This Warm-up gives students a chance to get an idea of how to add subjective values to a modeling situation. In later activities for this lesson, students model planning a trip using different methods of transit. One consideration for the methods of transit is which method is generally preferred. Students should keep their values in the table for an activity later in the lesson.

Display a table with additional columns, and invite 2–4 students to share their responses. Clarify their understanding by asking questions such as:

• “Why do you think this student gave this mode of transit a value greater than 1?” (They like using that mode of transit more than walking.)
• “Did anyone use a value less than 1? What does that mean?” (Yes. It means that they like walking more than that mode of transit.)
• “Could you rank Jada’s favorite methods of transit from least favorite to most?” (Yes. She likes riding the bus least, then riding a scooter a little more, then walking, then taking a car, then the train, and likes riding a bike the most.)
• “What considerations did you use to rank your enjoyment riding a bus?” (On the one hand, it is better for the environment to use public transit, and it is easy to ride without having to pay attention. On the other hand, it doesn’t go directly where I want, and I’d prefer not to be around so many people.)

In this activity, students use their insights from the unit to analyze and interpret a set of mathematical models and a set of data in context. Each situation involves more than two constraints, and can therefore be represented with a system with more than two inequalities.

Interpreting and connecting the inequalities, the graphs, and the data set (which involves decimals) prompts students to make sense of problems and persevere in solving them (MP1), and prompts them to reason quantitatively and abstractly (MP2).

Focus the discussion on the connections between the graphs and the inequalities, and on the last two inequalities for each trip. Ask questions such as:

• “How did you know which modes of transit each person used?” (By matching the coefficients in two of the inequalities to the values in the table.)
• “Why do you think Jada and Tyler both included the inequalities in which the variables are greater than zero?” (The distance on each mode of transit cannot be negative. They each used two modes of transit, so the distance on each must also not be zero.)
• “How do those inequalities affect the graph of the solution region?” (They limit the solution region to the first quadrant.)
• “Jada and Tyler each wrote five inequalities. Could all five form a single system?” (Yes) What does it mean to have a system with five inequalities?" (There are five constraints that must be met. The solutions to the system satisfy all five constraints simultaneously.)

This activity is designed to give students opportunities to use their understandings from this unit to perform mathematical modeling.

The travel context is familiar from the previous activity, but students are challenged to choose quantities, determine how to represent them, interpret and reason about them, and use the model they create to make choices. It also enables students to reflect on their model and revise it as needed (MP4).

Students are likely to want to use graphing technology because the information involves decimals and the inequalities written would be inconvenient to graph by hand. This is an opportunity for students to choose tools strategically (MP5).

Select groups to share their visual displays. Encourage students to ask questions about the mathematical thinking or design approach that went into creating the display. Here are questions for discussion, if not already mentioned by students:

• “What constraints did every group use?” (All groups used that their variables must be positive.)
• “How do the graphs of the various trips compare?” (They all have an overlapping region, but some use more constraints than others.)
• “Did anyone have to revise or change their model in order to come up with a solution they could use?” (Yes, we didn’t know what number to use for emissions, so we adjusted that number until it created a region that made sense.)
• “How did you use the graph to find a plan that works for the conditions of your trip?” (We looked at the overlapping shaded regions and tried to find a grid point in that region.)