In this sequence of small whiteboard questions, students are shown the contour graph of a function of two variables and asked to find the derivative. They discover that, without a function to differentiate, they must instead think of the derivative as a ratio of small changes. This requires them to pick two nearby points. Which two?

• The gradient is perpendicular to the level curves.
• The gradient is a local quantity, i.e. it only depends on the values of the function at infinitesimally nearby points.
• Although students learn to chant that "the gradient points uphill," the gradient does not point to the top of the hill.
• The gradient path is not the shortest path between two points.

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".

Students work in groups to measure the steepest slope and direction at a given point on a plastic surface and to compare their result with the gradient vector, obtained by measuring its components (the slopes in the coordinate directions).

Students use prepared Sage code to predict the gradient from contour graphs of 2D scalar fields.

This small group activity using surfaces relates the geometric definition of directional derivatives to the components of the gradient vector. Students work in small groups to measure a directional derivative directly, then compare its components with measured partial derivatives in rectangular coordinates. The whole class wrap-up discussion emphasizes the relationship between the geometric gradient vector and directional derivatives.