##
* *Reflection: Intrinsic Motivation
Name that Conic - Section 2: Investigate: Conics on Calc

I was getting a lot of questions from my students as they progressed into the second part of this activity where they needed to make generalization about the general form of each conic section. Students overall seemed unmotivated to find the generalizations, and I guess I don’t blame them. Why would they want to discover a short-cut for identifying what type of shape it is when it’s really not too difficult to just graph the general form of the conic section and then identify the shape using the calculator as a tool?

Once I noticed most of my students had completed the front side and grouped the equations on the back, I stopped the class and quickly read through the correct answers… while looking only at a blank page. I emphasized to students that they should be able to identify the type of conic sections they are looking at in general form within seconds, not the minutes it took to graph in their calculator. Their goal now was to find what rules I was using to quickly identify these. I encouraged students to find a rule that worked for all the given conic sections and to check it on some made up equations. I also told students we would be playing a speed game next. I think that helped up the ante.

Here are some sample responses from my students:

Conics on Calc, Student Response 1 – very vague answers with incorrect use of vocabulary

Conics on Calc, Student Response 2 –identifies the generalizations, but also some generalizations that aren’t always true (Also, uses ‘variable’ incorrectly… reminds me that I need to get my students writing more in math!)

Conics on Calc, Student Response 3 – Correct generalizations just not specific enough on Ellipses… a common issue for many of my students.

Conics on Calc, Student Response 4 – Three out of four generalizations correct, just missed that Ellipse again. I like how the student used the general form of a conic section to help describe the generalizations.

If there had been more time in class, I think this would have been a great opportunity for some class discussions. Did everybody’s generalizations work? Were they specific enough? Too specific? Must they always be true?

*Intrinsic Motivation: Motivating Students to Find Generalizations*

# Name that Conic

Lesson 2 of 13

## Objective: SWBAT graph conic sections using technology and identify the type of conic section when in general form.

## Big Idea: Students develop their own system for quickly identifying a conic section in general form and then race to name that conic first.

*60 minutes*

#### Warm-up: General Form

*5 min*

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.

*expand content*

#### Investigate: Conics on Calc

*35 min*

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?

*expand content*

#### Game time! Name that Conic!

*20 min*

**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.

*expand content*

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- UNIT 1: Basic Functions and Equations
- UNIT 2: Polynomial Functions and Equations
- UNIT 3: Rational Functions and Equations
- UNIT 4: Exponential Functions and Equations
- UNIT 5: Logarithmic Functions and Equations
- UNIT 6: Conic Sections
- UNIT 7: Rotations and Cyclical Functions
- UNIT 8: Cyclical Patterns and Periodic Functions
- UNIT 9: Trigonometric Equations
- UNIT 10: Matrices
- UNIT 11: Review
- UNIT 12: Fundamentals of Trigonometry

- LESSON 1: Cutting Conics
- LESSON 2: Name that Conic
- LESSON 3: Human Conics: Circles and Ellipses
- LESSON 4: Circles and Completing the Square (Day 1 of 2)
- LESSON 5: Circles and Completing the Square (Day 2 of 2)
- LESSON 6: Ellipses
- LESSON 7: Human Conics: Parabolas
- LESSON 8: Parabolas
- LESSON 9: Parabola Problem Partner Critiques
- LESSON 10: Hyperbolas
- LESSON 11: Non-Linear Systems of Equations
- LESSON 12: Conic Sections Test Review
- LESSON 13: Conic Sections Unit Test