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Scale 'n Bowl III - posted November 15, 1999

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

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Find a scaling ratio in the form of a fraction (e.g. 2/3) 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 was the most difficult part of the puzzle for students. Many of them were not able to finish the problem, and therefore did not submit a solution. By this time, students had gotten used to the simulation and question, so most of the difficulty seemed to lie in trying to manipulate fractions to make appropriate scaling ratios.

Through classroom observation, it was noted that students who were successful in finding the correct fraction developed interesting strategies. One common strategy was to increase/decrease the denominator incrementally, and then move on to the numerator. This worked very well. Students used this method spontaneously, and it seemed to improve their understanding of fractions.

However, students had to make lots of guesses to try to figure out the correct solution, and therefore did not write most of their thinking down. Those who did write their solutions tended to give very long answers.

Students also had a great deal of trouble in trying to put the fractions in order, which contradicts the proposition above that students developed a sense of fractional number lines. However, we did see a few students apply the strategy above to answer this question.

4 students received credit this week.

Jack C., age 13 - School of the Arts, San Francisco, CA
Carson D., age 13 - School of the Arts, San Francisco, CA
Wil D., age 13 - School of the Arts, San Francisco, CA
Jeanette G., age 13 - School of the Arts, San Francisco, CA

View most of the solutions submitted by the students above

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