Scale 'n' Bowl 2
posted November 8, 1999 as a Math Forum EPoW

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 percents as scaling ratios.

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.

(The applet for this problem is currently unavailable)

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Sample submitted solution:

From: Jack, age 13
Carson, age 13
School: School of the Arts, San Francisco, California

1. For each of the 4 percent alleys, please explain the scaling ratios you tried, why you tried them, and which one worked.
Alley #1 solution: We tried scaling the ball to a percent of 85 because we felt it was smaller than the original ball. This percentage made the ball smaller but not small enough to fit through the alley way. Next we tried 80% but this was too small. We then tried 81% and this worked.
Alley#2 solution: We tried nine numbers that were all to small so we finally came to the conclusion of 45%.
Alley#3 solution: We first tried cutting the ball's percentage down by half. This was too small. We then kept adding one percent to the number until 58. They were all too small. Eventually we came into the 60% area and found that the correct answer was 62.5%.
Alley#4 solution: The first thing we noticed is that the percentage would be more than one hundred percent so we scaled the percent to 130. This was barely too small; we then tried135%. This was too big. We went down the scale, 134,133,132.5, and we eventually found that the answer was 131%.

2. After you have bowled all 4 alleys, put the scaling ratios that worked in order from smallest to largest and tell us how you chose this order.
The order of the scaling ratios, smallest to largest are 45%, 62.5%, 81%, and 131%. We came to this conclusion by looking at the numbers and like numbers without pecents you can count in your head, and you know which number is higher.

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