Lesson 21

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.

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.

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.

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.

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.

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