#### A Math Forum Project

 ESCOT Problem of the Week: Archive of Problems, Submissions, & Commentary

Student Version

### Scale n' Bowl II - posted November 8, 1999

(Try Scale 'n Bowl I before you begin this problem.)

Find a scaling ratio in the form of a percent that will enlarge or reduce the blue ball to fit through the green bowling alley and move the two red pins.

1. For each of the 4 percent alleys, please explain the scaling ratios you tried, why you tried them, and which one worked.

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.

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Students were fairly successful with this part of the puzzle. They found percentages easier to manipulate than decimals. Again, the goal was to find scaling ratios in the form of percents that would correctly scale a bowling ball to fit through the alley and hit the pins.

When students noticed the second question, which asked them to put the ratios in order, they were successful at it.

As in the first part, students did not always list all of their guesses, and did not make the reasoning behind their guesses explicit. Therefore, it is unclear whether or not students' thinking was made explicit to them.

### Highlighted solutions:

 From: Carson D., age 13Jack C., age 13 School: School of the Arts, San Francisco, CA

```Page Percent Alleys

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

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%.

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.
------------------------------------
```

### 8 students received credit this week.

Kailea B., age 13 - School of the Arts, San Francisco, CA
Alan C., age 13 - School of the Arts, San Francisco, CA
Andrew C., age 15 - School of the Arts, San Francisco, CA
Jack C., age 13 - School of the Arts, San Francisco, CA
Carson D., age 13 - School of the Arts, San Francisco, CA
Kevin D., age 13 - School of the Arts, San Francisco, CA
Derek M., age 14 - School of the Arts, San Francisco, CA
Cindy Y., age 13 - School of the Arts, San Francisco, CA