pictorial models to problem solving

By: Erlyn M. Geronimo

WARM UP EXERCISE• Express the following in symbols:

• Y is 2 greater than x

• Maria’s age 3 years ago if she is m years old now

• The sum of two consecutive odd numbers if the smaller number is k

• The length L is twice the width W

• The value of a two digit number if the tens digit is x and the units digit is y

• The number c is two less than the product of a & b

USING PICTORIAL MODELS TO SOLVE WORD PROBLEMS

• Translating word problems into algebraic equations is a skill not easily acquired by students in the higher elementary and lower secondary school.

• Formulating and manipulating algebraic expressions require a level of abstraction which is not easily attained by students.

• The use of pictorial representations connects better with the intuitive perception of students, helping them understand relationships between the quantities involved in the problem and leading them to a strategy in solving it.

Using Venn Diagrams

• One of the pictorial models that are useful in finding relationships between sets is the Venn Diagram.

• Having a visual model of the sets in consideration provide an efficient way of determining cardinalities of said sets.

Example 1• A grade six teacher asked her class of 42 students

when they studied for her class the previous weekend. Their responses were as follows:

• 9 said they studied on Friday• 18 said they studied on Saturday• 30 said they studied on Sunday• 3 said they studied on both Friday and Saturday• 10 said they studied on both Saturday and

Sunday• 6 said they studied on both Friday and Sunday• 2 said they studied on Friday, Saturday, and

Sunday

Example 1• Assuming that all 42 students responded and

answered honestly, answer the following questions:

• How many students studied on Sunday but not on either Friday or Saturday?

• How many students did all their studying on one day?

• How many students did not study at all for this class last weekend?

Example 1

FridaySaturday

Sunday

24

8

127

16

2

Example 1

• Assuming that all 42 students responded and answered honestly, answer the following questions:

• How many students studied on Sunday but not on either Friday or Saturday?

• How many students did all their studying on one day?• How many students did not study at all for this class

last weekend?

1625

2

Example 2

• Every GOOP is a GORP.

• Half of all GORGS are GORPS.

• Half of all GORPS are GOOPS.

• There are 40 GORGS and 30 GOOPS.

• No GORG is a GOOP.

• How many GORPS are neither GOOPS nor GORGS?

Example 2• Every GOOP is a GORP.

• Half of all GORGS are GORPS.

• Half of all GORPS are GOOPS.

• There are 40 GORGS and 30 GOOPS.

• No GORG is a GOOP.

• How many GORPS are neither GOOPS nor GORGS?

GOOPS

GORPSGORGS

20

3010

20

10

Example 3

?

Example 3

Example 4• On a balance scale, two spools and one

thimble balance 8 buttons.

• Also, one spool balances one thimble and one button. How many buttons will balance one spool?

• On a balance scale, two spools and one thimble balance 8 buttons. • Also, one spool balances one thimble and one button. How many

buttons will balance one spool?

Singapore Model Method or Block Model Method

• The Model building approach to solving word problems was developed locally years ago by Hector Chee, a very experienced Mathematics teacher, and has been widely used in the teaching of math in primary schools in Singapore.

• Kids in Singapore are introduced to the method from as young as Primary One (the equivalent of Grade One).

Part-Whole Model

• In this model, a whole is divided into two or more parts.

• When the parts are known, we can find the whole by addition.

• When the whole and one part are known, we can find the unknown part by subtraction.

Example 5

• Donna spent PhP240 on a photo album. When she spent 3/8 of her remaining money on a novel; after which half of her money was left.

• How much did Donna spend on the novel?

• What fraction of her money did Donna spend on her photo album?

Example 5• Donna spent PhP240 on a photo album. When she

spent 3/8 of her remaining money on a novel; after which half of her money was left.

Novel Money LeftPhoto Album

240

2 240 1 120

Example 5• How much did Donna spend on the novel?• What fraction of her money did Donna spend on her

photo album?

Novel Money LeftPhoto Album

240

2 240 1 120

P360

2/10 or 1/5

Example 6• There are 240 cows and goats on a farm.• Three-fifths of the goats is equal to 3/7 of the

cows.• Find the difference in the number of cows and

goats in a farm.

GOATS:

COWS:

240

difference

Example 6• Find the difference in the number of cows and

goats in a farm.

GOATS:

COWS:

240

difference

12 240

1 20

2 40

40

Comparison Model

• In this model, two or more quantities are compared.

• If the two quantities are given, we can find their difference or ratio.

• If one quantity and either the difference or ratio is given, we can find the other quantity.

Example 7• JB, James and Joseph collected 242

soda cans for the school recycling campaign.

• James collected twice as many soda cans as JB.

• Joseph collected four times as many soda cans as James.

• How many more cans did Joseph collect than JB?

Example 7• JB, James and Joseph collected 242 soda cans for the school

recycling campaign. • James collected twice as many soda cans as JB.• Joseph collected four times as many soda cans as James. • How many more cans did Joseph collect than JB?

JB:

James:

Joseph:

242

Example 7• How many more cans did Joseph collect than JB?

JB:

James:

Joseph:

242

11 242 1 22

7 154

154

Example 8

• Claire, Pam and Tanya have 500 stickers among themselves.

• Pam has 5 more stickers than Claire.

• Tanya has thrice as many stickers as Pam.

• How many stickers has Tanya?

Example 8

C:

P: 5

T:

500

• Claire, Pam and Tanya have 500 stickers among themselves.

• Pam has 5 more stickers than Claire.

• Tanya has thrice as many stickers as Pam.

• How many stickers has Tanya?

5 5 5

Example 8

C:

P: 5

T:

500

• How many stickers has Tanya?

5 5 5

5 20 500+

5 480 1 96

3033 + 15

Before-After Model

• When a quantity or quantities change, a comparison is made between the new value(s) and the original value(s).

• This is sometimes combined with the comparison method in the more complicated word problems.

Example 9• Mike and Sarah had a total of PhP540.

• Mike spent PhP240. He now has three times as much money as Sarah.

• How much more money had Mike than Sarah at first?

Example 9• Mike and Sarah had a total of PhP540.• Mike spent PhP240. He now has three times as much

money as Sarah. • How much more money had Mike than Sarah at first?

Sarah:

Mike:

Before After

240

540 300

Example 9

How much more money had Mike than Sarah at first?

Sarah:

Mike:

Before After

240

540 300

4 300

1 75

390

Example 10

• The number of pupils who passed a mathematics test is 108 more than the number of pupils who failed.

• If 36 more pupils pass the test, the number of passers will be 10 times the number of failures.

• Find the number of pupils who took the test.

Example 10

Passed:

Failed:

Actual If…

• The number of pupils who passed a mathematics test is 108 more than the number of pupils who failed.

• If 36 more pupils pass the test, the number of passers will be 10 times the number of failures.

• Find the number of pupils who took the test.

36

36

108

Example 10

Passed:

Failed:

Actual If…

• Find the number of pupils who took the test.

10836

36

36 + 1082 + 11

180 2 + 9

9 180

220

1 20

56

164

2

Example 11One third of a number is 13 less than one-half of that number. What is the number?

Let x be the number

+ 13 =6

2x + 78 = 3x

x = 78

Example 11One third of a number is 13 less than one-half of that number. What is the number?

13

6

1

78

78

Acknowledgment

• Special thanks to A/P Flordeliza Francisco of Ateneo de Manila University for most of the examples used in this presentation.