We started definite and indefinite integrals in the Business Calculus course I teach this week. Most of this week was spent on indefinite integrals, but now that we have covered some rudimentary types (polynomials, exponential functions, etc) the curriculum calls for Riemann sums and definite integrals.
I started by throwing this up on the projector. Then I asked the students to pair up and estimate the area under the curve.
To set up the task yesterday, I had asked any students who are in the habit of reading ahead (I suspect none of them are) to refrain from reading the next section and anyone who had taken calculus before not to give anything away the next day. Today, after a few moments of awkward silence, I was pleased with how “loud” the class was as the students discussed the problem. Earlier we had done an activity where they attempted to evaluate indefinite integrals and I walked around to answer questions, but this time I stayed to the side and let them talk. I didn’t offer suggestions or hints, I just listened.
Some things I heard:
- “Well it has to be more than a half.”
- “How did you get that?” “Watch me now, I did math!”
- “You got .65? We got .62222.” (Another student went up and wrote “.62223” like they were on the Price-Is-Right, which I found very funny.)
- “Okay so there are 12 full squares and then…”
- “How long is that? .2?”
I know that isn’t a long list, but those are the things that stuck out to me at the time and what I remember now a few hours later.
I realized in retrospect that I should have hidden the algebraic part of desmos, as showing the equation threw at least one student off. However, most understood from my statement of the task that they were just to think about the area and not try to manipulate equations. Another nice thing that I didn’t think about when I designed the task but I realized once the students began is that the grid nature of the Desmos display provided some nice scaffolding for the students as they thought about how to estimate the area. I briefly considered hiding the grid but decided against it. It was allowing them to be more precise with their estimate and the squares will lead nicely into the rectangles for the Riemann sum tomorrow.
I wrote “Estimates” on the chalkboard and told the class to write their numbers up there once they finished. I realized right after I erased it at the end of class that I should have taken a picture for this post, but apparently the theme of today was, “Taylor will realize all the things he should have done with this lesson right after not doing them when it is too late.”
More evidence of this theme in a moment.
There were about eight or nine estimates written on the board, which is less than the amount of pairs that I knew I had. Oh well. I don’t like forcing college students to do things. When I am older and even more curmudgeonly I don’t know if that will mean that reluctance will get worse or that it will lessen because I won’t care if they get grumpy about it.
There were a few estimates that caught my eye. I attempted to scaffold the thought process for the class in the estimates that I chose and asked students to explain. The ones I thought important were:
- 1 because the area sits inside a square of area 1 and its a decent upper bound
- .62 because I heard the student who had that estimate explain his reasoning and he was very clear
- 2.45 because it was obviously too big, but still useful for developing sense of the problem
- .67 for two reasons: the first was that was the group who said “It is bigger than a half” and secondly because I had a hunch they guess .67 and got lucky since that is essentially the area
To continue the day’s theme, I also should have pointed out the guess “0.005” since it was clearly too small, but again I did not think about it until after the lesson was over.
In my mind, a good order for asking for explanations from students for these would have been:
- 0.005, a guess we know is too small
- 2.45, a guess we know is too big
- 1, since this is a good start for a reasonable bound
- .67, a guess from the “It’s bigger than a half” group (I was also right that they guessed. They thought it was funny they guessed correctly when I told them.)
- .62, because of the quality of the explanation that I overheard
Of course, this is not the order I went in. I jumped around a little at the time. I should have anticipated the types of guesses before the class and been looking for numbers of a similar type to these. Next semester, I suppose. But that is one reason why I blog. To remember what I learned teaching the lesson.
I had the student who noticed it was bigger than a half repeat his observation for the class and asked if everyone saw his reasoning.
No one would fess up to the guess of 2.45 even after I said our class was like Planet Fitness, a judgement-free zone.
Then I let the student who had come up with .62 go through his explanation of how he arrived at his number even though it was a little low. Essentially his argument went as so: “Okay so there are 12 boxes with area 0.04,”–he pointed them out on the screen and counted them –“so we have 0.04*12. And then you add and kinda guesstimate the remaining area and that’s how we came up with .62.”
The only clarifying question I asked was why the boxes had area 0.04, which he also explained. Overall I was very pleased not only with the explanations of the students I put on the spot but also the general conversation that happened. I talked breifly about how tomorrow we would learn a method for finding areas under curves using rectangles and then I dismissed.
Tomorrow when teach Riemann sums it will be a little more lecture/procedural as I introduce the idea and method, but I plan to have them work together to create the sums. Looking forward to it.