This is a classic–five years ago, at my past school, Drew & I sat around (with a few other colleagues, I don’t recall which ones) at our monthly meeting and talked about graphing, coming up soon in our Algebra 1 classes.
“They just can’t calculate slope from a table.”
“Well, my kids don’t seem to connect what the numbers mean to the rate of change. They don’t get what it means.”
“I wish they could just explain it in a sentence.”
And so on. We had some common concerns, but other things were more specific/personal pet peeves. We focused in on our students understanding of slope as a rate of change and some of the things we wanted them to be able to do:
 Calculate slope from a table
 Assign units to slope
 Describe the rate of change in a sentence
In retrospect, this lesson was probably also the start of my obsession with word problems and understanding what a problem actually means.
We decided that we wanted our students to be able to put together those pieces and then scaffolded the problems. There are three sets of cards, and we did each set on a different color (so we could easily see where each group was). To make sure students were finding what we were asking for, each group got an answer blank. With the students in groups of 4, we handed out the answer blank and then the first set of cards. Each student was told to take 1 (they are numbered 14) so that they could take a little more ownership–this would probably be a great place for each group member to have a role, but I’m not that good.
1. Ms. Cutter lives 75 blocks from Solorio and bikes to school.
Use the table to find her biking speed and
time required to ride to school.
minutes

blocks

0

0

1

5

2

10

3

15

4

20

5

25

6

30

7

35

Here’s the first problem. Stop it. Now.
I don’t care what the answer is, and I didn’t ask you for it. (This is actually worse with teachers than with students–teachers have no interest in following my directions because they don’t want to think about rate of change this way either, or at least not the ones I’ve presented to!) We wrote it like this because what the problem is asking shouldn’t always change our approach–jumping straight to an answer can take away understanding what you’re actually doing.
We start with determining which variable is independent/dependent and x/y, and then label the top and bottom of the fraction–units only, no numbers. Then we find the rate of change from the table (yes, my bike is jetpropelled–it makes a funny joke so I’ve never changed it.) Set 1 all has an intercept of 0, so these aren’t too tough. As soon as most of the class has the rate, fill that in on your fraction as well. The last thing we need to do is state the rate of change as a sentence. Nothing fancy, “She bikes five blocks in one minute” “Every minute she goes five blocks” or whatever. But say what it means.
Finish up Set 1, and start on Set 2.
5. Ms. Cutter bikes to Marquette Park to see the
tennis team and then bikes home.
Use the table to model her trip.
minutes after tennis

total blocks traveled

0

14

1

19

2

24

3

29

4

34

5

39

6

44

7

49

Same as the first, but now we’ve got an intercept. We’re still focused on the rate of change, so we start with our units again, but then discuss (first in groups then as a class) what has changed and how to handle it. You can use starting point or intercept here, but we want to get across that we can’t unride or uncarry or unpaint. It’s happened already, so it won’t be affected by anything else. The only change here is to fill in that starting point as well, although it doesn’t need to carry over to the sentence.
It’s getting a little bit harder now, and if you have a single period class, you’re also out of time now. Hand out some tables and have students find the rate of change and state it in a sentence for homework to reinforce what you’ve done today. On to Set 3–the students are realizing this was not nearly as easy as we thought but are also starting to understand what the rate is, how to find it and what it means.
The only new part of Set 3 is writing a function for the situation, which you can also skip and come back to later (I would still do the rest of the card).
I think this is a good framework. But I haven’t had the opportunity to discuss it with anyone in the five years since we wrote it, and I haven’t revised it either. I’d love to hear what changes you would make or how you would tweak it to make it even better. I would appreciate any feedback or suggestions you have in the comments. And if anyone is interested in the follow up lesson after this, I’d be happy to post about that later.
I’m going to be out of the country for a week, so I’m not ignoring your comments if I haven’t replied. I really would love to hear what people would do with this lesson.