Module-3-PSMM.docx

Modules in Problem Solving,Mathematical Investigation and Modeling
Mary June Tan-Adalla
Faculty, College of Education
Module 3
PROBLEM SOLVING HEURISTICS
Overview
Strategies are things that George Pólya would have us choose in his second stage of
problem solving (devise a plan) and use in his third stage (carry out the plan). In actual fact he
called them heuristics. These are the strategies that could be used in solving the problem. As you
solve more problems, you learn strategies and techniques that can be useful. But no single strategy
works every time. Some strategies might work for a number of problems but some might not.
There are a number of common strategies that the problem solver can use to help them
solve problems. This module presents several strategies that will be of value for solving
mathematical problems. Examples of how these strategies are applied in solving problems are also
found in this module.
Learning Outcomes
After learning this module, you should be able to:
 identify patterns from a systematic exploration of a problem situation and formulate
conjectures;
 make a diagram to clarify understanding of non-routine problems;
 collect and record data systematically and use logic in solving a problem;
 verify the correctness of a solution;
 produce alternative solutions and make connections among concepts;
 solve advanced (Olympiad level) multi-step problems in various topics from the secondary
curriculum;
 modify a problem, look for symmetry, or make it simpler;
 act out to solve a non-routine problem;
 make a list or table in solving problems;
 check solutions using alternative (or invented) solution methods;
 work backwards by reversing operations (or drawing deductions) after assuming the
conclusion in solving certain problems;
 use equation or formula to solve problems;
 apply guess and check to solve problems;
 set and effectively use notations in problems solving or proving;
 justify solutions using the pursue parity technique and coloring proof;
 contrast and compare multiple solutions to a problem.
3.1 Search for a Pattern and Formulate Conjectures
A problem can be solved by looking into the relationship of the elements of a list and
identifying the pattern. The ability to see patterns is important in mathematical problem solving.
Working with patterns also develops number sense. Looking for patterns is an important problem-
solving strategy because many problems are similar and fall into predictable patterns. A pattern,
by definition, is a regular, systematic repetition and may be numerical, visual, or behavioural.

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Discovering and extending patterns requires inductive and deductive reasoning. First,
conjecturing a general pattern based on examples is inductive reasoning. Then proving the
generalization requires deductive reasoning. Writing new examples of a pattern based on a general
rule is another use of deductive reasoning.
Looking for a pattern may be appropriate when:
A list of data is given.
A sequence of numbers is involved.
You are asked to make a prediction or generalization.
Information can be expressed and viewed in an organized manner, such as a table.
Let us solve the following problem using the 4-step Polya process.
Problem 1: Find the next term in the following sequence: 2, 5, 10, 17, ___.
Understand We are looking for the next number in the sequence 2, 5, 10, 17.
Plan The strategy is to look for a pattern and make a conjecture.
Do Each term is 1 more than a square number (1, 4, 9, 16).
Conjecture:
I conjecture that each term in the sequence will be 1 more than a
square number (induction).
If so, the next term is 25 + 1 = 26 (deduction).
Another way of making conjecture is:
I note that the difference between successive terms are 3, 5, 7. I
generalize that the differences between terms will increase by 2 each
time (induction).
The 5th term is 17 + 9 = 26 (deduction)
Check 1 + 1 = 2
2 + 3 = 5
5 + 5 = 10
10 + 7 = 17
17 + 9 = 26
Problem 2: Mae has written a number pattern that begins with 1, 3, 6, 10, 15, …. If she
continues this pattern, what are the next four numbers in her pattern?
Understand Given the conditions, the numbers are increasing. We are looking for a
number which is greater than 15.
Plan The strategy is to look for a pattern and make a conjecture.
Do Look at the numbers in the pattern.
3 = 1 + 2 (starting number is 1, add 2 to make 3)
Therefore the new numbers will be:
15 + 6 = 21
21 + 7 = 28
28 + 8 = 36
36 + 9 = 45

