Big Idea 1 from the 2016-2017 AP Calculus Curriculum Framework is Limits.
Enduring Understanding 1.1: The concept of a limit can be used to understand the behavior of a function.
Mathematical Practice for AP Calculus (MPAC) 2: Connecting Concepts
Students can connect concepts to their visual representation with and without technology.
Mathematical Practice for AP Calculus (MPAC) 4: Multiple Representations
Students can associate tables, graphs, and symbolic representations of functions.
Students can develop concepts using graphical, symbolical, verbal, or numerical representations with and without technology.
We begin calculus with a discussion of limits. I throw a dart at the dart board and ask what just happened.
You hit the bullseye.
(I really did this year – on the second try – after missing the board completely on the first try.) But how did the dart make it to the board? It had to go half the distance to the board. And then half the distance again. And again. And again. How did the dart make it to the board?
The other demonstration is at the recommendation of my grandfather, now 101, who taught high school mathematics for over 50 years. I take a measured piece of yarn (1 meter), cut it in half, and begin to make a pile of the “halves”. How long does the yarn measure in the pile if we put it end to end? It gets closer and closer to measuring 1 meter, but does it ever make it to 1 meter?
How do you provide opportunities for students to “see” limits?
I used to [ineffectively] wave my hands. Now we use technology. Students won’t use this type of visualization while they’re taking a test, but we find it invaluable while they’re learning.