Students are shown a topographic map of an oval hill and imagine that the classroom is on the hill. They are asked to point in the direction of the gradient vector appropriate to the point on the hill where they are "standing".
There is a hill in this classroom. The top of the hill is in the middle of the room at the ceiling. This topo map below describes the hill in the room. You are standing at some point on the topo map. Use your right arm to point in the direction of the gradient.
For this activity, the class is asked to stand from their seats. The students are told that they are all standing on an elliptical hill, represeted by the topo map, and one location of the classroom is selected as the top of a hill, typically in the center of the room. (If you are in a tiered lecture hall, then make use of the actual hill in the room, istead!)
The students are asked to close their eyse and point in the direction of the gradient.
Ask students to generalize the concepts in this activity to functions of three dimensions. Emphasize the understanding that the gradient is always perpendicular to the level curves (for two dimensions) or level surfaces (for three dimensions).
Reiterate the main points in "Student Conversations"