Unit 5, Lesson 5
More Graphs of Functions
Grade 8
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This Warm-up prompts students to compare four graphs. It gives students a reason to use language precisely (MP6). The activity also enables the teacher to hear the terminologies students know and how they talk about characteristics 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 ask them to indicate when they have noticed three graphs that go together and can explain why. Next, tell students to share their response with their group and then together find as many sets of three as they can.
Which three go together? Why do they go together?
Invite each group to share one reason why a particular set of three go together. Record and display the responses for all to see. After each response, ask the class if they agree or disagree. Since there is no single correct answer to the question of which three go together, attend to students’ explanations and ensure the reasons given are correct.
During the discussion, prompt students to explain the meaning of any terminology they use, such as “continuous,” “discrete,” “segment,” and to clarify their reasoning as needed. Consider asking:
During the discussion, avoid introducing the traditional names of
5 Practices
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The purpose of this activity is for students to identify where a function is increasing or decreasing from a graphical representation. In the previous activity, students focused more on single points. In this activity they focus on collections of points within time intervals and what the overall shape of the graph says about the relationship between the two quantities (MP2).
Monitor for groups who use different strategies for identifying increasing or decreasing intervals. Some strategies, ordered here from simplest to most involved, might be:
Tell students to close their books or devices (or to keep them closed). Arrange students in groups of 2, and display the first graph for all to see. If students are not familiar with time plots, explain that each point represents the value for one year, starting with the point for 1991. Ask students, “Does the graph show the amount of garbage produced as a function of the year, or the year as a function of the amount of garbage produced?” (amount of garbage produced as a function of the year)
Give groups 1 minute to think of a question that the information in the graph can answer. For example, “About how much garbage was produced in 2010?” (about 230,000 thousand or 230 billion tons) Invite 2–3 groups to each share their question, and ask a different group to give the answer.
Select students with different strategies for identifying intervals of increasing and decreasing, such as those described in the Activity Narrative, to share later. Aim to elicit both key mathematical ideas and a variety of student voices, especially from students who haven’t shared recently.
Scatterplot, horizontal, year, 1990 to 2015 by fives, vertical, garbage in thousands of tons, 190,000 to 270,000 by 20,000. Points start at 1991 comma 208,000, and increase steadily to 2007 comma 255,000 except for slight decreases at 1995 and 2001. Between 2006 and 2009, the points decrease, but increase again from 2009 to 2014.
A scatterplot, horizontal, year, 1900 to 2015 by fives, vertival, percentage recycled, 15 % to 27 % by threes. Beginning with 1991 comma 16%, points trend linearly upward to 1996 comma 22%, then linearly upward at a less steep rate to 2011 comma 26 %, then two points trend down.
The purpose of this discussion is for students to use and listen to the language used to describe intervals of increasing and decreasing for the graphs.
While discussing each graph, display for all to see. For the first graph, invite previously selected students to share how they identified when the amount of garbage produced was increasing or decreasing. Sequence the discussion of the strategies in the order listed in the Activity Narrative. If possible, record and display the students’ work for all to see.
Connect the different responses to the learning goals by asking questions such as:
Conclude the discussion by displaying the second graph and asking, “How might you generally describe this graph to someone who couldn't see it?” (The percentage of garbage that was recycled increased overall from 1990 to 2011 but began decreasing from 2011 to 2013.) Invite students to share their descriptions with their partner, and then select 2–3 students to share with the class.
Here is a graph showing the temperature in a town as a function of hours after 8:00 p.m.
The graph of a curve on the coordinate plane. The horizontal axis is labeled “time in hours after 8 pm” and the numbers 1 through 11 are indicated. The vertical axis is labeled “temperature in degrees Fahrenheit” and the numbers 45 through 60, in increments of 3 are indicated. The curve starts on the vertical axis at the point 0 comma 60, and moves downwards and to the right. It continues downward until reaching a minimum point of 8 comma 45, turns, and then moves upward and to the right, passing through the point 11 comma 57.
The graph of a function tells us what is happening in the context the function represents. In this example, the temperature starts out at
Help us improve by sharing suggestions or reporting issues.
The graph shows the temperature between noon and midnight in one day in City A.
Graph in a coordinate plane, horizontal, time in hours after noon, 0 to 12 by ones, vertical, temperature in degrees Farenheit, 50 to 60 by ones. The graph starts at 0 comma 50 and climbs slowly, the more quickly as it moves right, going through 2 comma 52 and 4 comma 57 before peaking at 5 point 8 comma 59. The graph then steadily declines as it moves right until it reaches the point 12 comma 52 point 5.
Students may not answer with a range of dates. They might instead list each year it increased. A list is acceptable, but be sure students see the connection between, for example, the list “1996, 1997, and 1998” and the same years stated as “from 1996 to 1998.”