The mathematical purpose of this activity is to give students practice recognizing situations that can be represented by inequalities and equations, and by systems of equations and inequalities.

There are many options for tailoring this activity to students’ needs. The information about each family can be separated into two problems so that students don’t need to work with systems of equations or inequalities; they can just express each constraint with a single equation or inequality. Students can also skip the graphing step for any situation that is not a system of inequalities. Or students can be given the option to represent the solutions to each situation in any way that shows what’s going on.

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

The goal of this discussion is for students to explain how they know that a particular constraint, such as the amount of time a family wants to spend at the amusement park, is best represented by an equation or inequality, and how they know. This supports students as they further connect real-life situations to symbolic and graphical representations of inequalities.

Select students to share their responses. Ask students to describe any equations or inequalities that could be used to represent the constraints.

Focus the discussion on which constraints were represented with inequalities, how students came up with the inequalities, and how students used graphs or reasoning to find ordered pairs that satisfied multiple constraints at once. Here are some questions for discussion:

• "Which constraints could be represented with inequalities?" (Both of the constraints for Clare's family, Jada's family's restriction to spend most of their time riding rides, and both of Diego's family's constraints can be represented with inequalities.)
• "How did you come up with the inequalities for these constraints?" (I thought of the limiting equation, then decided the direction of the inequality based on a coordinate pair that made sense in the situation.)
• "How did you find ordered pairs that had to satisfy more than one constraint?" (Using the graph or reasoning about the situation.)

It is important for students to leave this discussion feeling confident in recognizing a constraint in a given situation and how to represent that constraint with mathematical symbols.

This activity gives students an opportunity to practice interpreting regions of the plane created by two lines dividing the plane. Students select a point in each of the 4 regions, change the linear equations into inequalities that would shade a given region, and check that their point makes their inequalities true. As students decide what inequality symbols are needed to shade a region, they look for and make use of structure (MP7). In the associated Algebra 1 lesson, students explore the idea of a system of inequalities and regions that are shaded by both inequalities to represent the solution.

The purpose of the discussion is for students to recognize that regions in the plane can be described by several inequalities and points in that region will make both inequalities true.

If time permits, select another region for students to describe by modifying the equations into inequalities.

• Region B: and
• Region C: and
• Region D: and

Here are some questions for discussion:

• "The equations and are represented by parallel lines. How can you change the equations to inequalities so that the region between the two lines is shaded by both inequalities?" ( and would shade the region between the two lines.)
• "Is the point in the region that would be shaded by both inequalities and ? Explain your reasoning." (Yes, since and are both true.)
• "Without graphing, what is a point that would be in the region shaded by both inequalities and ? Explain your reasoning." (The point is in the shaded region since and are both true.)