An Important Prelude: On one of the early days in my calculus class, I have groups imagine a roadtrip. It’s one of the very best things I do, because it problematizes the idea of how something can be going at a particular speed at a moment in time. Like a speed involves a rate of change of position in a time interval, but we don’t have a time interval at a moment in time. So saying something like “we were going at 58mph at 2:03pm” suddenly goes from a statement kids accept to a problematic statement. And at that moment, we’re ready for calculus.
Setting the Stage Now: In calculus, I had gotten my kids to take tons of derivatives, and then taught them about antiderivatives and how to take them. But of course the whole time they were doing antiderivatives, they were asking “but why are we doing this?” I got this question a few times, but I just said “you’ll see very soon… probably next week… I just want to get some of this algebraic thinking out of the way so we can focus on concepts.” 
The Problem: I looked back at what I’ve done in the past about how to draw the connection between the antiderivative and signed areas together, and had a bunch of pretty terrible methods. I didn’t like any of them. So I put out a call on twitter while brainstorming myself.  I got a ton of responses, three of which stood out to me. The first one is something I kinda did at the start of the year already (before we did any calculus), so I’m excited for when my kids see the connection…
The second came was from the appropriately named @calcdave:
What’s incredible is that when I tweeted out asking for help, I had already started brainstorming … and it was so amazing that we had such similar ideas that I had to take a photo of my scratch work to send David:
I also got a DM from Brett Parker (@parkermathed) with a screenshot of how he digs into the idea. And what was his idea?
Yup. Driving in a car. Which of course got me thinking of the start of the year when I basically transitioned to derivatives using a road trip.
What I did: Part I
So I told kids we were going to put a pin in anti-derivatives for a short while to go on a short detour. And I used Brett’s setup, but changed it to be a followup from the roadtrip we had talked about earlier this year. We read the setup together, and then I gave each group a few minutes to talk about part (a).
Surprisingly, this was not an easy question for kids. Many didn’t instantly think distance=(rate)(time). Additionally they didn’t know what assumption to make about the speed for the minute that passed between 7:10 and 7:11. I emphasized the approximate part of the question, and really told kids they would need to make an assumption.
When we debriefed, most kids suggested Alex was probably driving 69mph for the minute, so they did .
Some suggested he was maybe going 70mph for most of the minute, so they suggested . And with that, I suggested that maybe he was going 68mph for most of the minute… We did the calculations for all three and saw they were pretty similar.
(We also had discussed that in that minute, perhaps Alex started at 70mph, went to 100mph, and then slowed down to 68mph… We just didn’t know. So we were making and using an assumption, but one that is pretty reasonable.)
Then kids in their groups went to the next three parts. And each group was assigned an assumption: average speed, left hand speed, or right hand speed.
The only errors I saw kids make in part (b) was not taking the different time intervals into account. Since we did one example which had a time interval of hour, some groups were using that time interval for everything. But pretty rapidly most kids got there.
In our debrief, I wrote out the calculation kids did for the left hand assumption, and then asked students what I would have to change to do the calculation for the right hand assumption. (That was written in yellow.) This question was a key question to ensure kids understood the difference.
Then we talked why this was an approximation? (We had to make assumptions about the speed of the car at all the times inbetween the times we were given.) And then kids said that to get a more accurate distance for Alex, we’d need more data.
And that’s precisely what they got when they flipped the sheet over.
I assigned some groups the left hand assumption, and some groups the right hand assumption. I thought with so many data points, kids would be like “argh! I have to do all these calculations!” But no, there were no audible groans (I don’t think…) And they plotted the points and then did the calculations. I told kids to write out all their calculations instead of just saying how far Alex drove in the first minute, the second minute, the third minute, etc. No everyone listened. Shame. They lost out. Because when we debriefed, we saw:
Yup. The fact we could factor out the 1/60th is key. It made this all go so much faster. Only a few kids noticed that when doing the calculations. And then we compared that with the right hand assumption:
The same answer! Why? Kids saw that the sum inside the brakets (after factoring out the 1/60) would be the same because the starting speed and ending speed were both the same.
So we’re done, right? NO. We had two more moves to make.
First, we looked at the graph.
And I asked: “if we were doing the left hand assumption, what would our velocity graph look like?” And we concluded that for each 1 minute interval, Alex was driving at a constant rate. So it would look like this:
Second, I asked: Between 7:05 and 7:06, how much are we assuming Alex went? (Kids answered ) Here’s where I did some talking. I could probably have asked kids to think geometrically and had them come up with this, but I was running out of time.
I said: “where do we see 67 represented on this graph?” (Kids said the height of the velocity graph from 7:05 to 7:06.) I then said: “where do we see the on this graph?” (Kids said the 1 minute length.) So I drew a rectangle. Slowly. And then shaded it in. Slowly. And turned around. Slowly.
Yes. Audible gasps from many. I then said: “What about 7:04 to 7:05? What was our calculation for the distance Alex traveled?” And after a few more, we saw:
Kids saw. Heck yeah. They got that the approximate distance that Alex traveled was algebraically calculated in one way. And then they saw that number had a graphical representation. It was awesome.
I left by showing the original velocity points graphed. Reminded them of our left hand assumption, and that it was just an assumption, though a pretty good one. And then I drew the curve below. And dropped the microphonic device and left.
All in all, this was a pretty detailed blow-by-blow. And since I did it in two different classes, and things unfolded slightly differently in each, this is an amalgam of what happened. But it’s a pretty solid recap of the story I wanted to tell and how it was told. (And it’s a testament to the help of my (twitter) frands.) Most of the time, students were working. But a lot of great conversations happened as a whole class. It is a long post, but the question we worked on was only one page [editable version to download: 2018-04-30 A Road Trip Reprise] And it probably only took 20 minutes total from start to finish. A really exciting 20 minutes for me.
Stay tuned for how I used the idea that @calcdave and I both stumbled upon to make the connection between area and anti-derivatives. Right now kids have seen that there is a connection between area under a velocity curve and the distance someone has traveled. There is still no connection to anti-derivatives. That’s coming up.
 To be honest, I made the decision years ago to do tons of antiderivatives before introducing the integral. I wanted all the algebraic work done and solid before we introduced the concept of the integral. I didn’t want kids messing with the idea of signed area, new notation, and tough antiderivatives all at the same time. I still kind of think this is the right decision, but some doubts have crept in. I just really hate when I have to say “you’ll see why… promise…”
 Apparently I asked this on my blog ages ago in 2009.