The purpose of this activity is to see a method of developing an equation to model a set of data.

This activity uses the Worked Example routine. In this routine, students analyze steps used to solve a problem and discuss specific steps with a partner and the whole class. During the discussions, students construct viable arguments to justify the steps taken by the problem solver (MP3). Later, students can apply these strategies to solve similar problems.

The purpose of this discussion is to explore a method of creating an equation to model a set of data.

Tell students to open their books or devices. Display the following questions, and give students 1–2 minutes of quiet think time to consider their responses.

• “What did the solver do first? Why?” (The solver drew a line that fits the data well. Visualizing the line makes it easier to identify a reasonable -intercept and slope.)
• “In Step 2, the solver identified two new points that are not part of the data. Why did they create those points?” (The points the solver identified are both on the line. One is the -intercept, and the other is another point on the line. The solver can use the two points to calculate the slope, or they could draw a slope triangle using those points.)
• “In the last step, the solver wrote an equation. How can they make sure it is a good model of the data?” (The can check by graphing the line and seeing how well it fits the data or by making a table of values to see if it is similar to the original data.)

Ask partners to compare their responses to each question and decide if either or both answers are correct. During this partner discussion, monitor to select groups who reason about the questions in different ways. Follow with a whole-class discussion, inviting 1–2 selected groups to share their answers for each question.

If time allows, consider displaying a second worked example next to the first that either:

• Shows a different method of solving a similar problem that students can compare and contrast with the first example (MP7)
• Shows the same method with an error that students can critique and correct (MP3)

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).

The purpose of this discussion is to compare the equation models students created. Display the scatter plot using technology. Graph some of the different equations students used as models. Here are some questions for discussion:

• “Which model seems to fit the data best?”
• “What do the models you created have in common?” (They have similar -intercepts. They have similar slopes.)
• “Does everyone need to have the exact same model?” (No, as long as the model fits the data well, the models will give similar answers to the follow-up questions and any analysis that is done.)

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. Students decide between different possible models for different sets of data, which is a part of modeling with mathematics (MP4). The first two data sets are more scaffolded because they ask students questions about the type of function that would fit best and the meaning of the numbers in the equation. The other data sets are less scaffolded because they ask students only to develop an equation to model the data directly.

Monitor for students who use these different strategies:

• 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

The goal of this discussion is to compare methods of creating models based on data sets.
Display 2–3 approaches from previously selected students for all to see. Use Compare and Connect to help students compare, contrast, and connect the different approaches. Here are some questions for discussion:

• “How does the vertical intercept and common difference or common quotient show up in each method?”
• “Why do the different approaches lead to similar outcomes?”
• “How do these different representations show the same information?”

Continue the discussion, focusing 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.