##
* *Reflection: Discourse and Questioning
Another Solution - Section 3: Wrap it Up

I sometimes just walk around and listen rather than collecting student work. In this lesson, I heard a student explaining to his partner:

**Student 1:** "Imaginary roots happen when an equation factors out to give answers that don't make sense, like when we got imaginary ohms for one problem."

**Student 2:** (His partner) "What about answers that aren't imaginary but still don't make sense like getting a negative profit?"

**Student 1:** "Some answers are like that, but most imaginary answers are not for real problems, while sometimes you can have just a negative answer, like when you owe money or when something happened back in time."

**Student 2:** "OK, so just because it doesn't work doesn't mean it's imaginary, but then how do we explain imaginary if Mrs. Rampy calls on us?"

**Student 1:** "We just say that it's any answer with an i in it!"

This dialogue is valuable to me because I can see that my students actually have some understanding that not all solutions to a problem are viable, but it also shows me that some are still not solid in what an imaginary number might mean in a real-world situation, so I need to continue to build that connection.

*Student Discussion*

*Discourse and Questioning: Student Discussion*

# Another Solution

Lesson 4 of 8

## Objective: SWBAT solve quadratic equations with real coefficients and complex solutions from real-world problems.

## Big Idea: Imaginary answers to real-world problems! How is that possible? Let your students explore that question for themselves.

*55 minutes*

#### Set the Stage

*10 min*

I begin today's class with the same equation on my board as yesterday, but this time I ask my students to use their graphing calculators to graph the functions to try to find factors. **(MP1)** Because this equation has imaginary roots, they will not be able to use zeros as they have for previous factoring-by-graphing activities. I ask for any suggestions about how the imaginary roots might relate to the graph and let my students discuss it for a few moments. Generally they come to the consensus that they can't graph imaginary roots on a real coordinate system. This sounds pretty simple, but I think it's important to let my students reach some of these conclusions independently rather than being told that it's true or reading it in their textbook.

I then ask if they think the equation could still have meaning like when we use a quadratic to represent the area of a rectangle. This also leads to interesting discussion and I again try to act primarily as a facilitator rather than the conductor. When my students have either reached a consensus or an impasse I tell them that today they get to work with some quadratic equations that have a real-world connection even though they have imaginary roots.

*note: lesson image courtesy of http://www.flickr.com/photos/hikingartist/4192576317/sizes/l/*

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#### Put it into Action

*40 min*

*You will need copies of the real world complex quadratics handout for this section of the lesson. *

**Teamwork ***25 minutes: *I tell my students that they will be working with their front partner to complete today's challenge. **(MP1) **I also advise them to prepare to present their work to the class for one of the problems, which I will randomly select when everyone is done. As they're working I walk around offering encouragement and assistance as needed. I anticipate that some students will struggle with writing appropriate equations for each problem. For them I try to ask questions like these that I might use for problem #1 "How is this problem like others you've seen? What does the starting height represent in a quadratic function/equation?" or for problem #3 "Since you have the basic equation, how can you rewrite it with just one variable?"

**Presentations ***15 minutes*: When all the teams have completed the problems I randomly select a team to present, then have them roll the die to see which problem they will present. I remind those students not currently presenting that I expect them to offer appropriate critique and questions to those presenting. **(MP3) **I continue selecting teams until everyone has presented.

My goal for this activity is to help my students become aware that they might get answers that are mathematically correct but that have no real meaning. I explain this a bit more in my video.

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#### Wrap it Up

*5 min*

To close this lesson I ask my students to explain to their partner in their own words (with appropriate math vocabulary) what they think imaginary roots represent. This gives them a chance to articulate what they've been working with and to really talk about something that is fairly abstract. I don't expect all of my students to really understand what imaginary numbers are, but I hope to give them at least a starting point for future math classes.

*expand content*

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- UNIT 1: First Week!
- UNIT 2: Algebraic Arithmetic
- UNIT 3: Algebraic Structure
- UNIT 4: Complex Numbers
- UNIT 5: Creating Algebraically
- UNIT 6: Algebraic Reasoning
- UNIT 7: Building Functions
- UNIT 8: Interpreting Functions
- UNIT 9: Intro to Trig
- UNIT 10: Trigonometric Functions
- UNIT 11: Statistics
- UNIT 12: Probability
- UNIT 13: Semester 2 Review
- UNIT 14: Games
- UNIT 15: Semester 1 Review