# 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. 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.