Lesson 12

This Math Talk refreshes students' knowledge about contraints and the values that meet them. Students are prompted to determine if four given combinations of raisins and walnuts meet a certain cost constraint. The reasoning elicited here prepares them to write and solve systems of linear equations in two variables later.

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. 

To explain their reasoning, students need to be precise in their word choice and use of language (MP6).

Display one problem at a time. Give students quiet think time for each problem and ask them to give a signal when they have an answer and a strategy. Keep all problems displayed throughout the talk. Follow with a whole-class discussion.

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 \(\underline{\hspace{.5in}}\)’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 \(\underline{\hspace{.5in}}\)’s strategy?”
• “Do you agree or disagree? Why?”

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 \(4(2.25) + 8(0.75) = 15\) 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 these likely strategies for completing the tables, from less precise to more precise:

• Guessing and checking
• Using the graph (by identifying the point with a given \(x\)- or \(y\)-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

Identify students who use each strategy and ask them to share later.

Arrange students in groups of 2 and provide access to graphing technology. Explain to students that they will now examine the same situation involving two quantities (raisins and walnuts), but there are now two constraints (cost and weight).

Give students 3–4 minutes of quiet time to complete the first set of questions about the cost constraint, and then pause for a whole-class discussion. Select previously identified students to share their strategies, in the order listed in the Activity Narrative. If one of the strategies is not mentioned, bring it up. 

Ask a student who used the graph to complete the table to explain how exactly the graph was used. For example: "How did you use the graph of \(4x+8y=15\) to find \(y\) when \(x\) is 2?" Make sure students recognize that this typically involves hovering over different points on the graph or tracing the graph to get the coordinates. 

Ask students to proceed with the remainder of the activity. If time permits, pause again after the second set of questions (about the weight constraint) to discuss how the strategies for completing the second table is like or unlike that of completing the first table. Alternatively, give students a minute to confer with their partner before moving on to the last question.

Some students might think that the values in the second table need to reflect a total cost of $15. Clarify that the table represents only the constraint that Diego bought a total of 2 pounds of raisins and walnuts.

The idea of finding an \((x,y)\) pair that satisfies multiple constraints should be familiar from middle school. If students struggle to answer the last question, ask them to study the values in the table. Ask questions such as: "If Diego bought 0.25 pound of raisins, would he meet both the cost and weight requirements?" and "Which combinations of raisins and walnuts would allow him to meet both requirements? How many combinations are there?"

Invite students to share their response and reasoning for the last question. Display the graphs representing the system (either created by a student or as shown in the Student Response). Discuss with students:

• “Can you find other combinations of raisins and walnuts, besides 0.25 pound and 1.75 pounds, that meet both cost and weight constraints?” (No)
• “How many possible combinations of raisins and walnuts meet both constraints? How do we know?” (One combination, because the graphs intersect only at one point.)

Explain to students that the two equations written to represent the constraints form a system of equations. We use a curly bracket to indicate a system, like this: 

\(\begin{cases}\begin {align} 4x + 8y &= 15\\ x+ \hspace{2mm}y&=2 \end{align} \end{cases}\)

Highlight that the solution to the system is a pair of values (in this case, pounds of raisins and walnuts) that meet both constraints. This means the pair of values is a solution to both equations. Graphing is an effective way to see the solution to both equations, if one exists.

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).

Keep students in groups of 2 and provide them with continued access to graphing technology.

Tell students that they will now look at several situations that also involve two quantities and two constraints. Their job is to represent each pair of constraints by writing a system of equations and then to find a solution to the system.

Give students 3-4 minutes of quiet time to think about the first question and then a minute to share their thinking with their partner before proceeding with the rest of the activity. Leave a couple of minutes for whole-class discussion. If time is limited, ask each student to solve two of the questions.

If students struggle to sort out the information in a problem, suggest that they start by drawing a picture or diagram to help understand the situation. Another idea is to start by creating one or more tables to list possible combinations of values and observe the relationship between the values. 

For students who need more scaffolding, ask questions such as “What equation can you write to say that the number of tables, long tables and round ones, is 25?” and “What do your variables represent?” Then, ask students to use the same variables to write an equation that says that there are 190 seats.

Select students who use different strategies to share their responses and reasoning. Start with strategies that are less systematic (such as guessing and checking) and move toward those that are more systematic (graphing or using algebra). If possible, display their work for all to see. After each student presents, ask if others solved it the same way.

If no students share an algebraic strategy, ask if anyone solved the problems without graphing. If so, invite them to share their rationale. Otherwise, it is not crucial to probe further at this moment.

If time permits, ask students to reflect on the strategies that were used. Some of them—graphing and algebraic solving—required writing equations. Others—guessing and checking, or using tables—did not. Discuss questions such as:

• “Why might it be helpful to write a pair of equations to solve these problems?”
• “Are there situation in which it might not be helpful to write equations?”