1. Students will derive the midpoint formula.
2. Students will derive the distance formula using the Pythagorean Theorem.
3. Students will apply the midpoint and distance formulas.
Activity Sheet (1 per group)
String (1 per group)
Rulers (1 per group)
Masking or Duct tape (1 roll per group)
Students will have worked with the Pythagorean Theorem.
The lesson will begin with a warm-up activity. Students will be asked to solve two problems in order to prepare for the following. In the first problem, students are given two points on a number line and asked to find the midpoint between those two points. Students are asked to come up with an "algorithm" for determining the midpoint between any two points on a number line. In the second problem, students are given the measures of the legs of a right triangle and asked to find the length of the hypotenuse. Students are asked to generalize this algorithm to find the distance between any two given points. The instructor will initialize a discussion once the students have had ample time to work on this activity. Students will be reminded to keep in mind their algorithms as they work on the following activities.
This activity requires some imagination on the part of the students and the teacher. The students are to imagine that the classroom is a square (or rectangular) island. They are going on an expedition (in their groups) to find the hidden treasure located on this island. The expedition group which gets closest to finding the site of the treasure wins. If you have GSP in your classroom, students could use it to make constructions. However, it is not necessary.
Island Treasure: Discovering the Midpoint and Distance Formulas
Imagine that our classroom is a square island (say The Island of Dr Moreau, or Blackbeard's Island, or Jurassic Park). A treasure map has recently washed ashore and been publicized. Everyone is trying to get to this island and find the treasure. You are commissioned by Bill Gates (as if he doesn't have enough money already) to go on an expedition to find the treasure. Use the map and directions below to find the treasure. But, do not share any secrets for there are other expeditions seeking the treasure as well.
1. Locate the center, C, of the island. Drive a stake at the center (ie. mark with tape). How did you find the center of the island?
2. Next find the distance between C and the Southeast corner of the island. How far is it? Find the center of this distance and drive a stake at this point, R.
3. Find the distance between C and the Southwest corner of the island. How far is it? Find the center of this distance and drive a stake at this point, S.
4. Find the distance between R and S. How far is it? Find the center of this distance and you have found the site of the treasure.
5. To avoid being robbed your expedition decides to not exhume the treasure. Instead you leave it behind and set up camp nearby until the armed rescue team can reach you to escort you to the ship. This rescue team is located at the Southwest corner of the island. Unlike you, they do not know anything about mathematics or the algorithms for finding the distance and midpoint between two points. You are to fax them a map showing how to find you. This map is to include the simplest directions possible. In your groups, complete the map and directions.