I intend for this Warm Up to take the students about 15 minutes to complete and for us to review as a class. In this Warm Up, I provide students with 6 graphs in the form f(x)=mx + b (see Warmup_Screen_Shot) using the Desmos online graphing tool. My students need to determine the initial (or starting) value of the function, as well as the slope of the line. For our purposes, the initial value is the y-intercept. With respect to the slope, I also want my students to describe the pattern. Is the line constantly increasing, constantly decreasing, or just constant.
An important objective of this lesson is for my students to be able to recognize situations that are constantly increasing, constantly decreasing, or just constant as circumstances that can be modeled with a linear function. I want my students to recognize that if the function has a constant rate of change, it is linear. I am always worried that my students will overgeneralize this to "relationships that are increasing or decreasing are linear." So, before the end of this lesson, I will remind my students that increasing or decreasing patterns that do not demonstrate a constant rate of change are not linear. We have seen these before, during our functions unit (see Intro of Parent Functions). I want my students to keep these in mind as we drill deeper into the concept of a linear function.
In our next activity, I have my students work with their table partners. At this point in the year the table partners are working in relatively homogeneous groups. The three problems on the Introduction worksheet generally take my students about 15 to 20 minutes to work.
In the first problem, I expect my students will find the 100 lbs for the starting amount easy to understand. Using a slope of -2/3 to represent feeding the deer 2/3 lbs of feed per day may challenge some of my students. As they work on this problem I am paying close attention to the graphs that the students are producing.
In the second problem my students generally enjoy writing a scenario to match the graph. The work of Student 1 and Student 2 is typical of what my students produce. Both of these students describe the time intervals in words, instead of parentheses notation, to indicate when the function was increasing, decreasing, or constant. These two students disagreed about whether the horizontal segment represented a constant speed of 10 m/s or that the bike had came to a complete stop. The interpretation of this segment of the graph is always a point of disagreement in my class.
In Problem 3, I ask my students to create a t-table. In this problem the students are examining a cost function. The initial $5000 fixed cost represents the y-intercept. The cost to produce each additional unit represents a constant rate of change and defines the slope of the cost function. My students generally find this type of function easy to understand. The most common mistake is the use the fixed cost as the slope. Some students calculate the cost of the first unit and then write an equation using this cost as the slope (i.e., slope = 5200).
After we discuss the Partner Practice problems, I want to go over the notation for describing increasing, decreasing, and constant intervals for a function. Since we have looked at different functions and I want my students to recognize that this is leading somewhere interesting, I assign students an Independent Practice with non-linear functions. The goal here is to identify increasing, decreasing, and constant intervals, which is easy enough even with complicated function like these.
Throughout this discussion I make sure that my students recognize that the graph actually represents a function. The multiple examples on this worksheet help my students to appreciate that functions increase and decrease in lots of interesting ways. Linear patterns are our current focus in this unit; this activity helps us to identify what is linear and what is not. I want to make sure that my students have adequate opportunities to think about the fact that only linear functions have a constant rate of change, which results in a straight line when the function is graphed. Again, horizontal segments are challenging, so we take time to discuss why they are linear: the difference between any two points is zero when we perform a slope calculation, so the rate of change is constant (see my Discussing the Piecewise Functions reflection for more about how I lead this activity).
If the discussion is going well (and it usually is because these graphs are interesting), we will informally discuss ideas like local maximum and local minimum. The opportunity to do this presents itself when a segment of a graph of the function increases and then decreases. My students will often come up with good ideas for how to describe these sections if I ask the right question like, "What is important to know with respect to describing the rate of change for this segment?"
The Independent Practice worksheet comes from the website listed below:
http://www.sausd.us/cms/lib5/CA01000471/Centricity/Domain/2330/Worksheet%206_Inc-Dec%20Functions.pdf (last accessed 12-16-15)