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
[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
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
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
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
You can use the Varying Motion
sketch projected to the front of the room to help guide the discussion
of the current graph as well as graphs created later in the lesson.
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
|m = ||rise|| or m = ||y2 – y1 |
|run||x2 – x1 |
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. 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
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
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.