This Math Talk focuses on values meeting a constraint. It encourages students to think about using values in an equation and to rely on what they know about cost in context to mentally solve problems. The strategies elicited here will be helpful later in the lesson when students write and solve systems of linear equations in two variables.

Many students are likely to approach the task by multiplying pounds of raisins by 4 and pounds of walnuts by 8, and then see if the sum of the two products is 15. Some students may arrive at their conclusions by reasoning and estimating. For example, seeing that a pound of walnuts costs $8, students may simply reason that the first combination is not possible. Or they may reason that the last combination is impossible because at $4 a pound, 3.5 pounds of raisins would cost more than $12, so it is not possible to also get 1 pound of walnuts and pay only $15 total. 

In describing their strategies, students need to be precise in their word choice and use of language (MP6).

Ask students to share their strategies for each problem. Record and display their responses for all to see. To involve more students in the conversation, consider asking:

• “Who can restate ’s reasoning in a different way?”
• “Did anyone have the same strategy but would explain it differently?”
• “Did anyone solve the problem in a different way?”
• “Does anyone want to add on to ’s strategy?”
• “Do you agree or disagree? Why?”
• “What connections to previous problems do you see?”

Conclude by reminding students that the given situation involves a cost constraint. Emphasize that only the third option, 2.25 pounds of raisins and 0.75 pounds of walnuts, satisfy the constraint. One way to check if certain values meet the constraint is by writing an equation and checking if it is true. For example, the equation is true. If we replace the weights of raisins and walnuts with other pairs of values, the equation would be false.

In this activity, students write and graph equations to represent two constraints in the same situation and use tables and graphs to see possible values that satisfy the constraints. The work prompts them to think about a pair of values that simultaneously meets multiple constraints in the situation, which in turn helps them make sense of the phrase "a solution to both equations."

Monitor for students who use these strategies:

• Guessing and checking
• Using the graph (by identifying the point with a given - or -value and estimating the unknown value, or using technology to get an estimate)
• Substituting the given value of one variable into the equation and solving for the other variable

Plan to have students present in order from less to more precise.

This activity allows students to apply what they learned about meeting multiple constraints simultaneously and to practice solving simple systems of equations in context. Students are given several problems that can be efficiently solved by writing and solving systems.

As students work, notice the strategies they use. Many students may choose to graph the systems because that strategy was used in the first activity, but some students may choose to solve the systems by guessing and checking, creating tables, or by reasoning algebraically. (In middle school, students learned the substitution method for solving systems of equations. Expect students to have varying degrees of facility with it at this point.)

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