Guess and Check Technique Based on Polya’s Four-step Model

By: Muhammad Safwan bin Shuhami &

Salihin bin Ahmad

Definition

Trial and error, or trial by error, is a general method of problem solving, fixing things, or for obtaining knowledge. "Learning doesn't happen from failure itself but rather from analyzing the failure, making a change, and then trying again."

Importance

All research mathematicians use guess and check, and it is one of the most powerful methods of solving differential equations, which are equations involving an unknown function and its derivatives. A mathematician's guess is called a "conjecture" and looking back to check the answer and prove that it is valid, is called a "proof." The main difference between problem solving in the classroom and mathematical research is that in school, there is usually a known solution to the problem. In research the solution is often unknown, so checking solutions is a critical part of the process.

Examples

Process Problem 1Marty did 2 of these activities. He paid for

them with a $10.00 bill. His change was $3.75. What 2 activities did Marty do? (Hint: Make a guess. Then check your guess.)

ActivityCostMovies-$3.50Putt-Putt Golf-$3.00Skating-$2.00Go-Kart Rides-$2.75

Understanding the Problem· How many activities did he do? (2)

· How much money did he have? ($10.00)    · What was his change? ($3.75)

Planning a Solution· How much money did he have? ($10.00) .What was the change? ($3.75)· How much did he spend? ($6.25)· If he saw the movies and golfed, how much

money would he have spent? ($6.50) .Did he do these 2 activities? (No, they cost too

much.)

Finding the AnswerGuess and Check· Try movies and skating—$3.50 + $2.00 =

$5.50. (too little)· Try movies and go-karts—$3.50 + $2.75 = $6.25. (correct)

The activities that Marty did are the movies and the go-kart rides.

Process Problem 2 I wrote 5 different numbers on 5 cards. The

sum of the numbers is 15. What numbers did I put on the cards? (Hint: Make a guess. Then check your guess.)

Understanding the Problem · How many numbers did I write? (5)

· What is the sum of the numbers? (15)· How many numbers on each card? (1)· Are any 2 numbers the same? (No, they are all different.)

Planning a Solution · Could 1 of the numbers be 15? (No, because

the rest would be 0 and we said that all the numbers were different.)· Select 5 numbers and check to see if their sum is 15.

Finding the Answer Guess and Check · Try 0,1,2,3,4—0 + 1 + 2 + 3 + 4 = 10 (too

little)· Try 1,2,3,4,5—1 + 2 + 3 + 4 + 5 = 15 (correct)

The numbers are 1, 2, 3, 4, and 5.

Process Problem 3 David's age this year is a multiple of

5. Next year, David's age will be a multiple of 7. How old is David now?

Understanding the Problem · What do we know about David's

age this year? (multiple of 5)· What do we know about David's age next year? (multiple of 7)

Planning a Solution · List some multiples of 5. (5, 10, 15, . . .) Multiples

of 7. (7, 14, 21, . . .)· Guess what David's age might be this year and add 1 to it to see if that number is a multiple of 7. (See solution.)

Finding the Answer Guess and Check · Try 10 and 11. (No, 11 is not a multiple of 7.)and

so on Make an Organized List Multiples of 5-5,10,15,20 Multiples of 7-7,14,21,28 David is 20 years old now.

QUESTIONS 1. Mary has 6 coins, which have a total

value of 67 cents. What combinations of coins could she have? Use denominations of 1¢, 5¢, 10¢ and 25¢.

Penny 1¢---->2 Nickel 5¢---->1 Dime 10¢--->1 Quarter 25¢--->2 Total=6 

2. Navigate your spaceship to the "Black Hole". The product of the numbers along your path must be 2592.

3. Find a set of 3 consecutive even numbers whose sum is 294.

•294 ÷ 3 = 98•98 – 2=96•98•98 + 2=100