Lesson 11
Skills for Modeling with Mathematics
These materials, when encountered before Algebra 1, Unit 5, Lesson 11 support success in that lesson.
11.1: Which One Doesn’t Belong: Four Graphs (5 minutes)
Warm-up
This warm-up prompts students to compare four graphs. 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 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 why a particular item does not belong, and together find at least one reason each item doesn’t belong.
Student Facing
Which one doesn’t belong?
A
B
C
D
Student Response
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Activity Synthesis
Ask each group to share one reason why a particular item does not belong. 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 one does not belong, 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, decreasing, linear, nonlinear, or quadrant. Also, press students on unsubstantiated claims.
11.2: Which Model? (20 minutes)
Activity
The purpose of this activity is to learn or remember a straightforward way to check if a relationship is linear or exponential (subtracting and dividing successive terms). Also, in the synthesis, there is an opportunity for students to rehearse creating a scatter plot, and recall how to recognize that a graph of a relationship might indicate that it is linear or exponential.
Student Facing
Consider each situation:
- A person starts with $24,000 in a savings account. Each month, she deposits an additional $2,000 in the account.
- A 30-year old puts $24,000 in a retirement account that increases by 10% each year.
- The value of a car depreciates by a factor of \(\frac45\) of the car’s value every year. The car initially cost $24,000.
- A farmer has stored 24,000 pounds of grain. His cows eat 4,800 pounds of grain per month.
Match each situation to one of these tables:
A.
| \(x\) | \(y\) |
|---|---|
| 0 | 24000 |
| 1 | 19200 |
| 2 | 15360 |
| 3 | 12288 |
B.
| \(x\) | \(y\) |
|---|---|
| 0 | 24000 |
| 1 | 26000 |
| 2 | 28000 |
| 3 | 30000 |
C.
| \(x\) | \(y\) |
|---|---|
| 0 | 24000 |
| 1 | 19200 |
| 2 | 14400 |
| 3 | 9600 |
D.
| \(x\) | \(y\) |
|---|---|
| 0 | 24000 |
| 1 | 26400 |
| 2 | 29040 |
| 3 | 31944 |
Student Response
Teachers with a valid work email address can click here to register or sign in for free access to Student Response.
Activity Synthesis
Questions for discussion:
- “How can you tell if the relationship is linear or exponential?” (See if there is a constant rate of change by subtracting successive terms and seeing if the difference is constant. See if there is a constant growth factor by dividing successive terms and seeing if the quotient is constant.)
- Look more closely at the table for the depreciating car. How could you compute that the growth factor is \(\frac45\)? (If you divide any two successive terms, like \(19,\!200 \div 2,\!400\) or \(15,\!360 \div 19,\!200\), you get 0.8 which is equal to \(\frac45\).
Demonstrate how to create a scatter plot using technology using the table of values about the farmer’s grain. Then, ask students to use technology to create a scatter plot using the table of values about the car depreciation. Display these graphs side by side and observe ways in which the graphs suggest that one would be better modeled with a linear function and the other with an exponential function.
If students have access to the digital version of the materials, the “Graphing Calculator” tool under Math Tools is recommended. Add a new table, enter the values, and then adjust the graphing window (under the wrench button) so that the scatter plot is visible. Here is an example https://www.desmos.com/calculator/gdxthrlgcz
11.3: Growth of a Small Business (20 minutes)
Activity
This activity is an opportunity to practice deciding between a linear and exponential model based on data given in a table. It is also an opportunity to practice creating a scatter plot using technology. The data is not perfectly linear or exponential, which is addressed in the launch.
Launch
Tell students that in this activity, they have an opportunity to decide whether a linear or exponential model is better for a given set of data and create a scatter plot and a model. Draw students’ attention to the word “approximately” in the task statement. In this case, the data can not be fit perfectly with a model. So, when they are checking for constant differences or quotients, they’re looking for a number that best encapsulates the differences or quotients they find.
Student Facing
Here are two sets of data representing the annual revenue of two different small businesses for the past ten years. One of them had growth that was approximately linear, and one of them had growth that was approximately exponential. The revenue is expressed in thousands of dollars.
Business A:
| year | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |
|---|---|---|---|---|---|---|---|---|---|---|
| revenue | 61.2 | 68.4 | 74.9 | 83.1 | 88.5 | 96.4 | 104.1 | 109.9 | 117.0 | 125.2 |
Business B:
| year | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |
|---|---|---|---|---|---|---|---|---|---|---|
| revenue | 40 | 47.9 | 57 | 70.1 | 82.4 | 99.5 | 118.9 | 144.1 | 172.0 | 205.8 |
- Which company’s growth could be modeled by a linear function, and which by an exponential function?
- For the company with exponential growth:
- What was the growth factor?
- What is an equation that represents the relationship between year and revenue?
- Use technology to make a scatter plot of the data and also graph your equation. If your equation does not look like a good model for the data, adjust it until you have a good model.
- For the company with linear growth:
- What was the rate of change?
- What is an equation that represents the relationship between year and revenue?
- Use technology to make a scatter plot of the data and also graph your equation. If your equation does not look like a good model for the data, adjust it until you have a good model.
Student Response
Teachers with a valid work email address can click here to register or sign in for free access to Student Response.
Activity Synthesis
Invite a few students to display their scatter plots and models. Because the data is a little bit messy, different students might come up with models that are slightly different from each other but still reasonable. Ask these students to explain:
- “How did you decide whether a linear or exponential model would be better?”
- “How did you determine the growth factor or rate of change?”
- “How did you construct your equation?”
- “How does the scatter plot and graph show that your model is a good fit for the data?”