# Engaging students: Ratios and rates of change

In my capstone class for future secondary math teachers, I ask my students to come up with ideas for engaging their students with different topics in the secondary mathematics curriculum. In other words, the point of the assignment was not to devise a full-blown lesson plan on this topic. Instead, I asked my students to think about three different ways of getting their students interested in the topic in the first place.

I plan to share some of the best of these ideas on this blog (after asking my students’ permission, of course).

This student submission again comes from my former student Avery Fortenberry. His topic, from Algebra: ratios and rates of change.

In this viral YouTube video a man asks his wife the question “If you are traveling 80 miles per hour, how long does it take to travel 80 miles.” The wife overthinks the question and instead of trying to calculate how long it would take using the information of 80 miles per hour and how that they were going to travel one hour, she tries to think of how quick the tires are spinning and estimating the speed using her speed in running. The couple later goes on to talk on the Comedy Central show Tosh.0 where the wife explains the reason she was confused was that she had not slept well the night before and she was stressed with just finishing her finals. This video stresses the importance of making sure people understand that 80 miles per hour means you travel 80 miles in one hour.

The history of a rate of change is interesting when you consider the history of calculus itself. An important concept of calculus is finding derivatives, which is finding the rate of change or slope of a line. Calculus’s discovery was credited to both Isaac Newton and Gottfried Leibniz who both published their work around roughly the same time. This caused a dispute between the two men and they both accused the other of stealing their work. While both contributed much to the world of mathematics, it was many of Leibniz’s concepts of calculus that we still use today such as his notation dy/dx used for derivatives. Despite that Leibniz died poor and dishonored while Newton had a state funeral.

One of my favorite websites is khanacademy.org. This website has helped me from when I was in high school all the way to now it is still helping me understand concepts I may not have fully understood in class. It is a valuable resource to use when teaching about rates of change because there are countless videos over rates of change and slope and derivative that explain in detail all the concepts of it. Also, it has multiple practice problems that help you practice and study for an exam. I even used it for this project to help refresh my memory on rates of change and I was also looking at its word problems to help think of a word problem on my own for the A1 section of this project. Khan Academy also teaches you by reviewing all difficult steps in problems so that you can understand all the concepts.

Resources:

http://www.uh.edu/engines/epi1375.htm