I begin this lesson by asking my students if they've ever participated in a "guestimating" event like trying to estimate the number of jelly beans in a jar. This leads to a general discussion of what it means to estimate after which I put a graph of the function f(x) = x^3 on the board and ask my students to independently estimate the rate of change for the domain from x=1 to x=2. (MP1) When they've had a few minutes to work, I randomly select students to post their responses on the board, four at a time, until all students have posted. I then have the class critique the answers with a focus on which ones fit the graph the best. (MP3) As we work through these critiques I ask for volunteers to explain how they determined their answers. I'm anticipating some students might say that they look at the slope between two points because that should be familiar from earlier classes. I hope that someone also suggests using the axes to estimate points if they aren't given. If these aren't mentioned I might ask a leading question like "How can you find points to work with if none are labeled on the graph?"
Instead of just having my students work independently as they practice estimating rates of change, I create a classroom friendly competition which encourages student engagement. I tell my students that they will each need a white board, marker, and eraser to write and share answers. I distribute the Estimate handout (except for the final page with all the equations and rates of change!) and review the rules of the game.
I ask if there are any questions, then begin the game, with each student working to estimate each rate of change. (MP1)
I close this lesson by giving each student a notecard and asking them to write three strategies/tips, that they would give to an absent classmate, for estimating rate of change accurately. (MP2, MP6)
If you needed to provide your class with more specific directions, you might have your students answer the following questions: