Hands-on Visualization in Preparation for Computer Graphics

Julie Barnes and Kathy Ivey, Western Carolina University

When we read the description of your session, it made us think of some of the hands-on activities that we do in class to help students truly ‘feel’ what a graph is telling them.  These are low tech ideas that mostly use no more than some string, masking tape, and children’s toys, and they can be modified for use in a wide range of mathematics classrooms.  They are activities that we do prior to using technology because it helps build a foundation for students to discuss the graphics when they do generate them.

Here are some examples of these activities:

Algebra Aerobics:  In precalculus or college algebra, we introduce translations by taping an x – y axis on the floor, placing students on the graph to form a function, and then having students walk around the function according to translation rules.   We act out things like f(x+1), f(x)+1, 2f(x), f(-x), etc.  Students have said that during exams they stop and think about how it feels to add something to a function before using the related graph.  In another related activity, we have sometimes taped a yarn function to the floor and walked on it to determine if it had an inverse, if it was continuous, if it was even, etc.

Spider Webs:  In Calculus 3, we put up a three dimensional axis system in the middle of the room with yarn.  We walk around the room talking about where we are, what a line or cylinder would look like, and how we would write an equation.  Finally, we plot cross-sections and vertical slices of f(x,y) = x^2 + y^2 by having the students hold yarn appropriately.  When we move to the computer images of these functions, students often talk about how it would feel to walk around on the graph.  Some students have said that when they look at a graph, they close their eyes and remember where they would be standing in the room before they can really understand what the graph is telling them.

Mountains:  When studying contour lines in Calculus 3, we ask students to use Play Doh to build what the surface looks like from simple contour line sketches.  We also bring in photographs of mountain regions and topo maps that correspond to the photographs.  Students have to match each photograph with its map.  Later when discussing computer graphics, students often talk about contours in terms of what a mountain looks like, or what it would feel like to model it with Play Doh.

Epsilon – Delta Games:  In Real Analysis, we tape a function to the floor and use yarn to make epsilon regions.  Then students create delta regions with more yarn.  The major benefit here is that even though we are focusing in on local properties, students can still see the big picture because they are standing on the full graph while zooming in with the yarn.  After experimenting with the yarn, students seem more able to point to graphs and talk about epsilon and delta regions more easily.

We have more activities like these, but we realize that talks in your session are only 10 minutes long.  If we are chosen to speak in your session, the majority of the talk would consist of showing photographs of students doing some of the above activities while explaining why the activity is useful.