Math Forum - Problem of the Week

Scale 'n Pop

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Introduction

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 helium-filled 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.

Questions:

1. What fractions popped the balloon in:

booth 1:

booth 2:

booth 3:

2. Review the sequence of fractions you used in booth 3. Explain your strategy for picking fractions in this sequence.

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.

4. What is the difference between increasing the numerator and increasing the denominator of a fraction?

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