Using prior knowledge of logarithms and the irrational number, e, students will be able to learn the concept of natural logarithms by making predictions for the true strain of a material in which an object is compressed.
1. Warm-Up: Ask students to write down their thoughts about what it means and what happens when a person is "strained". How about when a material is strained? How would one go about measuring the strain of a person? How about the strain of a material? Student answers will be discussed. (duration: 6 minutes)
2. Instruction: Once a student mentions or makes at least a vague allusion to some change to dimension of material in their response (such as a description of "stretching"), the concept of engineering strain will be discussed, i..e, it is a ratio of the change in length to the original length. The actual logarithmic formula for true strain will not be discussed yet, however. (duration: 3 minutes)
a. Have students make a prediction for: what the strain of a material would be if it were stretched 16 inches beyond its original length of 8 inches? If it were stretched only 8 inches beyond this original length? 4 inches? Discuss answers. Students should be able to calculate the predictions, "2", "1" and "0.5", respectively. (duration: 5 minutes)
b. Have students make a prediction for: what the strain of a material would be compressed to one-half of its original length of 8 inches? Students should be able to see that "0.5" is not a reasonable response, because it was already used to describe a 50% growth in length. Ask students how they would measure it, then. Students should therefore be able to conclude that compression yields a negative value for strain. Once that concept has been established, students should be able to estimate the strain for this situation to be "-0.5". (duration: 5 minutes)
c. Have students make a prediction for the resulting length if the strain value was -1. Have them explain whether or not their predicted value is reasonable. If students predict a length of zero, as expected, they should be able to conclude there is a problem with the formula they have been using, since if the object now had a length of zero, it would now cease to exist. This leads into the gist of the lesson for natural logarithms, for which a real-world concept can be found in the formula for true strain. (duration: 5 minutes)
4. Instruction: Once students have concluded that the previous algorithm for calculating strain had limitations due to the presence of the asymptote, it will be briefly mentioned that the formula they had been using was called engineering strain, which is linear in nature. However, because material is continually being stressed or compressed, the ratio of its change in length compared to its previous length will constantly be changing, as well, suggesting the curve of a logarithmic function. At this point, the equation for true strain will be discussed, where the base for the logarithm happens to be the irrational number, e. (duration: 5 minutes)
5. Classwork/Homework: Students will calculate a variety of problems involving missing variables for true strain, original length, and resulting length. (duration: remainder of classtime)
C. Assessment/Closure/ Follow-Up
Students will be shown a picture of a stress-strain curve (similar to the one below, from the Structural Fire Engineering Design page at the University of Manchester in the United Kingdom) and given a brief introduction to the concept of engineering stress. They will then be asked to analyze the labeled points of the curve, and give their opinion as to what exactly is going on at each point.
In order to view the PowerPoint Slides that actually encompass the lesson, click on the "Lesson Plan" attachment below.