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Conjecture:
I note that the difference between successive terms are 2, 3, 4, 5. I
generalize that the differences between terms will increase by 1 each
time (induction).
The 6th term is 15 + 6 = 21
The 9th term is 36 + 9 = 45 (deduction)
Check Check if the answers satisfy the conditions in the problem.
3.2 Make a Diagram
Many problem-solvers find it useful to draw a diagram or a picture of a problem and its
potential solutions prior to working on the problem. It allows the problem-solvers to understand,
visualize and “see” the problem.
Using diagrams may be appropriate when:
A physical situation is involved.
Geometrical figures or measurements are involved.
You want to gain a better understanding of the problem.
A visual representation of the problem is possible.
Problem 1: For his wife’s birthday, Mr. Cruz is planning a dinner party in a large
recreation room. There will be 22 people attending. In order to seat them, he needs to
borrow card tables the size that seats one person each side. He wants to arrange the
tables in a rectangular shape so that they will look like one large table. What is the
smallest number of tables that Mr. Cruz needs to borrow?
Lesson Assessment 3.1
Solve the following problems by looking for a pattern and making conjecture. Show the
solution by applying Polya’s 4-step process.
1. A three-sided polygon has no diagonal. A four-sided polygon has two
diagonals. A five-sided polygon has five diagonals. A six-sided polygon has
nine diagonals. How many diagonals are there in a seven-sided polygon?
2. Each triangle in the figure below has 3 dots. Study the pattern and find the
number of dots for 7 layers of triangles.

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Understand We are looking for the smallest number of tables. It is given that each
side of the table can accommodate only one person.
Plan The strategy is to draw a diagram.
Do There can be many ways of arranging the tables for them to look like one
big rectangular table. One is given below. It can seat 22 people but use
24 tables. We should look for another arrangement.
Drawing more arrangement of tables, we can see that if the tables will be
drawn in only one row, it can seat 22 people in just ten tables.
Check By counting the tables that can seat 22 people, we can now determine
the smallest number of tables.
Problem 2: Out of forty students, 14 are taking English and 29 are taking Chemistry. If
five students are in both classes, how many students are taking English only? How
many are in either class?
Understand We are looking for the number of students taking English only and the
number of students who are in either class. It is given that there are a
total of 40 students, 14 taking English, 29 taking Chemistry, and 5 who
are enrolled in both classes
Plan The strategy is to draw a Venn diagram.
Do Draw the universal set for the 40 students, with two overlapping circles
labelled with the total in each.
Since five students are taking both classes, put 5 in the overlap.

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Subtract 5 from 14, leaving 9 students taking English only.
Subtract again 5 from 29 leaving 24 students who are taking Chemistry
only.
Add all the numbers inside the circles. There are a total of 38 students
who are in either English or Chemistry class.
Check 5 + 9 = 14 students taking English
5 + 24 = 29 students taking Chemistry
9 + 5 + 24 + 2 = 40 total students
40-38 = 2 students who are not enrolled in either English or Chemistry
3.3 Use Logical Reasoning
Logical reasoning questions are designed to measure the ability to draw logical conclusions
based on statements or arguments, and to identify the strengths and weaknesses of those
arguments. It evaluates the ability to analyze, critically evaluate, and complete arguments as they
Solve the following problems by making a diagram. Show the solution by applying
Polya’s 4-step process.
1. A snail is at the bottom of a 10-meter well. Each day, he climbs up 3 meters.
Each night he slides down 1 meter. On what day will he reach the top of
the well and escape?
2. A survey was conducted in the University of Eastern Philippines about the
preferred social media sites of the 170 students. One hundred fifteen
students use Facebook, 110 use Instagram and 130 use Youtube. There
were 85 students that use Facebook and Youtube, 75 use Facebook and
Instagram, 95 use Instagram and Youtube, and 70 use all the three sites.
How many use Facebook only? How many use Instagram only? How many
use Facebook and Youtube but not Instagram?

