## 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

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Learning Outcomes