Connect the sign of a graph's slope to the increasing or decreasing behavior of a function and the value of the slope to the function's rate of change.
Use the concavity of a function's graph to describe the change in the function's rate of change and vice versa.
Quick Lesson Plan
Today students use a simplified model of the business cycle to discuss a function’s concavity. While students already have some intuition about a function’s slope representing a rate of change, today we are exploring the idea that a function’s concavity represents how the rate of change is changing.
In this activity students interpret various points on the graph and describe in context what is happening with GDP. Note that scales on the x- and y-axis have been purposefully left off so students can focus on the conceptual ideas of rate of change, rather than actually computing it. The questions are designed to get students connecting graphical features (slope, concavity, intervals of increasing/decreasing) with contextual observations (GDP increasing or decreasing as signaling expansion or recession, the rate at which GDP is changing over time). The margin notes will layer on the new vocabulary words.
Can you think of a time when the U.S. economy was at a peak? Can you think of a time when the U.S. economy was at a low point (trough)?
When is GDP changing quickly? When is GDP changing slowly?
What does it mean that GDP growth is slowing down? Does that mean GDP is decreasing?
Do the times at which GDP is changing the fastest correspond to the high and low points of the graph? Why or why not?
The greatest challenge students have with concavity is confusing “speeding up” and “getting faster” with “increasing rate of change”. Rates of change can be both positive and negative, and the absolute value of the rate of change gives the speed at which the quantity is changing. So if GDP was changing at a rate of -20 dollars per unit of time and then later GDP is changing at a rate of only -5 dollars per unit of time, the rate of change has increased since -5>-20 even though the rate of change has slowed down (GDP is not changing as quickly).
This is why looking at the concavity of a function is so helpful! Students generally don’t struggle to identify whether a portion of the graph is concave up (like a cup) or concave down (like a frown), so as long as students connect this behavior to whether a rate of change is increasing or decreasing they will be good to go. Steepness of a curve is referring to “how fast” and does not include direction.