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?
