A2 Driving to the Pyrenees

Introduction
A2 Driving to the Pyrenees is the second investigation in the acceleration unit. If you've not done so already, I recommend that you check out the first investigation first.

General Teaching Notes
In A1 students began to visualize slope as a representation of velocity. They imagined drivers on journeys with changing velocities and created graphs of distance versus time. They also graphed velocity as a function of time. These explorations continue in A2, with one twist: They are given the desired distances and are invited to invent the velocities, calculate the time each segment takes, and then graph. Some students will initially struggle with calculating time from distance and velocity. I am opposed to giving them D = rt because it draws them formulaic while they could instead still exploit their own experience of these relationships. If needed, I prefer to talk them through a couple examples with five minutes of leading questions at the beginning of class: Suppose you want to drive to the national park 100 km away. What kind of roads are those? How aggressive a driver are you? How fast would you go? If you went that fast, how long would it take? How fast would someone else here go? How long would that take? How did you get that answer?

1) Some students may choose to convert to minutes instead of keeping the table in decimal hours. You can suggest decimal hours, but I consider it an excellent exercise for them to labor through the minute conversions, if that is their preference. They also may hiccup at the total distance when they have to add -50 km. Let them work it out. Let them know that the second graph at the bottom is a graph of velocity vs. time, and have them label it appropriately. Students will have a variety of methods of graphing velocity, from something that looks like a bar graph, to a modified bar graph that shows some acceleration and deceleration, to simply graphing single points and then connecting the dots. Each attempt is a great opportunity to engage them in debate and discussion. I do not correct students much on the velocity graph, but I do go to groups and ask lots of questions. I want them to experiment and think about how to represent the velocity graphically.

I am, however, quite assiduous in pointing out mistakes on the distance graph. It's important to set an expectation that they graph as precisely as possible while acknowledging graphs are approximations. As math teachers we are comfortable when we see rough sketches because we already know the relationships implied by the graph, but students don't necessarily know that a steeper line corresponds to a bigger slope which is representing a bigger velocity which means the car goes further in less time. They are quite capable of understanding this, but they need to work it out as opposed to simply memorizing and implementing rise over run.

2) This repeats the drive done in number 1, but of course students can invent at least some parts of it, like how long Michael takes to eat lunch, and whether or not he forgets something at the gas station and has to go back for it. They could even invent a slightly longer or shorter route with different roads and so on. Some students may initially balk at inventing anything new--encourage them to be creative.