The Ellipse (Day 1 of 3)
Lesson 5 of 13
Objective: SWBAT explain the definition of an ellipse.
The students are ready to do activity 3 of the Wax Paper activity series. After completing two of these activities earlier in the unit, the students should be able to work this activity quicker than the circle and parabola activities.
After about five minutes I plan to start sharing the students' designs (wax paper ellipse). Some of the designs will be very circular while others will be more oblong. I ask the students why the pictures are different. The students should realize that the further the 2 points are from each other the more oblong the ellipse will be.
I continue displaying the designs while asking, "Why did the directions in the activity not make a circle? What is different with what we did with the circle?"
Definitions of an ellipse
Once the students have generated some initial ideas about the difference between an ellipse and a circle, we will work towards formalizing this explanation. I plan to share the definition of an ellipse from our course text (p. 742, Larson, Precalculus with Limits, 2nd ed.)
Then, to help the students better understand this definition, we will make a human ellipse. We accomplish this using a piece of string that is a few meters long. I pick two students to represent the foci of the ellipse. These students hold the ends of the string. The remaining students take turns grabbing a point on the string somewhere between the 2 ends. Each moves until they find a point where the string is taut. Then, the student stays at that point. Another student now takes hold of the string at a different point and follows the same procedure. We continue until each student takes a place on the ellipse. (The Making a Human Ellipse reflection includes a video demonstration of this activity.)
In my class, the Human Ellipse helps students who struggle with the idea that the sum of the distances to the focal points is constant. After using this activity to help students gain a concrete understanding of the definition, I share an abstract representation of what we just constructed (see labeling the figure) and define key features and parameters of an ellipse and the equation for this conic section. I use these models to show students how the sum of the distances in the definition is actually the length of the major axis (see value of ellipse constant). This is an interesting idea, because it helps explain why c^2=a^2- b^2.
I now share the standard equation of an ellipse. The students write the standard equations on their reference sheet. I explain that we can prove the formula, similar to the method for the parabola but we will not do that today. The process requires a lot of substitutions, so I do not want to use class time on this proof. Even though I did not prove the standard equation, I do like to show students how the formula for c^2 is found (see length relationship for ellipse).
Before closing the conversation, I will discuss with students how to determine and explain the relative values of the parameter a, b and c in the Standard Form Equation for an Ellipse. This discussion includes the use of the parameter a to determine if the ellipse's major axis is vertical or horizontal.
Teacher Note: At this time I do not discuss the eccentricity of the ellipse. I have found that students get confused when it is discussed at this stage of their work with conics. I wait until we convert the conic equations to polar form before covering this topic. I find that the wait helps to make the application of the concept more relevant to my students.
As we end the lesson, I once again display a labeled diagram of an ellipse. I ask the students, "What would need to change or what would need to be true if the ellipse was actually a circle?" I want my students to realize the major and minor axis would be equal in length, and, the two foci would be coincident at the center of the circle. After discussing the changes I explain that the circle is sometimes considered a special case of an ellipse.