## Varying Motion

• Lesson
9-12
3

This lesson helps students clarify the relationship between the shape of a graph and the movement of an object. Students explore their own movement and plot it onto a displacement vs. time graph.  From this original graph, students create a velocity vs. time graph, and from there create an acceleration vs. time graph.  The movement present and how to interpret each type of graph is emphasized through the lesson, which serve as an excellent introduction to building blocks of calculus.

To introduce the lesson, look at the Velocity of a Car Overhead as a class. The overhead shows a flat velocity vs. time graph, which should be used to scaffold understanding of more complex velocity vs. time graphs.

While examining the graph, ask students questions like those below. The terms displacement, velocity, and acceleration are crucial to understanding the lesson. These terms are defined later in the lesson, so do not stress the terms here. If you know your students are unfamiliar with the terms, modify the questions below to distance, speed, and change in speed.

• What is this graph showing?
[The graph is showing the velocity of a car vs. time.]
• What is happening to the velocity of the car?
[The car is moving at a constant velocity.]
• What is happening to the acceleration of the car?
[The car is not accelerating, meaning that it has zero acceleration.]
• What is happening to the displacement of the car?
[The displacement of the car is increasing over time.]
• How might you determine the car’s displacement after 1 hour? Look closely at the units on the graph to help you answer this question.
[By looking at the units, students may be able to determine that the units of the x-axis and the units of the y-axis, when multiplied, give you the units of distance, mi/hr × hr = mi. This can then be related to the area of a rectangle, which is equal to length × width, or 55 mi in this example.]

Graphs are not always nice straight lines. What happens if the graph line is a curve? How would we determine the area under the graph? Students can explore what the best way to estimate areas under a curve using the Estimating Areas overhead. A suggestion would be to have students work in small groups to attempt to figure out the areas under the graph for the curve provided. Groups could then share their methods with the class before moving on to the next part of this activity.

Some of these methods may include (but are not limited to):

• Subdividing the area under the graph into rectangles with the top of the rectangle passing through graph line at the midpoint of the top.
• Subdividing the area under the graph into rectangles with the top of the rectangle touching the graph line at the top, right vertex of the rectangle.
• Subdividing the area under the graph into regular geometric shapes and calculating the areas of each shape.
• Counting the approximate number of grid squares under the curve.

Before moving into the student activity, it is wise to point out the difference between distance and change in position, or displacement. This is discussed in the beginning of the Varying Motion activity sheet. Depending on the courses completed by students, they may not have an understanding that displacement has a measured value along with a direction. Distance, however, is simply the measured value without a direction. It is not possible to have a negative distance, as it is simply a number and unit, but it is possible to have a negative displacement, as you could be moving in a direction that you have defined to be negative. For example, if you define north as positive and move to a position south of your starting point, your displacement is negative.

Similarly, there is a difference between speed and velocity. Velocity is the rate of change of your displacement, so it has a measured value, unit, and direction. Speed is simply the rate of change of distance, so it has a unit but no direction. Read through the example on the activity sheet as a class.

Your students are now ready to collect their own data showing varying motion. Place students in groups of 3. Assign each student to the role of timer, walker, or marker, as defined on the activity sheet. Each group will need a long walking space like a hallway, gymnasium, or sidewalk outside.

Have students place a piece of tape on the floor, marking their starting line. As one person walks, the timer will call out 4 second intervals. The marker will place a piece of tape on the floor where they determine the walker’s foot is at the moment the timer calls out. The walker should vary the rate they are walking at, alternating between slower and faster walking speeds. It is important that the walker does not move so quickly that the marker can’t keep up. The walker should always walk in the same direction to avoid over-complicating later analysis. This should continue until 60 seconds have elapsed (or until they run out of walking room). If time allows, you can have one group demonstrate the procedure to ensure all students understand their individual roles.

As a group, students are to go back and measure the displacement from the starting line to each of the tape markers. Note that it is not the distance between marker tapes that is important; it is the distance from the starting line to each tape marker that must be measured. In the image below, the distances marked, 6 seconds, 12 seconds, and 18 seconds would be measured in feet or meters. The values are noted on the Varying Motion activity sheet.

After the class, returns to the classroom, they can begin work on the activity sheet questions. In Question 2, if students cannot identify the independent and dependent variables, you can tell them that time is the independent variable, while displacement is the dependent variable. However, first encourage them to realize this for themselves. The displacement depends on how much time has elapsed since the walker started moving, while time is independent of the student's displacement.

Once a displacement vs. time graph has been created, students can answer specific questions from their activity packages. In general, the displacement graph will be some kind of curvy line. A suggestion would be to use one group’s graph and create an overhead as an example for class discussion.

Begin the discussion by asking students how far the walker traveled in one minute. Draw a line from the beginning position of 0 to the final position. From there, ask them what the average velocity was in that one minute? They may answer in the form of a certain number of ft/min (or m/min) or simply suggest ways to find it. If students connect the ends of the graph with a straight line, the slope of that line represents velocity. Guide students to looking at the slope of the drawn line as being the average velocity. Point out that the displacement vs. time graph is not a straight line, and consists instead of curves that are more and less steep. Therefore, the slope of the line between the ends is only an approximation of the average velocity.

One observation from the displacement vs. time graph should be that the steeper the slope of the curve, the faster the person was walking during that time interval. Have students recall the slope of a line:

or

What is the significance of the slope of the displacement vs. time graph?

[If students are able to see the significance of statements like faster and slower, they can conclude that the slope is, in fact, equal to the velocity of the walker.]

Students may comment that their graph is a curve not a straight line. How can you calculate the slope of a curve?

