Elementary POW, September 23-27,1996
Elem POW Problems || Sept-Nov '96 Problems || Elem POW Main Page
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The Fortune Cookie Problem
I attended a Math forum Advanced Internet Institute this past
summer with math teachers from across the United States as well as one
teacher from Australia. We had dinner together one night at a Vietnamese
restaurant. At the end of the meal the waiter brought us each a fortune
cookie. On the front of the fortune was a saying and on the back was six
numbers that were supposed to be lucky for you. After sharing our sayings,
Gail asked, "What's it called in poker when you have three cards with one
number and two cards with another?" The group responded that Gail had a
"full house." Gail then announced, "Let's play poker with the numbers on
our fortunes."
I looked at my numbers 13 4 2 3 7 10 and knew I had a
terrible poker hand so I suggested, "Let's play a math game instead. Why
don't we play who has the most prime numbers?" to which Gail replied,
"That's okay with me." (I didn't know that Gail's numbers were 3 8 3 5
3 5 . Sheila then looked at her hand and said, "I'd like to play who has
the most multiples of 4," to which Sandy added, "I'd rather play who has
the most composite numbers."
Obviously, each of us named a math game that we thought would give
us the best chance of winning. I thought I'd surely win by playing most
prime numbers since I had 4 primes on my card; but I didn't know that Gail
had 5 prime numbers on her card.
Here's your challenge should you choose to accept it:
Suppose each of the four teachers opened a new fortune cookie. Help us to
decide which math game we should play to give us the best chance of
winning. Make sure you back up your decisions with sound mathematical
reasoning. You could do this individually but it would probably be more fun
to do it in a small group with your classmates or at home with your family.
If you do the problem as a group, give one set of numbers to each person in
the group. After reviewing only your own numbers, each person should
declare the math game they wish to play (It's okay if two people choose the
same game).
Once you've each committed to the game you wish to play, the
scoring begins. You might want to fill in a chart like the one below.
Teacher Numbers
------- --------
Ruth 22, 20, 2, 8, 3, 31 * Ruth's score
Game: Gail's score
Sheila's score
Sandy's score
Gail 11, 27, 31, 26, 24, 11 * Gail's score
Game: Ruth's score
Sheila's score
Sandy's score
Sheila 5, 6, 16, 5, 30, 22 *Sheila's score
Game Ruth's score
Gail's score
Sandy's score
Sandy 25, 25, 26, 12, 25, 5 * Sandy's score
Game Ruth's's score
Gail's score
Sheila's score
(If you're wondering where the numbers came from, you can write an easy
program for the TI-82 graphing calculator that will generate random
numbers).
Here are some questions for you to think about.
1. Based on your choice of a game for each teacher, which teacher's game
gave the highest overall score?
2. How did you decide on the math game you would play for each teacher;
what reasoning did you use in selecting each game?
3. If you solved the problem as a group, which one of you won the game?
4. If you could change your game after seeing the other player's numbers,
would you? If yes, what would your new game be and why?
This week's mentors are from
Mendocino Grammar School on the North coast of California.
They are 5th graders from Gail Lauinger's class.
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Correct Solutions Submitted by: Casey Gorish 6 grade Murray Middle School Ridgecrest, CA 93555 ssusd2@owens.ridgecrest.ca.us Mrs. Jameson 5th grade class Heights Elelmenntary School This week's problem solvers: Kristina, Kandice, Elizabeth, Kara Mrs. Bowerman Fourth Grade Kyrene de la Mariposa Tempe, AZ mbower@mar1.kyrene.k12.az.us Answers by entire class
I'd like to thank this week's "Visiting Math Mentors" from Mrs. Lauinger's
5th grade class at Mendocino Grammar Schhol, Mendocino, CA. They not only
answered the solutions sent to them but also provided us with a rubric for
evaluating solutions.
-Ruth
-------------------------------------------
We solved the problem first and discussed it. We read the solutions and
sorted them. Based on our work and the submitted solutions we made a rubric
for evaluating the solutions. This problem could be interpreted many ways.
There were six solutions submitted. We chose three as best meeting the
criteria.
thorough explanation
clear solution
makes math sense
high level of math thinking (use of primes, square numbers, factors)
organized presentation
Mrs. Lauinger's 5th Grade Class
Mendocino, CA
--------------------------------------------
Highlighted Solutions
Casey Gorish
6 grade
Murray Middle School
Ridgecrest, CA 93555
ssusd2@owens.ridgecrest.ca.us
Ruth wants to play who has the most numbers that are divisible by 2
because her numbers are 22, 20, 2, 8, 3, and 31.