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occur in ordinary language. The questions are designed to assess a wide range of skills involved
in thinking critically, with an emphasis on skills that are central to legal reasoning. Logic puzzles
are good examples of non-routine problems that can be solved by logical reasoning.
Logical reasoning may be appropriate when:
Problems presented can be solved using analysis and reasoning.
There is a need to draw a conclusion.
There is a little or no computation required.
Problem 1: Fred, Ted, and Ed are taking Mary, Carry and Terry to the JS promenade.
Use these clues to find out which couples will attend the JS Promenade.
1. Mary is Ed’s sister and lives in Kalayaan Street.
2. Ted drives a car to school each day.
3. Ed is taller than Terry’s date.
4. Carry and her date ride their bicycles to school everyday.
5. Fred’s date lives on Kasipagan Street.
Understand Conditions are already given in the problem.
Plan The strategy is to logical reasoning by creating a matrix logic.
Do Let us create a table like the one below. We can use an X to mark a
square that is not a valid conclusion and a ✔ to mark a square which is a
valid conclusion.
Look at the first clue. Since Mary is Ed’s sister, she’s not his date. Put
an X in Mary’s column beside Ed’s name. Look at the third clue. Since
Ed is taller than Terry’s date, he is not Terry’s date. Put an X in Terry’s
column next to Ed’s name. We can now deduce that Carry is Ed’s date.
We can also evaluate the other boys as Carry’s date. Put X’s in the
column for their names.
Look at the fifth clue, Since Fred’s date lives on Kasipagan Street,
Mary could not be Fred’s date since Mary lives in Kalayaan Street. Put
an X mark on Mary’s column beside Fred’s name. Therefore, Mary is
Ted’s date. What is left is the column for Terry next to Fred’s name. So,
Fred is Terry’s date.
Mary Carry Terry
Fred X X ✔
Ted ✔ X X
Ed X ✔ X
Check We can go back to the clues to determine the correctness of our answers
made through logical reasoning.
Problem 2: A bridge will collapse in 17 minutes. Four people want to cross it before it
will collapse. It is dark night and there is only one torch between them. Only two people
can cross at a time. Gabe takes a minute to cross. Gail takes 2 minutes. Gabrielle takes
5 minutes and Grant takes 10 minutes. How do they all cross before the bridge will
collapse?

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Understand Conditions are already given in the problem.
Plan The strategy is to logical reasoning.
Do Gabe and Gail cross first using up 2 minutes since Gail can cross in
2 minutes.
Gabe comes back using 1 minute to make it 3.
Gabrielle and Grant will cross next consuming Grant’s maximum
minute, making it 13 minutes.
The Gail crosses back over making it 15 minutes.
And finally, Gabe and Gail cross together to make it 17 minutes.
Check Sum up the total minutes used
2 + 1 + 10 + 2 + 2 = 17
3.4 Solve a Simpler Problem
Some solutions are difficult because the problem contains large numbers or complicated
patterns. Sometimes, a simple representation will show a pattern which can help solve a problem.
Breaking problems down into simpler cases can help to solve the problems. Set aside the original
problem. Modify the problem and work through a simpler related problem. Replace larger numbers
with smaller numbers to make calculations easier, and then apply same method of solving it to the
original problem. Look for a pattern that may be emerging.
Simplifying the problem is appropriate when:
Complex problems can be simplified.
Problems are related to another familiar problem.
Patterns are involved.
If a problem is confusing, the numbers can be rounded, or simpler numbers can
be used to help make a plan to solve it.
When a problem is too complex to solve in one step.
Solve the following problems using logical reasoning strategy. Show the solution by
applying Polya’s 4-step process.
1. Four married couples belong to a bridge club. The wives’ names are Kitty,
Sarah, Josie and Anne. Their husbands’ names (not in order) are David, Will,
George, and Frank. Will is Josie’s brother. Josie and frank dated for several
times, but then Frank met this present wife. Kitty is married to George. Anne
has two brothers. Anne’s husband is an only child. Determine who is married
to whom.
2. A farmer wants to cross a river and take with him a wolf, a goat, and a cabbage.
There is a boat that can fit himself plus either the wolf, the goat, or the cabbage.
If the wolf and the goat are alone on one shore, the wolf will eat the goat. If the
goat and the cabbage are alone on the shore, the goat will eat the cabbage.
How can the farmer bring the wolf, the goat, and the cabbage across the river?

