Give every student the handout Student worksheet - Conics on Calc and have students read through the first page and begin working on putting all of the equations in general form. Point out to students that the general form of a conic is written across the top of their handout. They need to make their equations look like this. No matter what calculator students are using they will need to first put all of the equations in general form. Students can work independently here. It is important that each student is comfortable converting conic sections to their general form. My students sailed through this part. A few students did ask if the terms needed to be in that exact order or if the whole equation just needed to be set equal to zero.
Next, students should begin to organize these general form equations into the table on the following page by categorizing them by their graph. Students will need to use a graphing calculator with an APP for conic sections to make this go smoothly. The purpose of this activity is not to see if students can solve these complex equations for y and graph using a normal graphing utility, but instead just to identify the shape of the conic section and eventually make generalizations about these general forms. So again, it is important that students have access to a conic sections graphing APP. We use the TI Nspire calculators which already have the APP built in. However, if you have any of the older TI calculators (TI-83, TI-84, TI-84+) you will need to install this APP on the calculator ahead of time which can be found on the Texas Instruments website.
Instructional Note: If students are working with a TI-84 or TI-84 plus graphing calculator the directions on how to access the conic sections APP are on the front side of the handout.
If your students are working with the TI-Nspire Calculators the APP is already built in to the calculator's graphing function. Watch the video below for a demonstration on how to find this. Note, that students won't know how to identify the conic section yet so they should be graphing these in general form using the 'Conic' option which can be found by opening a graphing page, then selecting menu, 3: Graph Entry/Edit, 2: Equation, 6: Conic. I choose to model the first problem with my students. It was helpful to talk about inputting a zero for the coefficient of the xy term since that term did not exist and you cannot delete it in the general form the calculator displays.
I predict that my students are going to try to take a short-cut here and just write the number of the problems on the table. However, it is important that students actually write the general form of the equation under each category. At the bottom of this worksheet students will be asked to summarize their findings by making generalizations about the general form of conic equations. So of course, this is posing another great opportunity for students to demonstrate Mathematical Practice 8: Look for and express regularity in repeated reasoning. Are students able to identify what all parabolas have in common in general form? What is the difference in the general form of a circle versus an ellipse? Difference between an ellipse and a hyperbola?
Preparation: This game needs to be prepped before students arrive. First, make one copy of pages 2 and 3 of the Name that Conic student answer sheets on cardstock. You will also need enough copies of the answer doc (page 1) for each student. Cut apart the Name that Conic cards and divide into 8 groups insuring that they get mixed up so that a variety of shapes are in each group. Label 8 envelopes or folders with the letters A-H. Then label 5 cards in each envelope with the letter and the number (ie. A1, A2, A3, …, B1, B2, B3, …). Place these 8 envelopes around the classroom. I plan to place some on tables of desks, my side counter, and I may just even tape some cards to the wall or magnet to the board. Just spread them out for crowd control.
Narrative: With about 20 minutes remaining of class (presuming all students have worked through the Conics on Calc activity), I am going to pass out the student handout Conics in General Form. This just summarizes the information that students hopefully have already discovered in the Conics on Calc activity. Next, students will be asked to find the 8 envelopes around the room and to use their findings to name the conic sections. I am going to probably set this up to be a race. The first few students to me with all correct answers will get a treat or homework quiz bonus.
This activity could also be done as a pass-the-envelope type activity if you don’t want children running around like crazy people. I Give each table of students an envelope and then set a timer for when they have to pass the envelope to the next table. The point of this activity is that students should be able to identify the conic quickly so just be sure they feel the pressure of time in some way.
Assign Homework 1 - Conic Sections from this unit for homework tonight.