Previously, students have learned that any point that is in the overlapping solution regions of the graphs of two inequalities is a solution to the system formed by those inequalities. In this activity, students take a closer look at whether points that are on the boundary lines are solutions to the system.

Focus the discussion on the points on the boundary lines and how students determined if they are or are not solutions to the system.

Highlight explanations that state that a solution to a system of linear inequalities must be a solution to every inequality in the system. If a point on the boundary line is not included in the solution set of one inequality (so the graph is a dashed line), then it is also not included in the solution set of the system.

This activity gives students an opportunity to determine and request the information that they need in order to use their understanding of systems of linear inequalities to solve problems that involve satisfying multiple constraints simultaneously.

The Information Gap structure requires students to make sense of problems by determining what information is necessary, and then to ask for the information that they need to solve it. This may take several rounds of discussion if their first requests do not yield the information that they need (MP1). It also allows them to refine the language that they use and to ask increasingly more precise questions until they get the information that they need (MP6).

Students may approach the problems in different ways, including by guessing and checking, but the problems can be efficiently solved by writing and solving systems of linear inequalities.

Every clue given in a data card can be written as an inequality. If is the number of children on a team, and  is the number of adults, the clue "a team with only adults is not allowed" can be expressed as , and "a team without adults is allowed" can be expressed as . But because other membership rules are more restrictive, and because thinking only about positive values of and is natural in this situation, it is not essential for these constraints to be represented. (The blank coordinate planes provided also show only the first quadrant, implying positive solutions.)

In an upcoming lesson, students will look more closely at the inequalities and .

Students in both roles may wonder if all clues on the data card constitute a “rule.” Those holding a data card may not know how to respond if or when asked, “What is the first rule?” or "What is one of the rules?" Those who are asking for information may not know if what is given counts as a rule. Clarify that a rule should be general enough to include multiple possibilities, rather than just one specific case. For example, "If there are 3 adults, there must be at least 6 children" is a specific case, rather than a general constraint.

After students have completed their work, share the correct answers, and ask students to discuss the process of solving the problems. Here are some questions for discussion:

• "Which membership rule does each shaded region on the graphs represent?" (Valid descriptions of the regions)
• "Can all the clues be written as an inequality? If so, what are they?" (Yes. For example on Data Card 1 the inequalities could be: where represents the number of adults and the number of children on the team.)
• "Should we graph them all? Why or why not?" (To solve the given problems, the inequalities do not necessarily need to be graphed, but it can be helpful to visualize what is happening in the situations.)
• "How does the first set of graphs help us answer questions about whether 6 adults and 8 children are acceptable?" (The point on the graph is not in the shaded region for all of the conditions, so it is not in the solution region.)
• "How do they tell us about the maximum allowable number of adults on a team?" (Looking at the solution region, the point that is the highest in the direction is , but a third of an adult does not make sense, so 5 is the maximum because 6 or more is not in the solution region.)
• "How does the second set of graphs help us find the minimum number of team members and the composition of the team?" (Points closer to the origin in the solution region of the graph could represent the minimum number of team members. In this case, the point is closest to the origin and it represents the smallest team with 6 children and 2 adults.)

Situation 1:

Situation 2:

Math Community

Invite 2–3 students to share the norm they chose and how it supported the work of the group or a realization they had about a norm that would have worked better in this situation. Provide these sentence frames to help students organize their thoughts in a clear, precise way:

• “I picked the norm ‘.’ It really helped me/my group because .”
• “I picked the norm ‘.’ During the activity, I realized that the norm ‘’ would be a better focus because .”