Unit 5, Lesson 1
Matching Up to Data
Integrated Math 3
1.1
Warm-up
Instructional Routines
Which Three Go Together?Materials
NoneActivity Narrative
This Warm-up prompts students to compare four data sets. It gives students a reason to use language precisely (MP6). It gives the teacher an opportunity to hear how students use terminology and talk about characteristics of the items in comparison to one another.
Launch
Arrange students in groups of 2–4. Display the data sets 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, tell each student to share their response with their group, and then together find as many sets of three as they can.
Activity
NoneWhich three go together? Why do they go together?
A
B
| (hours) | 0.03 | 0.12 | 0.22 | 0.3 | 0.93 | 1.02 | 1.28 | 1.55 | 2.17 | 2.77 | 5.7 |
|---|---|---|---|---|---|---|---|---|---|---|---|
| (degrees Fahrenheit) | 69 | 67.8 | 67.4 | 66.3 | 59.9 | 59.1 | 57.5 | 57.3 | 55 | 52.3 | 47.1 |
C
D
Student Response
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Building on Student Thinking
Activity Synthesis
Invite each group to share one reason why a particular set of three goes 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, ask students to explain the meaning of any terminology they use, such as “increasing” or “decreasing.” Also, press students on unsubstantiated claims.
Lesson Synthesis
Display the three graphs. Invite students to describe how to transform each graph to fit the points.
Here are some questions for discussion:
- “Which graph fits the data the best?” (All three can fit perfectly, for example, by translating the parabola up 2 and right 1.)
- “What would be a situation where each graph—linear, quadratic, and exponential—would be the best choice to model the relationship?” (Any explanation works so long as the rate of change in the scenario matches the graph chosen. For example: The exponential would be best if the situation was population growth over time. The line would be best if the situation was price per pound.)
- "When we fit a curve to data points, will the curve always go precisely through all of the points?" (Not always, but the shape of the curve should fit the general shape of the data.)
Student Lesson Summary
The data in the graph show the temperature , in degrees Fahrenheit, of a can of soda hours after it was put into the refrigerator.
Graph of a decreasing exponential function, h T plane. Horizontal axis, scale negative 12 to 18 by 2’s. Vertical axis, scale zero to 60 by 10’s. Function is discrete and has a horizontal asymptote at y = 35.
What if we want to build a function that fits this data set? One way to find a function that fits the data well is to start with a simpler function that has the same general shape as the data when graphed and transform it. What shape does this data form?
Let’s try an exponential decay function. We can get the right shape using a simpler equation like , but the graph doesn't fit where the data is. The graph of the function given by isn't represented by a simple equation, but it does fit the data. (What did multiplying by 45 and adding 36 do to the graph?) In this unit we will learn how to translate, reflect, and stretch graphs to fit data.
Graph of two decreasing exponential functions, h T plane. Horizontal axis, scale negative 12 to 18 by 2’s. Vertical axis, scale zero to 60 by 10’s. Blue function is T = (0 point 7)^h and green function is T = 36 + 45 times(0 point 7)^h.