Transcript Try the possibilities

TRY THE POSSIBILITIES
OTHER 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 𝑥 2 𝑥 - 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 ONE
2.
THE FACTORS FOR 2 ARE 1 AND 2. IF WE REMOVE 2, THE SUM IS 1. NOT THIS ONE EITHER
3.
THE FACTORS FOR 3 ARE 1 AND 3. IF WE REMOVE 3, THE SUM IS 1. NOPE
4.
THE FACTORS FOR 4 ARE 1, 2, AND 4. IF WE REMOVE 4, THE SUM IS 3. WRONG AGAIN
5.
THE FACTORS FOR 5 ARE 1 AND 5. IF WE REMOVE 5, THE SUM IS 1. NOT YET
6.
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, C
POSSIBILITIES
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
•[DOC]Solving Problems by Trial and Error
Dr. Antonio Quesada – Director, Project AMP
Project AMP
1
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 first
Page of a 3 page
document on how to solve
problems using the trial and
error method. It is a part of
the AMP Project through
the University of Akron under
the direction of Dr. Antonio
Quesada. To locate this
document, follow these steps:
1. Go to www.google.com
2. Type in the phrase
‘math problem solving trial
and error’
3. Look for and click on the
tab that says:
[DOC]Solving Problems
by Trial and Error