For the May afternoon CAMI meeting, we considered the case of the tethered goat. We observed goats tethered to fences and barns of various shapes and sizes and considered the grazing area of the goats under various conditions.
Initially, the group noticed and wondered many things in regards to our scenarios including, but not limited to:
- The length of the rope by which the goat is tethered to a fence
- The uncanny resemblance of the barren land to the color of ice cream
- The hexagonal shape of the corral, as well as the goats inability to go under or above it (of course the King of Pop received props for introducing the phrase “too high to go over, too low to get under”)
- The existence of a half cylinder hay bale, two large boulders, and three trees
- What is the pink shape?
- Why doesn’t the goat eat the rope (instead of the grass), get free, and escape?
- Why isn’t the bucket close enough for the goat to reach it in case he gets thirsty?
In order to experiment with the numerous variables, we primarily used this Jamboard to calculate the amount of grass the goat could eat using geometric relationships. One group explored patterns in the relationship between the length of the rope and the area the goat would be able to access. Their calculations took into account the number of “turns” around corners in the corral or barn. Another group examined the relationship between the area available to the goat and the shape of the barn or corral to which he is tethered. The groups’ explorations ventured beyond the Jamboard to Geogebra and Desmos. Additional challenge problems were provided for further independent exploration.
We encourage you to check out these scenarios, play around with them, and see if you can come up with some of your own!
Thank you to Patrick Honner, whose article, How to Solve Equations That Are Stubborn as a Goat (Quanta), inspired this meeting.
