Some students may think that the height must be known before they could find the missing area or base. Encourage them to look for a pattern in the table and to reason from there.

In an earlier lesson, students encountered relationships where two quantities that could vary were multiplied together to yield a constant. The focus there was on studying the relationships and describing them using equations—in any form. For example, if and are quantities that vary, and  is a constant, they might write , , or . 

In this activity, students encounter a similar relationship. The goal here, however, is for students to recognize—by repeatedly calculating the value of one quantity and then the value of the other quantity—that a particular form of equation might be handy for finding one quantity but not so handy for finding the other. 

When answering the first couple of parts of the first question (finding the length of a section given the number of volunteers), students can use various strategies to efficiently reason about the answers. They might draw diagrams, use proportional reasoning, or think in terms of multiplication (asking, for example, "8 times what number equals 2?"). They might also write an equation such as  , substitute the number of volunteers for , and then solve the equation.

As students progress through the parts, they will likely notice that some strategies become less practical for finding the value of interest. One strategy, however, will remain efficient: dividing 2 by the number of volunteers, or evaluating  , or , at the given values of .

Likewise, when answering the third question (finding the number of volunteers given the length of a section), students could start out with a variety of strategies and fairly easily find the number of volunteers. Later, however, the number of volunteers becomes a bit more cumbersome to find except when using division (that is, dividing 2 by the given length, or evaluating  , or , at the given values of ).

Identify students who use these or other approaches, and select them to share their strategies during discussion. 

Select students to present their strategies for solving either set (or both sets) of questions. Start with students using the least straightforward approach and end with those who wrote for the first set of questions (or for the second set of questions).

Emphasize that isolating the variable that we're interested in—before we substitute any known values—can be an efficient way for solving problems. Once we pin down the variable of interest first and see what expression is equal to it, we can simply evaluate that expression and bypass some tedious steps.

Highlight that isolating a variable is called “solving for a variable.” In road clean-up context, if we want to know the length of a road section each volunteer would be responsible for, we can solve for . If we want to know how many volunteers would be needed, we can solve for .

In this activity, students solve problems involving two quantities in non-proportional linear relationships. As before, they are prompted to reason repeatedly about the value of one quantity given the other, and to generalize the process by writing an expression (MP8). They then connect the work here to the idea of writing an equation and isolating a variable of interest.

Monitor for students who:

• Find the missing value by reasoning about the quantities informally.
• Write an equation and reason about the equations.
• Write and solve the equation for a variable.

For example, to find the number of minutes that have passed when Tank B has 18 liters left, students may:

• Reason that the tank has lost 62 liters because or . If each minute it loses 2.5 liters, then it would take or 24.8 minutes for it to lose 62 liters.
• Write and think: "I'm looking for a number that, when multiplied by 2.5 and subtracted from 80, gives 18," or "I'm looking for a number that, when multiplied by 2.5, is equal to ."

• Write and solve for this way:

  Or this way:

Plan to have students present in this order to move students from less to more formal arguments involving equations.

Regardless of the approach students take, the important idea to spotlight (for Tank B) is that finding the time at which the water in the tank reaches a certain volume can be done by subtracting that volume () from the original volume (80 liters) and then dividing it by 2.5. That process can be summarized by the expression: .

If we start out with the equation and perform allowable moves to isolate , we will end up with .

For students who struggle to write expressions for liters and liters, encourage them to revisit their previous three calculations. Some students may need to write out their work more carefully before noticing that they could perform the same steps using or in place of a number.

Invite previously selected students to share their methods for isolating the variable of interest. Sequence the discussion of the strategies by the order listed in the Activity Narrative. If possible, record and display their work for all to see.

Connect the different responses to the learning goals by asking questions such as:

• "How do you know that the expression gives us the liters of water in Tank A after m minutes?" (In both cases—when finding the liters of water in the tank after 4 minutes and after 80 seconds—we multiplied the minutes by 9 and then added the result to 124. So for m minutes, we'd multiply m by 9 and add that to 124.)
• "How do you know that the expression gives us the minutes at which the tank reaches  liters?" (When finding the minutes at which the tank reaches 151 liters, 191.5 liters, and 270.25 liters, we subtracted 124 from each number and then divided the difference by 9. So we can do the same for liters.)
• “From the description of the situation, is it easier to write an expression for finding the number of liters in the tank or an expression for the number of minutes passed?” (It is easier to write an expression for the amount of water left in the tank because the rate is given in terms of liters per minute.)

For the questions asking to find the number of minutes passed for a variable amount of water, demonstrate how to write an equation and solve for the variable of interest, if no students mention this method.

For example, in Tank A, we know the relationship between the liters of water in the tank, , after  minutes is: . To find the minutes after which the tank reaches  liters, we can isolate :