Effective questions when using dynamic graphs

Most graphing tools have a feature to add a variable constant in a way that can be changed dynamically. In Desmos and GeoGebra this is by using sliders; in Autograph it is via the constant controller. When the parameter is changed you can view its effect on the graph and this can be discussed with students. You can even animate the change so you don't need to do it manually!

This feature of being able to dynamically change a variable constant is a very powerful tool to help students build their understanding; however, to make the best use of it, it is useful to have some questions to ask to direct students' thinking.

Effective questions when using dynamic graphs

  • When I vary … how does it move and why?
  • For what value of … will …?
  • If I vary … how will it move?

Examples using the intersection of a line and a parabola

To exemplify this I'm going to use the intersection of the parabola y=x2 with the line y=mx+c. To do this I've plotted:

  • y=x2 
  • y=mx+c
  • (0,c) - this point isn't strictly necessary but it helps students "see" the motion of the line as vertical when c  is varied.


You can play with a dynamic version of this in GeoGebra (www.geogebra.org/m/hrthedxp) or Desmos (www.desmos.com/calculator/qweu6rky6k).
In all of the examples below I want the students to focus on the number of points of intersection of the line and the parabola.

When I vary … how does it move and why?

Question:
I've set m=2. When I vary c how does the number of points of intersection change and why?


For the responses to this I would expect students to be able to observe that the number of points of intersection can be 0, 1 or 2. I would hope that some would be able to link this to solving the simultaneously and using the discriminant of a quadratic equation. This would lead to directing them to thinking about the related equation x2−mx−c=0.

For what value of … will …?

Question:
I've set m=4. For what value of c will the line and parabola have a single (or repeated) point of intersection?
This is a less open question but when used as a follow-on from the previous type of question it allows for students to apply their understanding and obtain an answer. The answer can then be verified dynamically.

If I vary … how will it move?

Question:
I've set c=2. If I vary m how many points of intersection will there be between the line and the parabola? Will it be 0, 1 or 2 or will it be always 2?



This more demanding style of question requires the students to be familiar with dynamic graphs and should be used after they've experienced answering questions of the first two types a few times. Being able to predict how varying the constant will affect the graph is a very useful stage in supporting students with generalising.

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