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Problem 1: Justine bought her refrigerator for ₱12 575. This was ₱3 800 more than Caryl
paid for her refrigerator. How much did Caryl pay?
Understand Given the conditions, what Justine paid was higher than what Caryl paid
for the refrigerator. We are looking for a smaller answer than ₱12 575.
Plan The strategy is to think of a simpler problem. We can replace the numbers
with smaller units.
Do We can assume that Justine bought her refrigerator at ₱5. This is ₱2
more than Caryl paid for her refrigerator. How much did Caryl pay? The
answer is ₱5 - ₱2 = ₱3. Therefore, returning to the original problem, the
answer can be found by
₱12 575 - ₱3 800 = ₱8 775
Caryl bought the refrigerator at only ₱8 775
Check If Caryl paid ₱8 775 for the refrigerator, ₱3 800 more than ₱8 775 is
₱8 775 + ₱3 800 = ₱ 12 575.
Problem 2: A soccer team won 24 of 36 games in the first season. If the team had the
same ratio of wins to games in the second season, and they won 16 games, how many
games did they play in the second season?
Understand Given the conditions, what we are looking is the ratio equal to 24:36
where one of the given number is 16.
Plan The strategy is to think of a simpler problem. We can replace the given
ratio with smaller units.
Do The answer can be found by simplifying the ratio of 24:36 to 2:3, and then
cross-multiplying to find the total number of games played in the second
season.
2x = 48
x = 24
So, 24 is the total number of games played in the second season.
Check By applying the proportionality theorem, 24:36 is equal to 16:24 which is
also equal to 2:3.
Solve the following problems by having a simpler problem. Show the solution by
1. Ron and Alejandro went up the hill to pick apples and pears. Ron picked 10
apples and 15 pears. Alejandro picked 20 apples and some pears. The ratio of
apples to pears picked by both Ron and Alejandro were the same. Determine
how many pears Alejandro picked.
2. Determine the sum of the whole numbers from 1 to 3 000.

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3.5 Make a List or Table
Organizing information often makes it easier to solve problem. Recording ideas in a list can
help in determining regularities, patterns, or similarities between problem elements. Organizing
data into a table can be very helpful when solving problems. A table is an orderly arrangement of
data. When students have opportunities to design and create tables of information, they begin to
understand that they can group and organize most data relative to a problem.
Making a list or table may be appropriate when:
Listing the possible answers can help solve the problem.
Listing the given information can help identify patterns or similarities.
Problem 1: The telephone area codes in a certain country are three digit numbers. The
first digit cannot be 0 or 1. The second digit can only be 0 or 1. The third digit is not 0.
How many different area codes can start with digit 3?
Understand Given the conditions, the first digit can be 2, 3, 4, 5, 6, 7, 8, and 9.
The second digit can only be 0 or 1.
The last digit is 1, 2, 3, 4, 5, 6, 7, 8, and 9
Plan The strategy is to list the possible three-digit area codes.
Do Assuming that the first digit is 3, list all possible area codes with a digit of
0.
301 302 303
304 305 306
307 308 309
List all possible codes with 1 as a second digit.
311 312 313
314 315 316
317 318 319
There are 18 possible codes that begin with the digit 3.
Check Check if all answers satisfy the conditions in the problem.
Problem 2: What is the one’s digit of 7100?
Understand We are looking for the specific last digit of 7100.
Plan The strategy is to make a table.
Do
n 7n One’s digit
1 7 7
2 49 9
3 343 3
4 2 401 1
5 16 807 7
6 117 649 9
7 823 543 3
8 5 764 801 1
9 40 353 607 7

