##
* *Reflection: Developing a Conceptual Understanding
Investigation of Distance and Time Graphs Using a CBR - Section 4: Exit Slip

An extension that I wanted to add to this lesson to develop a deeper understanding for students was to compare d=rt + 0 to y=mx + b. If time permits in a future lesson, I will have students write the equations of the lines from the distance-time graph as a piecewise function. With m=r and d=y and t=x. This extension would also help students not only see the relationship between the two formulas but also review how to write domain and range for the restricted line segments.

*Possible Extension of this lesson*

*Developing a Conceptual Understanding: Possible Extension of this lesson*

# Investigation of Distance and Time Graphs Using a CBR

Lesson 8 of 13

## Objective: SWBAT analyze and draw the graphs of distance over time, as well as determine the increasing, decreasing, and constant intervals.

## Big Idea: To match graphs to a student walking in real time to develop a conceptual understanding of distance over time and compare it to speed over time.

*45 minutes*

#### Warm Up

*10 min*

I plan for this Warm Up to take about 10 minutes. I guide this Warm Up to introduce students to the technology that we will use today, the CBR 2. CBR stands for Calculator Based Ranger, a small motion detector that connects to the TI graphing calculators. I am having to use a backing called a cradle for the TI-Nspire Cx with a USB connection. (I had to modify this lesson due to the fact of only having one cord to run the CBR on the TI-Nspire Cx.)

I ask for a student volunteer for this Warm Up. I have provided students with two graphs in the Warm Up. I ask students to predict the necessary walking directions for the volunteer to match the motion shown on the graph. A common mistake that students make on their predictions is that the student will walk at the same angle as the line. In fact, the student walks straight out from the CBR, and the slope indicates the rate at which they walk. It is the student volunteer's job to walk to create the same graphs on the CBR. They may take guidance from the audience. I have students raise their hand for feedback or input. It takes them a few minutes to get a feel for the idea that the slope of the graph is determined by speed and direction, rather than position.

*expand content*

#### Partner Work

*15 min*

After the Warm Up, students complete a Partner Activity on Distance and Time. I intend for the Partner Activity to take about 15 minutes. In pairs, students explore the four graphs without the CBR. Students again have four graphs to make predictions for the walking directions that will create the graph. Students are welcome to use the CBR at the front of the room to verify their predictions. Some students are still struggling with the idea that the graph does not represent the exact path that the student walks.

I purposely plan the circle for the fourth graph, to test if students walk in a circle to create the graph. Some students do predict this, but most students realize that time must move forward, and it is not possible to loop back. At the end of this activity, we discuss how the circle is not possible, and that a vertical line is not possible because time is continuous.

At the end of this activity, students should realize that a line segment with a positive slope is moving away from the CBR, and that a negative slope is moving toward the CBR. However, the student walks in a straight line. The slope indicates the rate at which the student is walking which we discuss as a whole class in the next activity. A horizontal line segment means that the student has stopped, the distance is not increasing, but time continues to increase.

*expand content*

#### Class Activity

*15 min*

For today's class activity I have my students identify the increasing, decreasing, and constant intervals for Stephanie's bike ride. While students are working independently, I walk around to monitor their progress. Most of my students do well on this activity. After reviewing the Warm Up, students seem to understand the distance-time graph well. One common mistake that my students make is to identify the intervals using the y-axis instead of the x-axis. Another source of confusion is with the use of interval notation.

After my students identify the intervals of Stephanie's bike ride individually, I ask my students to compare their answers with their table partner for a couple of minutes. Then, I will randomly call on groups to share with the class:

- During which intervals can we say that Stephanie's speed increased?
- During which intervals can we say that Stephanie's speed decreased?
- During which intervals can we describe Stephanie's speed as constant?

In the second part of this activity. I have students sketch a graph describing Stephanie's speed over time, based on their work with the position v. time graph. This task is challenging. My students are typically unfamiliar with speed-time graphs. If necessary, I model the first line segment from left to right. I model the problem in the video below.

#### Resources

*expand content*

#### Exit Slip

*5 min*

I give today's Exit Slip as a formative assessment. I want to assess each student's ability to write a story that matches a distance-time graph. **Students are asked to write a story about their journey to school and create a distance-time graph to match the story.** I provide students with extra graph paper if needed for their graph which they may attach to the Exit Slip. The students may complete the Exit Slip as homework if needed. Most of my students show a pretty good understanding of how to model their trip home using a distance-time graph. They also show an understanding of decreasing intervals, increasing intervals, and constant intervals. Here is an example of a Student Journey Home as they explained it took them 60 minutes to get home.

*expand content*

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- UNIT 1: Introduction to Functions
- UNIT 2: Expressions, Equations, and Inequalities
- UNIT 3: Linear Functions
- UNIT 4: Systems of Equations
- UNIT 5: Radical Expressions, Equations, and Rational Exponents
- UNIT 6: Exponential Functions
- UNIT 7: Polynomial Operations and Applications
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- UNIT 9: Statistics

- LESSON 1: First Day of School
- LESSON 2: Introducing Functions
- LESSON 3: Identifying Functions and Providing Rationale
- LESSON 4: Domain and Range of Graphs Using Set Builder Notation
- LESSON 5: Domain and Range of Graphs Using Interval Notation
- LESSON 6: Evaluating Functions Using Function Notation
- LESSON 7: Evaluating Graphs and Equations Using Function Notation
- LESSON 8: Investigation of Distance and Time Graphs Using a CBR
- LESSON 9: Introduction of Parent Functions
- LESSON 10: Transformations of Parent Functions
- LESSON 11: Preparing for Partner Presentations on Transformation of a Parent Function(Day 1 of 2)
- LESSON 12: Partner Presentations on the Transformation of a Parent Function(Day 2 of 2)
- LESSON 13: Mastery of the Function Unit