In our Thursday session at the 2024 NCTM Annual Conference, Karen Sleno and I presented on the topic of Rates of Change. This is an overarching idea of mathematics that begins in earnest in Algebra 1. But why do we usually only associate “rates of change” with Calculus? Because in previous classes, the topic of rates of change is often reduced to the skill of finding a slope, presenting students with only a single and often limited perspective on this important concept. Instead of thinking of how two quantities covary, students just parrot back “rise over run” and apply a formula with x1 and x2 and y1 and y2 (an approach that often goes awry because the calculation is divorced from its conceptual meaning, leaving room for lots of computational errors).
We think it’s time for a change! Thinking about how fast something is changing is a topic that is accessible to all our students. From a heart rate to a marathon pace, from pumping gas at the gas station to riding a roller coaster, students encounter rates all the time! How do we connect students’ experiences navigating the world with the mathematical definitions, notation, and concepts we want them to learn? With EFFL lessons, of course!
While we could have started with Jenin downloading songs from Spotify (an Algebra 1 lesson introducing rate of change), in this workshop we focused our attention on AP Precalculus and AP Calculus content.
Activity 1: How Much Does It Cost to Rent a U-Haul?
Through these lessons from the first unit of AP Precalculus, students learn to describe the rates of change of linear and quadratic functions. Students study the average rate of change of both function types, making use of graphs, tables, equations, and verbal descriptions of the two changing quantities.
In the workshop, we gave an overview of Activity 1 and then did a deep dive into Activity 2, giving participants the chance to take off their teacher hat and experience this lesson from a student perspective.
Activity 3: Geri’s Greeting Cards
In Unit 4 of AP Precalculus, students begin by reviewing arithmetic and geometric sequences, then jump into this lesson that explores the differences between linear and exponential functions. Instead of treating these as separate topics covered in siloed units, students compare and contrast two scenarios related to greeting cards and note the differences in growth patterns. Ultimately, students learn to differentiate between functions where the outputs grow additively and functions where the outputs grow multiplicatively.
The slide below summarizes students’ learning up to this point.
Activity 4: Can a Human Break the Sound Barrier?
We ended the workshop in Calculus-land, studying Felix Baumgartner’s record-breaking freefall to earth from a helium balloon in the stratosphere in 2012. While students at this point are familiar with finding the rate of change over an interval, we now turn our attention to finding the rate of change at an instant. This of course requires finding the average rate of change over small intervals containing that instant. Students improve their estimates by using smaller and smaller intervals, and the margin notes highlight the structure of a difference quotient. In the final part of the activity, we use the variable “h” to represent the length of the interval, and find the average rate of change between the instant in question and a point that is h units away from that instant. When we evaluate the limit as h approaches 0, we get the exact instantaneous rate of change. Three cheers for the powers of Calculus!
Big Take-aways:
Research (such as the studies discussed in Jo Boaler’s new book “Math-ish”) support the teaching of big ideas rather than a checklist of standards. Additionally, when “teachers give the students questions and tasks before they teach the methods they need to solve them” higher outcomes are attained and “researchers conclude that this happens because students get a greater opportunity to struggle—to think about and draw from the knowledge they have already developed.” (Boaler, p. 72)
To make this transition to a richer, student-centered approach to teaching mathematical concepts, we will need to:
Create opportunities for students to engage with concepts authentically
Use real life contexts to make the mathematics relevant
Offer opportunities for students to talk with each other about ideas and to make conjectures
Cultivate a safe space where all opinions and thoughts are valued.