Select students to explain what each graph means in the context of flag raising and give an argument for why it is or is not realistic. Consider asking partners to illustrate their response together—one partner explaining what is happening with the flag and the other acting out the action of the flag raiser.

Discuss with students which graph is the most realistic and why. Also discuss whether using certain units of measurements would make an unrealistic graph more realistic, or vice versa. (For example, none of the graphs would be realistic if the time is measured in days or months and the height in kilometers or centimeters.)

Next, solicit some explanations for why the vertical line does not represent a function. If not mentioned in students' explanations, bring up a key point: A vertical line can be interpreted to mean that for one particular input (3 units of time, in this case), there is an infinite number of possible outputs. Therefore, it cannot represent a function.

Some students may have trouble starting their graphs because they don’t know what upper limits to use for the axes. Ask them to watch the video clip again and try to gather some information that may help them decide on the upper limits. Assure them that some estimation and decision making are necessary.

Select previously identified students to share their graph and explain their drawing decisions. If time permits, display the graphs for all to see and briefly discuss:

• How the graphs are alike and how they are different.
• Key features such as intercepts, maximum, minimum, and intervals when the function increases, remains constant, or decreases.

If time permits, invite a couple of students to share how they used their graph to estimate the flag’s average rate of change. Emphasize that the average rate of change (which would vary for different graphs) represents how fast, on average, the flag was moving from the time it started being raised until the time it reached the top of the pole.

Some students might mistakenly think that when the pools are “full,” the water in each pool has reached the same height. Remind students that the two pools have different heights, so it takes different heights of water to make them full.

Invite or select students to share their graphs and the assumptions and decisions they made as they were graphing. Display graphs that correctly represent the same situation but look different because of variation in assumptions.

Discuss questions such as:

• “How would the vertical values of the two graphs compare when the pools are full?” (The vertical value of the large pool would be up to 12 inches greater than that of the small pool because the large pool is 12 inches taller than the small pool.)
• “When each pool is being filled by one hose and at the same rate, should the two graphs have the same slope? Why or why not? If not, which graph has a greater slope?” (No. The water in the smaller pool rises more quickly because the area of the pool is smaller, so the slope for that graph should be greater.)
• “How does the graph of the large pool change when one hose is moved from the small pool to the large pool? Does its slope increase, decrease, or stay the same?” (The slope of the graph for the large pool doubles because water is now rising at twice its previous rate. The graph for the small pool stays constant, as the water is no longer increasing in height.)

Some students may mistake the horizontal axis on the graph to represent horizontal distance rather than time. Because the movement of the bouncing ball is primarily up and down (except toward the end, when it begins rolling), these students might sketch a graph that is essentially a series of overlapping vertical segments. Ask them to revisit the input variable and what the horizontal axis represents. Then, ask them to plot some points for different values of .

Discuss how students created a graph from the movement of the dropped ball, and discuss the connections between the graph and the situation. Ask questions such as:

• “Which parts of the ball’s movement did you find easier to plot more precisely? Which parts require estimating?” (The position of the ball when it was released and the points when it hit the ground were easier to plot because they were more distinct and obvious. The points when the ball reached a peak and changed direction required a bit of estimation. The points between the peaks and the bounces were mainly estimated.)
• “In this case, the horizontal intercepts are also the minimums of the graph. Why is that?” (The ball could not have a height that is less than 0. The lowest point it could be is 0 feet.)
• “What happened to the output value, , as the input value, , increased?” (Overall, the output decreased as increased, but there were intervals—whenever the ball bounced up—in which the height of the ball increased as time increased.)

If time permits, discuss students’ responses to the last two questions.