#### A Math Forum Project

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

Student Version

### Scale n' Bowl I - posted November 1, 1999

Find a scaling ratio in the form of a decimal (e.g. 0.5) 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 5 alleys, please explain the scaling ratios you tried, why you tried them, and which one worked.

2. After you have bowled all 5 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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This puzzle explored the concept of scaling ratios in decimal form.

Students were fairly successful in generating successful answers to the puzzle. The first part of the puzzle consisted of several bowling alleys and a bowling ball that students were to scale so it would fit through the alleys. Students were asked to keep track of their guesses and explain why they made them. Although some students were able to do this, many simply guessed until they arrived at the solution, and then tried afterward to remember which guesses they had made. This defeated the purpose of the puzzle, which was to make the guess-and-check strategy explicit. Other students, however, did note their guesses as they went along, and wrote why they had made these guesses.

In the second part of the puzzle, students were asked to put the decimal scaling ratios they found in order. Students had more difficulty with this task.

In terms of making changes to this puzzle, it would be beneficial for students' guesses to be automatically generated and noted in their solution boxex so that they could look back and see their thinking. It's very tedious for students to write each answer as they go along.

Here are further comments on specific solutions:

Carson and Jack have excellently described the reasoning process behind their guess-and-check decisions, but the description of how they put the decimals in order is poor, although they at least refer to the difficulty of putting 0.8 and 0.35 in order.

Georgia and Gregory give noninteresting explanations of their guess-and-check process, but give a precise explanation of how they put the decimals in order. They also state explicitly one of our main objectives, which was to get kids to get a sense of order in decimals, fractions, and percents by understanding their effects on size when used as multipliers (or scaling ratios). That is, numbers greater than 1 enlarge an object, 1 keeps things the same size, and numbers smaller than 1 shrink an object. If every student who participated understood that, we would be quite happy.

Paul and Rachel give another excellent explanation for their ordering of the numbers:

"2. .35, .8 ,1.4 ,2.,3.... .35 is the smallist because .35 means it is 35/100 of a whole. .8 is 80/100 and 1.4 is 1 whole and 40/100 of a whole. 2 is two wholes. and 3 is 3 wholes."

### Highlighted solutions:

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

```Page Decimal Alleys

alley #1 results: At first we tried different scaling ratios to determine how
big the number would have to be in order for the blue ball to pass through the
green rectangles and bump the red balls.  We figured out that the correct
scaling ratio was 2.

alley #2 results:The green rectangles and the red balls were farther apart
than in alley #1 therfore we first changed the scaling ratio to 4.  This was
too big so we tried 3 and that worked.

alley #3 results: We first changed the scaling ratio to 1.5 because the area
between the rectangles seemed smaller than in alley #1.  The ball got stuck so
we then tried 1.3 but this was too small.  As a result we found that 1.4 was

alley #4 results:Since the blue ball was already larger than the area between
the rectangles and we knew that the scaling ratio of the ball was at 1 we then
made the sclaing ratio 0.7.  The ball passed right through everything barely
missing the red balls so we increased the scaling ratio by 0.1.  0.8 was the

alley #5 results:  in alley 5 we observed that the area between the rectangles
were smaller than the other alleys.  As a result, we put 0.3, but this was too
small.  then we put 0.4, but this was too large.  Our final, correct answer
was 0.35.

conclusion:  Alley #5 has the smallest scaling ratio of 0.35.  Alley #4 has
the next smallest scaling ratio of 0.8.  Next came alley #3 with a scaling
ratio of 1.4.  Next alley#1 with a scaling ratio of 2.  The biggest scaling
ratio came from alley #2 with a ratio of  3.

We placed the decimals in order from least to greatest like this since o.35 is
smaller than .80.  This is so because 35 is smaller than 80, and so on.

------------------------------------
```

 From: Georgia D., age 13Gregory T., age 13 School: School of the Arts, San Francisco, CA

```Page Decimal Alleys

Part 1:
Alley One: We did twenty times two and we got forty. When the ball went
through the alley, it knocked the pins. Our previous guess was twenty times
three and it was too big, so we reduced it.
Alley Two: We did twenty times three because in our previous game we had
guessed three and it was about the size of this problem to hit the balls.
Alley three: We did twenty times 1.45 and got a ball radius of 29 which hit
the pins. We stated with 1.5 and it was too big, we got smaller and smaller
until the ball was right.
Alley four: We got twenty times .8 and the ball radius was 16 which hit the
balls. We started with .68 and got bigger until we got the right radius.
Alley five: We got twenty times .35 with the ball radius of 7. We started with
.32 and got bigger.

Part 2:
Our smallest answer was .35 then .8 then 1.45 then 2 then 3. Because as we
look at the ball size it gets smaller with the smaller numbers and bigger
witht the bigger numbers.
------------------------------------
```