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The one’s digits repeat in a cycle of four: 7, 9, 3, and 1. Every fourth
power has a ones digit of 1. Since 100 is divisible by 4, 7100 will have a
one’s digit of 1.
Check A calculator can be used to generate the equivalent of 7100.
3.6 Guess and Check
Guess and check gives students opportunities to engage in some trial-and-error approaches
to problem-solving. It should be understood, however, that this is not a singular approach to
problem-solving but rather an attempt to gather some preliminary data. This is also called
guesstimate, trial-error or grope-and-hope.
Guess and check may be appropriate when:
There are limited numbers of possible answers for testing.
You want to gain a better understanding of the problem.
You can systematically try possible answers.
There is no other obvious strategy to try.
Problem 1. Carmela opened her piggy bank and she found that she has ₱ 135. If she had
only 10 centavo, 25 centavo, ₱1, ₱5 and ₱10, how manycoins of each kind did she have?
Understand We are not specifically given the number of coins for each denomination.
As long as the sum is ₱135, we can have many options.
Plan The strategy is to guess and check whether our guess is correct.
Solve the following problems by making a list or table strategy. Show the solution by
1. Ana and Mara are good friends. When Ana got the flu, her doctor wrote a
prescription for twenty 30 mg pills, and told her to take two pills a day, one after
breakfast and one after dinner. Three days later, Mara also got the flu. Her
doctor prescribed thirty 20 mg pills, and told her to take a pill every four hours
between 9:00 and 21:00. Whose medicine was finished first?
2. Marvin forgot the pin number of his ATM card. He remembered that he had
rearranged the digits from his house number to program the 4-digit code. If his
house number is 1256, what are the possible codes that he can try in the
automatic teller machine?

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Do We can assign our guess for the number of coins.
10 centavo – 50 coins
₱1 – 30 coins
₱5 – 20 coins
₱ 10 - 4 coins
Multiplying each denomination with the number of coins and adding
them, we found out that we have a total of ₱185. We have missed the
target of ₱135. So, we assign again guesses for the number of coins for
each denomination until we arrive at the correct total. Let us guess again.
₱1 – 10 coins
₱5 – 18 coins
₱ 10 - 2 coins
Here, we are able to have a total of ₱135. So, this is one correct answer.
There are still a lot of possible combinations of the coins that you can try.
Check .10 x 50 = 5
.25 x 40 = 10
1 x 10 = 10
5 x 18 = 90
10 x 2 = 20
135
As long as the sum of the coins is ₱135 and all coins are represented
then the guess is considered as correct.
Problem 2. The sum of the ages of a father and his son is 100. The father is 28 years
older than his son. How old are they?
Understand The sum of the ages is 100.
The difference of the two ages is 28.
Plan The strategy is to guess and check whether our guess is correct.
Do We can assign our guess for the numbers.
Guess 1: Try 60 and 40
60 – 40 = 20
Since we want a difference of 28, the numbers should be further apart.
65 – 35 = 30
The difference is too big, so they should be a little closer.
64 – 36 = 28
Therefore, the father is 64 years old and his son is 36 years old.
Check .64 + 36 = 100
64 – 36 = 28

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3.7 Act it Out
Act it out is a math problem solving strategy where you use concrete objects (things that
you can touch and hold) to represent people or items in the given math problem and move them
along as you read the word problem to help you “see” what's going on in the problem.
Acting a problem out may be appropriate when:
Counting objects can help to solve problems.
Moving objects can help to solve problems.
Problem 1. The figure below shows 9 matchsticks arranged as an equilateral triangle.
Rearrange exactly 5 of the matchsticks to form 5 equilateral triangles, without leaving
any stray matchsticks.
Understand The condition is to rearrange exactly 5 matchsticks to form 5 equilateral
triangles.
Plan The strategy is to act it out.
Do The figure will look like the one below.
Check Count the total number of equilateral triangles. There are 4 smaller
triangles and 1 bigger triangle.
Solve the following problems using guess and check. Show the solution by applying
1. A kindergarten class is going to a play with some teachers. Tickets cost 5 pesos
for children and 12 pesos for adults. The number of tickets sold amounted to
163 pesos. How many children and teachers went to the play?
2. In a farm, there are some pigs and chickens. If there are 87 animals and 266
legs, how many pigs are there in the farmyard?

