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PROBLEM SOLVING

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POLYA’S 4-STEPS IN PROBLEM SOLVING
George Polya (1887 – 1985) is best known for his book “How to Solve It” which lays out the strategies in problem – solving which is not
only good for Math but for any other disciplines.
1. Understand the problem – state the problem in your own words; find out what is already known (given information); find out what is
expected to solve the problem
2. Devise a plan – break down the problem into smaller parts; list down available information; list down missing information; produce a
table or chart; look for a pattern; write an equation; and produce an organized list of all possibilities.
3. Carry out the plan – keep an accurate record of attempts and their results; revise the plan if necessary
4. Review the solution – once the solution is found, check it again to see if this is the needed output
Sample problem:
a. There are 480 marbles in a box whose color is either black or white. If there are 46 more black marbles than white, how many black
marbles are there? How many white marbles?
Understand the problem:
What is asked? How many black marbles and white marbles are there?
Given: there are 480 marbles and there 46 black marbles than white
Devise a plan:
What is the operation to be used? Subtract, division and addition
Write the number sentence:
a. (480 – 46 = n) / 2 → white marbles, then divide the difference by 2 then add 46 to quantify black marbles
Carry out the plan:
Solution: 480 – 46 = 434/ 2 = 217 white marbles, 217 + 46 = 263 black marbles
Review the solution:
217 + 263 = 480; therefore, there are 217 white marbles and 263 black marbles
ACTIVITY 1 (6pts):
Problem: A bag of kapeng barako and a mug together cost P310.00 The bag of kapeng barako costs P125 more than the coffee
mug. What is the cost of a bag of kapeng barako and the cost of a mug? Write your answer using AGONSA.
Determining the number of Gray Tiles
Problem: The white sections of the floor are square and the larger white square has exactly 8 tiles more on each side than the smaller
one. If there are 10,000 tiles needed to cover the entire floor, how many gray tiles are required?
Asked: How many gray tiles are required?
Given: The larger white square is 8 tiles more than the smaller one; 10,000 tiles
Operation: multiplication, addition, subtraction
Number Sentence: (x + x + 8)2 or (2x + 8) (2x + 8) = 10,000
Solution: √(2đť‘Ą + 8) 2 = √10,000
2x + 8 = 100 → 2x = 100 – 8 → 2x = 92 → x = 46
Substitute 46 to all the x, then we have the lengths 46 and 54
Gray section:
46 x 54 = 2484, multiply it to 2 since there are two gray sections
2484 x 2 = 4,968 tiles
White section:
10,000 – 4,968 = 5,032 tiles
Answer: there are 4,968 gray tiles and 5,032 white tiles.
PROVING A CONJECTURE
Situation: Prove that the following procedure always produces a number that is thrice the original number.
Procedure: Pick any number. Multiply it by 9, add 6 to the product, divide the sum by 3, and then subtract 2.
Understand the problem:
the challenge is to produce a proof that using the given conjecture will produce a number 3 times its original number
x → 3x
Devise a plan
Procedure
Mathematical form
Pick any number
x
Multiply it by 9
9x
Add 6 to the product
9x + 6
9đť‘Ą + 6
Divide the sum by 3
3
9đť‘Ą+6
Subtract 2
-2
3
Carry out the plan:
9đť‘Ą+6
Prove that 3 – 2 = 3x
Proof:
9đť‘Ą+6
3
–2=
3(3đť‘Ą+2)
3
–2→
9đť‘Ą+6
3
– 2 = 3x
ACTIVITY 2 (6pts):
Devise a plan by proving the given conjecture in mathematical form
Conjecture: Pick a number. Multiply the number by 5, add 15 to the product, divide the sum by 5, and subtract 3.
MAKING PREDICTIONS BASED ON PATTERNS
Problem: The price of 30 apples is P450.00, How much will it cost if you will purchase 45 pieces of apples?
Asked: How much will it cost if you will purchase 45 pieces of apples?
Given: 30 apples = P450; 45 apples
Operation: division and multiplication
Number Sentence: (P450/30) x 45 = N
Solution: 450/30 = 15; 15 x 45 = P675.00
Answer: 45 pieces of apples cost P675.00
ACTIVITY 3 (6pts):
Solve the given problem using AGONSA method.
Problem: In an education field trip, Grade 3 section A collected a total amount of P40,250. If each person paid P1,150, how many
students will join the field trip?
DETERMINING THE NTH TERM OF THE SEQUENCE
Problem: In a networking company, they started with 6 persons. Each day 5 more members are added. What is the total population of
the company after 20 days?
Formula in arithmetic sequence: An = a1 + (n-1) d
An – nth term
a1 – first term
n – number of terms
d – common difference
Asked: What is the total population of the company after 20 days?
Given: 30 apples = 6 person, 5 members a day, 20 days
Operation: division, multiplication
Number Sentence: An = 6 + (20-1) 5
Solution: an = 6 + (19)5 → 6 + 95 = 101
Answer: The total population of the company after 20 days is 101.
ACTIVITY 4 (6pts):
Solve the given problem using AGONSA method.
Problem: A student is saving his money in a piggy bank. Every week, he puts P20 in it. How much will she save after 35 weeks?
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