Lesson 25

This warm-up prompts students to compare four descriptions of trips to an amusement park. It gives students a reason to use language precisely (MP6). It gives the teacher an opportunity to hear how students use terminology and talk about characteristics of the items in comparison to one another.

Arrange students in groups of 2–4. Display the descriptions for all to see. Give students 3 minutes of quiet think time and then time to share their thinking with their small group. In their small groups, ask each student to share their reasoning why a particular item does not belong, and together, find at least one reason each item doesn't belong. To allow all students to access the activity, at least one item has a non-mathematical reason it does not belong. Encourage students to find reasons based on mathematical properties.

Monitor whether students notice for which families, and which questions, a range of solutions is possible.

Ask each group to share one reason why a particular item does not belong. Record and display the responses for all to see. After each response, ask the class if they agree or disagree. Since there is no single correct answer to the question of which one does not belong, attend to students’ explanations and ensure the reasons given are correct. During the discussion, ask students to explain the meaning of any terminology they use, such as “answer,” “solution,” or “inequality.” Also, press students on unsubstantiated claims.

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).

Explain to students that they are going to be representing some constraints. Ask students to reflect back on other situations they studied involving constraints, and some of the ways they represented them. (When we fit a budget or the riddles, we wrote equations and used tables. We made graphs, tables, and wrote equations and inequalities.)

If no student mentions it, remind students of the strategies they developed to represent constraints graphically:

• trying points and color-coding them
• making a systematic table
• writing equations and inequalities based on the situations

Explain to students that they get to make choices about which strategies and representations they use to create their graph.

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 to 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 which satisfied multiple constraints at once. Possible discussion questions:

• "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. 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.

Arrange students in groups of 2. Display the graph for all to see. Gesture to show the whole of region A which includes parts of both the second and third quadrant.

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: \(y > x + 1\) and \(y > \text{-x}-2\)
• region C: \(y < x + 1\) and \(y > \text{-x}-2\)
• region D: \(y < x + 1\) and \(y < \text{-x}-2\)

Some questions for discussion:

• "The equations \(y = x\) and \(y = x + 5\) 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?" (\(y > x\) and \(y < x + 5\) would shade the region between the two lines.)
• "Is the point \((3, 2)\) in the region that would be shaded by both inequalities \(y \leq 2x - 3\) and \(y \geq 1-x\)? Explain your reasoning." (Yes since \(2 \leq 2 \boldcdot 3 - 3\) and \(2 \geq 1-3\) are both true.)
• "Without graphing, what is a point that would be in the region shaded by both inequalities \(x + 2y > 3\) and \(5x < 3y + 1\)? Explain your reasoning" (The point \((0,2)\) is in the shaded region since \(0 + 2 \boldcdot 2 > 3\) and \(5 \boldcdot 0 < 3 \boldcdot 2 + 1\) are both true.)