In this activity, students continue working with the data from the Warm-up as they consider the two functions represented by the data. Expressions for each function are purposefully left out of this activity to help students focus on the meaning of the new function created by combining the functions in context.

Students are building skills that will help them in mathematical modeling (MP4). They don't decide which model to use, but students have an opportunity to reason about data in context and see how to combine known functions into new functions in order to view the story told by the data in a different way.

Monitor for students who are considering the scale of the numbers as they graph the data for . For example, while a change of almost 200 million books from 2010 to 2016 seems large, it is a change of less than 7% of the total. The decrease from 2,730 million to 2,700 million is a change of less than 2%.

If students are unsure how to set up and label the axes to graph the values of , consider saying:

• “Tell me how the labels for this graph compare to those of the axis for the graph of .”

• “How could the values for in the table help you set up the scale for the vertical axis?”

The purpose of this discussion is for students to understand that we can combine functions to make new functions and what that can mean for a specific context.

Invite students previously identified to share their explanations for the plot of , displaying their graphs for all to see to help students illustrate their ideas. Here are some questions for discussion:

• “What does mean in this context?” (In 2010 the average number of books bought per person in the US was about 8 books.)
• “Compare with . Describe how they are alike and how they are different.” (They both dip in 2011 and then go up, but then doesn't change much while appears to decrease. This is because the population is increasing while the number of books sold stays about the same, so the denominator is getting large while the numerator is about the same which means the quotient, , decreases.)
• “Let’s say is the average cost of a book, in dollars, in the US. What would the function defined by tell us?” (The average amount of money spent per person in the US on books in year .)

Conclude the discussion by asking students which graph they think is most useful for tracking book sales. After a brief quiet think time, invite students to share their reasoning. Depending on perspective, such as a local bookseller versus a large publishing company, both graphs can be useful for planning for future sales.

This activity focuses on combining two functions to make a new function. Here, students sketch the graph of the function defined as the sum of two original functions. To help students make generalizations about the process, in each question one of the starting functions is always the same while the other is either constant, linear, or quadratic (MP8).

Monitor for students who approach the activity in different ways based on how precise they make their sketch, such as:

• Making a rough sketch by observation.
• Measuring the distance of each function from the -axis in a few locations and then combining those heights to determine the -value of the new function at specific values of .
• Making a table of values for and the other function, as in the previous activity, in order to find the sums before making a sketch of the new function.

Also look out for possible incorrect approaches due to students not considering the numerical values of the functions and what it means to add, such as by:

• Sketching a curve that is halfway between the two given graphs.

The purpose of this discussion is for students to share how they made their sketches, focusing on how the numerical value of the new function is defined by the sum of the numerical values of the two original functions. 

Invite previously selected students to share their approaches. Sequence the discussion of the approaches by the order listed in the Activity Narrative. If possible, record and display their work for all to see.

Next, invite previously identified students who took an incorrect approach to share in the order listed in the activity narrative. Where possible, identify what function the student did graph.

Connect the different responses to the learning goals by asking questions such as:

• “How did making a table before sketching change the accuracy of the graph?”
• “What patterns did you notice between the different graphs?”
• “What do you think a graph of E minus the other function would look like?”

If time allows, ask students to choose one of the questions and sketch the graph of a function defined by the difference between and the other function.