Some students may struggle to express a number like 296,600,000 in millions, or they may think that a quantity like 2,320 million doesn’t quite make sense. Ask them to write the following quantities as numerals: 1 million, 10 million, 100 million, and so on, then use their list to help figure out how to say 296,600,000 in millions or to write 2,320 million as a numeral.

Students may think they do not have enough information to sketch a graph of . Encourage them to read through the activity to identify points that must be part of the graph of , but are not written in form.

Select students to share their responses and sketches of the graph of .

Make sure students can interpret a statement given in function notation, such as or  , in terms of the situation and articulate it completely. For instance, students might say, “The output is 1,860 when the input is 15” or “When the input is years, the output is 1,000 million,” or some other variation that doesn’t convey the quantities fully. Push them to refine their interpretation so that it is clear that means “1.86 billion people owned a smartphone in the year 2015” and means “A billion people owned a smartphone years after the year 2000.”

When discussing possible graphs of , acknowledge that a graph of could be drawn in various ways. The information we have is limited to the four input-output pairs, so what happens between the points is up for interpretation. But it would make sense, based on the context, for the graph to show very little change before year 2010 and then a rapid increase afterward.

The purpose of the activity is to familiarize students with function notation and its use in context. If time allows, these additional questions about modeling a context may be discussed.

• “The output of is defined in terms of ‘millions of people’ instead of individual persons. What might be a reason for this?” (If the unit is “persons,” the scale would show very long numbers with 9 or 10 digits each, which would be much harder to read and might lead to mistakes. Using “in thousands” or “in millions” strategically makes it possible to use simpler numbers. It makes it easier to draw attention to important characteristics of a graph.)
• “The input is defined in ‘years after 2000.’ Could we have instead used calendar years, such as 2002?” (Yes.)
• “What might be a reason to choose ‘years after 2000’ as the unit?” (It allows us to write smaller numbers. Another reason is that sometimes knowing a duration is more useful than knowing a particular point in time at which something happens. For instance, in the drone activity, knowing the number of seconds that had passed before the drone landed was more useful than knowing that it landed at, say, 2:45 p.m. In this activity, the year 2000 might be significant in the development of smartphones, so measuring time with that starting point might be useful.)
• “Would it make sense to use a line of best fit based on the 4 known points?” (Not in this case. The function should contain these points, not just model the information with a line. Also, it doesn’t make sense for this information to follow a linear pattern.)

Some students may think there is not enough information to accurately graph the function. Assure them that this is true, but clarify that we are not after the graph, but rather a possible graph of the function based on the information we do have.

Select previously identified students to share their interpretations of the inequalities in the first question and to show their graphs. Ask them to explain how each statement is evident in their graph. Discuss questions such as:

• “Why might it be true that ?” (The heat is turned off at or after 15 minutes, or the pot of water is taken off the stove.)
• “You and your partner agreed on what each statement meant. Are your graphs identical? If not, why might that be?” (Function was not not completely defined. We have information about the temperature at some points in time and how they compare, but we don’t have all the information about all points in time.)