[To be able to determine velocities at given times, students must create tangent lines to the graph at certain points. By definition, a tangent line is a line that touches a curve at one point and whose slope is equal to that of the curve at that point.]

Tangent lines can be taught or reviewed using the Drawing a Tangent Line overhead. It may be helpful to use the overhead in conjunction with the Varying Motion Sketch (you will need Geometer's Sketchpad).

To draw a tangent line:

• choose a point on a curved graph.
• use a straight edge and touch the point on the curve.
• adjust the angle of the straight edge so that it is ‘equidistant’ from the curve on either side of the point in the immediate vicinity of the point.

The tangent line will have a large slope if they chose a point where the curve is steep; and have a small slope if they chose a point where the curve is more level. A tangent line lets students assign a number to the steepness of the curve. Slope is a measure of steepness.

Have students determine the units of slope. They should note that these units are ft/s (or m/s if data was measured in meters). These are the units of velocity. Because the velocity was calculated at a specific second, or instant in time, it is called an instantaneous velocity.

As students continue the activity sheet complete Questions 5 to 8, they will create a velocity vs. time graph, where time is still the independent variable and velocity is dependent. Again, the velocity graph will be some kind of curvy line. Using a student group example on overhead, allow students to make observations. One observation by students could be that the rates of change in the velocity graph are not as large as the rate of change in the displacement graph. This is because the walker was a human being and not a motorized vehicle, and so it is difficult to create a wide range of velocities. Another observation is that there are times when the slope is negative, which did not occur in the displacement vs. time graph. This is because the walker was slowing down, or decelerating in that time period.

An additional question to explore is what the slope of the line is representing on a velocity vs. time graph. The slope is representing the acceleration, or change in velocity, of the walker. Therefore, a tangent line at a point on the curve would represent the instantaneous acceleration.

The final portions of the activity sheet have students create and analyze an acceleration vs. time graph, much like they did for velocity vs. time, and finally analyze the area under the curve. In the Measuring Area section, as students analyze the units and see that they are multiplying the independent and dependent variables, they will realize that the area under the curve represents the displacement of the walker. They can reflect back on how to do this from the Estimating Areas overhead from the beginning of the lesson. Once students have calculated the area under the curve for various times, they can then compare their calculated displacements to the displacements they read directly from the displacement vs. time graph.

As a final reflection, have students share their answers for Question 15 on the activity sheet. If the numbers for the area under the graph and the reading of displacement are not exactly the same, what might account for the difference? The readings taken directly from the graph and the calculated values for change in displacement will not be exactly the same because tangent lines were drawn as an estimate of instantaneous velocity to generate the velocity vs time graph, and then the area under that curve was estimated to calculate a displacement. These two estimates would have created error.

### In the Classroom

Assessment Options

1. Ask students to fill in the following table, leaving the words in brackets blank:
 Type of Graph Reading Directly Slope of the Graph Area under the Graph Displacement vs Time [Displacement] [Velocity] [N/A] Velocity vs Time [Velocity] [Acceleration] [Change in displacement] Acceleration vs Time [Acceleration] [N/A] [Velocity]

2.  Have students complete a journal reflection including answers to the following:

• What math concepts were completely new to me in this lesson?
• What concepts did I find easy to understand?
• What ideas do I not understand fully?
• What can I do to help me increase my understanding of these ideas?

Extensions

1. Students can calculate the area under the graph for the acceleration vs. time graph to confirm that it does, in fact, generate similar numbers to those found on the velocity vs. time graph.
2. Use the Varying Motion sketch to have every student estimate the slope of a tangent line when x = 5. Compare these values as a group. This comparison could be turned into a box-and-whisker plot, using the Mean and Median applet. This will emphasize that error is built into estimating tangent lines and slopes from a curved graph.
Mean and Median Applet
3. Have students walk backwards during some parts of the activities. Explore what happens to the graphs. This would allow students to see negative velocities and displacements within certain time intervals.

Questions for Students

1. Describe a situation where a person has positive displacement, positive velocity, and negative acceleration.
[Answers may vary. Referring to their activity, if a person was moving forward but slowing down, they would be in this situation. A real world example of this would be the 1st leg of the 100m sprint relay team after they have handed off their baton. Runners must stay in their leg and will often continue running in a positive direction behind the next runner, slowing down as they do.]
2. Can a person have a negative displacement, positive velocity, and positive acceleration? Describe how this could have happened during this activity.

[This is possible. The explanations may vary. Referring to their activity, if a person started behind the start line and moved forwards, increasing their speed, they would be in this situation. A real world example of this would be the 2nd leg of a 100m sprint relay team — they start running before their start line, and accelerate before getting the baton and crossing the line that marks the end of their hand-off zone.]

Teacher Reflection

• How did the students demonstrate understanding of the materials presented?
• What were some of the ways that the students illustrated that they were actively engaged in the learning process?
• Did students explore a variety of ways to determine the area under the curve? What methods seemed to be used most efficiently?

### Other Related Resources

• Success Story

### Learning Objectives

Students will:

• Collect and graph data.
• Use slopes of tangent lines to create graphs of instantaneous velocities and instantaneous accelerations.
• Use the area under a graph line to calculate velocities and displacements at specific moments in time.

### NCTM Standards and Expectations

• Analyze functions of one variable by investigating rates of change, intercepts, zeros, asymptotes, and local and global behavior.
• Approximate and interpret rates of change from graphical and numerical data.
• Apply informal concepts of successive approximation, upper and lower bounds, and limit in measurement situations.

### Common Core State Standards – Practice

• CCSS.Math.Practice.MP5
Use appropriate tools strategically.
• CCSS.Math.Practice.MP7
Look for and make use of structure.