In this Warm-up, students look for a relationship between two quantities by interpreting a verbal description and analyzing pairs of values in a table. They then use the observed relationship to find unknown values of one quantity given the other and to think about possible equations that could represent the relationship more generally (MP8).

The work here reinforces the idea that the relationship between two quantities can be expressed in more than one way, and that some forms might be more helpful than others, depending on what we want to know. In this context, for instance, if we know the area of the parallelogram and want to know its base length, the equation  is more helpful than .

Invite students to share their responses and explanations. Then focus the whole-class discussion on the second question. Discuss with students:

• "Are the two equations we chose equivalent? How do you know?" (Yes. There is an acceptable move that takes one to the other. If we divide each side of by 3, we have , which can also be written as . If we multiply each side of by 3, we have or .)
• "If we know the base, which equation would make it easier to find the area? Why?" (. The variable for area is already isolated. All we have to do is multiply the base by 3 to find the area.)
• "If we know the area, which equation would make it easier to find the base? Why?" (. The variable for the base is already isolated. We can just divide the area by 3 to find the base.)

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. 

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 .