## Investigation.docx - Section 4: Investigation

# Motion - Distance vs Displacement

Lesson 7 of 17

## Objective: SWBAT explain and represent the difference between distance and displacement in context.

## Big Idea: How far did you run? Students will distinguish between distance traveled and displacement, then represent each with integrals.

*56 minutes*

#### Warm-Up + Homework Review

*25 min*

To begin today's class, I will ask students to sketch plausible position, velocity, and acceleration functions for this story – stack your graphs vertically on 3 separate coordinate axes:

**NASA is launching a shuttle into outer space. There is a 10 second countdown, then the shuttle’s engines ignite and propel it upward along a vertical line perpendicular to the earth’s surface. The velocity steadily increases for 150 seconds until it reaches its maximum cruising speed of 17,320 mph.**

Before letting students begin the warm-up, I will demonstrate the use of the CBR Virtual applet that students will use on tonight’s homework.

**Technology Note**: As part of the homework assignment, students will be e-mailing screen captures from the applet to me. Students will need a computer with Java installed; you might advise students to use a computer in the school library after school today or before school tomorrow if their computer at home will not load the applet.

Usually there are a few students who do not complete this part of the homework, but most students will have engaged with the applet and will be more readily able to participate meaningfully tomorrow when we build on students’ experiences with the applet. By assigning student use of this applet, students gain experience that supports **SMP #4 modeling** and **SMP #5 using appropriate tools strategically**.

As students work on the warm-up, I will circulate paying attention to how students **attend to precision (SMP #6). **I will remind students to label their axes appropriately, including units. I watch for students who do not recognize the misalignment of the units of time in the problem, from 150 seconds to 17,320 miles per hour. If I see this occurring, I might ask or discuss with students whether they prefer converting the units into seconds or into hours, and why – although either conversion is mathematically viable, converting to seconds would keep the numeric values in the problem smaller and therefore more convenient to work with.

Either as students are working or when we go over the warm-up together, I ask students to suggest ways to improve the accuracy of their sketched position function (**SMP #2: reason abstractly and** **quantitatively**). I hope that some students will observe that they could calculate the instantaneous rates of change at various points on the position function using y-values from the velocity function, and, they might compute the area accumulated under the velocity function to determine the y-values for the position function.

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#### Setting the Stage

*3 min*

I launch this activity by standing still at a starting point in the front of the room. I take 5 steps forward (what other number of steps would I take? Of course FIVE!). I come to a stop. Then, I take 5 steps backward. I ask this question verbatim: “How far did I travel?”

The intentional ambiguity in this question will stimulate controversy among students, which is a great way to initiate a class discussion and ultimately leads into today’s lesson on distinguishing displacement and distance traveled. Mathematically-speaking, displacement is easiest to deal with, simply because we can let the positive and negative signs work themselves out. On the other hand, total distance is tricky because all segment lengths traveled are positive. Does this sound like any particular math concept we know about? Of course, absolute value.

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- UNIT 1: Back to School
- UNIT 2: Limits and Derivatives
- UNIT 3: Formalizing Derivatives and Techniques for Differentiation
- UNIT 4: Applications of Differentiation, Part 1
- UNIT 5: Applications of Differentiation, Part 2
- UNIT 6: The Integral
- UNIT 7: Applications of Integration
- UNIT 8: Differential Equations
- UNIT 9: Full Course Review via Motion
- UNIT 10: The Final Stretch - Preparing for the AP Exam

- LESSON 1: Limits and l'Hospital
- LESSON 2: Know Your Limits
- LESSON 3: Local Linearization, 1st and 2nd Derivative Tests, and Computing Derivatives
- LESSON 4: Derivatives Algebraically and Graphically
- LESSON 5: The Calculus of Motion
- LESSON 6: Motion - Velocity on Intervals
- LESSON 7: Motion - Distance vs Displacement
- LESSON 8: Motion - With Multiple Derivatives
- LESSON 9: Motion and Optimization
- LESSON 10: Calculus and My Car's Dashboard
- LESSON 11: Rockin' Related Rates
- LESSON 12: Meet My Friend Riemann
- LESSON 13: Cookies and Pi
- LESSON 14: Accumulate This!
- LESSON 15: Wait, the Interval Width Varies?
- LESSON 16: Integrating Areas to Get Volumes
- LESSON 17: More Areas and Volumes