Students will be able to represent graphically the difference between velocity and speed functions, and compute total distance and displacement.

Original function, 1st derivative, 2nd derivative → Position, velocity, acceleration. Let’s investigate some motion applications.

30 minutes

2 minutes

I begin with 1-2 minutes of full-class practice with the Multiple Derivative flashcards – having these cards in mind will assist students later in the lesson with motion. As always, have students provide clear verbal explanations for several flashcards to promote **SMP #6 attending to precision**.

25 minutes

I will begin by today's investigation by projecting the optimization problem “Find the minimum distance between the point (–2,0) and f(x) = arctan(x).” Before letting students dive right into an algebraic approach to solving this problem, it is helpful to lead a brief discussion about what the problem is asking us to do (find the minimum distance, not only the (x,y) coordinates), which piece of information is the constraint (f(x) = arctan(x)), and what equation we want to differentiate and set equal to zero to find its minimum (the distance function).

The GeoGebra applet (Minimize Distance to Function.ggb), though helpful for students to visualize the problem, does not really assist with the algebraic solution. It motivates the need to define the point on the function in terms of variables since that point can move anywhere along the function for purposes of finding distances to the point (–2,0). Treating this optimization problem as yet another exercise to solve with a quick, snappy algebraic-only approach is not likely to be retained by students. To galvanize students’ learning and to aid with retention, I plan to ask students to complete a **quick-write** summarizing the solution process for these types of problems, in general.

My plan is that my students will complete the Speed-Velocity and Distance-Displacement Graphically worksheet in class. The front side (direction and speed) will be completed today. The back side (distance and displacement) will be completed tomorrow. The front side takes less than 10 minutes to complete, but longer if you invite kinesthetic demonstrations. If there is enough time in class, I will ask students to “act out” the athlete’s running drill. This physical modeling is very worthwhile and helps students develop proficiency with SMP #4 modeling. I will approach this activity by giving students time to think about the velocity graph provided at the top of the worksheet, circle two descriptors on each interval, and then be prepared to act out her motion in front of the class. If students are uncomfortable or uncertain about this part, I may provide time for them to consult with a partner.

Today, students should complete Questions #1-4 individually. I will be on watch for students who misinterpret the motion terminology in each sentence. I find that turning their attention back to the velocity graph and the circled descriptors on each interval at the top of their sheet helps them understand how to successfully complete each sentence. I will also remind students about their Multiple Derivative flashcards, because the properties of the original function, first derivative, and second derivative correspond directly to the position, velocity, and acceleration functions.

** DIFFERENTIATION**: If students struggle to understand questions #5-6 in particular, I will direct them to the descriptors they circled on each interval, refer to the kinesthetic demonstrations either by other students or by me, and note that the x-axis on a velocity graph means the velocity = 0, which is a stopped position, and therefore any object approaching a stopped position must be slowing down. These teacher moves support students in SMP #1 making sense of motion problems.

5 minutes

A nice way to wrap up today's work on the motion worksheet is to have them plan and re-demonstrate the athlete’s motion, this time sketching a number line on the board (in 1 meter units) and being more precise with the specific locations where the athlete changes directions.

If short on time, ask students to explain to a neighbor why a particle with negative, decreasing velocity is actually speeding up. Call on one or a few students to share their findings with the whole class.

Tonight's Homework is listed in the word version of this narrative. If you are interested in learning more about my approach to closing lessons, please Watch my video discussing strategies for closing lessons.