Some students may be unclear what it means to write an equation that "makes it easier to find the number of vertices (or faces)." Remind them of the work in an earlier activity. In the post-parade clean-up activity, for instance, they wrote the equation to quickly find the length of the road section that each volunteer would clean up, , if there were volunteers. They wrote to quickly find the number of volunteers, , if each volunteer were to clean up miles.

This activity encourages students to write an equation in two variables to represent a constraint and then to solve for each of the variables. One motivation for rearranging the equation is to find an expression that, when entered into a calculator or computer, can then be used to quickly find the value of one quantity given the value of the other.

The digital version of this activity includes instruction for using GeoGebra spreadsheets to perform calculations. As they use spreadsheet technology to create mathematical models, test them, and solve problems, students engage in aspects of mathematical modeling (MP4).

When finding the number of cars, , given trucks, students may arrive at the equation  by generalizing the calculation that they performed when the number of trucks was a numerical value. Likewise, in the last question they may arrive at by using some numerical values for and generalizing the process.

While this strategy is expected and perfectly reasonable, make sure that students also see that we can arrive at the same expression for and for by rearranging the equation .

Highlight that we can solve for when we know the number of trucks and want to compute the number of cars, and solve for when we know the number of cars and want to find the number of trucks.

This activity gives students another opportunity to write and rearrange equations in two variables and to do so in context.

Unlike in previous activities, students no longer start by computing one quantity given numerical values of the other quantity and then generalizing the process. Instead, they are prompted to write one equation, solve for each variable, and interpret the solution. Students also articulate why one model might be more helpful than the other under a certain circumstance. Along the way, students reason quantitatively and abstractly (MP2) and engage in aspects of modeling (MP4).

Making spreadsheet technology available gives students an opportunity to choose appropriate tools strategically (MP5).

Some students may have trouble sorting out the given information because of unfamiliarity with the context or with certain terms. Ask them to explain the setup of the problem as best they understand it, and then point out any information that might be missing.

Verify that the equations students wrote do correctly isolate each variable. If students wrote expressions in different forms for a variable (for instance, and  for ), discuss how the expressions are equivalent.

If time permits, consider asking students to use their equations to answer these questions:

• "How many people could be hired if the department is resurfacing 16 miles of roads?" (17 people) "3 miles?" (47 people)
• "How many miles of roads could be resurfaced if the department is hiring 2 new workers?" (22.5 miles) "If no new workers are hired?" (About 23.37 miles)

Make sure students understand that solving for makes it possible to quickly find the number of people that the department could hire given some miles of road to be resurfaced (and while sticking to the budget). Similarly, solving for makes it possible to quickly find the miles that could be resurfaced given any number of new hires.