Student handout: The Hillside (Updated)

Problem-Solving 2023
Students work in groups to measure the steepest slope and direction on a plastic surface, and to compare their result with the gradient vector, obtained by measuring its components (the slopes in the coordinate directions).
What students learn
  • The gradient is a geometric representation of the steepest slope and direction.

Warm-up: Imagine you are standing on the side of a tall hill. List three things you would want to know about your position.

On your Mark: The surface represents the hill's height \(h\). If \(x\) and \(y\) (and \(h\)) are measured in feet, measure the steepest slope possible at the blue dot. Include units.

Steepest slope: \(\underline{\hspace{2in}}\)

Get Set: Measure \(\frac{\partial h}{\partial x}\) and \(\frac{\partial h}{\partial y}\) at the blue dot. Then form the vector \(\frac{\partial h}{\partial x}\,\boldsymbol{\hat{x}} + \frac{\partial h}{\partial y}\,\boldsymbol{\hat{y}}\). Include units.

\(\frac{\partial h}{\partial x} = \underline{\hspace{.75in}} \hspace{.5in} \frac{\partial h}{\partial y} = \underline{\hspace{.75in}} \hspace{.5in} \frac{\partial h}{\partial x} \,\boldsymbol{\hat{x}} + \frac{\partial h}{\partial y} \,\boldsymbol{\hat{y}} = \underline{\hspace{.75in}} \,\boldsymbol{\hat{x}} + \underline{\hspace{.75in}} \,\boldsymbol{\hat{y}}\)

Go: Find the magnitude of \(\frac{\partial h}{\partial x} \,\boldsymbol{\hat{x}} + \frac{\partial h}{\partial y} \,\boldsymbol{\hat{y}}\).

Challenge: Rotate the surface some amount on the grid. Redraw the \(x\) and \(y\) directions extending through the blue dot, and redo On your Mark, Get Set, and Go. Did anything change? Stay the same? Why?

Copyright 2014 by The Raising Calculus Group

Learning Outcomes