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Scale 'n' Bowl 3
posted November 15, 1999
as a Math
Forum EPoW
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Introduction: In the Scale 'n' Bowl puzzle series, students were
given instructions to find a scaling
ratio that will enlarge or reduce
the blue ball to fit through the
green bowling alley and move the
two red pins. For each of the alleys,
they were asked to explain the scaling
ratios tried, why they tried them,
and which one worked. After bowling
all alleys, students put the scaling
ratios which worked in order from
smallest to largest, describing
how they chose this order.
In this part of the series, students used fractions as scaling ratios.
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Where's the Math: The four-part
problem maintains the same graphical
representation in each section,
but requires the use of first decimal,
then percentage and fractional notation
to manipulate the size of the bowling
ball. Students may start with trial
and error guessing, but soon discover
strategies to narrow the input values
to get the necessary ratio. Discovering
strategies for finding "in
between" ratios scaffolds students
to tackle the more complex notation
of fractions in this part of the
problem series, when they are challenged
to explore the relationship between
the numerator and the denominator.
Each of the problems in the series
requires that students place the
ratios in ascending order, encouraging
them to revisit their problem-solving
strategy and solidify their developing
understanding of numerical representations
of the concept of scale.
Standards: Number
& operations, measurement,
geometry
Role of Components: The
Scale 'n' Bowl problem series uses
Geometer's
Sketchpad to animate scaling
and bowling actions based on values
inputted into number
entry fields (labeled with ESCOT
labels) and triggered by a button
panel. Instructions are displayed
via the HTML
viewer, and solutions are entered
in a
text entry box. Users reset
the alley with an Action
menu. Javascript
handles communication among components.
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(The applet for this problem
is currently unavailable)
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Sample submitted
solution:
1. For each of the 5 alleys, please
explain the scaling ratios you tried, why
you tried them, and which one worked.
Alley#1: On our first bowl we thought
that if we scaled the ball to 2/3 of it's
original size it would go through the green
tube and hit the balls, we were right.
Alley#2: In our attempts, we scaled
the ball in many different sizes. They included:
1/2, 3/4,2/5, 3/5, and 2/6. Finally, we
tried 3/7 and this was the correct answer!!!!
Alley#3: We scaled the ball to many
different sizes. And through a lucky guess
I found that the fraction was 8/15.
Alley #4: We tried several different
scaling ratios. Some were 4/3, 5/3, 5/4,
7/4 and finally 6/4 (3/2), which was correct.
We tried improper fractions because we observed
that the ball had to be bigger than its
original size!!!
Alley#5: The ball needed to be enlarged
so again I tried 6/4 as our fraction. This
was barely to small so I made the fraction
7/4 and this was correct!!!!!
2. After you have bowled all 5 alleys,
put the scaling ratios that worked in order
from smallest to largest, and tell us how
you chose this order.
The scaling ratios from smallest to largest
are 1/3, 3/7, 8/15, 2/3, 3/2, and 7/4. We
put them in this order by visualizing and
knowing how big each one would be in our
head.
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Reflections:
This was the most difficult part of
the Scale 'n' Bowl puzzle series for students. Many of them were
not able to finish the problem, and therefore
did not submit a solution. By this time,
students had gotten used to the simulation
and question, so most of the difficulty
lie in trying to manipulate fractions to
make appropriate scaling ratios. Through
classroom observation, it was noted that
students who were successful at finding
the correct fraction developed interesting
strategies. One common strategy was to incrementally
increase/decrease the denominator, and then
move on to the numerator. This worked very
well. Students used this spontaneously,
and it seemed to improve their understanding
of fractions. However, students had to make
lots of guesses to try to figure out the
correct solution, and therefore did not
write most of their thinking down. Those
who did write their solutions tended to
have very long answers. Students also had
a great deal of trouble trying to put the
fractions in order. This counters the proposition
above that students developed a sense of
fractional number lines. However, we did
see a few students apply the strategy above
to answer this question.
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