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Scale 'n' Bowl 1
Posted November 1, 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 decimals 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.
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:
From: Georgia,
age 13
Gregory, age 13
School: School of the Arts, San Francisco,
California
1. For each of the 5 alleys, please
explain the scaling ratios you tried, why
you tried them, and which one worked.
Alley One: We did twenty times two
and we got forty. When the ball went through
the alley, it knocked the pins. Our previous
guess was twenty times three and it was
too big, so we reduced it.
Alley Two: We did twenty times three
because in our previous game we had guessed
three and it was about the size of this
problem to hit the balls.
Alley three: We did twenty times
1.45 and got a ball radius of 29 which hit
the pins. We stated with 1.5 and it was
too big, we got smaller and smaller until
the ball was right.
Alley four: We got twenty times .8
and the ball radius was 16 which hit the
balls. We started with .68 and got bigger
until we got the right radius.
Alley five: We got twenty times .35
with the ball radius of 7. We started with
.32 and got bigger.
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.
Our smallest answer was .35 then .8 then
1.45 then 2 then 3. Because as we look at
the ball size it gets smaller with the smaller
numbers and bigger with the bigger numbers.
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Reflections:
This puzzle explored the concept of
scaling ratios in decimal form. The students
were fairly successful in generating successful
answers to the puzzle. The first part of
the puzzle consisted of several bowling
alleys which students had to scale a bowling
ball to fit through. Students were asked
to keep track of their guesses and explain
why they made them. Although some students
were able to do this, many simply made guesses
until they arrived at the solution, and
then tried to remember which guesses they
had made afterwards. This defeated the purpose
of the puzzle, which was to get students
to make their guess and check strategy explicit.
Others, however, did note their guesses
as they went along, and wrote in why they
had made these guesses. In the second part
of the puzzle, students were asked to put
the decimal scaling ratios they found in
order. Students had more difficulty with
this task. In terms of making changes to
this puzzle, it would be beneficial for
student guesses to be automatically generated
and noted in their solution box so that
they could look back and see their thinking.
It's very tedious for them to write each
answer as they go along.
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