave her 7 more books. How many books did Ann have

altogether?

Part-Whole Model ?

John has 20 marblesHe gave 3/5 of it to Peter.How many marbles did John give to Peter?

Part-Whole Model?

John has 20 marblesHe gave 3/5 of it to Peter.How many marbles did John give to Peter?

5 unit 20 marbles

1 unit 4 sweets

3 units 3 x 4

= 12

John gave 12 marbles to Peter.

20

Comparsion Model Show the relationship between 2

quantities when they are compared

E.g. compared by showing the difference

Comparsion Model (Try it)

Alice had 3 books.She had 9 books less than Beth.How many books did Beth have?

Comparsion ModelAlice had 3 books.She had 9 books less than Beth.How many books did Beth have?

3Alice

?

Beth

9

What do you think is the common mistake made by many students?

3 + 9 = 12

Comparsion Model (to find the difference)

Jess had 12 beads and Ken had 4.How many more beads had Jess than Ken?

Comparsion Model (to find the difference)

Jess had 12 beads and Ken had 4.How many more beads had Jess than

Ken?

4Ken

?

Jess

12 – 4 = 8

What do you think is the common mistake made by many students?

12

Model drawing promotes conceptual understanding via visual representations rather than “cue words” method.

More than use addition Less than use subtraction

Ann’s age is twice the age of Bill.Bill’s age is 3 times the age of Carol.If there total age is 70, What is the age of

Bill?

Comparsion Model

Ann’s age is twice the age of Bill.Bill’s age is 3 times the age of Carol.If there total age is 70, What is the age of Bill?

70

A

B

C

10 units 70 1 unit 7

3 units 21

Bill’s is 21 years of age.

Change/ Transforming Model

This type of model can be used to solve complex problems

The parts can be transformed into smaller units.

This type of model is useful for tacking problems which involve before-and-after situations.

At first, Sara had 4/7 of the number of marbles Jack had. When Sara received 36 marbles from Jack, both had the same number of marbles.

(a) How many more marbles did Jack have than Sara at first?

(b) How many marbles were there together?

At first, Sara had 4/7 of the number of marbles Jack had. When Sara received 36 marbles from Jack, both had the same number of marbles.(a) How many more marbles did Jack have than Sara at first?(b) How many marbles were there together?

S

J

Before

S

J

- 36

+ 36

(a)3 units 36

1 unit 12

6 units 6 x 12

72

Jack has 72 more marbles than Sara.

(b) 22 units 22 x 12 marbles

They were 264 marbles altogether.

After

S

J

- 36

+ 36

(a)1 ½ parts 36

1 part 24

3 parts 24 x 3

72

Jack has 72 more marbles than Sara.

(b) 11 units 264 marbles

They were 264 marbles altogether.

After

Three halls contained 9,876 chairs altogether. One-fifth of the chairs were transferred from the first hall to the second hall. Then, one-third of the chairs were transferred from the second hall to the third hall and the number of chairs in the third hall doubled. In the end, the number of chairs in the three halls became the same. How many chairs were in the second hall at first?

Hall 1 (Before)

Hall 2

Hall 3

After

Hall 1

After

Before

Hall 2

Hall 3

Hall 1

Hall 2

Hall 3

Hall 1

12 units 9876 (M1)1 unit 9876 ÷ 12 = 823

5 units 5 x 823 (M2)= 4115 (A1)

There were 4115 chairs in the second hall at first.

Hall 2

Hall 3

Hall 1

Thank you