Problem Solving, Mathematical Investigation and Modelling - Topic 1

PROBLEM SOLVING,
MATHEMATICAL
INVESTIGATION, AND
MODELLING
Problem
Solving

PROBLEM SOLVING
a process in which real-world problems are
being solved using mathematical techniques
refers to mathematical tasks that have
potential to provide intellectual challenges
for enhancing students’ mathematical
understanding and development
to solve a real-world problem, it has to be
constructed such that it can be expressed
using mathematics (modelling)

POLYA’S PROBLEM SOLVING
TECHNIQUES
Understand the problem
Devise a plan
Carry out the plan
Look back

UNDERSTAND THE PROBLEM
• Do you understand all the words used in
stating the problem?
• What are you asked to find or show?
• Can you restate the problem in your own
words?
• Can you think of a picture or diagram that
might help you understand the problem?
• Is there enough information to enable you to
find a solution?

DEVISE A PLAN
• Guess and check
• Look for a pattern
• Make an orderly
list
• Draw a picture
• Eliminate
possibilities
• Solve a simpler
problem
• Use a model
• Consider special
cases
• Work backwards
• Use direct
reasoning
• Use a formula
• Solve an equation

CARRY OUT THE PLAN
Make sure that the correct values and
equation are used
If the method failed, discard and choose
another

LOOK BACK
Check if the solution makes sense
Reflect on the process: Can it be
simplified?
Use the method to solve other similar
problems

EXAMPLE
A hat and a jacket cost ＄100. The
jacket costs ＄90 more than the
hat. What are the cost of the hat
and the cost of the jacket?

Understan
d the
Problem
After reading the problem for the first time,
you may think that the jacket cost ＄90 and
the hat cost ＄10. The sum of these costs is
＄100, but the cost of the jacket is only ＄90
more than the cost of the hat. We need to find
the two dollar amounts that differ by ＄90
and whose sum is ＄100.
Devise a
Plan
Write an equation using 𝒉 for the cost of the
hat and 𝒉 + 𝟗𝟎 for the cost of the jacket: 𝒉 +
𝒉 + 𝟗𝟎 = 𝟏𝟎𝟎
Carry out
the Plan
2ℎ + 90 = 100; 2ℎ = 100 − 90
2ℎ = 10; ℎ = 5

PROBLEM SOLVING,
MATHEMATICAL
INVESTIGATION, AND
MODELLING
Translation of
Statements to
Symbol

TRANSLATION OF STATEMENTS
TO SYMBOLS
It can be really frustrating to spend
fifteen minutes solving a word
problem on a test, only to realize at
the end that you no longer have any
idea what “x” stands for, so you have
to do the whole problem over again.

TO SYMBOLS
Certain words indicate certain mathematical
operations. Below is a partial list.

TO SYMBOLS
Note: “less than” construction is
backwards in the English from what it
is in math.
Ex: translate “15 less than x”, the
temptation is to write “15 – x”

TO SYMBOLS
Note: order is important in the “quotient/ratio
of” and “difference between/of” constructions.
Ex: a problems says “the ratio of x and y”,
it means “x divided by y”, not “y divided
by x”
a problem says “the difference of x and
y”,
it means “x – y”, not “y – x”.

TO SYMBOLS
a. Translate “the sum of 8 and
y” into an algebraic expression.
This is translated to “8 + y”.

TO SYMBOLS
b. Translate “4 less than x” into
an algebraic expression.
This is translated to “x – 4”.

TO SYMBOLS
c. Translate “x multiplied by 13”
into an algebraic expression.
This is translated to “13x”.

TO SYMBOLS
d. Translate “the quotient of x
and 3” into an algebraic
expression.
This is translated to “
𝒙
𝟑
”.

TO SYMBOLS
e. Translate “the difference of 5
and y” into an algebraic
expression.
This is translated to “5 – y”.

TO SYMBOLS
f. Translate "the ratio of 9 more
than x to x" into an algebraic
expression.
This is translated to “
(𝒙+𝟗)
𝒙
”.

TO SYMBOLS
g. Translate “nine less than the
total of a number and two” into
an algebraic expression, and
simplify.
This is translated to “(n + 2) – 9”,
which then simplifies to “n – 7”.

