First Day of AP Calculus: A Complete Plan

A question a stopwatch cannot answer

Before naming a single term, put students in front of a real falling object and ask a question no single stopwatch reading can answer: how fast is it falling at the exact instant one second has passed. A stopwatch only ever times an interval, never an instant. Answering that question honestly is what the whole course is going to spend on, so it is worth building the first day around it rather than around a syllabus.

Drop a tennis ball, or anything safe and reasonably heavy, from a measured height: a stairwell landing, the top of the bleachers, or a chair on a table all work. If the drop point measures out near 100 feet off the ground, the height in feet after $t$ seconds follows the same free-fall model used everywhere in physics, built from the constant pull of gravity near the surface, $g = 32$ ft per second per second.

Students do not need to trust this formula on faith. It comes from the drop they just watched, and nothing about it depends on a textbook, so it is fair game to hand out before a single vocabulary word appears.

The ten-minute hook: shrink the interval until the answer holds still

Ask for the average speed of the fall between one second and some later moment. Average velocity on an interval is a slope: the change in height divided by the change in time.

Put students in pairs, hand each pair one value of $\Delta t$, and give them the formula and four minutes to compute their row by hand. Collect every answer on the board at once, ordered from the widest interval to the narrowest.

What the year is about

This year answers two questions the hook just raised, without naming them yet. First: how fast is something changing at one exact instant, which is the derivative, covering how it is defined and computed through several more units, including the composite and implicit rules that follow once functions get tangled together. Second: how much has piled up over a stretch of time or space when the rate of change keeps shifting, which is the integral, covering both the computation and its applications to area, volume, and motion. BC sections add three more pieces: sequences of numbers and the series they build, motion and curves described by something other than $y$ as a function of $x$, and a few extra integration techniques and models. Every method the course teaches, from the chain rule to substitution to related rates, is a faster way to do exactly what today's hook did by hand: watch a quantity shrink or grow and read off what it is approaching.

The syllabus, in one paragraph

Grades come from three buckets: unit tests worth the largest share, free-response practice scored the way the AP exam scores it worth the next share, and daily homework and practice completion worth the rest; every unit test allows one no-penalty retake within a week once the practice set for that unit is complete. Bring a graphing calculator every day starting next week, a notebook kept only for this class, and a habit of showing work by hand, since more than half of the AP exam allows no calculator at all. Late homework loses nothing the first two times each quarter and is a zero after that. Visit tutorial hours before an assignment is late, not after; the schedule and every policy above will also live on the class page so nobody needs to remember it from today.

The full fifty-minute period

The hook above is ten minutes on its own. Here is where the rest of a fifty-minute period goes so nothing gets rushed at the end.

No-tech variant

None of this needs a computer, a projector, or a class set of calculators. Every piece has a version that runs on paper.

• No projector: write $h(t) = 100 - 16t^2$ on the board and have each pair compute its row with long multiplication instead of a calculator; the numbers involved, such as $1.1^2$, are small enough to do by hand in under a minute.
• Shorter period than fifty minutes: compress the one-paragraph content overview to a single spoken sentence and send the fuller paragraph home as a one-page handout to read that night.
• No safe drop point in the building: swap the live drop for a photo sequence taken in advance, since even a basic phone camera can time a short fall to within a tenth of a second, or skip the drop entirely and hand out the height data already computed for a hypothetical 100-foot fall.
• No falling object at all: run the identical shrinking-interval idea on a car trip instead, using odometer or trip-timer readings a driver already has, and ask for speed at one exact minute mark rather than an average over the whole trip.

The first homework

The homework repeats today's idea at a new instant, so the first graded work of the year is the same skill practiced once more, not a jump to something unfamiliar.

1. Using the same model, $h(t) = 100 - 16t^2$, find the average velocity on $[1.5, 1.5+\Delta t]$ for at least three shrinking values of $\Delta t$, down to $\Delta t = 0.01$.
2. In one sentence, state what number the averages are walking toward, and explain why a smaller $\Delta t$ should land closer to the true instant rather than farther from it.
3. Try the same idea on the secant to tangent walkthrough: drag the interval down until the line settles, read off the number, and check that it matches the hand computation from question one.
4. Optional, for early finishers: open the limit method chooser, pick any problem there, and note in a sentence whether the same idea of a value being approached still explains the algebraic method it reveals.

Where this leads next

The next few class periods turn today's shrinking-interval arithmetic into the formal tools of how to find limits and into the cases where a function's pieces do not line up, covered in continuity and discontinuities. The Unit 1 hub collects both alongside practice. Later in the year, once every unit is on the board, how to study for AP Calculus AB turns this same course overview into a weighted study plan.