In this investigation, students start by looking at real-world data, which means that it will not perfectly fit a function. In order to understand what this data is showing, students will use the online graphing calculator Desmos (see the explanation about how to use this). First, they use desmos to find a function to describe the Stoping Distance of a car that was travelling at a given speed. Once they have created this function using the method of curve fitting in Desmos, they find a different function to determine the original speed of the car if we know the Stopping Distance. This is the inverse of the original function, and this is a radical function.
The option for differentiation is to introduce the additional challenge of finding a function for the Reaction Distance, or the distance the car travels while the driver is realizing that they need to stop. This is a linear relationship. Then they can find the Braking Distance function. The sum of these two functions is the Stopping Distance function.
To keep the lesson simple for some students, you can focus just on the Total Stopping Distance Problem Sentence, only requires them to consider one function. If you want to introduce more of a challenge, have students work on the Braking Distance plus Reaction Distance Problem Set. Either way, the goal of the lesson is for students to generate a real-world inverse function and to interpret the meaning of this function.
In terms of scaffolds, I model the process of fitting a curve to a data set using sliders in Desmos by doing this on a projector. This is the only whole-class scaffold I provide. As students work, I circulate and ask them: What does this function mean? How does it behave? What are the inputs and the outputs?
As always, the purpose of the closing is to help call students’ attention to the big ideas of the lesson. Often during these kinds of investigations, students get caught up in the details of adjusting the sliders and dealing with the technology, which makes the closing all the more important so that students know what the point of this lesson is.
To illustrate the first problem, give examples of two different data tables showing the inverse functions and ask students to describe the meaning of the inputs and outputs. If they have not already articulated this, now is the time to identify these functions as quadratic functions and radical functions. Though the closing may seem repetitive, asking students questions about the same big idea in different ways gives them the chance to develop a deeper understanding of today's lesson.