Sheila wants to play who has the most numbers that are divisible by
2 also because her numbers are
5, 6, 16, 5, 30, and 22.
Gail wants to play who has the most numbers that
are odd because her numbers are 11, 27, 31, 11, 24, and 26. Sandy wants
to play who has the most numbers that are divisible by 5 because her
numbers are 25, 25, 26, 12, 25, 5.
Ruth has the best chance of winning who has the most numbers that
are divisible by 2 because since 22, 20, 2, and 8 are even numbers, Ruth's
score is 4 points, Sheila's score is 4 points, Gail's score is 2 points,
and Sandy's score is 2 points.
Sheila has the best chance of winning who has the most numbers that
are divisible by 2 because since 22,30,16,and 6
are even numbers, Ruth's score is 4, Sheila's score is 4, Gail's score is
2, and Sandy score is 2.
Gail has the best chance of winning the most
numbers that are odd because since 11,27,31, and 31 are odd, Gail's scoreis
4, Ruth's score is 2,Sheila's score is 2, and Sandy's score is 4.
Sandy has the best chance of winning the most numbers that are
divisible by 5 because since 25,25,25, and 5 are divisible by 5, Sandy's
score is 4, Ruth's score is 1, Sheila's score is 3, and Gail's score is 0.
Based on my choice of a game for each teacher, Ruth's, Gail's, and
Sheila's games were tied for the highest overall score of 12. I decided on
a mathgame for Ruth becuase she had more even numbers than odd, Gail had
more odd than even numbers, Sheila had more even than odd numbers, and
Sandy had numbers that were divisble by than those that were even. If I
could
change my game after seeing the other numbers I would not.
Mrs. Jameson 5th grade class
Heights Elelmenntary School
This week's problem solvers: Kristina, Kandice, Elizabeth, Kara
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
We rewrote the problem in our own words:
We thought the problem said...
Four teachers went to a restaurant and got fortune cookies with 6 lucky
numbers on them. So they decided to play a game using their lucky numbers.
Each wanted to play a game in which she would win. Can we create a fair
game that gives everyone the best chance of winning?
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
This was our strategy:(We tried to think of more than one way) The four of
us each wrote 6 random numbers, and traded slips of paper. We went around
the circle, telling what game we wanted to play. Some of us wanted to play
games like multiples of 2, 3, factors or multiplying to get the highest
score, or having the most double digits.
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
We solved it! We'll show you how we did: We each made up a game for our
numbers on our papers so each of us would win. We each wrote down in words
how we could win and why.
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
We checked our work by trying...many different games. We found that
everyone suggested a different game so she would win.
Elizabeth 41,8,30,10,06 (multipling all numbers)
Kandice 4,7,1,0,7,8 (most multiples of two.)
Kara 6,18,41,5,10, and 80 (having the most double digits
Tina 3,9,12,15, 50 (most multiples of three)
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Our answer makes sense because...
After trying all the games, the person who suggested using a game of
multiples of 2 was the winner of the game because there were more
possibilities.
Mrs. Bowerman
Fourth Grade
Kyrene de la Mariposa
Tempe, AZ
mbower@mar1.kyrene.k12.az.us
Answers by entire class:
Ruth would want to play multiples of 2 because she would have four numbers
that fit that category. She might want to play even numbers, because there
are four of those on her card. We knew that even numbers were all
multiples of two, so we could use either category.
Gail would want to play how many numbers have two digits because she has
all six of her numbers being in double digits. Sandy would want to play
how many twos are in your numbers because she would have five numbers with
the number 2 in them.
Sheila would also want to play even numbers because she would have four of
them. She would not know that she was tying with Ruth.
1. Gail would be the winner with six 2-digit numbers if they played that
game. We weren t sure if any of the other people had two digit numbers, so
we looked for the category that would give us the most points.
2. We decided on the best game for each person after talking about all the
categories that we could think of. We then tried to find the one category
that would give each person the highest total.
3. We worked in groups at first, then shared all our ideas with the entire
class.
4. We would have changed the game for Sheila because she tied with Ruth
for even numbers. We might have played factors of thirty, and then she
would have four numbers for that category.
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