Bugs on your windshield – An introduction to definite integrals

Considering how tired I was this morning on the first day back to school, I could only imagine how the students might be feeling. Today was the first day of our definite integrals unit in Calculus, and I decided to start off class today nice and easy with the following question:

Suppose I pay you to clean the classroom according to the following plan. I’ll give you $400 for the
first hour, $200 for the second, $100 for the third, and so on. If it takes you 6 hours to clean the room,
how much do you make?

They joked about the silliness of the plan and what they would do given this opportunity. Then they got down figuring out the solution. They were a bit rusty and many assumed there was something complicated going on, so some started recalling geometric series and writing functions involving 2^x. These students quickly gave in to peer pressure and just calculated the total directly. It’s always interesting to see how more experienced students decide not to take the simplest route (though in high school, I think I was one of them.)

The other warm-up question I gave for the day was the following:

The images below represent the windshield of the bus during one of Mr. Weinberg’s trips in New Zealand.
The windshield initially had no bugs on it.

The students were a bit annoyed at having to do this, but they got a much needed review of approximating derivatives. Most students used a central difference, with only a couple using just a forward or backward difference. The fact that they did both was really useful during discussions later on about using left, right, and midpoint calculations for integrals.

As tends to occur with my students, especially at this point in the year when they know most of my ideas don’t come out of nowhere, they demanded to see some of the pictures I took. I was, of course, happy to oblige:

I was able to show them a few more actual scenic pictures, which kept things light as they needed to be before diving into the tedium of calculating areas under curves manually.

The rest of the lesson went great and was essentially unchanged from last year, with the exception of using the following data table instead of a table of velocity vs. time:

Originally I was going to start the lesson with this, but added the second warm-up activity when I thought it might seem a bit too contrived to just throw a table like this at them without any feasible way of actually generating it. I also gave the warning that though the values in this table was made up (though some thought that it seemed completely in character for me to actually take the pictures every hour for the purposes of Calculus), it would be possible to generate such a table using the procedure they followed in the warm-up question.

We talked about how we might estimate the total number of bugs during each two hour interval if we knew the rate and assumed that rate was constant. The left hand and right hand sums came straight out of this. A couple students immediately thought about averaging together the two rates to do midpoint, and later on that led very nicely to a visual discovery of the trapezoidal rule. When we looked at what this process then meant graphically, most students seemed to find the overall concept pretty simple.

The mechanics of doing a left/right/midpoint sum with a function initially appeared more complicated, but having them set up the calculation using a table to organize the values (as with the smash rate table) made a big difference.

Overall, I think the students last year got along fine with the more traditional introduction finding displacement from a table of velocity vs. time data. They got the concept fine, as did the students this year when I showed them how it was really the same as what we did. I think it made a difference to be able to introduce the topic in a more quirky way that grabbed their attention slightly more than something that was just plain easy to understand.