In my Precalc class we were studying average rates of change. As you do. Anyway, the formula looks something like .

This lesson was an eye-opener for me. I had mostly 10th and 11th graders in the class, and I casually drew a distance v. time graph on the board, and drew two dots on it. I was going to talk about how the slope of the line between the two points was equal to the average speed experienced by an object moving from one point to the other.

I admit that I was naive. I really thought I could say something like “the average speed experienced by an object moving from one point to another. Oh btw the ‘points’ are in space-time, not, like, space, ok? So anyway…”

It turned out, as you may have guessed, that my students were not very comfortable with the idea of a space-time graph. “Riley, why did you write an x on the y-axis?” and “What happened to the x-axis?” are representative of the actual quotations I wish I had written down at the time.

This was super-sad for me, because the only thing interesting about the average speed of anything is that it can be represented as the slope of a line through space-time. Yes, a cop might be able to prove you were speeding if there were two speed traps a known distance away from each other and the cops write down license plate numbers and times, right, but I try to avoid that problem: they can’t catch every type of speeding that way, and it seems like maybe, instead of writing down thousands of license plates and times, they should just use the *radar gun* next to their stop watch.

So, I became determined that my students would understand space-time. Maybe we wouldn’t get to factoring that tiny percentage of quadratic polynomials that are factorable, but then maybe I’d just let them use a calculator on their exams. Here’s what I did:

I marked up the sidewalk in 1-meter increments, from zero to 60 meters. I put my students in groups of five – a big enough group that they can support each other in yelling, so they can all yell at once with no one having to yell by themselves. I gave each group of students a stop watch, and I had one myself. Already, you’re thinking, “oh, this is going to be great!”

I said to the class, “BEHOLD SPACE, class!” and indicated the length of the sidewalk. I said to the class, “BEHOLD TIME, class!” and indicated the timers. I moved to the 5-meter mark and said, “BEHOLD, I am at ‘5 meters’ in space, class!” I looked at my stop watch exaggeratedly and said, “BEHOLD, I am at ’50 seconds’ in time, class!”

And then they understood space-time, and we went back inside.

Haha, no, not yet. Though yelling imperatives about understanding does usually work well for me, in this case the activity called for something a little more interactive. I actually said, “Group 1, please move as a group to the 15-meter mark, and be there at 120 seconds.” This was very explicit and gave them about 30 seconds to move 20 meters. “Don’t miss the time!” I called as they were in transit.

To Group 2, I said, “Group 2, please move as a group to the 18-meter mark, and be there at 140 seconds!” This gave them much less time to essentially do the exact same thing that Group 1 had been done.

At this point I have 2 groups on the line and 3 groups in a big clump near 0, still. I whip out my portable dry erase board, and draw the graph with the two points in space time that groups 1 and 2 had hit. I made sure to leave plenty of room for the next five minutes on the graph.

“Group 3!” I commanded. “Please move to the point 10 meters, 200 seconds! Please move as a group!” This was a slightly different format of instruction, but it was close enough to the earlier instructions that they got it.

“Group 4!” I’m at the top of my voice now, trying to invoke the spartan leader from “300.” “Please move as a group to 5 meters, 215 seconds!!” They have to run to get there, but it’s a short distance so that I won’t embarrass anyone who still doesn’t quite get it.

Now there are 4 groups on the line. There has just been running, and there is still giggling. I call for everyone’s attention and bring my voice back down so that they bring their voices back down. “Here are the two points that groups 3 and 4 just hit,” I said, graphing them carefully because I’m going to reference it later. “I have a special instruction for you, group five.” Everyone is paying attention now – this is suspense. “When I say go, please move to the point 7 meters, 260 seconds. Once you have hit that point, please say ‘here we are!’ This is a new part of the instruction.” They move to (7m, 260s), and say “here we are,” but softly. I say, “everyone, when I count to 3 please say ‘here we are’ as loud as you can! One, two, three!” and about half of them say “here we are.” I say, “one, two, three!” and more of them say it a little louder. “One, two, three!” and most of them say it really loudly.

Here’s the most memorable part of the activity. After the last loud “here we are!” I call on group 5 again. “Group 5, please move to 18 meters, 300 seconds,” making them hurry but not run. “When you get there please yell ‘here we are’ as loud as you can.” While they were in transit, I moved to group 4 and said, “Group 4, please move to 23 meters, 310 seconds, and when you get there please yell ‘here we are!’ as loud as you can!” Moving quickly down the line, and before group 4 was done, I addressed group 3, and said, “group 3, please move to *40 *meters, 310 seconds. *You don’t have much time!*” and they were off.

I kept moving down the line until each group had gotten a couple of instructions. My last instruction to each group was “please move to the classroom, 500 seconds. Please sit as a group.”

So I just took 20 minutes to run around on the sidewalk (getting out there and synchronizing the watches takes a while). Here are questions the students can answer fluently now:

- What is it like moving from (5m, 200s) to (20m, 210s)?
- Which is harder, moving from (10m, 150s) to (15m, 155s), or moving from (10m, 150s) to (20m, 155s)?

