# Rate of Change

Multiple Representations

Henri Picciotto

If a concept is important, we should teach it more than once, and preferably in more than one way. Rate of change is important! Is there anything to add to the oft-repeated "rise over run" mantra? Yes! Whether you're teaching middle school, or calculus, or anything in between, you should be able to find some useful ideas by clicking these links.

1. A puzzle based on an unsolved problem: No Three on a Line.
2. A geoboard lesson in Geometry Labs 10.2 (possibly preceded by 10.1).
3. Lessons in Algebra: Themes, Tools, Concepts (ATTC), especially in chapter 8. Here is a list of all the relevant lessons: 2.9, 3.8, 4.4, 4.5, 4.8, 4.11, 5.C, 6.8 , 8.1-8.A, 8.8, 8.9, 9.2, 9.A, 9.C, 10.3, 10.6, 10.8, 11.3, l2.A, 12.5, 12.8. Many involve “real world” scenarios and multiple representations.
4. Some of the ATTC lessons involve function diagrams. Go to the Function Diagrams home page for a lot more along those lines, including a lesson about operations for middle school, all the way to a good way to visualize the chain rule. Once you get the basic idea, see also Kinesthetic Function Diagrams.
5. Slope triangle puzzles in a fun applet: Stairs
6. Make These Designs using y = mx + b in an electronic grapher.
7. A different visualization of rate of change in a set of related applets, culminating in a challenging set of exercises: Doctor Dimension.
8. "Slope Angles", an introduction to trigonometry in Geometry Labs. Do Lab 11.2 before 11.1.
9. Pattern Block Trains — starts easy, gets quite challenging.
10. For teachers and math nerds: formal proof of y = mx + b using basic geometry.