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Scale 'n' Pop
Posted December
4, 2000 as a Math Forum EPoW
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Introduction: In this problem,
students resize a balloon by scaling
it with fractional scaling factors.
The balloon must be small enough
to fit between two walls, yet be
big enough to be popped by a pair
of nails.
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Where’s the Math: This
problem allows students to investigate
the properties of fractions. The
applet allows a visual representations
of basic properties of fractions,
and makes it easier to see things
like the fact that a smaller numerator
makes the fraction smaller, but
a smaller denominator makes the
fraction bigger. The problem also
illustrates how fractional scaling factors can be
used to change the size of an object. The
data table displays that scaling
and illustrates conversion between
fractions and decimals.
Standards: Geometry,
measurement,
number
& operations
Role of Components: A SketchpadBean
displays the balloon, barriers,
and the nails. The original diameter
and scaled diameter are displayed
with non-editable NumberEntry
components; the scaling fraction
is displayed with an editable NumberEntry
component. All other components
are standard Swing components: the
lane chooser uses JRadioButtons,
the buttons are JButtons,
and the labels are JLabels.
The history is displayed in a JTextArea.
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Try the applet!
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Sample submitted solutions:
(No single submitted
solutions got credit for this problem. Here
are some parts of solutions submitted by
different students.)
1. What fractions popped the balloon
in: (various students)
booth 1: 2/3
booth 2: 3/8
booth 3: 5/2
2. Review the sequence of fractions
you used in booth 3. Explain your strategy
for picking fractions in this sequence.
(submitted by Katrina at Vista
Middle School)
I first tried 1 but it was too small.
then I picked 2/1 and it was still too small.
So then I picked 4/1 and that was too big.
So I decided to go in the middle of 2 and
4 which is 3 and I chose 3/1 and that was
still too big, so I picked 5/2.
3. Last year a student playing this
game in a different booth found that 3/2
made the diameter too small and 5/3 made
the diameter too large. Recommend a strategy
the student might use to find a fraction
that will pop the balloon.
(submitted by Sarah at Taipei
American School)
I recommend that the student find the least
common denominator for the two fractions,
and then go to the next smallest common
denominator fraction in between the previous
ones. I would repeat this until I found
the right fraction.
4. What is the difference between increasing
the numerator and increasing the denominator
of a fraction?
(submitted by Lauren at Taipei American
School)
Increasing the numerator makes the balloon
bigger and increasing the denominator makes
the balloon smaller. This is because when
you have the denominator increased you make
the numerator be worth less than it started
out as.
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Reflections:
We're sorry to say that no one got credit
for this EPoW.
However, most students who submitted answers got the first
question right, that is, naming the fractions
that popped the balloons.
For the second question, in describing the strategy
used to get those fractions, many students said that
they used the "guess and check" method. It's fine to
use guess and check, but an explanation is needed as to what
was guessed and why, what information came from that guess, and
how guesses were chosen before finally hitting on an answer.
That's the value of guess and check.
Many students started to answer question 3, but didn't give
enough detail.
Likewise, most responses correctly explained that increasing the numerator makes the balloon bigger and increasing the denominator makes the balloon smaller. We were looking for something more detailed, however.
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