options: Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph.

For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity.

Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes.  For example, if the function h(n) gives the number of person-hours it takes to assemble n engines in a factory, then the positive integers would be an appropriate domain for the function.
problem_activity: Activity
In this activity, students continue to compare two functions by studying graphs and statements in function notation, as well as interpreting them in terms of a situation. They also revisit the meaning of a solution to an equation such as
\(C(t) = 20\)
, both abstractly and in context, and apply what they know about average rate of change to compare the trend shown by each graph.
Additionally, the activity draws students’ attention to the point where the two graphs intersect, its meaning in context, and its corresponding representation in function notation.
Launch
Ask students how many of them have a landline phone at home and how many only have cell phones. If students are unfamiliar with landline phones, explain as needed.
Display the graphs for all to see. Discuss questions such as:
“How would you describe the trends in phone ownership over the years?” (Cell phones are increasing in use and landlines are decreasing.)
“How would you describe the shape of the graph of each function?” (Both are roughly linear. The value of
\(H\)
is decreasing over time, so the graph slants downward. The value of
\(C\)
is increasing over time, so the graph slants upward.)
“In what year did about 60% of homes have a landline?” (2012)
Speaking, Reading: MLR5 Co-Craft Questions.
Use this routine to help students compare and interpret two functions. Display the student task statement and graph, leaving out the questions. Ask students to write down possible mathematical questions that could be asked about the situation. Invite students to compare their questions before revealing the activity’s questions. Listen for and amplify any questions involving the overall shape of the graphs, trends over specific time intervals, and the intersection point of the two graphs. The process of creating mathematical questions, without the pressure of producing answers or solutions, prompts students to make sense of the given information and to activate the language of mathematical questions. This work helps to prepare students to process the actual questions.
Design Principle(s): Maximize meta-awareness; Support sense-making
Engagement: Develop Effort and Persistence.
Encourage and support opportunities for peer interactions. Prior to the whole-class discussion, invite students to share their work with a partner. Display sentence frames to support student conversation, such as: “One thing that is the same is . . .”, “One thing that is different is . . .”, “_____ represents _____.”
Supports accessibility for: Language; Social-emotional skills
Student Facing
\(H(t)\)
is the percentage of homes in the United States that have a landline phone in year
\(t\)
.
\(C(t)\)
is the percentage of homes with
only
a cell phone. Here are the graphs of
\(H\)
and
\(C\)
.
[IMG]
Estimate
\(H(2006)\)
and
\(C(2006)\)
. Explain what each value tells us about the phones.
What is the approximate solution to
\(C(t)=20\)
? Explain what the solution means in this situation.
Determine if each equation is true. Be prepared to explain how you know.
\(C(2011) = H(2011)\)
\(C(2015) = H(2015)\)
Between 2004 and 2015, did the percentage of homes with landlines decrease at the same rate at which the percentage of cell-phones-only homes increased? Explain or show your reasoning.
Student Response
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Student Facing
Are you ready for more?
Explain why the statement
\(C(t) + H(t) \leq 100\)
is true in this situation.
What value does
\(C(t) + H(t)\)
appear to take between 2004 and 2017? How much does this value vary in that interval?
Student Response
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for free access to Extension Student Response.
Anticipated Misconceptions
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.
Activity Synthesis
Focus the discussion on the meaning of equations such as
\(C(t) = 20\)
and
\(C(2015)=H(2015)\)
, 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:
\(C(t) = 20\)
means “in year
\(t\)
, 20% of homes relied only on cell phones” and, based on the graph of
\(C\)
, the value of
\(t\)
that makes that statement true is 2008.
\(C(2015) = H(2015)\)
means “in year 2015, the percentage of homes with only cell phones and the percentage of homes with landlines are equal.” We know this is true because at
\(t=2015\)
the two graphs intersect, which also means they share the same output value.
The average rate change for
\(C\)
is positive because, in the measured interval, the value of
\(C\)
increased overall. The average rate of
\(H\)
is negative because the value of
\(H\)
decreased overall.
An average rate of change of 3.8% per year for
\(C\)
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
\(C\)
is 4% per year, while the average rate of change for
\(H\)
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.)


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