Keeping All Her Balls in the Air

On Monday, Anne will finish her gig as a Math Interventionist. She has really enjoyed this assignment and will be sorry to see it end. My totally unbiased perception is that she has done quite well in this assignment. I do all of my research across the dinner table from her, listening to her tell me about her day. Today, Anne showed me the following story problem that she plans on giving to her students. The question is real, but since she didn’t give me the answer key, I had to take my best guess at it.

Sam and Ben are playing in a ball pit at the Fun Zone and of course, they start thinking about math. They ask the manager about the dimensions of the ball pit and he says 10’ wide x 12’ long x 6’ deep. Sam says, “Oh, then it is 120 square-feet.” Ben says, “No way!” Sam says, “Is too!” and they start pelting each other with balls. Can you end their disagreement?

1. Take advantage of this opportunity to turn this conflict into a teachable moment. Explain that if Sam was speaking of the area of the ball pit then he is correct. Then ask Ben if he can compute the volume of the pit.
2. Explain to these students that the volume (V) of a ball can be calculated using the formula: V=4/3πr³, where r is the radius of each of the balls. If each ball is 3″ in diameter, then the volume of each ball is a little more than 14 cubic-inches. Have Ben and Sam count the balls in the pit. The number of balls in the pit times 14 in³ will yield an approximation of the volume of the pit and it will keep those little buggers busy for a while.
3. Have Ben and Sam compute the parabolic trajectory of each ball they want to throw at each other, before they can throw it. If they somehow manage to pull off this feat, then have them using Euler’s Method, compute the transition from a parabolic trajectory to an elliptical orbit. Explain to them that this calculation will allow them to throw the balls at each other even harder than before. Remind them to show their work, no Hidden Figures.
4. Fill the pit with water, using the flow rate calculate the volume of water that it takes to displace all of the balls and fill the pit. And then let the next teacher fish out Ben and Sam.