Lesson 19

Provide access to graphing technology. It may be desirable for students to work in groups of four, so that they can view all four graphing windows on four different devices at the same time.

If needed, demonstrate using technology to graph the function and set the graphing window for the first given window. Then, allow students to set the remaining graphing windows and answer the questions.

The main purpose of this activity is to learn or revisit how to specify a graphing window using the available technology. Once students show proficiency with that, quickly move on to the next activity.

In this activity, students get more practice adjusting their graphing windows. Then, they explore why someone might want to choose a different graphing window to tell a different story.

Display the 4 graphs for all to see. Ask students to think of at least one thing they notice and at least one thing they wonder. Give students 1 minute of quiet think time, and then 1 minute to discuss the things they notice and wonder with their partner, followed by a whole-class discussion.

After demonstrating or asking a student to share how to adjust the window to display each version of the graph, focus discussion on the last three questions. Emphasize that all of the graphs are showing the same information, but the choice of graphing window makes the information look either harmless or alarming. Questions for discussion:

• “What are some reasons graphing window C might make people worry?” (It appears very steep, like the concentration of the harmful chemical is shooting up.)
• “What are some reasons someone might decide not to worry about graphing window C?” (The time span is 80 years, so perhaps the trend can be interrupted before then. Also, we would need more information about how concentrated the chemical needs to be before it is harmful. Maybe 8 parts per billion isn’t actually anything to worry about.)
• “What are some reasons graphing window B might look like nothing to worry about?” (It appears very flat, like the concentration of the chemical is hardly increasing at all.)
• “What are some reasons someone might decide to dig deeper, if they were only presented with window B?” (It only shows 8 years, so someone might wonder what happens after that. Also, if you knew the model was exponential, you would suspect it might shoot up eventually.)

In this activity, students choose a graphing window to tell a particular story.

Continue access to graphing technology. Students may be ready to dive into the activity without much assistance, or may need to engage in a notice and wonder to familiarize themselves with the context and the information encoded in the equation and the graph.

Invite selected students to share their responses. Questions for discussion:

• “What adjustments to the graphing window make the cost seem lower? What adjustments make the cost seem higher?” (The graph in the second question make the cost seem lower, because the graph looks quite flat and close to the horizontal axis. The graph is the first question make the cost seem higher, because the graph looks quite steep.)
• “For each situation, how did you decide whether to make the graph look flatter or steeper?” (I thought about whether the situation suggested the function should be increasing slowly or quickly.)
• “What were some strategies to make the graph look flatter?” (Keep the domain the same or smaller, and make the range larger.)
• “What were some strategies to make the graph look steeper?” (Keep the range the same, and make the domain larger.)
• “In most graphing technology, you can zoom in or zoom out. Why might you want to set a specific graphing window, instead of zooming in or out?” (Zooming in or out often doesn’t change the overall shape of the graph, because the domain and range are being scaled by the same amount. Choosing a specific graphing window allows you more control over what is shown.)