• Interpreting derivatives at points on a surface
• Measuring derivatives at points on a surface
• Ranking points on contour maps by the magnitudes of directional derivatives at each point

Estimated Time: 30 min

See handout on activity page.

Students should be able to:

• Read values from a contour map
• Approximate the derivative as a ratio of small changes

• Tabletop Whiteboard with markers
• Surface Kit
• Enlargements of surface's Contour Map in flexible, dry-erasable plastic sleeves
• A handout for each student

None unless this is the first time students have used the plastic surfaces.

This activity is a good one for students to practice presenting different parts of their solutions. Valuable conversations that can be add on an individual group basis or as part of the presentations are:

• Does the “height” of the surface actually represent height or some other quantity?
• What are several different ways of observing whether a particular partial derivative is positive, negative, or zero?
• What are the advantages of the measuring tool? What are the limitations?
• How can you tell qualitatively when a partial derivative is large vs. small?
• Some students may conflate the value of the function with its derivative. Others may think that the derivative is large when the slope is changing by a lot. Ask students to distinguish between these concepts.
• If you are using handouts for the contour maps, students may discover that the contour maps can be folded to make an approximation of a surface. This can be tricky for the third contour map (using something made of cloth and using multiple people can help!).

Some students may not draw their origins and label their axes without prompting.