# Lessons One of the guiding principles for this project was that the models should be designed with specific in-class student interactions in mind. That is, the models and lessons should be designed in tandem. Below are some sample curricular materials for lessons in which the models were used.

Instructors: Please email Stepan Paul (spaul@math.harvard.edu) or Janet Chen (jjchen@math.harvard.edu) for additional materials.

## Sample Problems

Below are PDFs of several problems designed to be given to students along with the 3D models. For instructor notes, solutions, and source files, please email spaul@math.harvard.edu. See Models page for information about the models used in each lesson.

• Introduction to multivariable functions (PDF, instructor notes, solutions lesson reflection). Uses the function graph model, K'Nex kit, dry-erase markers.
• Traces of a saddle (PDF). Uses the saddle surface, K'Nex kit, chalk.
• Surface from level sets and a trace (PDF). Uses K'Nex kit.
• Match the solids to descriptions in cylindrical and spherical coordinates (PDF). Uses the three solids.
• Partial derivatives from a graph (PDF). Uses the function graph model, K'Nex kit, dry-erase markers
• Triple integral over a solid (PDF). Uses the parabolic solid, K'Nex kit, chalk.
• Triple integrals in cylindrical and spherical coordinates (PDF). Uses the three solids.
• Tangent line to a graph (PDF). Uses the function graph model, K'Nex kit, dry-erase markers
• Parameterize the paraboloid (PDF). Uses the paraboloid surface, K'Nex kit, chalk.
• Review problems (PDF). Uses the parabolic solid, K'Nex kit, chalk.
• Introduction to local and global extrema (PDF). Uses the function graph, K'Nex kit, dry-erase markers.
• Introduction to Lagrange Multipliers (PDF). Uses the function graph, K'Nex kit, dry-erase markers.
• Stokes' Theorem Example (PDF). Uses the paraboloid surface, chalk.
• Stokes' Theorem Example 2 (PDF). Uses the saddle surface, K'Nex kit, chalk.