Lisa Brock
Assignment 2
Doing Demonstrations Using
Graphing Calculator
When teaching students, is
it better to graph 5 graphs at once or produce the graphs in sequence, adding
one at a time?
Producing the graphs one
at a time is better for student understanding than producing a bunch of graphs
at once. Say a teacher would like
students to explore the equation y=(x-h)2+1 for different values of h. Let h=-2, -1, 0, 1, 2. If all 5 graphs at once, the students see the following
picture:
This graphic is rather
confusing to a student in Algebra I.
Remember, the students do not have as much mathematical experience as
the teacher. To really develop understanding
using this software, it is essential to add one graph at a time. Let’s look at how a teacher might
introduce the graphs one at a time.
Begin by letting h=0. The
equation becomes y=(x-0)2+1. Graph this equation.
Ask the students what they
notice about the graph. The may notice
that the vertex is on the y-axis or that the parabola is concave up. Now let h=1. The
equation becomes y=(x-1)2+1.
Before graphing this equation, ask the students to predict what will
happen to the graph. Now graph the
equation.
The students can now see
that changing h to 1 shifts the
graph to the right one unit. Have
students compare this result to their prediction. Have students hypothesize as to why this occurs.
Now let h=2. The
equation becomes y=(x-2)2+1. Before graphing this
equation, ask the students to predict what will happen to the graph. The students should suspect that the
graph will shift one more unit to the right. Now graph the equation.
And the graph does shift
another space to the right. Now
let h=-1. The equation becomes y=(x-(-1))2+1 or y=(x+1)2+1.
Before graphing this equation, ask the students to predict what will
happen to the graph. The students
will probably suspect that the graph will shift one unit to the left compared
to the red parabola. Now graph the
equation.
The graph moves one space to the
left of our original parabola. Now
let h=-2. The equation becomes y=(x-(-2))2+1
or y=(x+2)2+1. Before graphing this equation, ask the
students to predict what will happen to the graph. The students will probably suspect that the graph will shift
one more unit to the left. Now
graph the equation.
Producing the graphs one
at a time yields the same picture as graphing all 5 graphs at once. When the graphs are produced one at a
time, the students get the chance to make predictions and the time to ponder
how changing the equation affects the graph.
A teacher can repeat this
process with the equation y=a(x-h)2+k by varying a or k. Once the students are comfortable with how a, h, and k
affect the graph of y=1(x-0)2+0. By investing time in
the students understanding these effects, students will be able to quickly
graph equations of this form. The
position of the graph will make sense to the students. Students can also quickly determine the
number of real roots a quadratic has by knowing where the parabola would be
positioned on the graph.
That kind of understanding
is not developed when the different values for h are graphed all at once.
Note that this process can
be done to explore other types of two variable equations. Graphing calculator also graphing in
3-dimension, so this process can also be done to explore three variable
equations.
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