A continuing series of posts on how I implement topics in my classroom. Last year we looked at Pre-Calculus. Let's explore some Calculus, shall we?

### Objective

Students understand the relationship between an original function, its derivative, and its second derivative. Students understand that a value of zero on one graph implies meaning about the equivalent location at steps "above" and "below" that graph.

### Progression

I found great success with curve sketching, a subject I found difficult as a student, through the early introduction of integrals. I establish an "up" and "down" narrative about the role of integrals and derivatives. I find it helpful to introduce this with a bit of physics and introduce the connections between position, velocity, and acceleration.

Eventually, the "up" and "down" visual looks like this:

I do a little physics demonstration with a tennis ball, discussing its velocity and direction and various moving away or towards a table. It's a remarkable "OHHHHHHHHHHH" kind of moment that I wish your average physics class included.

Units are a critical part of the discussion. Later you need students to understand which operator is necessary based on interpreting a lot of language. Unit awareness helps with some of this. Being given information noted in meters/second with an answer requested in meters should be a huge flag that going "up" or integrating is appropriate. See: What Helps Me?

### Tasks

This takes a lot of repetition and discussion to get right. I find this Desmos graph helpful. I place an emphasis on values vs. behaviors, a tricky idea conceptual. For example: a piece of a function can have positive values yet negative behavior. Say that piece belonged to a graph labeled f'(x). What graph can we construct from those values? What graph depends on the behaviors? Catching that difference takes a minute.

Primarily we spend a lot of time drawing families. Students are given a starting graph at any point in the family.

Eventually we arrive at the idea that my pattern of behaviors at one level, say f'(x), should be the values of my graph at the f''(x) level. If we moved "up" the family, the relationship is inverted.

### Interpretation

Curve sketching is great and all, but it means nothing if you can't interpret the drawings. Sequence wise, I go Riemann Sums > Curve Sketching > Integration > Curve Interpretation. At this point we do a lot of heavy vocabulary lifting: minimums, maximums, points of inflection. concavity, and justifications for the appearance of all these things. Justifications in Calculus are relatively simple, but teaching students to make simple arguments is strangely difficult.

At this point we've reached December and students are ready for a concluding task:

### Going Further

Soon enough, I've introduced the concept of algebraic integration and the +C terms that follow. Students will start asking questions throughout curve sketching. When going "up" a level, many will ask how they know where to put the graph along the y-axis. I play a little dumb and say the answer is coming soon.

Once you've established +C as a class norm and they understand the idea of what that constant represents, you get into the numerous possibilities with what the graph one level "up" could look like, and bam, you're ready for slope fields and differential equations.

Early access to integrals saves so many headaches later. The idea has had time to marinate in their minds. It's hard to talk about slope fields if kids aren't square on the base concept on why there could be so many options in the first place.