Intent In this activity, the first of a series of activities that support the curve-fitting students will be doing to predict the period of a 30-foot pendulum, students sketch graphs from In-Out tables.

Mathematics The data sets in this activity are nonlinear, so the graphs that students create—once they have determined the proper scales for the axes—will contain data points that they will then connect with smooth curves. Each of the data sets will have a turning point at which the graph changes from increasing to decreasing. However, the turning point might not be right at one of the data points in the table, so students will be required to guess the location of the maximum values of these nonlinear functions.

Progression Students will create and analyze two graphs on their own and then share their results in their groups and with the whole class.

Approximate Time 20 minutes for activity (at home or in class) 15 minutes for discussion

Classroom Organization Individual, then groups, followed by whole-class discussion

Doing the Activity Before students begin, it may be helpful to discuss how they might choose scales that will allow them to see the graphs clearly.

Discussing and Debriefing the Activity Have students compare their answers and share how they decided on the price that would maximize profit. If anyone connected the dots with line segments rather than with a smooth curve, ask whether prices between those given in the tables would necessarily have profits on those line segments. Students should recognize that they cannot tell for sure what price would give the maximum profit. If they just choose the price among those listed that maximizes profit—$18 for a CD, $40 for a CD player—ask whether a price slightly higher or lower might result in a greater profit. (There is no definitive answer to this question.) Students don’t need to formulate equations to describe the data here; that idea will be pursued later in the unit. [link to math maps]