I enter my Algebra II classroom two minutes before class, open my computer and plug it into the SmartBoard. By the time it powers up, most of my students have entered the room and are sitting down and chatting. I pull up the day’s SmartBoard and I get started. The day before was exhausting, and I was in a cranky mood then. (My Algebra II kids didn’t see this, because I gave them a test that day.) I tell my kids we all have bad days, but that when I was thinking “argh, bad day!” I started thinking of all the good that I have, and I thought of my wonderful Algebra II class. (Which they are.) So I wanted to let them know that. They liked hearing that. I liked saying that. It was a nice 30 seconds.

I then pointed to the SmartBoard

And we got started. I talked about how we’ve done so much algebraic manipulation and solving so far. Absolute value equations, exponent rules, radical equation, inequalities. And we’ve done some baby graphing (lines, crazy functions which we used our calculator to graph). But today, I said, was going to be a turning point in our course — and graphing would be the emphasis.

I introduce the discriminant, and we talk about where we’ve seen that (answer: in the quadratic formula). I tell them they will soon see the use of it. But first we should get familiar with it. [1] We calculate it for a few quadratics. And then I asked them “so what? what does this thing tell you?”

(Silence.)

I move on and say, “Okay, we just calculated a discriminant of -11 for a quadratic equation. Tell me something.”

I didn’t have them talk in partners, and when I got more silence, I highlighted the discriminant in the quadratic formula:

And then I asked “what is something mathematical you can tell me about the quadratic if the discriminant is -11?”

A few hands went up, and then I should have had them talk in partners. But I didn’t. I called on one, who said “there will be .” “What do you mean?” “The solutions will have imaginary numbers.” “Right!”

I then go on to explain it in more detail to those who still don’t see it. And then I explain how the two zeros are going to be complex (because they have a real part and an imaginary part). I see nods. I feel comfortable moving on.

I then ask “what happens if the discriminant is 10?”

I call on a random student whose hand is not raised, who answers “they will be real.” I ask for clarification, and they said “the solutions will be real.”

So I go to the next SmartBoard page and I start codifying our conclusions:

I’m hankering for someone to ask the obvious next question, and indeed, a student does. “What happens if the discriminant is 0?”

And we discuss, and realize there will only be one real solution. This gets added to the chart.

I then ask them to spend 15 seconds thinking about this — what they just learned. To see if it makes sense, or if they have any questions. Just some time.

I’m not surprised (in fact, I’m delighted) when a student asks: “Can you ever have a discriminant equal zero?”

I suddenly realized that for some of my kids, we’re now in the land of abstraction. There is this new thingamabob with a weird name, the *discriminant*, and the students don’t know what it’s for or why we use it. We’ve been talking about s, s, and s and even though we’ve done a few examples, it isn’t “there” for the kids yet.

I throw on the board. He nods approvingly. Then I ask what the solution or solutions are for that equation, and they find the one real solution. Which gets repeated twice when we factor.

I then give them 5 minutes to check themselves by asking them to do the following 3 problems:

I walk around. Two students are actually doing the quadratic formula. So I go up to the board and underline the things in blue — and ask “do you need the full force of the quadratic formula to answer THIS question?” (Secretly I grimace, because who the heck cares if they use the QF or use the discriminant to answer the question? But if I’m teaching something, I want my kids to practice it.)

When we all come back together, I project the answers

And I get called out (rightfully so!) on improper mathematical language (imaginary vs. complex). So I fix that. I’m feeling slightly guilty about asking the two students to use the discriminant instead of the QF to answer the question, because who cares!, and so I tell the class that the discriminant is just a short way to tell the number and nature of the solutions, but don’t worry if you forget it, because you can always pull out the big guns: the quadratic formula. Which will not only tell you the number and nature of the solutions, but also what the solutions are!

I have them make a new heading in their notes

And I have them work with a desk partner to solve three quadratic equations using any method they like (they only know factoring, the quadratic formula, and completing the square).

They get the right answers, for the most part. The ones who aren’t getting it right are having trouble using their calculator to enter in their quadratic formula result. I want to move on, because of time, so I tell them that we can go over calculator questions in the next class but I want them to put those aside so we can see the bigger picture now.

They then are asked to graph the following three equations on a standard window:

We also talk about the difference between the two things they are working with:

We then look at the graph:

At this point, I haven’t pointed out the *x-*intercepts, but I asked students to see if they can relate the two questions. I grow a bit impatient, and I point to one *x*-intercept, and then the other, for the equation. Hm. They start to see it. We then look at the equation. Finally, the equation. Which doesn’t have any x-intercepts. Which confused some.

Someone said “that’s because the solutions are complex.” I pointed and said something like “yeah!” and then tried to explain. Some students got confused, because we did plot complex numbers on a complex plane, so they were like “you can plot complex solutions too!” I tried to address their concerns, by saying that the *x*-intercepts show us the solution when . But the *x*-axis is a REAL number line, not a complex number line. I don’t think they *all* got what I was saying.

We codify what we know:

This all took about 30 or 35 minutes.

I somehow totally forgot to do something key: bring the discussion back to discriminants. I didn’t ask them “so what can a discriminant tell you about a graph of a quadratic?” It might be obvious to us, but I guarantee you that only a few kids would actually be able to answer that after our lesson.

We spend the remaining 15 or 20 minutes on graphing quadratics of the form and by hand. The students were working in pairs. Then at the end we make some observations as a class.

Class ended, and then I had more work to do.

The point of this post is two fold:

1) I’m in a teacher funk. You can see it in this class. I didn’t work backwards. I gave them what they needed to know (the discriminant), and then motivated it second. I did lots of teacher centered things. I rarely let them discuss things with each other. Blah. Especially for something conceptual, not good. Not terrible, not good.

2) Teaching is exhausting. Anyone who teaches knows that even in a non-interesting lesson like this, a teacher has to constantly be thinking “what do my kids get?” and “do I need to say that again and reword it?” and “do I address the calculator issue of 2 kids when 15 seem to be okay?” Basically, every 10 seconds is a choice that needs to be made, a thought about how to adapt, where to go, what to do.

[1] Honestly, personally, I think the whole idea of the discriminant is stupid and I would have no problem doing away with it. It’s a term with very little meaning and almost no use. But I am asked to teach it, so I do.