Today we are discovering the last conic section, the hyperbola. As with my other conics lessons the students begin by doing the Wax Paper Activity (Activity 4). As I move around the room I notice that some student put the second point inside the circle instead of outside the circle.
Once most the students have a good design, I ask if they see a design like they did on the previous wax paper activities. Some students will say a parabola because they only see one side of the design. I show a students work and show the students how there are two parts of this design. Students that have very wide hyperbolas will struggle seeing the 2 sides. Since I had mentioned all 4 conics the students realize that we are making a hyperbola.
Once students see the shape I begin questioning students about the construction:
When I moved around the room I have identified shapes that are opening wider and some that are narrow. The students who made these designs are asked to share. "Why these are shaped differently." (Wax Paper Hyperbola) I want students to see how the placement of the 2 points changes the shape of hyperbola.
Now that we have created hyperbolas, I share a definition (Larson, Precalculus with Limits, 2nd ed., p. 751). We read the definition as a class and then compare this definition to the definition of an ellipse. "How are these definitions alike and different?" I want students realize an ellipse has a sum of the distances to the focal points is a constant, while for a hyperbola difference of the two distances is a constant.
As part of my literacy instruction, students should use contextual clues from the definition to realize that the equation will be difference instead of a sum. When the students give their hypothesis I have them explain why they made this prediction.
After making the predictions I put the standard equation of a hyperbola on the board.(Larson, Precalculus with Limits, 2nd ed.,p. 751)"Does the equation look like you expected?" Students record this information on their conic sections reference sheet.
Now that the formula is on the students reference sheet, I want to see how the parameters and the key features are connected. I begin by looking back at the definition. I use the hyperbola diagrams from Larson, "Precalculus with Limits, 2nd ed., p.751. As shown on the diagram we discuss how the definition says that the difference in the distances is a constant. I label the constant by the parameter "c". I then mark a different point on the graph and label the distance to each foci. We write the difference equation and I then ask "What does this difference equivalent to?" Some students say c while others will say the first difference. "If I found another point and found the distance to foci from that point what would the difference be equivalent to?"
Students struggle with understanding that the differences are equal not the distances. Once students understand this we can discuss how this might be used to find the standard equation.
I now move to the parameters h,k,a,and c. "Notice how the standard equation says the center is at (h,k), where would the center be for a hyperbola?" Once students reason that it is half way between the foci I go to the second slide that has the key features labeled. We talk about the line that connects the 2 foci is called the transverse axis and that all key features are on this axis.
We also look at the 2 standard equations and compare the standard equation of an ellipse. "What do you notice about the terms of the hyperbola equation to the terms of the ellipse equation when we change the orientation?" Understanding that the parameter a squared is always under the positive term will help students determine the orientation. If students realize which variable, x or y, the parameter a squared is the denominator for will help students identify the orientation of the graph. This is similar to what happened with the ellipse.
We finally look at the equation that connects the parameters a, b and c. "How is this equation different than the ellipse equation?" Many students say this is the Pythagorean Theorem. I let the students know that the Pythagorean Theorem is used to prove this relationship but it is a long proof.
At this points students want to know what the parameter b represents for the hyperbola. I ask students to tell me what the parameter b represented for an ellipse. The parameter b for the hyperbola will work like the ellipse. It is the the distance perpendicular to the transverse axis. There is not a point but the parameter does help find the equation for the asymptotes.
I share the definition for the asymptotes of a hyperbola from the text. I draw a sketch to illustrate how the asymptotes help us to think about and recreate the shape of the hyperbola. I ask the students how the slopes of the 2 asymptotes are related. Seeing how they are opposites helps with writing the equations.
To help my students understand how to find the asymptotes, I begin by first reviewing the point slope form of an equation. I ask students to consider what we know about the slope of a line and I use their ideas to help them understand how to find the slope for the asymptotes. We write the equation by using the center for the point since both asymptotes go through the center. We then find the slope by using the square root of the constant under the y divided by the square root of the constant under the x. Of course the only difference in the two asymptotes by doing the problem this way is that one slope is positive and the other slope is negative. Generally, my students understand this process. By using the point slope form of an equation and the position of the parameters a^2 and b^2, my students can find the asymptotes without memorizing a formula.
As the class ends I again want students to connect what we are learning to a real world use of hyperbolas. I again share the website Occurence of a Conic. This site again has examples that students have seen of could see if they visited the site. Some students have seen the nuclear reactor that is in Missouri and others have seen the one in Kansas so this connects with the students.
We look at each example I ask students if they have seen hyperbolas in other places. We will look those up on the Internet so students see what the students are explaining.