# Rate of Change

I often promote the idea that if a concept is important, we should teach it more than once, and preferably in more than one way. Rate of change is one such concept. It can be approached in various ways from middle school to calculus. Is there anything to add to the oft-repeated “rise over run” mantra? Yes! Here are ten ideas from this blog, and from my website.

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. Many, many 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.

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 + 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. Pattern Block Trains — starts easy, gets quite challenging.

9. Slope Angles, an introduction to trigonometry in Geometry LabsDo Lab 11.2 before 11.1.

10. For teachers and math nerds: formal proof of ymxb using basic geometry.

You may not have time to check all of this out today. That’s OK, there’s no rush. This is a topic you’ll be teaching for a long time. Diversify your toolbox!

— Henri