21.1: Which One Doesn’t Belong: Lists (5 minutes)
This warm-up prompts students to carefully analyze and compare lists of numbers. In making comparisons, students have a reason to use language precisely (MP6). The activity also enables the teacher to hear the terminology students know and how they talk about characteristics of linear change or exponential change.
Arrange students in groups of 2–4. Display the lists 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.
Which one doesn’t belong?
A: 81, 85, 87, 90, 93, 96
B: 81, 78, 75, 72, 69, 66
C: 10, 13, 16, 19, 16, 13
D: 81, 27, 9, 3, 1, \(\frac13\)
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 linear, exponential, rate of change, or growth factor. Also, press students on unsubstantiated claims.
21.2: Holy Agave! (15 minutes)
The purpose of this activity is to review how to create a scatter plot using technology and come up with a function to model the data. This work is similar to the work in the associated Algebra 1 lesson, but includes more explicit instructions and steps toward analyzing data and choosing an appropriate model (MP4).
Display the table of data for all to see, and ask students whether they think a linear or exponential model would be a better fit for this data. After a minute of quiet think time, ask a few students to share their reasoning.
If needed, demonstrate using graphing technology to input the given data and create a scatter plot.
If the graphing technology platform you are using allows it, consider showing students how to create sliders. In Desmos, simply type \(y=mx+c\) in the expressions list, and then click “all” to add sliders for both \(m\) and \(c\). You may need to adjust the maximum value of one or both sliders to be able to use them to create a good model for this data.
In the spring, an agave plant sends up a flower spike. Here are some data collected from an agave plant in a garden in Tucson, AZ, starting on April 2:
|day||height in inches|
- Use graphing technology to create a scatter plot, using days as the first coordinate and height as the second coordinate.
- Would a linear or exponential model be a better fit for this data?
- Create a function that is a good model for the data. If you chose an exponential model, start with the equation \(y=a \boldcdot b^x\) and select values for \(a\) and \(b\). If you chose a linear model, start with the equation \(y=mx+c\), and select values for \(m\) and \(c\).
- Graph your equation on that same coordinate plane as your scatter plot. Adjust the numbers you used in the equation to improve your model.
- Explain what each number in your equation means in this situation.
- Use your model to predict the height of the flower spike on day 10.
- Describe any limitations on the domain of the function modeling the data.
Display the scatter plot using technology. Explore some of the different equations students used as a model. Discuss which model is the better fit, and what makes a model a good fit. Invite students to share their responses to the last three questions.
21.3: Let’s Model Some Stuff (20 minutes)
This activity is an opportunity to practice some of the discrete skills used in modeling data in two variables with a function. The first two items are more scaffolded and the last two are much less so.
Data set A: The height of some buildings, in feet, and the number of floors in each building. Would a linear or exponential model be a better fit?
- Which equation would be the better model for the data, where \(x\) represents number of stories and \(y\) represents height of the building in feet?
- \(y= 21.5 \boldcdot (11.5)^x\)
- What is the meaning of the 11.5 and the 21.5, in this situation?
Data set B: The “enlarge by 25%” feature on a copy machine is used several times on a photo. The width of the photo in centimeters is measured after each copy is made. Would a linear or exponential model be a better fit?
- Which equation would be the best model for the data, where \(x\) represents the number of the copy and \(y\) represents width of the photo in centimeters?
- \(y=10 \boldcdot (0.25)^x\)
- \(y=10 \boldcdot (1.25)^x\)
- What is the meaning of the two numbers in the equation for the model?
- Data set C: The height of a different agave plant over time. Come up with an equation that would be a good model for this data.
day 0 1 2 3 4 5 6 7 8 height in inches 34 44 52 61 68 74 83 91 97
- Data set D: A person used a computer simulation to roll number cubes, and count how many rolls it took before all of the cubes came up sixes. This table shows the results. Come up with an equation that would be a good model for this data.
number of cubes 1 2 3 4 5 number of rolls 5 29 140 794 3,861
Focus the discussion on how students decided on the model to use for data sets C and D.
Ask students what features they would expect in a good linear or exponential model. Important responses include:
- the population predicted by the model is close to the actual data
- the general trend predicted by the model respects the data
Ask students how they went about fitting the line or curve for their linear or exponential models. Possible responses include:
- plotting the points and using a ruler to sketch a line that fits the data well, and then finding an equation for the line
- finding a line or exponential curve that goes through two of the data points, and then using it find the slope or the factor of growth
- finding an average of differences over equal intervals, and then fitting that line through the first data point
- finding an average of factors of growth over equal intervals, and then fitting an exponential function through the first data point