### Summary

In this lesson, students will learn the relationship between position, velocity, speed, and acceleration to solve real-world problems involving motion along a line. Students are expected to know the chain rule and how to find higher-order derivatives for both power and trigonometric functions before beginning this lesson.

### Essential Question(s)

How are position, velocity, and acceleration related?

### Snapshot

**Engage**

Students recall average rate of change through engaging with a real-world example.

**Explore**

Students run simulations, modeling vertical and horizontal motion, and see how they relate to their corresponding position and velocity graphs.

**Explain**

Students complete guided notes with the class and formalize their understanding of how position, velocity, speed, and acceleration are related.

**Extend**

Students apply what they have learned in order to sketch and match graphs of velocity and acceleration curves given position graphs.

**Evaluate**

Students demonstrate their understanding of the relationship between position, velocity, speed, and acceleration by answering a free response question.

### Materials

Guided Notes handout (attached; one per student; printed front/back)

Guided Notes (Teacher Guide and Model Notes) document (attached; for teacher use)

AP Calculus Free Response – Motion handout (attached; one per student; printed front only)

Sample Responses (attached)

Coloring instruments (6 colors per student; highlighters, colored pencils/pens, etc.)

Pencils

Paper

Student devices with internet access

### Engage

5 Minute(s)

Provide students with your session code. Then, have students go to student.desmos.com and enter the session code.

Direct students’ attention to **screen 1** and then read the following prompt as the Bell Ringer:

*An officer pulls you over and tells you that you were going 85 mph, and the posted speed limit is 80 mph. You think of a great idea—if you can quickly calculate your average speed, you might be able to convince the officer that your average speed was below the speed limit, and, therefore, he should let you off. You notice on your odometer that you have traveled 100 miles in the past 45 minutes. What was your average speed?*

Have students calculate the average speed and type their answer and reasoning into screen 1.

Use this prompt to discuss average rates of change compared to instantaneous rate of change. Listen for misunderstandings to see if students need a quick refresh on instantaneous rates of change.

As time allows, ask for volunteers to share how they found their average speed and whether they think they could use this idea to get out of a ticket. (Students will also have the ability to see their peers’ responses in the Desmos Classroom activity.)

Use **screen 2** to display the lesson’s essential question and **screen 3 **to identify the lesson’s learning objectives. Review each of these with students to the extent you feel necessary.

### Explore

15 Minute(s)

Instruct students to find a partner or assign students partners. On the Dashboard, press the orange plus sign three times to allow students to progress to **screens 4–6**. Have students press the "Run" button and work with their partner to observe the relationship between the van’s movement and the graphs of position and velocity. Then they are to read the graphs to answer questions about the direction the van is moving and the initial velocity.

As students finish screen 6, bring the class together for a whole-class discussion. Ask for volunteers to share how they found the initial velocity and what they noticed about the graphs when the van was moving to the right. Use this time to correct any misconceptions.

On the Dashboard, press the orange plus sign four times to allow students to progress to **screens 7–10**. Have students press the "Run" button and work with their partner to observe the relationship between the hot air balloon’s movement and the graphs of position and velocity. Then they are to read the graphs to answer questions about the direction the balloon is moving and the maximum height.

As students finish screen 9, bring the class together for a whole-class discussion. Ask for volunteers to share how they found the maximum height and what they noticed about the graphs when the balloon was moving up. Use this time to correct any misconceptions.

### Explain

30 Minute(s)

**Screen** **10** indicates that students are to set aside their Desmos Classroom activity to complete the Guided Notes with the class. Give each student a copy of the attached **Guided Notes** handout. Introduce or review the definitions of *position*, *velocity*, and *acceleration*.

Give each student 6 coloring instruments. Students could share the 6 highlighters, markers, or colored pencils/pens. Have students use four of the colors to color-code the slope and *y*-values of the given velocity function: *positive acceleration* (slope), *negative acceleration,* *positive velocity* (*y*-values), and *negative velocity*.

Then have students use two colors on the graph of speed to indicate a *positive* slope (increasing speed) and *negative *slope (decreasing speed). Ask students to see if they could use the velocity graph to determine when the speed of an object is increasing or decreasing, without using the graph of the speed curve.

Help students see that when the signs of the *velocity *and *acceleration* are the same, the object speeds up, while the object slows down when velocity and acceleration have different signs.

Continue to complete the handout as a class.

Once finished, have students add this to their math notebooks if that is a classroom norm.

### Extend

30 Minute(s)

On the Dashboard, click the orange "Stop" button; now students can complete the Desmos activity at their own pace.

**Screen 11** previews the work of screens 12–25 where students will examine the relationships between a particle’s position, velocity, and acceleration.

On **screens 12–17**, students are seeing motion that can be modeled by a linear position function. On **screens 18–25**, students are seeing motion that can be modeled by a quadratic position function.

Screen 12 starts with asking students to describe the motion of a green dot. Then screen 13 has students sketch the position-time curve to model the movement of the dot. Students receive feedback on screen 14 with the correct position-time graph. They are now to sketch the velocity-time curve and explain their thinking. Students receive feedback on screen 15 and are asked to sketch the graphs if the dot were moving twice as quickly and explain their thinking.

Using the Card Matching strategy, students are to match position-time graphs with their corresponding velocity-time graphs on screen 16, where they will see "Well done!" at the top of their screen if they match the cards correctly. After successfully completing the card match, students review screen 17 that transitions them into the next set of motion problems.

If students do not correctly match the cards on slide 16, they will see, "Go back to the previous screen and try again."

### Evaluate

15 Minute(s)

Once students have completed screen 25, direct their attention to **screen 26**. Give each student the **AP Calculus Free Response – Motion** handout and direct students to work independently so that you can individually assess what students have learned from this lesson. This question is intended to be solved without the use of a calculator.

Give students 10–15 minutes to answer the free response question. Then use the attached **Sample Response** slides to review the sample response and scoring guidelines with the class. Be sure to help students understand how to earn points on exam questions.

### Resources

CollegeBoard AP. (2010).

*AP Calculus AB 2010 Scoring Guidelines Form B*. Retrieved June 22, 2022, from https://secure-media.collegeboard.org/apc/ap10_calculus_ab_form_b_sgs.pdfK20 Center. (n.d.). Bell Ringer and Exit Tickets. Strategies. https://learn.k20center.ou.edu/strategy/125

K20 Center. (n.d.). Card Matching. Strategies. https://learn.k20center.ou.edu/strategy/1837

K20 Center. (n.d.). Desmos Classroom. Tech Tools. https://learn.k20center.ou.edu/tech-tool/1081

Memed_Nurrohmad. (February 28, 2019). Car [Illustration]. Pixabay. https://pixabay.com/illustrations/car-transportation-vehicle-bus-4025379/