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Problem 2. A plumber has to connect a pipe from a storage tank at the corner, S, of the
roof to a tap at the diagonally opposite corner, T, in the figure below. Find the number
of paths for the pipe if the pipe can only run along the edges of walls A, B, or roof C.
Understand The condition is to connect a pipe from point S to point T where the pipe
can only run along the edges of walls A, B, and C.
Plan The strategy is to act it out.
Do The movement from point S to point T is showed in the given illustration,
So, there are 6 numbers of paths for the pipe if the pipe can only run
along the edges of walls A, B, and C.
Check Count the total number of movement from point S to point T.
3.8 Use an Equation or Formula
Most word problems can be solved using formula or equation. Students can sometimes
make sense of a problem by changing the written problem to a number sentence. Using a formula
or equation is a problem-solving strategy that students can use to find answers to math problems
involving geometry, percent, measurement, or algebra. To solve these problems, students must
choose the appropriate formula and substitute data in the correct places of a formula.
Solve the following problems by acting it out. Show the solution by applying Polya’s 4-
step process.
1. In the given figure, ten circles form a triangle. What is the least number of circles
you need to change position in order to turn the triangle upside down?
2. Starting from his house, Zack cycled the following path: 2 km North, 3 km East,
3 km North, 8 km West, 1 km North, 6 km West, 10 km South, 7 km East, and
finally 4 km North. How far was Zack from his house when he completed the
above journey?

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Information can be translated to mathematical sentences.
The problem can be solved by using a formula.
Problem 1. An architect is designing a room that is going to be twice as long as it is
wide. The total square footage of the room is going to be 722 square feet. What are the
dimensions in feet of the room?
Understand We are looking for the dimensions in feet of a room given the area and
conditions for its dimension.
Plan The strategy is to use a formula or equation.
Do The formula for the area of a rectangle is: A=l(w),
where l= length and w= width. From the situation, we know the length is
twice as long as the width. Translating this into an algebraic equation,
we get:
A=(2w)w
Simplifying the equation: A=2w2
Substituting the known value for A:
2w2 = 722 Divide both sides by 2.
w2 = 361 Take the square root of both sides.
w = 19
2w = 38
The width is 19 feet and the length is 38 feet.
Check Use the formula for the area of a rectangle.
A = l(w)
722 = 19 x 38
722 = 722
Problem 2. One number is 10 more than another. The sum of twice the smaller plus three
times the larger is 55. What are the two numbers?
Understand We are looking for two numbers where one number is greater than the
other and the sum of twice the smaller and three times the larger is 55.
Plan The strategy is to use a formula or equation.
Do Let x be the smaller number
x+10 be the larger number
2x + 3(x+10) = 55
2x + 3x + 30 = 55
5x = 55 – 30
5x = 25
x = 5 is the smaller number
5 + 10 = 15 is the larger number
So, the numbers are 5 and 15.
Check First condition: 15 – 5 = 10
Second condition: 2(5) + 3(15) = 55

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3.9 Work Backward
Most problems are given to you with a set of conditions. Then you must find a solution. It’s
frequently helpful for students to take the data presented at the end of a problem and use a series
of computations to arrive at the data presented at the beginning of the problem.
Working backwards may be appropriate when:
The final result is clear and the initial part of a problem is obscure.
A problem proceeds from being complex initially to being simple at the end.
A direct approach involves a complicated equation.
A problem involves a sequence of reversible actions.
Problem 1: Agnes spent two-thirds of her money at the supermarket. Then she spent
three-fourths of her remaining money at the department store. Then, she went home
with only ₱250 left in her wallet. How much money did she have before going to the
supermarket?
Understand We are looking for the original amount of money that Agnes had before
she had her expenses.
Plan The strategy is to work backwards.
Do Starting with ₱250, let us count. Before she was left with ₱, three-
fourths was spent. So, ₱250 is one-fourth of the amount when she arrived
at the department store. Then, 1 000 is the whole amount when she
arrived at the department store.
Before she was left with ₱ 1 000, she spent two-thirds of an amount
in the supermarket. So, ₱ 1 000 is one-third of the amount when she
reached the supermarket. She had ₱ 3 000 when she arrived at the
supermarket.
Check We could now start with ₱ 3 000. Two-thirds of the amount, which is ₱ 2
000 was spent at the supermarket. She was left with ₱ 1 000. Three-
fourths of the amount, ₱750 was spent in the department store. She was
left with ₱250.
Solve the following problems using formula or equation. Show the solution by applying
1. Aria is 4 times as old as Aurvel. Three years ago, Aria was 7 times as old as
Aurvel. Find their present ages.
2. The length of a rectangle is 4 times greater than its width. If the length is
increased by 2 and the width is increased by 3, the area is increased by 58
square units. What is the original dimension of the rectangle?

