This document discusses using pictorial models like Venn diagrams and block models to help students solve word problems algebraically. It provides examples of using these models to represent relationships and operations in word problems involving sets, fractions, comparisons, and changes over time. Venn diagrams are used to find quantities within sets, while block models visually partition quantities to set up and solve equations. The examples illustrate how translating problems pictorially can help students develop algebraic skills needed to solve more abstract word problems.

1. By: Erlyn M. Geronimo

2. 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

4. 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.

5. 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.

6. 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

7. 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?

8. Example 1
Friday
Saturday
Sunday
2
4
8
12
7
16
2

9. 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?
16
25
2

10. 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?

11. 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
30
10
20
10

12. Example 3
?

13. Example 3

14. 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?

15. • 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?

16. 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).

17. 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.

18. 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?

19. 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 Left
Photo
Album
240
2 240 1 120

20. Example 5
• How much did Donna spend on the novel?
• What fraction of her money did Donna spend on
her photo album?
Novel Money Left
Photo
Album
240
2 240 1 120
P360
2/10 or 1/5

21. 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

22. 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

23. 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.

24. 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?

25. 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

26. Example 7
• How many more cans did Joseph collect than JB?
JB:
James:
Joseph:
242
11 242 1 22
7 154
154

27. 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?

28. 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

29. Example 8
C:
P: 5
T:
500
• How many stickers has Tanya?
5 5 5
5 20 500+
5 480 1 96
3033 + 15

30. 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.

31. 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?

32. 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

33. Example 9
How much more money had Mike than Sarah at first?
Sarah:
Mike:
Before After
240
540 300
4 300
1 75
390

34. 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.

35. 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.
3
6
3
6
108

36. Example 10
Passed:
Failed:
Actual If…
• Find the number of pupils who took the test.
108
3
6
3
6
3
6 + 1082 + 11
180 2 + 9
9 180
220
1 20
56
164
2

37. Example 11
One 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

38. Example 11
One third of a number is 13 less than one-
half of that number. What is the number?
13
6
1
78
78

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