8_24

Buried treasure problem:
Among his great-grandfather’s papers, José found a parchment describing the location of a hidden treasure. The treasure was buried by a band of pirates on a deserted island which contained an banana tree, a coconut tree, and a gallows where the pirates hanged traitors. The map looked like the accompanying figure and gave the following directions.

“Count the steps from the gallows to the banana tree. At the banana, turn 90° to the right. Take the same number of steps and then put a spike in the ground. Next, return to the gallows and walk to the coconut tree, counting the number of steps. At the coconut tree, turn 90° to the left, take the same number of steps, and then put another spike in the ground. The treasure is buried halfway between the spikes.”

José found the island and the trees but could not find the gallows or the spikes, which had long since rotted. José dug all over the island, but because the island was large, he gave up. Devise a plan to help José find the treasure.

While working on the problem: On a separate sheet of paper, write all the things you have used in the solution of this problem whether you know why they are true or not. Also, write all the terms you used.

All students approached the problem in a similar manner. Pictures were drawn, and on occasion several. Those who tried to keep the positions of the trees the same noticed that it seemed that the position of the gallows was not terribly important. We then used GeoGebra (to be found on the [|www.geogebra.com]) to represent the problem. We constructed three points: A, B, and C to represent the gallows, banana tree and coconut tree, respectively. The instructions demanded we walk from the gallows to the banana tree, so we constructed a **segment** between points A and B. We are to, then, turn **90°** to the right, and walk the same number of steps. To accomplish this we constructed a **perpendicular line** to the segment AB through point B. In order to find Stake_1 we constructed the **circle** with **center** B and radius a (which was the **distance** between points A and B). After some discussion, we settled on the definition:

//Definition// //1//: A circle is the set of all **coplanar points** that are **equally distant** from a given point called center. It was noted that if we did not specify that the points were coplanar, then the same definition would in fact in **3 dimensions** describe a **sphere**. It should be, however, noted that discussion contained questions of whether it would be necessary to include descriptors such as **continuous,** and **infinite**. The popular opinion was that such inclusion would be unnecessary due to the word ALL in the definition. The words that entered the vocabulary were: **line, function, measure, location, size.** The last three were an attempt to define the word **point**, and have lead to the conclusion that this a word that should stay ** undefined .** We found Stake_1 by finding the intersection of the constructed circle and the perpendicular though B to AB. In order to justify the existence of the intersection point, the following statement was made: //Statement 2:// Given a line l, ha point B on l, and a positive number a, there is a point C on line l that is **a away** from B. In a similar manner we constructed the Stake_2, and used the midpoint tool to find the treasure. By manipulating the gallows (try below)
 * Unanswered question 1: ** If a term is undefined, how can we talk about it and its properties? Does not "undefined" mean that we do not know what it is?

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we saw that the position of the TREASURE does not change. The conjecture some people made seem to be confirmed, but an important question remains:
 * Unanswered question 2: ** How will you **PROVE** that the position of the gallows does not matter. Also, how will you find the treasure?

Our next topic included discussion of definitions of point, **line** (notable words: **beginning, end, infinite, direction, has two points, contains 180° angle, curved, arc, straight**), **geometry.** The scribe was under impression that there was a consensus that a line is straight, infinite, contains many points. Which then leads us to assignments for Wednesday:

Take a sheet of paper, bean can lined with white paper, and a ball of sorts (tennis, golf, something you can draw on). On each of those decide what a straight line is. Draw some examples. Think of how you can check that something is straight.
 * The experiment is to be performed for Wednesday. Write up, in a form of an essay, is due on Monday:
 * Fill out the survey on [|webct.utah.edu]
 * Read the syllabus which is both on webct and [|syllabus3100f09.pdf]
 * Bring your laptop with GeoGebra