Lesson 6

This warm-up prompts students to compare four images. It gives students a reason to use language precisely (MP6) and gives the opportunity to hear how they use terminology and talk about characteristics of the items in comparison to one another. To allow all students to access the activity, each item has one obvious reason it does not belong. Encourage students to move past the obvious reasons and find reasons based on mathematical properties.

Arrange students in groups of 2–4. Display the images 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 rate of change, slope, or linear function. Also, press students on unsubstantiated claims.

The purpose of this activity is for students to understand the difference between actual data and information predicted by the linear model. This prepares students to understand that a line of best fit, while representative of the data, is different from the actual data and to start thinking about what differences occur between the two. This is useful for when students learn about residuals in the associated Algebra 1 lesson.

The purpose of this discussion is for students to understand that the actual data can differ from a linear model even when the model is a good one. Discuss how students used the graph and table to answer the questions. Here are sample questions to promote class discussion.

• “Is the table more helpful in answering questions about the actual data or the values estimated by the linear model?” (the actual data)
• “Which is more helpful in answering questions about the values estimated by the linear model: the graph, the equation, or the table?” (The graph that displays the linear model and the equation that represents the linear model. The graph is helpful in finding how many times the model and the actual sales were equivalent. The equation is helpful in finding when there was a difference of at least 25 between actual sales and the amount of sales predicted using the linear model.)
• “When asked about estimates, how do you use the graph to answer the question?” (The line on the graph shows every estimate made using the linear model \(y=1,\!000-16x\). Finding a point on the line that corresponds to specific information gives additional information about estimated values.) 
• “When asked about what was actually sold, how do you use the graph to to answer the question?” (The plotted points show how many sales were actually made when the eyeglasses were \(x\) dollars.)
• “How can you tell when the linear model and the actual data are equivalent?” (The plotted points are directly on the line when the two are equivalent.)
• “How can you tell when the linear model predicts a sales amount that is more than the actual sales amount?” (If the linear model predicts sales that are more than the actual amount, then the line will be above the plotted points.)

The purpose of this activity is for students to practice interpreting a linear model and understanding the difference between the actual data and the linear model predictions. Students can use the equation and graph to complete the table, which allows them to synthesize the graph, equation, and table to understand that they represent the same information. Then, students use the graph, equation, and completed table to interpret the data and predictions.

The purpose of this discussion is for students to interpret data and predictions using a linear model that is represented in three ways: using an equation, a graph, and a table. Discuss how students used the representations to answer the questions. Here are sample questions to promote class discussion:

• “How can the linear equation be used to complete the table with estimated amounts spent on gas?” (To use the equation to complete the table, substitute the given value for \(x\) and solve for the estimated \(y\).)
• “How can you use the graph to complete the estimated amount spent on gas?” (To use the graph to complete the table, find the point on the line associated with the given \(x \) to find the estimated \(y\)-value. Be sure to use the coordinates from the line, not the plotted points on the scatter plot.)