TO SYMBOLS
h. The length of a football field is 30
yards more than its width. Express the
length of the field in terms of its width w.
Whatever the width w is, the length is 30
more than this. Recall that “more than”
means “plus that much”, so you’ll be
adding 30 to w. The expression we’re
looking for is “w + 30”.

TO SYMBOLS
i. Twenty gallons of crude oil were poured
into two containers of different size. Express
the amount of crude oil poured into the
smaller container in terms of the amount g
poured into the larger container.
There are twenty total gallons, and we've
already poured g gallons of it. How many
gallons are left? There are 20 – g gallons left.
The answer we want is "20 – g".

PROBLEM SOLVING,
MATHEMATICAL
INVESTIGATION, AND
MODELLING
Number
Relations
Problem

NUMBER RELATION PROBLEMS
Number relation problems are types of
problems which deals with comparison
of one or more unknown numbers,
integers, counting numbers,
consecutive numbers, consecutive odd
and even integers. Finding the required
number is the ultimate goal of this
problem type.

Example 1: The sum of twice a number
and 5 is 9. What is the number?
Use the following relationship to set up
an equation:
[twice the number] + 5 = 9
Step 1: Let x represent the number.

Step 2: Write an algebraic expression
for “twice the number” in terms of x,
twice the number is 2x.

[twice the number] + 5 = 9
Step 3: Use the relationship to set up an
equation.
2x + 5 = 9

Step 4: Solve the equation for x.
2x + 5 = 9 ;
2x = 4 ;
x = 2
So the number is 2.

Example 2: Four times a number
decreased by10 is equal to the sum of
twice the number and 2. Find the
number.
Use the following relationship to set up
an equation:
[four times the number decreased by 10] = [the sum of twice the
number and 2]

Step 2 : Write an algebraic expression
for “four times the number decreased
by10” in terms of x.
four times the number decreased by 10 =
4x - 10.
Write an algebraic expression for “the
sum of twice the number and 2” in terms

[four times the number decreased by 10] = [the sum of twice the
number and 2]
equation.
4x - 10 = 2x + 2

4x - 10 = 2x + 2
2x = 12
x = 6
So the number is 6.

Example 3: The sum of two consecutive
integers is 71. Find the numbers.
As the two unknown numbers are
consecutive, the larger integer can be
expressed as x + 1.
[the smaller integer] + [the larger integer] = 71
Step 1: Let the two consecutive
numbers be x and x + 1.

[the smaller integer] + [the larger integer] = 71
equation.
x + x + 1 = 71

Step 3: Solve the equation for n.
x + x + 1 = 71
2x + 1 = 71
2x = 70
x = 35
So the two consecutive integers are 35
and 36.

TRY ME - NUMBER RELATION
PROBLEMS
Example 4: Twelve less than four times
the first of two consecutive integers is
three times the second. Find the
integers.
Example 5: The sum of three
consecutive integers is 126. Find the
three numbers.

PROBLEMS
Example 4: Twelve less than four times the
first of two consecutive integers is three
times the second. Find the integers.
Step 1: Let the two consecutive numbers be x
and x + 1.
consecutive, the larger integer can be
expressed as x + 1.
[twelve less than four times the first] = [three

Step 2 : Write an algebraic expression
for “twelve less than four times the first”
in terms of x.
twelve less than four times the first = 4x -
12.
Write an algebraic expression for “three
times the second” in terms of x.

[twelve less than four times the first] = [three times the
second]
equation.
4x -12 = 3(x + 1)

Step 4: Solve the equation for n.
4x -12 = 3(x + 1)
4x -12 = 3x + 3
4x - 3x = 3 + 12
x = 15
So the two consecutive integers are 15
and 16.

PROBLEMS
Example 5: The sum of three
consecutive integers is 126. Find the
three numbers.
consecutive, the second of the two
consecutive integers can be expressed
as x + 1, and the third is x+2.
[first integer]+[second integer]+[third

PROBLEMS
Step 1: Let the three consecutive
numbers be x a x + 1, and x + 2
[first integer]+[second integer]+[third
integer] = 126
equation.
x + x + 1 + x + 2 = 126

x + x + 1 + x + 2 = 126
3x + 3 = 126
3x = 123
x = 41
So the three consecutive integers are 41, 42,
and 43.

THANKS!
Does anyone have any question?
tmcc.francisquarre@gmail.com
+639 053 483 516

Problem Solving, Mathematical Investigation and Modelling - Topic 1

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