# Interpreting Graphs

I decided my Algebra students need a break from factoring.

(Actually, I decided that I need a break, because if I have to grade anymore quizzes that continue to show no understanding of a concept that we have been working on since January, I might explode.)

Our school currently has a grant funded by NMSI, the National Math and Science Initiative. The goal is to increase our enrollment in AP classes, as well as our performance on AP exams. The part of this that applies to me, as a non-AP teacher, is that we have training and resources geared toward teachers of lower levels to help introduce concepts that will be vital in calculus and statistics. These activities are usually pretty interesting, and well-scaffolded, so I’ve enjoyed using the few that I’ve done with Algebra students. They also provide AP-style multiple choice and free-response questions that are scaled down for the appropriate level.

I found one activity on interpreting distance graphs and another on interpreting rate (speed) graphs and decided to give them a try. We started Distance Graphs two weeks ago. Because of the nature of the assignment, I’m requiring students to answer using complete sentences (like you would on the AP exam) and provide explanations for their answers (like you would on the AP exam).

The actual interpreting of the graphs went surprisingly well. I was impressed by the accuracy of their answers to the interpretation questions. They know how to read graphs and they can use a fair amount of that knowledge to make inferences about the situation. For example, no one tried to convince me that because the graph went up and back down again it meant the person was going up and down a hill. They also made the connection between slope and speed really well.

The explanations for the questions (explain how you know, or explain how you figured this out), however…didn’t go well. At some point, I said if anyone else tried to tell me “because the graph shows it”, I was going to strangle them. (Because they are teenagers, of course the immediate reaction to that was to spend the next 10 minutes ONLY saying “because the graph shows it, Miss!” and then laughing like my impending aneurysm is the most hilarious thing in the world. The main culprit only stopped when I asked the class, “So when I kick A out of class in a minute because he’s DRIVING ME CRAZY, will his distance from me be increasing or decreasing?”)

We completed the first assignment where we analyzed distance graphs, and they turned them in. I scored them out of 2 points for each question – 1 point for the correct answer and 1 for a good explanation. My class average was 54%, and the highest score was 83%. (I was unpleasantly surprised by the number of students who didn’t even get the questions right that I had written out on the board.)

“So guys, I graded your Distance Graphs worksheets yesterday. They’re pretty terrible. I mean this in the most loving way possible, but your explanations suck.”

They laughed.

I acknowledged that I know why their explanations were terrible – they’ve never really had to do this before. Most of them have never been asked to write explanations for anything in math class before. They nodded. I said this is why we’re working on it. I know you’re not good at it, so let’s learn how.

Tangent: I was once again reminded of how much my students appreciate it when I acknowledge these kinds of weaknesses openly and make it clear that I’m not trying to hold them to an unreasonable expectation. I’m saying that I knew you weren’t going to be good at this, and I knew it’s not because you’re just refusing, it’s because you haven’t learned it. Without that moment of honesty regarding their side of this situation, there’s no way I would have been able to get them to complete the next activity we did.

So I passed back their graded worksheets and we quickly skimmed through the problems so I could describe what kinds of explanations I had been looking for. Then I passed out a 10-question multiple choice “quiz” (another source from NMSI, so the problems are intended to be similar in style to AP multiple choice questions). I gave them a few minutes to go through and answer the questions and then we checked answers. I was clear that their score on the assignment was in NO way related to the number of questions they got correct at this moment.

After we reviewed the answers, I instructed the class to choose 5 of the 10 problems and write an explanation for the correct answer. I encouraged them to check with me before turning in their assignment so I could make sure their explanations were good enough.

I was impressed. Granted, not all of my students actually had me check their work before turning it in (probably not even half did), but the ones that did were surprisingly good. In most cases, I make suggestions to improve their explanations – make them more clear or more specific – but the foundation that was there was solid.

Yesterday, we started an activity on interpreting rate graphs (as opposed to distance graphs). We talked about how the interpretation of this graph is different, since the y-axis is speed not distance. I had to remind the students of this fact over and over again, which I expected because they are used to reading distance graphs and not rate graphs. Honestly, I think I expected to have to do it more than I did.

At the end of class yesterday, I told two of my classes this:

“You guys are doing really well with your explanations! I’m really impressed by how much improvement I’ve seen from that first assignment a couple of weeks ago. This makes me really happy for two reasons. First, I’m proud of you – I get to see you learning and improving, and that makes me happy for you. The second reason is actually a selfish reason – it makes me feel like a crappy teacher when you aren’t learning. That’s what’s been going on the last 13 weeks of school – you haven’t been showing me that you’ve been learning anything, and I’ve felt like a bad teacher because of it.”

Whew.

I’m looking forward to finishing this activity with two of my classes tomorrow, so I can tell them what I got to tell one class today – you are doing calculus! Using the slopes in a distance graph to find the speed, and using the rate and time in the speed graph to find the distance – this is what the AP Calculus class has been doing this year! Granted, they are using harder graphs and equations, but this is how it starts!

I love blowing their minds with this kind of information.

After we finish these, I need to find a good activity for them to do more independently and summatively. They might be doing a poster project. Just need to find a good problem set.