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Problem 2: Angelo and Keith went to a computer shop with his brother, Keith. He spent
₱50 for tickets. His brother gave him ₱18 as his share for their expenses. Later, they had
some snacks for ₱90. At the end of the day, he had ₱25 left. How much money did he
have at the start?
Understand We are looking for the original amount of money that Angelo had before
he had his expenses.
Plan The strategy is to work backwards.
Do Starting with ₱25, let us count.
₱25 + ₱90 = ₱115
₱115 - ₱18 = ₱97
₱97 + ₱50 = ₱147
So, Angelo has ₱147 at the start.
Check We could now start with ₱ 147 – ₱50 = ₱97.
Then, ₱97 + ₱18 = ₱115
₱115 - ₱90 = ₱25
Angelo was left with ₱25.
3.10 Pursue Parity
Parity appears in problem solving and can prove basic, but interesting results. Parity refers
to whether a number is even or odd. While this may seem highly basic, checking the parity of
numbers is often a useful tactic for solving problems, especially with proof by
contradictions and casework. The parity of a number depends only on its remainder after dividing
by 2. An even number has parity 0 because the remainder after dividing by 2 is 0, while an odd
number has parity 1 because the remainder after dividing by 2 is 1.
Here are some basic properties of parity.
 Formal definition of “even” and “odd”: An integer is even if and only if it is of the
form n = 2k, where k is an integer; it is odd if and only if it is of the form n = 2k + 1,
where k is an integer. Any integer is either even or odd, but not both.
 Changing + into − or vice versa does not affect the parity. Another “obvious” fact
that is surprisingly useful. For example, an expression of the form a1 ± a2 ± · · · ±
an, with integers ai and arbitrary ± signs will have the same parity a1 + a2 + · · · +
an.
Solve the following problems by working backwards. Show the solution by applying
1. At a supermarket sale, 2 more than half of the numbers of oranges were taken
out from a box. Later, 2 fewer than half of the remaining numbers of oranges
were taken out again. If 20 oranges were left in the box, how many oranges
were there at first?
2. Georgina and Francheska were trying to decide on what pet they would buy.
At the pet store, they saw four more rabbits than birds. They saw one-half as
many birds as kittens. There were one-third as many kittens as puppies. There
were 36 puppies. How many animals did they see altogether?

47
 A sum of integers is odd if and only if there are an odd number of odd terms. Yet
another “obvious” fact with lots of applications.
 A product of integers is odd if and only if all factors are odd.
 Some of the arithmetic rules of parity that are extremely useful:
even ± even = even
odd ± odd = even
even ± odd = odd
even × even = even
even × odd = even
odd × odd = odd
Using parity maybe appropriate when:
Verifying whether an equality is true or false by using the parity rules of
arithmetic to see whether both sides have the same parity.
A problem involves proving of some properties of integers.
Problem 1: Ten balls numbered 1 to 10 are in a jar. Jack reaches into the jar and
randomly removes one of the balls. Then Jill reachesinto the jar and randomly removes
a different ball. What is the probability that the sum of the two numbers on the balls
removed is even?
Understand We are looking if the sum of the two numbers on the balls removed is
even given the above conditions.
Plan The strategy is to apply parity.
Do We find that it is only possible for the sum to be even if the numbers
added are both even or odd. We will get an odd number when we add an
even and odd. We can use complementary counting to help solve the
problem.
There are a total of 90 possibilities since Jack can choose 10 numbers
and Jill can pick 9. There are 50 possibilities for the two numbers to be
different since Jack can pick any of the 10 numbers and Jill has to pick
from 5 numbers in the set with a different parity than the one that Jack
picks. So the probability that the sum will be odd is 50/90 = 5/9.
Subtracting this by one gets the answer 4/9.
Check For the sum of the two numbers removed to be even, they must be of the
same parity. There are five even values and five odd values.
No matter what Jack chooses, the number of numbers with the same
parity is four. There are nine numbers total, so the probability Jill chooses
a number with the same parity as Jack's is 4/9.
Problem 2: The numbers 1 through 10 are written in a row. Can the signs ‘+’ and ‘-‘ be
placed between them, so that the value of the resulting expression is 0?
Understand We are trying to prove if putting ‘+’ and ‘-‘ signs from the numbers from 1
through 10 arranged in a row will result to 0.
Plan The strategy is to apply parity.

