Calculating Instantaneous Rate of Change (Lesson 9.2 Day 2)
Unit 9 - Day 4
Understand that limits turn an estimate of the instantaneous rate of change into the exact value of the slope
Set-up and evaluate a limit expression that gives the slope at a single point
Write the equation of a tangent line to a curve at a given point
Quick Lesson Plan
After first being introduced to the limit definition of slope at a point yesterday, students will spend time engaging in purposeful practice in today’s station activity. To prepare, print the station cards on cardstock paper and cut them so that each station is on a separate piece of paper. Hang up the cards around the room.
Stations can be run with a timer, where the group spends a certain amount of time at each station and then moves to the next station, but be aware that some stations will take much longer than others (like A,B, and H). The algebraic manipulation required to calculate an instantaneous rate of change can be difficult for some students. We recommend having students travel in pairs through the stations at their own pace.
There are two versions of the recording sheet in the same document. One version is more scaffolded and offers some step-by-step guidance in calculating the instantaneous rate of change in stations A, B, and H. The other version simply has blank boxes. We gave students the option to choose whatever version they wanted, thus allowing for differentiation.
Be prepared for questions about the algebraic manipulations required in Station B. While students focused on polynomial functions in 9.2 Day 1, today students explore how the definition of the derivative is applied to rational functions. Students may need reminders about how to add fractions of unlike denominators and how to simplify fractions within fractions.
A note about vocabulary: It is important that students understand that the terms “instantaneous rate of change”, “slope of the curve”, “slope of a tangent line”, and “derivative” (lesson 9.3) can all be used interchangeably. We purposefully mix these phrases into our activities so students can connect these concepts across different representations.