I was kicking myself while grading Algebra I tests and realized I had not sufficiently emphasized how to interpret average rate of change for my students…I can’t count that question against them, for sure! So now I’m on a mission to fix that. In the process, I discovered that my biggest mistake was failing to carefully distinguish between the words, slope, rate, rate of change, average rate of change, and constant ratio. I’m not a big fan of drilling vocabulary. I prefer to teach it like some foreign language courses do: immersion. But students need to make sense of and clearly distinguish between those words if they are to truly understand the concepts behind them.
Slope: rate of change between two points. It is consistent on a line.
Rate of change: how one quantity changes with respect to another. On a non-linear curve, the rate of change, changes. Depending upon which two points on the curve we choose to calculate the slope, we can get dramatically different rates of change.
Average rate of change: Since the rate of change continuously changes along a non-linear segment of a curve, we cannot say, “The slope or rate of change is ____.” In calculus we would say, “The slope of the tangent is ____.” In algebra, we say, “The average rate of change on this interval or between these two points is ____.”
Rate vs. constant ratio. If the moles in my backyard double every week and I begin with three, 3(2)x implies the constant ratio is 2. If I say the growth rate is 50%, I would write 3(1.5)x; and the constant ratio of a 50% growth rate is 1.5.
Now that I see what all I have neglected, it makes sense why I hadn’t put all that together. Seriously, am I the only one? I see my boo boo. I’ll fix it. I can defend my ignorance.
Oops 1: Khan Academy did not properly link a related session to their search filter
Fix 1: I put in a trouble report
Oops 2: I failed to reflect on (and then teach) all the connections.
Fix 2: I designed a Peardeck to orchestrate a whole class discussion with informal assessment. I can share that if you tweet me @LaneWalker2.
In order for students to be able to sort out the difference between slope, rate of change, average rate of change, rate, and ratio such they can respond to random questions in contexts, this analytical skill we will need to revisit this more than once. So…
Fix 3: Have regular warm-up questions involving slope, rate, rate of change, average rate of change, and constant ratio.
This will probably take many weeks to fix, but it is doable. In the meantime, my confession here feels great.