Students encounter percentages as the output of a function for the first time in this activity. Some students might think that the output of the functions here must be number of homes, and that they cannot estimate any output values because only percentages are known. Clarify that percent is the unit used in this case, as we are studying how the proportion of the two groups (rather than the actual number of homes in each group) changed over time.

Focus the discussion on the meaning of equations such as and , and on the meaning of the average rate of change of each function.

Select students to share their responses. Highlight the following points, if not already mentioned in students’ explanations:

• means “in year , 20% of homes relied only on cell phones,” and based on the graph of , the value of that makes that statement true is 2008.
• means “in year 2015, the percentage of homes with only cell phones and the percentage of homes with only landlines are equal.” We know this is true because at the two graphs intersect, which also means they share the same output value.
• The average rate change for is positive because in the measured interval, the value of increased overall. The average rate of is negative because the value of decreased overall.
• An average rate of change of 3.8% per year for means that the percentage of homes relying only on cell phones grew by about 3.8% each year.
• An average rate of change of -4% per year means the percentage of homes that used only landlines fell by 4% each year.

If time permits, discuss with students:

• “The average rate of change for is 4% per year, while the average rate of change for is -3.8% per year, which is very close to -4%. Why might the rate at which one increased be so close to the rate at which the other decreased? Could it be a coincidence?” (One possible explanation is that as people relied more on cell phones, they relied less on landlines, and they discontinued using landlines around the same time they acquired new cell phones. These people essentially went from one group to the other.)

Focus the discussion on how students made their matches. Ask students to explain how parts of the descriptions and features of the graphs led them to believe that a pair of representations belong together.

Next, invite students to share their graph of the fourth TV show. Display the graphs for all to see, and discuss how the graphs are alike and how they are different. Because each graph is created using the same description, they should share some common features. If they look drastically different, solicit possible reasons. (Possible explanations include differences in interpretation of the description or in the choice of scale for the vertical axis, errors in reading the description, and plotting errors.)

If time permits, ask students,

• “How are the graphs or the functions in this activity different from those in earlier activities?” (They show points rather than lines. The input is episode number, while in other cases, the input is time. Each function is shown on a different coordinate plane.)
• “How is the work of comparing functions here like comparing functions in earlier activities?” (They all involve comparing the output values of points on a graph, interpreting the points, making sense of verbal descriptions, and some estimating.)
• “How is the work of comparing functions here different from in earlier activities?” (Here we are comparing function values across three separate graphs, which is trickier than when the graphs are all on the same coordinate plane. Figuring out which function has a greater or lesser value was also easier when the functions were represented with lines or curves. It is harder to do with a set of points, especially when they are not in the same image.)

In this activity, students compare graphs and statements that represent functions without a context. Because no concrete information is given, students need to rely on their understanding of function notation and points on a graph to make comparisons and to interpret intersections of the graphs.

Previously, students saw equations such as and interpreted them in terms of a situation. Here, they begin to reason more abstractly about statements of the form and relate them to one or more points where the graphs of and intersect. They see that a value of that makes this equation true, or a solution to the equation, is the input value of such an intersection.

Display the graphs of and for all to see. Invite students to share their responses to the first set of questions. As they point out the greater function value in each pair, mark the point on the graph and write a corresponding statement in function notation: , , , and . Emphasize that the function value that is greater in each pair has a higher vertical value on the coordinate plane for the same input value.

Next, discuss if there are values of that make true and how we can tell. Point out that earlier we wrote because the values of and are equal when the input is 4. This means that and are coordinates of the same point.

So we can interpret an equation such as to mean that the values of and are equal when the input is , and that must be the horizontal value of the intersection of both graphs, which is a point that they share.