I plan to begin today's lesson with an equation for a conic section displayed on the board. I will encourage my students to work on the problem, letting them know that they can talk with each other as they work, so that they can make sure that they correctly interpret the equation.
At this point, I am asking students to determine which type of conic is represented by the equation. I feel it is important for my students to be able to identify the type of conic by inspecting the structure of the equation. There are many times in my calculus class that I ask the students to determine what a graph will look when they have an equation. Once the students determine the equation is for a circle the students need to identify the center and radius. Once that is done the students can sketch the graph.
I decided to use a circle in today's Bell Work because most students are able to reason how to graph the circle without my assistance. After this warm-up, I'll know who may need more assistance.
Once the students begin to master the task, I will ask a student to share a sketch with the class. Some students may ask how to accurately plot a circle. If they do, I will explain that I generally find four points and then make the circle. I assume the my picture will not be perfect. I point out how the directions say "a sketch". I want my students to understand that a sketch should look somewhat like a circle. I tell them that a sketch that accurately communicates the center and radius is often as much as is really needed. If an exact circle is required, then a tool needs to be used, like a compass or a computer application.
Throughout today's lesson we will work on how to sketch conic sections. The students have used key features such as the center, vertices and foci to write equations, but they also need to be able to visualize how the equation can be represented as a graph.
I begin the conversation by asking the students how we could graph the circle from the Bell Work using a graphing calculator (see resource page 1). The students know they will need to solve for y first, but some students have trouble just moving (x-3)^2 over as an entire term. I sometimes need to look at a simpler problem to help understand the algebra process. Another issue students have is they forget to consider the need to take both the positive and the negative square root of the number. A quick reminder is usually sufficient.
I now ask, "How can we put this into our calculators?" Some students will try and just put the positive equation into y=. When they graph they will only see a semi-circle. This serves as a further reminder of the need to consider both square roots. There is usually at least one student who recognizes the need to use Y1 and Y2 and graph the circle as separate functions.
Once both equations are entered, if the students use the standard calculator window, the graph still does not look like a circle. To make the graph look more circular I have the students go into the ZOOM menu and square the window. I discuss how the calculator screen is not square so there are not the same number of pixels between each point in the x and y directions in standard mode. The square mode has equal pixels between each point in both directions.
Another issue students will have with graphing the circle is that the graphs of the two equation do not meet (visually). If they are troubled by this, we will look at a table to verify that there are points that are not viewed on the graph. The graph not connection is a technology issue dealing with the pixels the calculator. If a student has a newer TI-84 or an Inspire calculator these issues may not exist.
Once these issues are resolved, I will ask the students what we should features we should identify when we make a sketch of a conic section (see page 2 of resource). I anticipate students will list the features we identified when we were writing equations. I then have the students look at their reference sheet and determine the key features of each conic before turning them loose on a new example.
I will give the students 2-3 minutes to work before I stop them and ask for the key features. After we agree on the features, I will give the students 1-2 minutes to place the points and sketch a graph. For an ellipse the students may not remember to find the minor axis. I will remind them to do this and explain how the minor axis enables a more accurate sketch.
The final example (page 2) is a hyperbola. I will most likely spend more time on this task than the firsts two. I expect to discuss how to use the asymptotes to generate a more accurate sketch.
I now give the students Graphing Conic Sections worksheet. This worksheet has eight problems. Four problems are in standard form and four are in general form. Some of the problems have numbers that require students to estimate the placement of the key points. Again we discuss we are sketching so we can estimate the points. The sketch should give us a good idea of the shape and position of the graph on the coordinate plane.
As the students work, I move around the room helping students. Some students need clarification on understanding the equations while others need help fixing algebra mistakes.
As we end the class I bring the class back together to ask the following questions.
These questions help me assess where my students are with graphing conics. I will be able to work with the students on ideas or other students clarify any confusions they have.