These questions are relevant to them because they’re still breathing hard from running. They’re interested because they still remember giggling when I asked the first group to run and they still remember scoffing when I asked them to move 20 meters in 5 seconds.

I do the teacher-at-the-front-of-the-room thing to practice graphing a couple of these points, and reading from the graphs. If I were omnipotent I would have had worksheets so that the groups could practice them themselves. But after we practice graphing points for a few minutes, we get to the whole point of the lesson.

There are two sets of two points on the board, one in red and one in blue. One is labeled group 1, and the other group 2. The kids now understand intrinsically what is being represented. I haven’t put any numbers on this particular graph, but I ask anyway: “which group had to run faster?” and it’s freaking obvious to them. I’m not saying they couldn’t have figured it out before we spent an entire class period on the setup, but they don’t have to figure this out. They see those points and they *know*. I say, innocently, “what’s the slope between these two points?” and we figure it out pretty quickly, having dredged up the slope formula the day before. I get in a couple of words about units, blah blah blah, and then I have them calculate the average speed for 10 different situations. We run up against the bell, so I let them go before we get to compare answers, but they got it. Everyone.

Everyone got it, and what’s interesting is that it’s not because it was useful or relevant to anything outside of class. They were immersed in it because it was fun, and because I made them move, and because they were involved in the exercise quite literally with their entire being. We talk about pseudocontext sometimes as if problems with genuine context can’t also be crap. Here’s a situation with no context – we created our own context – that was more valuable than a hundred interesting and relevant word problems.

This is fanstastic. I have kids out taking photographs of their world and the gradients they see. The one I loved the most was a series of photos of shoes, from flip flops to stilettos. I think when I get back after the Christmas break, I’ll take my classes outside and do your activity. They’ll love it.

Do they put the pictures of shoes on an axis? Have you seen that xkcd cartoon with all the fruits on an easy vs. delicious graph?

Let us know how the lesson goes if you do it! 😀

The latest xkcd cartoon: http://xkcd.com/833/ seems relevant to this post.

I wish I could have been there just to watch the coordination of that many groups 🙂 I wonder if I could talk any of our developmental math teachers into doing that…

Average Rates of Changes! These are incredibly important to understand the Fundamental Theorem of Calculus. I’m not going to delve into it now, but most kids think of a mean when you say average rates of change and many Calculus students have no idea what an average rate of change actually is. I like the speeding trap idea-but there is so much more that it is useful for. It helps you understand derivatives and integrals, and a quick google search will show you how much those matter!

Yes, average rate of change is an important concept. Taking calculus and physics at the same time changed the way I view myself and the world around me.

They’re still kind of boring! Applications are many steps beyond the skill, if you see what I mean. Unless, of course, you run up and down the sidewalk 😀

This is incredible! I love this idea, and I may have to use it in my own classroom.

I have to admit, I’m sometimes scared to introduce a position function like x(t), knowing that kids will reel at the use of “x” as a dependent variable. This comes up when I teach parametric equations in Precal, but perhaps it could be introduced as early as Algebra 2 or even Algebra 1. Algebra 1 students have no idea how important the concept of slope is, and I always feel like we could do more to emphasize it.

Also, in a simple way, you’ve demystified the space-time concept which seems like something reserved for advanced physics.

Parametric equations is a perfect place to talk about space-time, though. You can watch your calculator graph the equations x(t)=5t+4, y(t)=3 and watch how quickly it plots it, compared to x(t)=t+4, y(t)=3. I like fixing the y coordinate, just so they can wrap their head around what the parametric equations are doing. Then later we can do crazy stuff like projectile motion or even lissajous curves. The calculator does a nice job visualizing all this.

But going out to the sidewalk sounds like a lot of fun too. I’ll definitely have to try that. Thank you for the enlightening post!

I like the sidewalk especially since it’s so much more concrete (so to speak). I’d worry that parametric functions on a calculator wouldn’t isolate the issue – especially since “t” on a calculator can pass at an arbitrary rate. A different window setting will make x = 3t move slower than x = t, and then you’ve got a LOT of explaining to do 😉

Thanks for the comment! I hope you subscribe! 🙂

I do teach my kids about the TSTEP on the calculator. Regardless, if you graph both of the aforementioned equations simultaneously, you can see the relative difference regardless of the TSTEP. That is, plot:

x1(t)=5t+4

y1(t)=3

x2(t)=t+4

y2(t)=5 <–(so it's not on top of the other one)

Be sure to use "Simul" mode so they graph at the same time.

I'll do the same thing with projectile motion, too. This time, I'll fix x(t)=3 or some other constant, and let the projectile motion happen in y(t)…like y(t)=-16t^2+50t+2. Then we can watch it happen vertically, like it's supposed to be. Students sometimes get lulled into thinking that a parabola describes the shape of the ball's path in 2-space (which IS true, but it's often not what we're plotting) rather than in space-time.