Objectives: To develop the concept of different rates of change and the language of concavity.
Preliminaries/Lead-In: I like to talk to them about why we do this activity because it looks quite different to what they see when they read Section 1.2 in their text. Here's what I say: In this course we will be characterizing different types of functions by describing how they change (grow or decline). You already have a vocabulary for doing that - we just need to begin to be more careful about the words we use to describe change and think hard about what those words are really saying. So, we're going to work in a tangible setting today - considering different hypothetical scenarios for the growth (or decrease) of world population over time. We're not going to define and calculate "rates of change" as we will learn to do in our text in Section 1.2 (and revisit in class after this activity). We are just going to read very carefully normal English sentences written that describe a particular hypothetical growth scenario, and interpret it by coming up with possible population data and population graphs that fit the description written. This is a great place to start - with language you are familiar with - then I'd like you to go home and read Section 1.2. We'll talk about Section 1.2 in class as well - there we define some very specific calculations we can make to describe growth/decline of a function numerically.
Suggested Procedures: On the first page we ask why we would look at a graph of population over time - I will use this as an opportunity to talk to them again about what questions we can answer with a graph. I will highlight that really the only two types of questions are "What will the population be in the year ....." and " When will the population reach .....". After that I will let them get started on the activity. I will pull it back to the whole class if there seems to be widespread confusion but otherwise I will probably let them work until more or less everyone has come to the end of 5. Then I'll ask for some examples on the elmo of what they came up with and introduce the language of concavity - attaching it to the description of how rate of change is changing.
Wrap-Up/Take-Away: Don't worry about defining rate of change yet. It's better to come back after this activity and define rate of change numerically - maybe using some data from this activity - and relating behavior of rates of change with the shape of the graph from that activity. This actually jumps ahead into Ch. 2, but we have a good opportunity to note that relationship now.
Possible Homework: I will probably send them home work 9 - 11 and also to finish up the earlier questions if anyone is still working on them. I grade this one pretty closely since they had a good amount of time in class to work together and check with each other (and me). I do talk to them about how they are making their predictions - their data should establish a pattern - follow that pattern out to 2070. This is the idea of using a model to predict, so good conversation to have.
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