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## Surfaces: Instructor's Guide

### Main Ideas

- Introduction to graphs of functions of 2 variables.

### Students' Task

*Estimated Time: 15–30 minutes*

Students work in groups to interpret features of their model surface as properties of the function of which the surface is the graph.

### Prerequisite Knowledge

- None

### Props/Equipment

- Tabletop Whiteboard with markers
- A handout for each student

### Activity: Introduction

A good introduction to this activity is a SWBQ asking about representations of functions of several variables. Among the desired responses: graphs, level sets, tables of data, symbolic representations.

It is worthwhile to have students match their surfaces to the corresponding contour map on their own and determine the correct placement prior to beginning the activity proper. This can easily take 15 minutes.

### Activity: Student Conversations

Some students will wonder where to put the surface on their contour map, and how to orient it. That's the point!

### Activity: Wrap-up

Discuss what is needed to answer the questions: An origin, a choice of $x$ and $y$ directions, and a scale, collectively referred to as a choice of coordinate system.

The point of the (optional) “Get Set” part of the activity is to introduce the notion of *traces* (this word does not need to be used), and a possible notation for them, such as $f(x,3)$ and $f(2,y)$.

### Extensions

This activity leads naturally into the Park activity, especially if the last part (“Go”) is omitted. An introduction to integration may be appropriate prior to the Park activity.