SWBAT understand the relationship between the linear coefficient and the solutions of a quadratic equation. They will represent the relationship graphically and look for important features and trends.

Students will rely on the structure of the quadratic equation, as well as dynamic graphical representations, to understand complex roots more deeply.

10 minutes

I will begin class with a very brief recap. of the previous lesson. For this it will be helpful to have this GeoGebra applet up and running again*.* We'll be reminded of the overall effect that the value of *b* has on the parabola and its roots.

Next, I'll ask everyone to get out their handout from yesterday so that we can check the answers. I expect that everyone has completed the first page, so I'll call on students to read off their solutions one by one while I confirm them with the answer key. For now, I'll focus on the table values alone since these may be compared quickly and easily.

10 minutes

Now comes one of the hardest parts of the assignment: describing the motion of the roots verbally.

I'll begin by asking for a volunteer to read what he or she has written for the beneath the first number line. The rest of the class should listen carefully and offer a critique. Together, we'll come up with a good verbal description, and I'll be sure to write it on the board. This one will then serve as a model for the others.

Part of the purpose of this assignment is for students to learn the difference between an *overly vague* description ("the points move further away from each other") and an *overly detailed* description ("first, one point is 0.54 and the other is 7.46, then the one is 0.63 and the other is ..."). We're aiming for a nice balance that describes the motion qualitatively (faster, slower, closer, further, greater, less, etc.) but comprehesively. I've tried to give an example of what this might look like in my answer key. (More **MP 6**!)

Once this description is "perfected", I'd use the applet again and this time show the real roots. We should be able to see precisely the motion that has been described.

15 minutes

I expect that many students already began this during the previous lesson, but now I want everyone to turn their attention to the complex roots of the function. Working in small groups or individually, they need to carefully plot the roots in the complex plane, and they should see that they all appear to lie on the circumference of a circle!

Do your best to help them really see the beauty of this fact! Why on earth should they move in a circle?! Maybe it's just "round" but not really a circle ... Express disbelief, wonder, and awe and you'll carry them along with you. Once they're hooked, use this excitement to motivate them to *prove* that the path is circular.

To do this, they'll probably need some help getting started. For instance, you might ask, "Supposing it is a circle, where does the center appear to be?" When they answer that it's at the origin, ask, "Ok, so what would you have to prove in order to conclude that the roots always lie on a circle centered around the origin?" They should answer that they'd have to show that the roots are all the same distance from the center. To this, I respond, "Great! Now let's see if you can do it." (**MP 1 & 3**)

[I give a simple proof in my answer key, but for your information (or for a precocious student) I highly recommend this page at cut-the-knot.org.]

For the record, I don't expect all of my students to be able to come up with a proof that the path of the complex solution is circular. But I *do* expect all of them to *see* that it is circular and to ultimately gain an intuitive sense of *why*.

20 minutes

Now, to discuss and make sense of what we've seen, I'll have students come to the board to give their explanations. They should use the GeoGebra applet with the complex roots shown in order to make things more visually clear.

Please see this video for some details.