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Acting Out The Gradient: Instructor's Guide

Main Ideas

  • Geometric understanding of the directionality of the gradient vector.
  • Clarify misconceptions about the phrase “the gradient always points up hill”

Students' Task

Estimated Time: 5-10 minutes

Prerequisite Knowledge

A rudimentary understanding how a gradient acts on a scalar field.

Props/Equipment

Activity: Introduction

If the instructor stands on a chair or table in the center of the room. The top of the hill would be above the instructor's head. If this is not an easy thing to do, pick a point or student in the center of the room. Declare the top of the hill to be above the point you selected.

  1. 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, and one location of the classroom is selected as the top of a hill.
  2. The students are asked to point in the direction of the gradient.
  3. Many students will commonly point upward and towards the top of the hill, despite the gradient only lying in the x-y plane.
  4. Some students will also point parallel with the floor, but in the direction of the top of the hill. Depending on where they are on the ellipse, the gradient is not always the direction of the top of the hill.

Activity: Student Conversations

  1. After the students all point initially, ask which students are pointing at the top of the hill. Clarify that they are actually pointing upward or directly at the top of the hill. Then ask these students to lower their arms. Then ask a student who's pointing correctly why they're not pointing directly at the top of the hill. A follow up question to another student pointing correctly is: “Why is your arm parallel to the floor and not pointing at an angle?”

Activity: Wrap-up

The understanding that the gradient is always perpendicular to the level curves(for two dimensions) that they lie on. This is regardless of where the global maximum is of the function.

Extensions

This activity is part of a sequence of activities which address the Geometry of the Gradient. The following activities are included within this sequence.

  • Follow-up activities:
    • Navigating a Hill: In this small group activity, students determine various aspects of local points on an elliptic hill which is a function of two variables. The gradient is emphasized as a local quantity which points in the direction of greatest change at a point in the scalar field. This activity emphasizes the gradient as a local quantity and requires students to perform calculations on a given scalar field.
    • Visualizing Gradient: This activity uses a Mathematica (or Maple) worksheet to visualize the relationship between scalar fields and the gradient.

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