TRY THE POSSIBILITIESOTHER NAMES FOR THIS STRATEGY ARE ‘TRIAL AND ERROR’ OR

‘GUESS AND CHECK’. WE USE THIS STRATEGY WHEN A STRUCTURED OR LOGICAL APPROACH DOES NOT EXIST. WE ALSO USE THIS

APPROACH WHEN A STRUCTURED OR LOGICAL APPROACH DOES EXIST BUT WE HAVE NOT YET LEARNED HOW TO UTILIZE THESE

APPROACHES. IT IS NORMALLY AN INEFFICIENT METHOD BUT IS GOOD TO KNOW IN CASE WE CAN’T FIGURE OUT A PROBLEM IN A MORE

EFFICIENT MANNER.EXAMPLE: IF - 32 = 0 AND X IS A ONE DIGIT WHOLE NUMBER, WHAT NUMBER IS IT?SOLUTION: IF WE BEGIN WITH THE NUMBER 1, WE WILL FIND THAT 1, 2, AND 3 WON’T WORK, BUT THE NUMBER 4 WILL. SO, THE ANSWER IS 4.THIS SLIDE SHOW FOLLOWS THE SAME PATTERN AS THE ONES BEFORE IT, WITH PROBLEMS, SOLUTIONS, AND RESOURCES FOLLOWING.

Try the possibilities to discover the answer that works.

1) Find a number between 400 and 450 that is divisible by 2, 3, 5, and 7.2) When the children visited the farm and saw all the goats and chickens, they

counted 39 heads and 108 legs. How many were goats and how many were chickens?

3) Find the smallest prime number that is larger than 300.4) A perfect number is one in which the number itself is equal to the sum of

its factors (excluding the number itself). One example of a perfect number is 28. The factors of 28 are 1, 2, 4, 7, 14, and 28. If we remove 28 and add the other factors, we get 1 + 2 + 4 + 7 + 14, which equals 28. Find the only one digit perfect number.

5) Maria invited three other couples to a dinner party. After she puts the invitations in the mailbox, she realized she didn’t check to see if the right invitations were placed in the right envelopes. How many ways could she have put them in the envelopes so that at least one person got the right invitation?

1) FIND A NUMBER BETWEEN 400 AND 450 THAT IS DIVISIBLE BY 2, 3, 5, AND 7.

ONE SOLUTION OF MANY: WE NEED A NUMBER BETWEEN 400 AND 450, BUT THIS GIVES US 50 NUMBERS TO TRY. WE CAN CUT THIS DOWN BY LOOKING AT THE NUMBERS WE ARE TESTING. IN ORDER FOR 2 TO DIVIDE THE NUMBER, IT MUST BE EVEN; AND IN ORDER FOR 5 TO DIVIDE THE NUMBER, IT MUST END IN 0 OR 5. SINCE THE NUMBER MUST BE EVEN, THOUGH, IT MUST END IN 0. OUR POSSIBILITIES NOW BECOME 400, 410, 420, 430, 440, AND 450. NOW WE JUST CHECK TO SEE WHICH OF THESE CAN BE DIVIDED BY 3 AND 7. 3 GOES INTO 420 AND 450. OF THESE TWO CHOICES, ONLY 420 IS DIVISIBLE BY 7. THEREFORE, THE ANSWER IS 420.

2) WHEN THE CHILDREN VISITED THE FARM AND SAW ALL THE GOATS AND CHICKENS, THEY COUNTED 39 HEADS AND 108 LEGS. HOW MANY WERE

GOATS AND HOW MANY WERE CHICKENS?

WE COULD SOLVE THIS USING A SYSTEM OF EQUATIONS, BUT THIS PROCESS MAY BE BEYOND OUR STUDENTS. IN THAT CASE, WE CAN TRY THE POSSIBILITIES UNTIL WE FIND THE ONE THAT FITS. THERE WILL BE MANY WAYS TO GO ABOUT THIS, BUT HERE IS ONE OF THEM. WE KNOW WE HAVE 39 HEADS, SO THE NUMBER OF GOATS + THE NUMBER OF CHICKENS = 39. I WILL BEGIN BY LOOKING AT 20 GOATS AND 19 CHICKENS. THIS WILL GIVE 118 LEGS. THIS IS TOO MANY, SO WE WILL TAKE THE NUMBER OF GOATS DOWN TO 19. 19 GOATS AND 20 CHICKENS WILL GIVE 116 LEGS. NEXT I WILL TRY 17 GOATS AND 22 CHICKENS. THIS GIVES 112 LEGS. NOW I WILL TRY 15 GOATS AND 24 CHICKENS. THIS GIVES 108 LEGS, WHICH IS WHAT WE ARE LOOKING FOR. THE ANSWER IS 15 GOATS AND 24 CHICKENS.

3) FIND THE SMALLEST PRIME NUMBER THAT IS LARGER THAN 300.

IN ORDER FOR A NUMBER TO BE PRIME, THE ONLY FACTORS IT HAS ARE 1 AND ITSELF. SO, TO CUT DOWN THE POSSIBILITIES, WE CAN REMOVE ANY EVEN NUMBER (2 WILL GO INTO IT) AND ANY NUMBER THAT ENDS IN 5 (5 WILL GO INTO IT). OUR FIRST POSSIBILITY IS 301. 3 WILL NOT DIVIDE THIS NUMBER, BUT 7 WILL, SO IT IS NOT PRIME. NEXT, WE WILL TRY 303. 3 DIVIDES THIS NUMBER, SO THAT IS NOT PRIME. THE NEXT NUMBER WE WILL TRY IS 307. 3, 7, 11, 13, AND 17 WILL NOT DIVIDE THIS NUMBER, SO IT IS PRIME. OUR ANSWER IS 307.

