This Warm-up prompts students to make sense of a problem before solving it by familiarizing themselves with a context and the mathematics that might be involved (MP1).

Invite several partners to share one question with the class and record responses. Ask the class to make comparisons among the shared questions and their own. Ask, “What do these questions have in common? How are they different?” Listen for and amplify language related to the learning goal, such as “These questions ask whether the number of books sold and population are related” or “These questions ask how one quantity changes in relation to another.”

Reveal the question “How many books were sold per person?” and give students 1–2 minutes to compare it to their own question and those of their classmates. Invite students to compare similarities and differences by asking: “How do your questions relate to combining functions?”

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%.

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.