This example illustrates
how using a dynamic geometrical representation can help students develop
an understanding of vectors and their properties, as described in
the Number and Operations
Standard. Students manipulate a velocity vector to control the
movement of an object in a gamelike setting. In the first part, Components
of a Vector, students develop an understanding that vectors are composed
of both magnitude and direction. In this part, Sums of Vectors and
Their Properties, students extend their knowledge of number systems
to the system of vectors.
Task
Today you will be
directing an airplane, much as you directed the car in the previous
part. You will notice a red vector representing wind on the screen.
Use the blue vector to direct the airplane to catch the hurricane. How
does having a wind change the game?
Now pretend that
you are able to control the wind. By adjusting the red wind vector,
"blow" the airplane to catch the hurricane. Make one or more observations.
Now turn off
the Show Hurricane feature. Start the plane and explore the following:
Turn on the Show
Vector Sum option. A black, "sum vector" will appear that
you cannot directly control. Start the plane and begin moving it around
the screen using the red or the blue vectors. What relationship does
the sum vector have to the plane? How does adjusting the red and blue
vectors affect the sum vector?
Look at various
lengths and angles of the three vectors. Can you find a pattern? What
happens when you increase the length of one of the vectors? Increase
its angle? In what cases can you exactly predict the values for the
sum vector from the values for the red and blue vectors?
Using their midpoints,
arrange the three vectors so that they form a triangle. Adjust the length
of one of the vectors and again form a triangle. What does the triangle
that is formed tell you about the relationships among the three vectors?
In what way is the
sum vector a sum?
Adjust
the red and blue vectors so that the plane is stationary. What do
you notice about their directions and magnitudes? Find other values
for the two vectors that keep the plane stationary.
Try the following:
Adjust the red vector
so that its magnitude is about 5 and its direction is close to 45º.
Adjust the blue vector so that its magnitude is about 3 and its direction
is close to 90º. What are the magnitude and direction of the sum,
displayed at the lower right of the screen?
Now
reverse the values so that the blue vector has magnitude 5 and direction
45º and the red vector has magnitude 3 and direction 90º.
What are the magnitude and direction of the sum?
Try
interchanging other values for the red and blue vectors and make an
observation. What do you observe? How does it relate to another property
you've seen before?
Discussion
High school students
should not only explore more formally the properties of number systems
that they have already encountered in the lower grades but also experience
how those properties extend into new systems, such as vectors. (See the
Number and Operations Standard.)
Building on the experiences students had with a single vector in the first
part, this part gives them a second vector, representing the velocity
and direction of the wind, and asks them to consider how the two vectors
combine to affect the movement of the plane.
The third task allows
a more formal look at vector addition by showing the vector representing
the sum of the two velocity vectors. Students can see how changes in one
of the summands affect the sum; for example, as the angle of the red vector
is increased, the angle of the sum vector also increases. However, the
relationship between the summands and the sum is not easily seen unless
the summands have the same angle measure (in which case the lengths are
added) or inverse measures (in which case the lengths are subtracted).
In the fourth task,
students can notice that vectors with opposite directions and the same
magnitude cancel each other out. In other words, when "added," they
result in the identity, implying that they are inverses of each other.
Likewise, students should note in the fifth task that if the values of
the two vectors are interchanged, the same sum results. Thus, vector addition
is commutative.
Take Time to Reflect
Turn on the grid. What
additional questions could students explore about vectors
or vector addition if the grid is present?
Arrange
the vectors tail to tail by placing all three vectors so they
share a common initial point. How could this arrangement be
used to represent vector addition?
Arrange
the vectors head to tail by placing a velocity vector and
the sum vector so they share a common initial point and then
place the other velocity vector so its initial point is at
the terminal point of the other velocity vector. How could
this arrangement be used to represent vector addition?
Which arrangement
do you prefer: tail to tail or head to tail? How do the two
diagrams differ in the conceptual understanding of vectors
and vector addition that a student might perceive?
How might
the diagram of each arrangement contribute to students' development
of force?
Which
diagram emphasizes instantaneous rates of change? Discrete
rate of change? Forces acting on an object?
