I always have that one student in school who wants to know why it is important to learn about slope. After all, we never ever use it so why do we have to learn it.
I always ask them why they post the grade on steep hills in the mountains? Why do truckers need to know that they are about to head down a road that is 6% grade? That usually has students stopping to think. Unfortunately, it doesn't always work with students who have very little experience driving down a real road.
The use of slope, grade, or pitch is a wonderful way to introduce the rate of change. Instead of saying the mountain has a 6% grade, put that in actual circumstances such as if you travel 100 feet, you will go down 6 feet. That gives it the real world context that is often missing when you discuss slope. Slope and rate of change need to be discussed together to establish the relationship.
This article "Making Sense of Slope" has lots of great real world applications that show a variety of contexts of slope. It has everything from the cost of a watermelon, rate of earnings based on a per hour rate, growth of cable television, etc. The article is very detailed with diagrams to show the rate of change. It even addresses the question of a rate of change that is not constant.
This site has a lovely 35 page packet including the slides with great real world information including information on different types roofs with links to sites that explain how to calculate pitch. It includes information on road grades for roads, curbs, ditches, wheelchair ramps and it brings in the Americans with disabilities act. In addition, it addresses avalanches, skiing, and related topics. At the end, there is a great worksheet for people to calculate rise over run for a specific gradient.
I plan to take a week to assign students a chance to create a presentation such as a video or slideshow to show real life examples of rate of change. They can use examples found around town such as the microwave tower for undefined slope, or the foundation of someone's house for a zero slope. It will require them to apply their knowledge of slopes to create the presentation.