
ESCOT Problem of the Week: Archive of Problems, Submissions, & Commentary 
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A carnival is in town! It offers games of chance and games of skill. You decide to try your skill at a game called "Scale 'n Pop." Here's how it works.A heliumfilled balloon is tied to a rod. Just above it, between two boards, is a gap that leads to a pair of sharp nails. The distance between the nails is a little less than the width of the gap.
To win the game, you must figure out how to make the balloon small enough to fit through the gap, yet still large enough to touch (and be popped by) the nails. You can inflate or deflate it so that its diameter is a fraction of its original size. (That fraction may be greater than or less than 1.)
To Do and Notice
To scale your balloon, type a fraction into the box and then click on the Scale button; your balloon's new size will be displayed. When you think your balloon is the right size to reach the nails, click on the Release button. If your balloon doesn't pop, click on the Reset button, try a different fraction, and then scale and release it again. Repeat those steps until you win the game.
Applet
Click to open Scale 'n Pop applet
Questions:
 What fractions popped the balloon in:
booth 1:booth 2:
booth 3:
 Review the sequence of fractions you used in booth 3. Explain your strategy for picking fractions in this sequence.
 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.
 What is the difference between increasing the numerator and increasing the denominator of a fraction?
Teacher Support Page We're sorry to say that no one got credit for this week's problem.
Most of you got the answers to the first question right, that is, naming the fractions that popped the balloons. Good job!
For those of you who didn't get it, here they are:
Booth 1: 2/3
Booth 2: 3/8
Booth 3: 5/2
For the second question, in describing the strategy you used to get those fractions many of you said that you used the "guess and check" method. It's fine to use guess and check, but you need to explain what you guessed and why, what the check told you, and how you chose your next guess until you got your answer. That's the value of guess and check. To just say that's what you did doesn't help others understand how to solve the problem.
Here is a decent explanation by Katrina Elsfelder 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."
Many of you started to answer question 3, but didn't give enough detail. Sarah Fredrickson at Taipei American School gave a pretty good explanation: "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."
Most of you 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; for example, as Lauren Kaplan at Taipei American School said, "This is because when you have the denominator increased you make the numerator be worth less than it started out as."
As with all the PoWs, we're looking for clear, detailed explanations to answer the questions. That way other people can read them and learn from them.
Alex Acosta, age 15  Westside High School, Houston, TX
Lee Bobbitt, age 15  Westside High School, Houston, TX
Jasmine Branch, age 17  Westside High School, Houston, TX
Jerson, Jr. Cometa, age 18  Westside High School, Houston, TX
Donielle Miller, age 17  Westside High School, Houston, TX
Michelle Shadwell, age 17  Westside High School, Houston, TX
Mark Vuong, age 18  Westside High School, Houston, TX