Matthy Matthew: Good morning. How's it going?
Struggling Stephen: Not so well. I came in to talk to you about last week's exam.
MM: Okay, let's take a look.
Student pulls out exam, revealing a score of 36% F. MM flips through to look at SS's mistakes.
SS: I really struggled with the graphing problem. Can we go over it?"
MM: Sure. The inequality is x - 3y > 6. We need to try and get the y alone, so we subtract x from both sides. What are we left with on the left?
SS: Three y.
MM: No, we have negative three y.
SS: Why is it negative?
MM: In the original problem, it was x minus three y. The coefficient on the y is negative three. That term is negative.
MM: So we're left with -3y > 6 - x. What do we do next? How can we get the y alone?
SS: We subtract three. No, we subtract negative three. No, we add three.
MM: No. The y is being multiplied by negative three. The opposite of that is dividing by negative three, so if we want to get the y alone, we need to divide by negative three.
MM: So we're left with y on the left hand side. What do we get when we divide six by negative three?
MM: No, we get negative two. A positive divided by a negative is a negative.
MM: So we're left with y < -2 + (1/2) x. We have to flip the sign because we divided by a negative.
MM: So, we start our graph by plotting the y-intercept. What's the y-intercept?
SS: One half?
MM: No, it's negative two. The y-intercept is the constant term: the one without the x.
SS: Oh. Where do we put it?
MM rolls eyes as he loses patience, since graphed at least fifteen examples in class using the y-intercept.
MM: We look on the y-axis and go down two units to negative two.
MM: Now, what's the slope?
SS: One half?
MM gets just a little too excited, considering the student had already made four mistakes on the problem. Standards have clearly been lowered.
MM: So we go up one unit and to the right two units, and that gives us another point. And we keep going until we have a nice line.
MM: Now, we need to shade one side of the line. Do we shave above or below?
MM: No, we need to shade below here. Since y is less than negative two plus one half x, we shade below. Because we want smaller values of y, and smaller values of y are on the bottom of the plane.
SS: I have a question.
SS draws a "greater than" sign and a "less than" sign. Student points to "greater than" sign.
SS: This sign means "less than," right?
MM sighs. We spent four days on inequalities, looking at dozens and dozens of them, and still this kid doesn't know which is which.
FAST FORWARD FIVE MINUTES
MM helps SS solve an absolute value equation.
SS: I don't think we actually covered these in class.
MM: Yes, we did. We spent a day on them.
SS: Well, they weren't in the study guide.
MM pulls out a copy of the study guide and leafs through it.
MM: Here. There are actually three problems in the study guide addressing these problems. I believe we went over one of them during the review session.
I understand that some students come in with weak backgrounds, and many struggle with the material. But these are some of the most basic things we've covered over and over, and it's exhausting when students totally flub the first month and then think that they can just fill in a small gap here and there and be prepared. And then they go and accuse me of testing them on something we didn't cover in class.