Lesson 9

In this warm-up, students are given a simple equation in three variables and are prompted to rearrange it to pin down a particular variable. In each question, only the value of one variable is given, so students need to manipulate the equation even when some quantities are unknown. The work here prepares students to rearrange other variable equations later in the lesson.

As students work, look for those who substitute the given value before rearranging and those who first isolate the variable of interest before substituting. Invite them to share their approaches during class discussion.

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 \(\ell = \frac{2}{n}\) to quickly find the length of the road section each volunteer would clean up, \(\ell\), if there were \(n\) volunteers. They wrote \(n=\frac{2}{\ell}\) to quickly find the number of volunteers, \(n\), if each volunteer were to clean up \(\ell\) miles.

Select previously identified students to share their responses and strategies. Record and display for all to see the steps they take to rearrange the equations. Emphasize how each step constitutes an acceptable move and how it keeps the equation true.

Make sure students see that we can either substitute known values into the given equation before rearranging it, or we can rearrange the equation first before substituting known values. In the examples here, it doesn't matter which way it is done. Ultimately, we were solving for \(V\) in the first question and for \(F\) in the second question.

Explain that there will be times when one strategy might be more helpful than the other, as students will see in subsequent activities.

This activity encourages students to write an equation in two variables to represent a constraint and then 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.

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, \(c\), given \(t\) trucks, students may have arrived at the expression \(\dfrac {21,600 - 7.5t}{3.6}\) by generalizing the calculation they performed when the number of trucks was a numerical value. Likewise, in the last question they may have arrived at \(t = \dfrac{21,600 - 3.6c}{7.5}\) by using some numerical values for \(c\) and generalizing the process.

While this strategy is expected and perfectly reasonable, make sure students also see that we can arrive at the same expression for \(c\) and for \(t\) by rearranging the equation \(3.6c + 7.5t = 21,\!600\).

Highlight that we can solve for \(c\) when we know the number of trucks and want to compute the number of cars, and solve for \(t\) 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).

Familiarize students with the idea of organizational budgets, if needed. Explain that every governmental agency has a budget (not unlike a budget for a pizza party) and needs to decide how to allocate a limited amount of funds. Ask students to think of examples of expenses that a city's Department of Streets might have (road repairs, new street construction, additional workers, new equipment, and so on). Given a set budget, spending more on one thing means having less to spend on something else, so the agency would need to weigh their options and prioritize their needs.

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 correctly isolate each variable. If students wrote expressions in different forms for a variable (for instance, \(\dfrac {4,\!907 - 210m}{90}\) and \(\dfrac {1,\!962,\!800 -84,\!000m}{36,\!000}\) for \(p\)), 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 \(p\) 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 \(m\) makes it possible to quickly find the miles that could be resurfaced given any number of new hires.