How did you spend your summer vacation?
Ismael, Queequeg and Ahab are three high school buddies. They have rented a small sailing sloop for a two week whale poaching excursion. Unfortunately, on the fifth day of their odyssey they encounter a tropical storm and the boat sinks. As members of their championship swim team, they take the challenge to stroke their way to a small island which is conveniently located within sight of the mishap. They lie in the sane long enough to give dramatic effect, then walk to a point above the high tide mark and begin to assess their situation.
What do you know? Questions for characters? What do you need to know?
After a quick exploration, the three adventurers determine that the island is unpopulated, very small, has only a few coconut palms, an artesian freshwater spring flowing from the one rocky outcrop in the center of the island. Ismael is vertically challenged, standing exactly one meter in height. Queequeg is exactly twice Ismael’s height and really should go out for the basketball team if they ever get back to the school. Ahab is the class president and begins to organize the group. “If we are going to survive this dilemma we need a shelter. Ismael, look around and report back to me in 30 minutes. It is a good thing your digital watch is still working.”
Ismael returns in 30 minutes and takes them to an opening in the rocky outcrop. The width of the opening is wide enough to allow them to fit inside with room to spare, but there is no roof overhead. Ahab turns to Queequeg, “Queeq, you are the physics genius here. How can we cover this opening to keep us out of the sum and rain?”
Queeq has made some measurements. “The opening is between 15 and 16 Ismael’s wide. We will need to chop down a tree that is tall enough to cover it without wasting any wood.” “Since there aren’t many trees on the island, we need to know the exact distance of the opening,” Ahab announced, “Ismael, go back to the opening and make the measurement using smaller increments. Also, we need to measure the height of the trees and I’m not sure how we can do that.” “Don’t worry, I have a simple plan,” replied Queeq.
Part 4a (small lab)—Students will be given a meter stick and asked to divide the meter stick into 100 subdivisions.
Part 4b (second lab)—After a group discussion about a possible plan introduce a lab exercise:
1. Students will be given five sticks of different lengths. They will go outside and hold the sticks vertically (using a string to suspend them) and measure the length of the shadow cast by the stick.
2. A graph of length of stick vs. length of shadow will be constructed.
3. Data will be shared between classes and the data plotted on one graph.
4. A line of best fit will be drawn for each data set.
5. A slope will be calculated for each data set.
6. An equation will be written using the slope-intercept form of a linear equation.
7. A name will be generated for the slope.
The tangent function will be introduced. An additional drawing will be made using a protractor and graph paper. Comparison will be made with a trig table to discover the relationship between angle and slope. Differences in class data will be discussed and hypotheses generated about a cause for these differences.
The three castaways are sitting under their beautifully constructed shelter. Ismael’s watch is no longer working. Food and water have been plentiful, but they were getting homesick. As they watch the shadows of the trees on the white sandy beach, the boys begin to think of ways to insure they are discovered. Ismael remembers that the cruise ship his father owns sails near them in late December. With limited wood for a signal fire or a raft construction, they wonder how they can know when the ship will pass.
What have you learned? How can you extend this?
Part 5b: Activity with a sundial and the analemma. Extrapolate the data from the class data in the shadow lab. Introduce the photographs of the analemma. How could you construct a device to reproduce the analemma shape without a camera?
Part 5c: Use a protractor to measure the height of the flag pole. (reinforce tangents)