Have students work in pairs on rate_of_change_linear_warmup. The objective is for students to find the common difference in the table of data describing someone's driving. Students may have different strategies for finding the common difference. At the end of the warmup, take a few minutes to have different students share out their thinking (MP3). As an extension, have students write a function that would be used to find the output (miles) based on the input (hours).
Today's investigation is about bicycle racers who are riding up the Haleakala trail in Hawaii. To begin, see if any students know or can approximate how fast a bike racer can actually ride. Here is an article from wikipedia regarding typical speed. Lance Armstrong (maybe not the best example in light of recent events) once averaged approximately 24 mph in the Tour de France. A normal rider may travel at 12 mph. Students should keep in mind that the 38 miles in this investigation is almost entirely uphill.
This investigation will require students to make many assumptions (MP2) about the speed that a bicyclist can ride. The point here is to give them some background information so they can make an informed decision for their assumptions.
Students will work on the rate_of_change_investigation in pairs. Students will take the first few minutes with their partner to understand the problem (MP1) The goal is for each pair to make a poster on chart paper that shows their thinking about how fast each of these bike riders was traveling on their journey to the top of the volcano. Students can use the diagram on the worksheet as well as the questions to focus their thinking and identify pertinent information to include on their posters.
The most important part of this investigation is that students determine how long it will take Cliantha to reach the top because she is riding at a constant rate. Most students will base this on the launch information. Five hours is a pretty good estimate for how long it will take to reach the end of the 38-mile trail. If that is their estimate then (5, 38) would be the coordinates of point B. With this in mind, students can use a ruler (MP5) to determine the coordinates of points A and C. They can use proportional reasoning to determine what portion of the 5 and 38 each point is located at. Approximate locations could be (2, 26) for point A and (3, 13) for point C.
Students can use their approximations to justify (MP3) a story for each of the three bike riders. In doing this students will develop the concept of slope as a rate of change. Students will make connections between the steepness of a line and the rate of change that it represents.
Rate_of_change_linear_closing is a very straight-forward ticket out the door. Coupled with your observations from the posters that students completed you should have a good idea where students stand in their development on the concept of slope. Students should begin to understand that the slope of a line is represented by the rate of change written as a ratio (rise/run). Graphically, this ratio can be seen in how steep a line is. Encourage students to pull information from the Haleakala question when writing their answers to this ticket out. Students do not need to write an abstract definition. They should be encouraged to connect the concept to something concrete.