4) A PERFECT NUMBER IS ONE IN WHICH THE NUMBER ITSELF IS EQUAL TO THE SUM OF ITS FACTORS (EXCLUDING THE NUMBER ITSELF). ONE EXAMPLE OF A PERFECT NUMBER IS 28. THE FACTORS OF 28 ARE 1, 2, 4, 7, 14, AND 28. IF WE REMOVE 28 AND ADD THE OTHER FACTORS, WE GET 1 + 2 + 4 + 7 + 14, WHICH EQUALS 28. FIND THE ONLY ONE DIGIT PERFECT NUMBER.

HERE IS ONE SOLUTION: I WILL BEGIN WITH THE NUMBER 1 UNTIL I FIND THE NUMBER THAT IS PERFECT.1. THE ONLY FACTOR FOR 1 IS 1. IF WE REMOVE THAT, THE SUM IS 0. NOT THIS ONE2. THE FACTORS FOR 2 ARE 1 AND 2. IF WE REMOVE 2, THE SUM IS 1. NOT THIS ONE EITHER3. THE FACTORS FOR 3 ARE 1 AND 3. IF WE REMOVE 3, THE SUM IS 1. NOPE4. THE FACTORS FOR 4 ARE 1, 2, AND 4. IF WE REMOVE 4, THE SUM IS 3. WRONG AGAIN5. THE FACTORS FOR 5 ARE 1 AND 5. IF WE REMOVE 5, THE SUM IS 1. NOT YET6. THE FACTORS FOR 6 ARE 1, 2, 3, AND 6. IF WE REMOVE 6, THE SUM IS 6. BINGO!SO, THE ANSWER FOR THIS PROBLEM IS 6.

5) Maria invited three other couples to a dinner party. After she puts the invitations in the mailbox, she realized she didn’t check to see if the right invitations were placed

in the right envelopes. How many ways could she have put them in the envelopes so that at least one person got the right invitation?

HERE IS ONE WAY TO SOLVE THE PROBLEM. CALL THE OTHER COUPLES A, B, AND C. NOW, LIST ALL THE WAYS MARIA COULD HAVE STUFFED THE ENVELOPES. HERE WE GO!

THE WAY THEY SHOULD HAVE BEEN STUFFED: A, B, CPOSSIBILITIES

A, B, C (3 RIGHT)A, C, B (1 RIGHT)B, A, C (1 RIGHT)B, C, A (0 RIGHT)C, A, B (0 RIGHT)C, B, A (1 RIGHT)

OF THE 6 TOTAL POSSIBILITIES, 4 OF THEM GIVE AT LEAST 1 RIGHT INVITATION. SO, THE ANSWER IS 4 WAYS.

MORE TRY THE POSSIBILITY PROBLEMS AND RESOURCES ONLINE

• HTTP://WWW.STUDYZONE.ORG/TESTPREP/MATH4/D/GUESSCHECK4L.CFM• HTTPS://WWW.YOUTUBE.COM/WATCH?V=VWTWXE2DQUA• HTTPS://WWW.TEACHERVISION.COM/MATH/PROBLEM-SOLVING/48896.HTML• HTTPS://WWW.YOUTUBE.COM/WATCH?V=WIPYQUPV9WY• HTTP://WWW.IXL.COM/MATH/GRADE-5/GUESS-AND-CHECK-PROBLEMS• HTTPS://WWW.YOUTUBE.COM/WATCH?V=BSRCT7TCIO8• HTTP://PRED.BOUN.EDU.TR/PS/PS3.HTML

Project AMP Dr. Antonio Quesada – Director, Project AMP

Problem Solving

Activity: Using Trial & Error to Solve Problems

Team members’ names:___________________________________________________ Goal: In this activity you will learn how to use trial and error and a table to organize your work to solve word problems. Let us start with an example. Example. Joe has $10 more than his friend Paul. Together they have $40. How much money does each one have? First Approach to solve the problem. The trial & error method consists of guessing what the answer might be using an initial educated guess, and subsequently refining your next guess by taking into consideration the results obtained. First, you need to organize your work. For this it is recommended that you use a table. The headings for the table consist typically of the names of the variables involved. What will you choose for the headings in this problem? _____________________ The headings of the columns in the table that follows is appropriate for this problem.

Joe’s money Paul’s money Total money

Next, you need to make an educated guess. For instance, would it be reasonable to guess that Joe has $50? ________ Why? _______________________________________ What is the largest amount of money that Joe can have? _______ What is the smallest amount of money that Joe can have? _______ So, an educated guess is one that does not contradict the information given in the statement of the problem. Once you make an educated guess, you then proceed to fill all the entries of the table keeping in mind the relationships about them established in the problem. For instance, if you guess that Joe has $20, how much money will Paul have? ________ (Did you keep in mind the difference in the amount of money that Joe and Paul have? If not, guess again the amount that Paul will have.) How much will they have together in this case? ________

Note: this is the firstPage of a 3 page document on how to solveproblems using the trial anderror method. It is a part ofthe AMP Project throughthe University of Akron underthe direction of Dr. Antonio Quesada. To locate this document, follow these steps:1. Go to www.google.com2. Type in the phrase ‘math problem solving trialand error’3. Look for and click on the tab that says:[DOC]Solving Problems by Trial and Error

•ur  ...•[DOC]Solving Problems by Trial and Error