On the Boardwalk
"It isn't fair!", Jasmine announced to her class one
Monday morning. "I used up $10 worth of quarters playing a
boardwalk game over the weekend at the shore, and I only won once.
And all I got for winning was a lousy stuffed animal!"
Ms. Buffon often told her class that mathematics was all around
them, and had encouraged them to see the world with the eyes of a
mathematician. So she wasn't surprised that Jasmine shared this
incident with the class.
"Please explain why you thought there was mathematics
here," Ms. Buffon asked Jasmine.
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The students:
recognize the role that mathematics can play in explaining and
describing the world around them.
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"Well, first of all, I threw the quarters onto a platform
which was covered with squares, you know, like a tile floor, so that
reminded me of geometry. And as I was throwing my quarters away, one
after another, I was reminded of all the probability experiments that
we did last year, you know, throwing coins and dice. It wasn't
exactly the same, but it was like the same."
"Those were very good observations, Jasmine," said
Ms. Buffon, "you recognized that the situation involved both
geometry and probability, but you didn't tell the class what you had
to do to win the game."
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connect previously learned mathematics to the current
situation.
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"Oh, you just had to throw the quarter so that it didn't touch
any of the lines!" Ms. Buffon asked Jasmine to go to the board
to draw a picture, explaining to her that not everyone will visualize
easily the game she was talking about.
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use different forms of communication to define a problem and
share their insights.
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Every other Monday, Ms. Buffon began her geometry class with a
sharing session. Sometimes the "mathematics situations"
that the students shared did not lead to extended discussions, in
which case Ms. Buffon continued with the lesson she had prepared. But
she was prepared to use the entire period for the discussion, and even
carry it over into subsequent days, if the students got interested in
the topic.
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are afforded the opportunity to fully explore and resolve
mathematical problems.
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"Why didn't you think the game was fair?" she asked
Jasmine. Jasmine repeated what she had said earlier, that she should
have won more often and that the prizes should have been better.
Other students in the class were asked to respond to the question, and
after a lively interchange, they decided that for the game to be
really fair, you should get about $10 in prizes if you play $10 in
quarters; but, considering that they were having fun playing the game,
and considering that the people running the game should get a profit,
they would be satisfied with about $5 in prizes. Jasmine listened to
the conversation intently, and chimed in at the end "That lousy
bear wasn't worth more than a dollar or two!"
Moving the discussion in another direction, Ms. Buffon said
"Now that we understand that it is possible to explain
'fairness' mathematically, let us investigate Jasmine's game
to see if it really was unfair. What do you think were Jasmine's
chances of winning a prize?"
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explore questions of fairness, geometry, and
probability.
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This question evoked many responses from the class, and after some
discussion the class agreed with Rob's comment that it all depended on
the size of the squares. Jasmine did not know the actual size of the
squares, so the class agreed that they might as well try to figure out
the answer for different size squares. Dalia pointed out that this
looked like another example of a function, and Ms. Buffon commended
her for making this connection to other topics they had been
discussing.
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are encouraged to make connections to other topics within
mathematics.
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Returning to her previous question, Ms. Buffon suggested that the
students do some experiments at home to help determine the probability
of winning a prize. Each pair of students was asked to draw a grid on
poster board, throw a quarter onto the poster board 100 times, and
record the number of times the quarter was entirely within the lines;
to simplify the problem, quarters that landed off the grid were not
counted at all. Different students chose different size grids,
ranging from 1.5" to 3.5", at quarter inch intervals.
After school, Ms. Buffon visited the Math Lab where she spent some
time trying to find materials related to this problem. When she
looked under "probability" in the indexes of various
mathematics education journals, she was led to several articles
discussing geometric probability, which she learned is a branch of
mathematics which addresses problems like Jasmine's game. With these
resources available to her, Ms. Buffon no longer feels that she has to
have all the answers, and can entertain discussions about mathematical
topics with which she is unfamiliar. Tomorrow she will be able to
tell the class what she has learned!
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model problems and conduct experiments to help them solve
problems.
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The next day the students reported on their results, and Ms. Buffon
tabulated them in the following chart, and, at the same time, plotted
their results on a graph:
Size of Squares |
Number of Wins |
1.25 |
5 |
1.5 |
10 |
1.75 |
18 |
2 |
25 |
2.25 |
30 |
2.5 |
34 |
2.75 |
40 |
3 |
45 |
3.25 |
48 |
3.5 |
54 |
"Do you see any patterns here?", Ms. Buffon asked the
class. They all agreed that, as expected, the larger the size of the
squares, the more frequently Jasmine would have won the game.
"What do you think the size of the squares were on the boardwalk
game?" Ms. Buffon asked next. Everyone agreed that the squares
were most likely smaller than
1.25" since her one prize out of forty quarters corresponded
to 2.5 wins out of a 100 games, which was lower than obtained for the
smallest squares in the experiment. Turning to the graph,
Ms. Buffonasked "What would have happened if we tried the
experiment with squares smaller than 1.25?" The students
laughed, one after another, as they realized that if the size of the
squares were small enough, you would never win the game. "Well,
then, what would have happened if we tried the experiment with larger
and larger squares?" Looking at the graph, the class found this
a difficult question, but Fran broke the group's mindset by saying
"Yeah, suppose the squares were as big as this room?" Then
everyone realized that if the squares were larger and larger, you
would become almost certain to win the game. "Dalia, do you
remember your comment yesterday, that it sounded like we were working
on a function?" Ms. Buffon asked. "Would you sketch the
graph of that function for the class?" Ms. Buffon made a mental
note to discuss this problem with her precalculus students, since she
had many questions to ask them about this graph.
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collect, analyze, and make inferences from data.
recognize connections between numerical patterns and
functions.
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"Well, we've gotten a lot of information by using experimental
methods about Jasmine's game; let's see if we can figure out the
probability theory behind it as well." At this point, Ms. Buffon
was eager to tell her class what she had learned at the Math Lab.
However, being aware that the students will grasp the solution method
better if they have an opportunity to discover it for themselves, she
asked the class to discuss the following question in their study
groups: "How can you tell from the position of the quarter
whether or not you would win the game?" Going from group to
group, Ms. Buffon listens to the discussions. When the groups have
discovered that to win the game, the center of the quarter must be
sufficiently far from the closest border, she gives the groups their
next task - to describe where in the square the center of the
quarter must be.
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formulate and test mathematical conjectures.
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By reasoning in different ways, the groups all arrived at the same
picture involving a smaller square inside the original square, and at
the same conclusion - that you win if the center of the quarter
lies inside the smaller square. With this information, the students
are able to calculate the probability for any particular size of the
square, and even to write an equation for the function whose graph
they sketched earlier.
Having found the probability of winning the game, Ms. Buffon
planned to return to the question that began this whole discussion
- whether Jasmine's game was fair. But that was the topic for
another day.
Note: Ms. Buffon realized that the graph of the function was not
linear, as depicted earlier, even though the data seemed to indicate
linear growth. With her precalculus students, she would have them
translate the above situation into the equation
y=(x-d)2/x2, where d is the diameter of the
quarter. Then she would have them graph the function, enabling them to
discover that although the graph appears to be linear, in reality it
increases at a decreasing pace, and goes asymptotically to the line y
= 1.
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construct a pictorial model to represent the problem.
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