I plan for this Warm up to take about 10 minutes for the students to complete, and 10 minutes to review.
When comparing the work of two students, both students solve the problem using a t- table, that started with 1 second, but with different distances for y. The response of student 1 shows the daughter has ran 45 meters in 1 second. Adding the 40 meter head start and the 5 meters she runs in 5 seconds, this answer is correct. In the response of student 2, the distance of the father is 0 meters in 1 second and the daughter has ran 40 meters in 1 second. This answer does not represent the starting point correctly, and therefore the intersection is incorrect. It is clear that student one has a better understanding of the y intercept or initial point than student two. Student one has written the correct equation to the table for the father's time and distance of y=9x. The equation shows a initial point of 0 and and increasing situation of 9 meters per second.
I ask the following questions to clarify the initial point or the y intercept in the problem for students that get stuck on the warmup. I want to make sure to keep them moving forward in a productive struggle with the problem.
After discussing the y intercept and slope intercept form, the students recognize the initial point for the father is (0,0) and the initial point for the daughter is (0,40). The rate of change increases at 9 meters per second for the father and at 5 meters per second for the daughter.
In conclusion we discuss the importance of working from the correct initial point because it will change the point of intersection. It is important to be precise, Mathematical Practice number 6.
The Power Washer Problem I introduce after the Warm Up is from a website called Math Dude created for Montgomery County Public Schools in Rockville, Maryland. I use the video to introduce the problem, and I pause the video after the problem is presented, without revealing the answer.
The problem presents two different cost functions for Dave to rent a power washer to clean his deck. I have students solve the problem with their assigned partner. (I also considered presenting the problem myself, then having the students view the video after working the problem.) I provide a worksheet for the Power Washer problem that has 2 tables and one graph, but the students are allowed to use any method, as long as they can share their reasoning.
The students preferred using the table method for this problem. I presented the table method to build from what they already know, and introduced using substitution to solve this problem. Both of the equations were in slope intercept form, so I modeled substituting one of the expressions for y, and solving for x as shown in the video below.
Source for the Math Dude Video:
I use this Exit slip as a formative assessment on solving a system of equations for an increasing and a decreasing situation. Most of the students prefer using the table methodl, but a few students used the substitution method that I presented on the Power Washer problem.
The second problem also presents the student with an equation that would be a decreasing situation. We also discuss examples in the closure that would be a decreasing situation. I ask the class, What do we know about the slope of a decreasing situation? The class answers, that the slope will be negative in decreasing situations, and positive in increasing situations. Then I ask, "What is the sign of the y-intercept in decreasing situations or increasing situations?" As a class, we discuss that it depends on the initial value or the starting amount, it could be 0, negative, or positive.