SWBAT interpret the average rate of change of a quadratic function in terms of the velocity and acceleration of a projectile. SWBAT make use of the structure of the quadratic equation to compare projectile motion under a variety of conditions.

Projectile motion provides context for average rates of change in the context of velocity and acceleration. What goes up, must come down!

5 minutes

During this brief collaborative session, the students have just one task:

*Compute average velocities for each 1/2-second interval from 0 - 3.5 seconds and record them in a table.*

See the solutions document in the resources for an example of what this table might look like. Video: BL What Goes Up Day 2, the big idea

Watch out! Many students may simply take the change in the height as the average velocity. Remind them that this change occurred over a *half*-second. What would be the change over a *full* second? That's the velocity.

As I find the first groups completing the task, I make a point of guiding those students to the conclusion that the *average velocity* on the interval *t* = 0 to *t* = 0.5 is an estimate of the *actual velocity* at *t* = 0.25 (the midpoint of the interval). This will be crucial for understanding the average acceleration correctly! It might be helpful to suggest that they try graphing the velocity data, if they do, they will see that the velocity is a linear function. A graph will help students make sense of their calculations and also begin to see that the velocity itself is changing at a constant rate - the force of gravity!

Of course, the point of all of this is to lead students to see the meaning behind the coefficients in the displacement function. Ultimately, they will be able to identify the constant term with displacement, the linear term with velocity, and the quadratic term with acceleration.

5 minutes

In this section, the students and I have a brief conversation about acceleration to prepare them to answer the remaining questions. The dialogue might go something like this:

What is *acceleration*?

[Students should recall that acceleration is the rate of change of velocity. You can calculate average acceleration on a given interval by comparing the change in the velocity to the change in time.]

Is the velocity of the stone changing over time?

[Yes, it's decreasing.]

Okay. Let's use the average velocities we've computed to find some average rates of acceleration.

[Ok, well the velocity changes by a constant amount. I guess that means the acceleration is constant?]

How much does the velocity change each second?

[It goes down by 8. No, by 16. No, by 32. Huh?]

Wait a minute! Think carefully about how much time it's taking to change. What is the *per second* rate of change?

[Well, since the displacement is given every 0.5 sec, the average velocities are given per 0.5 sec., so the average accelerations are given every 0.25 sec.! The velocity changes by -32fps every second!] *See the Solutions document for more details on this!*

Great. What does that mean?

[Um.]

Well, Isaac Newton says that the stone would just keep moving in a straight line, and at a constant velocity, unless something acted on it. So, is something acting on the stone?

[Yes, gravity! The acceleration is the force of gravity! And *that* explains why it's constant.]

Good. From here you should be able to answer the final question. When you think you're done with it, let me know. If your answers are correct, I'll give you the next part of the assignment.

(Optionally, you might consider graphing the average velocities to see that they lie on a straight line with a constant slope. It's a nice visual confirmation of what the students are seeing numerically. It also forces you to assign each average velocity to a specific time, allowing you to see more clearly the actual interval over which the velocity is changing.)

20 minutes