Concept Attainment

by Kristi Dolan

Title of Lesson:  The Triangle Inequality Theorem

Lesson Objective:  The object of this lesson was to have the students come up with what the Triangle Inequality Theorem said through the use of the Concept Attainment Model of Instruction.

Brief Summary of Lesson:  The Triangle Inequality Theorem states:  "The sum of the lengths of any two sides of a triangle is greater than the length of the third side." In other words, in order for three sticks of various lengths to possibly form a triangle, you should be able to add any two of the lengths together, and that sum must be greater than the remaining length. Obviously, if the sum of the two shortest lengths is greater than the longest length, then you do not need to test any other possibilities.

I tried this method in three different classes. I am using my second class for my summary. I first wrote the words "Triangle Inequality Theorem" on the board. I then told the students that I was going to show them examples in which this theorem had been applied, and a triangle existed. I would also give them examples in which the theorem had been applied, and a triangle did not exist. Their job was to figure out what the theorem actually said. I started out with a positive example. I had the students tell anything they observed about the set of numbers, and I listed their observations on the board. The first positive example was "3, 4, 5." Some observations were "they go up by one;" "three squared plus four squared equals five squared;" and "two of them are odd numbers." I then added another positive example:  "2, 7, 8," and all of the original observations were crossed out. The students were stumped and asked for an example where a triangle did not exist. I gave them "3, 4, 7," as a negative example. They were still stuck and asked for another negative example. I wrote "1, 4, 4." Someone made the observation that "3+4=7 and 1+3=4."  They were on the right track! I gave another positive example, "1,3,3," and another negative example, "18, 21, 45." The students started adding up the two smallest numbers from the positive side, and I wrote the sums beside the examples. All was quiet for a couple of minutes, and I heard someone say, "Well, all of those (pointing to the positive examples column) are larger than the third number." I wrote this in the Observations column, and everyone began to test that idea with both the positive and negative examples. Someone else said, "Hey! The totals of those (pointing to the negative examples column) are equal to or less than the third length!" I then gave another positive and another negative example so that they could test their theory again, and then asked for someone to tell me what they thought the theorem said. A student said, "If the two smaller sides added together is greater than the third side, then it is a triangle, and if the total is equal to or less than the third side, then it cannot be a triangle. Voila! We then opened our books and read the theorem out loud.

Once the concept had been grasped, I asked the students, "What if you were only given two lengths. Could you figure out what the third length could possibly be? Would the third side have a range of possible solutions?" The students immediately figured out the range of possible solutions when I gave them "4 and 12" as an example (the answer is 8<x<16). I ended the lesson with the students applying the theorem to a few examples and then assigned some practice for homework.

Evaluation of Lesson Effectiveness and Suggestions for Improvement:

I felt that this method was very effective in teaching this concept. I really wanted to use it when I taught about parallelograms and/or polygons, but we have not reached that part in the book yet. I honestly did not think this method would work very well with this topic, but I was pleasantly surprised with the results. Our principal observed me during my class too. He loved it! He said it was refreshing to see something other than writing notes on the board and lecturing. He thought it was effective, also, and said they would probably never forget that concept! Let's hope he's right!