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* *Reflection: Complex Tasks
The Rock Problem - Section 2: Share

When teaching this lesson I almost feel like there are two layers to it. We have the conceptual layer of what happens are you throw a rock into the air; this layer does not have any numbers or values attached to it and we are discussing things on a purely conceptual level. The second layer has to do with this specific function and graph; for this layer we discuss the height, velocities, time, etc. While teaching I felt that we kept going back and forth between the two layers. The conceptual layer let us make sense of what we needed to do with the second layer.

For example, once we figured out that the velocity would decrease for the entire path of the ball since gravity was constantly acting on it, we could go back and look at the function and get actual values.

*Two Layers*

*Complex Tasks: Two Layers*

# The Rock Problem

Lesson 10 of 13

## Objective: SWBAT relate first and second derivatives to velocity and acceleration.

*50 minutes*

#### Launch and Explore

*25 min*

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.

#### Resources

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#### Share

*20 min*

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: **

**#9 - The second derivative of the position function:**

**#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.

#### Resources

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#### Summarize

*5 min*

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.

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- UNIT 1: Functioning with Functions
- UNIT 2: Polynomial and Rational Functions
- UNIT 3: Exponential and Logarithmic Functions
- UNIT 4: Trigonometric Functions
- UNIT 5: Trigonometric Relationships
- UNIT 6: Additional Trigonometry Topics
- UNIT 7: Midterm Review and Exam
- UNIT 8: Matrices and Systems
- UNIT 9: Sequences and Series
- UNIT 10: Conic Sections
- UNIT 11: Parametric Equations and Polar Coordinates
- UNIT 12: Math in 3D
- UNIT 13: Limits and Derivatives

- LESSON 1: The Limit of a Function
- LESSON 2: Techniques for Finding Limits
- LESSON 3: The Tangent Line Problem - Day 1 of 2
- LESSON 4: The Tangent Line Problem - Day 2 of 2
- LESSON 5: The Power Rule
- LESSON 6: Formative Assessment: Limits and Derivatives
- LESSON 7: Derivatives and Graphs
- LESSON 8: The Second Derivative
- LESSON 9: Maximizing Volume - Revisited
- LESSON 10: The Rock Problem
- LESSON 11: Unit Review: Limits and Derivatives
- LESSON 12: Unit Review Game: The Row Game
- LESSON 13: Unit Assessment: Limits and Derivatives