The Rock Problem

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SWBAT relate first and second derivatives to velocity and acceleration.

Big Idea

Slopes of tangent lines take on new meaning when given in the context of a falling rock.

Launch and Explore

25 minutes

I love today's lesson! First and second derivatives can be very abstract to students and today we give them meaning and students are presented with a context that make them seem very real. Today's lesson start by giving students this worksheet and having them work on it with their table group. It is rather lengthy, so I want to give them plenty of time to work on it and digest the concepts.

While they are working, there are a few key ideas that I watch out for. I want to make sure that students are on the right track and are not getting tripped up by simple ideas. Here are a few things that I monitor:

  • Average velocity: For questions #1-5, student must find the average velocity over a specific interval. If students get stuck, I ask them how we measure velocity to remind them that we just calculate distance divided by time. 
  • Understanding instantaneous velocity: Students may not understand this concept, so I liken it to when a police officer clocks someone using a radar gun - it is the speed at precisely one instant (theoretically).
  • Calculating the instantaneous velocity: I don't expect all students to figure out #6, but I at least hope they can see that as the interval gets smaller, we are approaching a better estimate to the instantaneous velocity.
  • Finding the velocity at the highest point: It may seem strange to students that velocity can be zero when the rock is at the highest point, but the rock must stop at some point when switching directions.
  • The first derivative: If students do not pick up on the fact that the first derivative is giving velocity, then I ask them to decide what unit the function is measured in to see if that will help.


20 minutes

To being our discussion we go through our answers to questions #1-5 from the worksheet and I choose one student to explain their process to the class. I make sure that during our discussion, students understand that we really used the slope formula to figure out the average velocity. When it comes time to answer #6, it will hopefully be clear that we need to find the slope of the tangent line at t = 1.

The questions on the back are a little more involved. In the videos below I highlight some key points and suggestions I make when discussing some of these questions.

#8 - The derivative of the position function:

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 #9 - The second derivative of the position function:

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#10 - Finding the time when the speed of the rock is 30 feet per second: Make sure that students realize that the speed is 30 feet/sec when the velocity is 30 feet/sec or -30 feet/sec. Thus, the rock will hit 30 feet/sec on the way up and on the way down.


5 minutes

This lesson is always demanding, but I find that my students seem to be really engaged and we have some great discussions. To close the lesson, I ask students to think about the following to summarize everything that we learned:

  • How are the position function, velocity function, speed function and acceleration function related?
  • For each of the position, velocity, and acceleration functions, what is x-axis measuring? The y-axis?

Finally, here is a homework assignment to reinforce the work we did with velocity and acceleration.