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# 5.3: Trains in Motion

Difficulty Level: At Grade Created by: CK-12

This activity is intended to supplement Algebra I, Chapter 4, Section 8.

ID: 11194

Time required: 15 minutes

## Activity Overview

In this activity, students will make observations about the motion of two objects. They will compare and contrast this motion and consider how it corresponds to a graph representing distance as a function of time. As an extension, students will explore the relationship between the slope and the rate.

Topic: Linear Equations & Functions

• Motion, distance = rate \begin{align*}\cdot\end{align*} time
• Slope, graphically expressing data

Teacher Preparation and Notes

• This activity can serve as a nice introduction to using formulas and graphs to represent a real-life situation.
• The student worksheet provides instructions and questions to guide the inquiry and focus the observations.
• Be sure to add the program to each student’s graphing calculator.

Associated Materials

In this activity, students use the program TRAINS to view two graphs. Make sure that students have loaded the program before beginning the activity. The first graph is of two trains leaving a train station. The second is the distance-time graph of this motion.

## Problem 1 – Observe Motion

After selecting the first option of the program OBSERVE MOTION, students will see two lines and a circle that travels each line. The circle represents the trains. Explain to students that even though the lines (tracks) stop, in the real world the train tracks would continue on. Also, students should pretend that there is track connecting the first train to the station.

When students press TRACE, three values will appear at the bottom of the screen. The \begin{align*}T-\end{align*}value represents time and the \begin{align*}X-\end{align*}value represents distance. Students can disregard the \begin{align*}Y-\end{align*}value.

Students can press \begin{align*}2^{nd}\end{align*} [MODE] to return to the Home screen after answering the questions on the worksheet. Pressing ENTER right away will return them to the program menu.

1. Answers will vary, but the following question should complement their answers.

2. When \begin{align*}t = 0\end{align*}, train \begin{align*}1\end{align*} is at \begin{align*}80 \ km\end{align*}.

3. Train \begin{align*}2\end{align*} starts at the origin.

4. Train \begin{align*}2\end{align*} is moving faster.

5. Train \begin{align*}2\end{align*} is traveling at \begin{align*}120 \ km/hr\end{align*}.

6. Train \begin{align*}1\end{align*} traveled \begin{align*}80 \ km\end{align*} in one hour.

7. Train \begin{align*}1\end{align*}’s speed is \begin{align*}80 \ km/hr.\end{align*}

8. They are at the same distance from the station at \begin{align*}240 \ km\end{align*}.

9. This occurs at \begin{align*}2 \ hours\end{align*}.

## Problem 2 – Distance-Time Graph

After selecting the second option of the program DIST VS TIME, students will see two lines will positive slopes, which intersect each other.

When students press TRACE, two values will appear at the bottom of the screen. The \begin{align*}X-\end{align*}value represents time and the \begin{align*}Y-\end{align*}value represents distance. In the upper right corner of the screen, \begin{align*}P1\end{align*} or \begin{align*}P2\end{align*} will appear. Students will determine that \begin{align*}P1\end{align*} represents train \begin{align*}1\end{align*} and \begin{align*}P2\end{align*} represents \begin{align*}P2\end{align*}.

Students should understand that the intersection point of the two lines is the time at which the trains are equal distance from the station.

Pressing \begin{align*}2^{nd}\end{align*} [MODE] will return students to the Home screen.

10. Train 2 has a steeper slope.

11. The slope is the speed or rate of change of distance.

12. a) \begin{align*}y-\end{align*}intercept of train \begin{align*}1\end{align*} is \begin{align*}80\end{align*}.

b) \begin{align*}y-\end{align*}intercept of train \begin{align*}2\end{align*} is zero.

13. This indicates their starting location.

14. train \begin{align*}1:d = 80t + 80\end{align*}, train \begin{align*}2:d = 120t\end{align*}

Also acceptable would be \begin{align*}y = 80x + 80\end{align*} and \begin{align*}y = 120x\end{align*}

15. \begin{align*}80t + 80 = 120t \rightarrow 80 = 40t \rightarrow t = 2 \ hr\end{align*}

Extension: Problem 2 – List of \begin{align*}d = r \cdot t\end{align*} Data

This extension introduces a numeric and graphical exploration. Students begin by determining what the next time and distance values would be in the list.

16. \begin{align*}5 \ hr\end{align*}, \begin{align*}25 \ km/hr\end{align*}

Students then are instructed to go to the data lists and change the data depending upon a different \begin{align*}r\end{align*} value. They are instructed to observe both the graph and the data when the \begin{align*}r\end{align*} value is changed.

17. When \begin{align*}r\end{align*} gets larger, the slope gets steeper.

When \begin{align*}r\end{align*} decreases, the slope is less steep.

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