48
Do Remember the arithmetic rules of parity.
even ± even = even
odd ± odd = even
even ± odd = odd
There are five even and five odd numbers from 1 through 10. Adding an
even number does not change the parity of a sum. Five odd numbers add
up to an odd number.
Let us take a number n :
n − n=0
Two numbers can only result in 0 on subtraction if they are equal.
In the given problem, if we add the two numbers , we get 2n.
We see 2n is even number. But sum of all numbers up to 10 is:
n∗(n+1)/2=10∗11/2=55
So it’s not even ,so it’s not possible to get 0 by any combination of ‘+’ &
‘-’ .
Check Try to place the symbols in a row of numbers from 1 to 10, the result is
always an odd number. For a sequence to be summed to zero
irrespective of the sign they have, they have to be even when summed
up.
Feedback
How did you go on so far with this module? Were you exhausted seeing a lot of problems
solved using the different strategies? Were you able to understand the discussion and the
examples provided in each strategy? Those strategies are helpful in solving non-routine problems.
Mastering these strategies will equip you with skills and knowledge as you learn higher
mathematics. This will eventually help you when you become a teacher.
Are there items remained unclear? Well, you can always go back and study the examples
given before you try to answer the lesson assessment. Keep practicing and you will surely master
it. You can also log on to the links provide in the suggested reading section of this module.
Solve the following problems by applying parity. Show the solution by applying Polya’s
4-step process.
1. A box is filled with 75 white beads and 150 black ones. There is a pile of black
beads near the box. Remove two beads from the box. If one is black, put
back the other (white or black). If both are white, put in a black one from your
pile. Each time one repeats this process, there will be one less bead in the
box. What will be the color of the final bead left in the box?
2. Of 101 coins, 50 are counterfeit, and they differ from the genuine coins in
weight by 1 gram. Peter has a scale in the form of a balance which shows the
difference in weight between the objects placed on each pan. He chooses
one coin, and wants to find out whether it is counterfeit. Can he do this in one
weighing?

49
Summary
To aid you in reviewing the concepts in this module, here are the highlights:
 The following are the common problem solving heuristics that the problem solver can
use in solving problems:
 search for a pattern and formulate conjecture
 make a diagram
 use logical reasoning
 solve a simple problem
 make a list or table
 guess and check
 act it out
 use an equation or formula
 work backward
 pursue parity
Suggested Readings
If you want to learn more about the topics in this module, you may log on to the following
links:
https://www.ck12.org/book/ck-12-algebra-basic/section/3.8/
https://brilliant.org/wiki/modular-arithmetic-parity/
https://faculty.math.illinois.edu/~hildebr/putnam/training19/invariants1.pdf
https://web.ma.utexas.edu/users/olenab/s12-PutnamParitySols(1st).pdf
https://artofproblemsolving.com/wiki/index.php/Parity
References
Ballado, R. & Adalla, M. J. (2017). Problem Solving. Quezon City, Philippines: Great Books Trading.
Engel, A. (1998). Problem solving strategies. New York, USA: Springer-Verlag.
Foshay, R., Kirkley, J. (1998). Principles for Teaching Problem
Solving. http://www.plato.com/pdf/04_principles.pdf
Larson, L.C. (1983). Problem solving through problems. New York, USA: Springer-Verlag.

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