 From: Christina C., age 13Helen Y., age 13 School: School of the Arts, San Francisco, CA

```Page Decimal Alleys
For alley #1, we plugged in 20 for the scaling ratio because we wanted to know
how big the ball was going to be. So, we saw that the ball was enormous and
couldn't fit through the alley. Then we tried smaller numbers like 2 for the
scaling ratio and we notice that the ball went in through the alley and
knocked off the 2 pins.
For alley #2, we first tried to use the scaling ratio of 5, but it was too big
to fit through the alley. Then we tried scaling ratio of 3 and found out that
it fit through the alley and knocked off the 2 pins.
For alley #3 we saw that the alley was smaller than alley #1, so we had to use
numbers that were between 1 and 2. We tried decimals of .25 and .5 and we saw
that 1.5 was too big for the alley, so we tried going down by tenths. We put
1.4 and it knocked off the two pins.
For alley #4, we saw that the ball with scaling ratio of 1, was too big to go
through the alley. So we tried numbers smaller than one. We started by going
down in quarters and figured that .75 was too little to knock out the pins.
Then, we tried .8 and it went right through the alley and knocked off the
pins.
For alley #5, we saw that the ball of a scaling ratio of one was too large to
even fit through the alley. So, we started out with a quarter (.25) and saw
that it was too small to knock off the pins. So, we went on by fifths and we
saw that .3 didn't work. We tried .35 and it went through the alley and
knocked off the pins.
The scalling ratios that worked were .35, .8, 1.4, 2, and 3 because .35 is
smaller than .8. .8 is smaller than 1.4 and 2 is smaller than 3.
------------------------------------
```

 From: Keturah B., age 13Jessica D., age 13Leah R., age 13Akira M., age 13 School: Frisbie Middle School, Rialto, CA

```Page Decimal Alleys
After we read the instructions, we started with alley one.  The first decimal
we tried is 0.5.  We tried this because we read it in the instructions.  0.5
was too small.  The next decimal we just guessed.  It was 3.0 next, but it was
a little to large.  We figured that since 3.0 was just a little to big 2.0
might work.  We were right.  The first decimal we tried for alley two was 1.0.
We tried 1.0 just to see if it would work.  It was really small.  Then we
tried 10.0 it was to big of a number to even try.  We finally tried 3.0 and it
worked.  We saw a pattern so we tried 4.0 for alley 3, but it couldnŐt fit.
The next numbers we tried were 1.0, 1.2, and 1.3.  They were all to small so
we tried 1.4 and it was perfect.  For alley 4 we started with 1.0 and finally
got down to 0.8 which worked.  We guessed alley five correct.  It was .35.

The smallest decimal was 0.35, then 0.8, 1.4, 2.0, 3.0.  We looked at the end
number in most decimals, but for 0.35 and 0.8 we looked  at how many numbers
there were.
------------------------------------
```

### 27 students received credit this week.

Erica A., age 13 - Frisbie Middle School, Rialto, CA
Lauren (roy) A., age 13 - School of the Arts, San Francisco, CA
Reina A., age 13 - Frisbie Middle School, Rialto, CA
Amy B., age 13 - Issaquah Middle School, Issaquah, WA
Keturah B., age 13 - Frisbie Middle School, Rialto, CA
Miguel B., age 13 - Frisbie Middle School, Rialto, CA
Christina C., age 13 - School of the Arts, San Francisco, CA
Jack C., age 13 - School of the Arts, San Francisco, CA
Jamecia C., age 13 - Frisbie Middle School, Rialto, CA
Carson D., age 13 - School of the Arts, San Francisco, CA
Georgia D., age 13 - School of the Arts, San Francisco, CA
Jessica D., age 13 - Frisbie Middle School, Rialto, CA
Paul G., age 15 - School of the Arts, San Francisco, CA
Lindsay J., age - Issaquah Middle School, Issaquah, WA
Octavius J., age 13 - Frisbie Middle School, Rialto, CA
Sophia K., age 13 - School of the Arts, San Francisco, CA
Sam L., age 13 - Frisbie Middle School, Rialto, CA
Akira M., age 13 - Frisbie Middle School, Rialto, CA
Kathy R., age 13 - Frisbie Middle School, Rialto, CA
Leah R., age 13 - Frisbie Middle School, Rialto, CA
Jasmine S., age 13 - Frisbie Middle School, Rialto, CA
Norma S., age 13 - Frisbie Middle School, Rialto, CA
Rachel S., age 17 - School of the Arts, San Francisco, CA
Gregory T., age 13 - School of the Arts, San Francisco, CA
Robert V., age 13 - Frisbie Middle School, Rialto, CA
Alicia W., age 13 - Frisbie Middle School, Rialto, CA
Helen Y., age 13 - School of the Arts, San Francisco, CA