Lesson 8

This warm-up prompts students to carefully analyze and compare the properties of four graphs. Each graph represents a function relating time and temperature. No scales are shown on the coordinate axes, so students need to look for and make use of the structure of the graphs in determining how each one is like or unlike the others (MP7).

In making comparisons, students have a reason to use language precisely (MP6), especially recently learned terms that describe features of graphs.

Arrange students in groups of 2–4. Display the graphs for all to see. Give students 1 minute of quiet think time and then time to share their thinking with their small group. In their small groups, ask each student to share their reasoning as to why a particular item does not belong, and together, find at least one reason each item doesn’t belong.

Ask each group to share one reason why a particular item does not belong. Record and display the responses for all to see. Encourage students to use relevant mathematical vocabulary in their explanations, and ask students to explain the meaning of any terminology that they do use, such as “intercepts,” “minimum,” or “linear functions.”

After each response, ask the class if they agree or disagree. Since there is no single correct answer to the question asking which one does not belong, attend to students’ explanations and ensure the reasons given are correct. Also, press students on unsubstantiated claims. For example, if they claim that the vertical intercept of graph C is different than all the others, ask them how they know (as none of the vertical axes have a scale).

Prior to this point, students have had multiple opportunities to interpret graphs of functions in terms of a situation. In most cases, students were given some information about the units of measurement of at least one quantity, which helped them reason about the functions being represented.

In this activity, students encounter graphs without units on either of the axes. They interpret the graphs in the context of raising a flag and make a case for whether the graphs are realistic (which may depend on the units chosen for the axes). Students also see for the first time a graph that is a vertical line and think about why this graph does not represent a function.

As students relate features of graphs to behaviors of quantities in a situation, they reason quantitatively and abstractly (MP2). In explaining and discussing why a graph is or is not realistic, students practice constructing logical arguments and critiquing the reasoning of others (MP3).

Select students to explain what each graph means in the context of flag raising and give an argument as to 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 to act 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 as to 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.

Previously, students analyzed several graphs to see if they could represent the height of a flag being raised in a ceremony. In this activity, students watch a video of a flag being raised. They sketch a possible graph to represent the height of the flag as a function of time, and then use their graphs to estimate the rate of change of the flag.

Students’ graphs do not need to be very precise, but should capture key features of the situation. To do so, students need to make some estimates. For example, they need to gauge the starting height of the flag, the height of the pole, the heights at which pauses happened, whether the flag was moving at a constant rate between the pauses, and so on. Students also need to make some decisions about the graph, for instance: the scale to use on each axis, whether the graph should be discrete or continuous, and so on. Along the way, students engage in aspects of modeling (MP4).

If desired and if possible, provide each student or each group with access to the video so students can replay the clip a few times, as needed. As students work, identify those who make different decisions for their graph and can explain their rationale. Ask them to share their work later.

If time is limited, focus work time and discussion on the graphing portion (the first question).  

Tell students that they will watch a video of a flag being raised and their job is to sketch a graph that represents the height of the flag, in feet, as a function of time, in seconds. Explain to students that their graphs do not need to be precise and some estimations are required, but the graphs should reasonably capture the movement of the flag.

Before playing the video, ask students to think about what information or quantities to look for while watching the video. If possible, record and display their ideas for all to see.

Here are videos that show the same clip of a flag being raised, played back at full speed and half speed. Show one or more of the videos for all to see. Students may wish to see the slower version a few times to help them sketch a graph.

Full speed:

Half speed:

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.

This optional activity allows students to practice creating graphs of functions based on verbal descriptions. Students sketch graphs that could represent the heights of water in two pools, each as a function of time since the pools start to be filled.

The modeling demand in this activity is higher than in previous activities, as students are given less quantitative information and therefore need to make more assumptions and decisions (MP4). For example, they need to consider whether and how the shape and size of each pool affect the rate at which the water rises, how much longer it would take to fill the large pool compared to the small pool, how moving a hose from one pool to the other changes the graph of each function, and so on.

Students are not expected to draw precise graphs. What is more important is that the graphs of the two functions are comparatively correct and reasonable in terms of the situation, that students consider the quantities carefully, and that they can explain their decisions.

As students work, look for different assumptions and decisions students make that result in a variety of graphs. Consider asking them to share their graphs and reasoning during the whole-class discussion.

Arrange students in groups of 2. Ask students to read the activity statement and be prepared to ask any clarifying questions about the task.

After answering students’ questions, give students a few minutes of quiet time to sketch the graphs for the first situation and then time to discuss the graphs with their partner. Tell partners to discuss their assumptions about the situation and the reasonableness of the graphs based on those assumptions. Ask them to revise their graphs based on each other’s feedback.

Consider selecting a few students to share their graphs and graphing decisions before the class continues to the second and third situations. Discuss questions such as:

• “The water in both hoses flows at a constant rate. Did you assume the constant rate in one hose to be the same as the constant rate in the other hose?”
• “Did you assume that the water for the two hoses was turned on at the same time?”
• “What assumptions did you make about the shape and size of each pool? Did you assume the pools to have the same area for their base and just have different heights, or that they have different areas?”
• “What assumptions did you make about how water was rising in each pool?”

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 would the graph of the large pool change when one hose was moved from the small pool to the large pool? Would its slope increase, decrease, or stay the same?” (The slope of the graph for the large pool would double, because water is now rising at twice its previous rate. The graph for the small pool would stay constant, as the water is no longer increasing in height.)

This optional activity gives students another opportunity to represent the quantities in a situation with a table and a graph, identify key features of the graph, and interpret those features in terms of the situation.

The following are some videos that show the same clip of a ball being dropped, but played back at different speeds. Show one or more of the videos for all to see. Before showing the videos, ask students to think about what information or quantities to look for while watching the videos.

• At the original speed: 

• At half speed: 

• At quarter speed: 

If possible, allow students to replay the videos several times to help them sketch as accurate a graph as possible. Alternatively, give students individual access to the interactive applet, if available.

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 began 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 \(t\).

Discuss how students translated the movement of the dropped ball into a graph and on 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 original 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, \(h\), as the input value, \(t\), increased?” (Overall, the output decreased as \(t\) 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.