Lesson 18

In this activity, students recall the meaning of inequality symbols (\(\lt\), \(\gt\), \(\leq\), and \(\geq\)) and the meaning of “solutions to an inequality." They are reminded that an inequality in one variable can have a range of values that make the statement true. Students also pay attention to the value that is at the boundary of an inequality and consider whether it is or isn't a solution to an inequality.

Give students 1–2 minutes of quiet work time. Follow with a whole-class discussion.

Draw students' attention to the last inequality (\(30 \geq h\)). Make sure students see that, even though the symbol is read "greater than or equal to," it doesn't mean that we're looking for values that are greater than or equal to 30.  The statement reads "30 is greater than or equal to \(h\)," which means \(h\) must be less than or equal to 30.

Next, ask students how they know whether each of those numbers (the 50, 20, and 30, or the boundary values) is a solution to the inequality. Emphasize that we can test those boundary values the same way we test other values—by checking if they make the statement true.

Display these equations in one variable for all to see: \(h = 50\), \(h =20\), and \(30 = h\). Discuss with students how these equations are different from the inequalities in one variable (aside from the fact that the symbols are different). Highlight the idea that there is only one value that could make each equation true, but there is a range of values that can make each inequality true.

This activity prompts students to interpret several inequalities that represent the constraints in a situation. To explain what the letters in the inequalities mean in the given context, students cannot simply match the numbers in the verbal descriptions and those in the inequalities. They must attend carefully to the symbols and any operations (MP6), and reason both quantitatively and abstractly (MP2). 

The work here also engages students in an aspect of mathematical modeling (MP4). Although students do not choose the variables to represent the essential features of a situation, they think carefully about and explain how the given models do represent the key features of the given situation.

As students discuss with their partners, listen for those who could interpret the inequalities clearly and accurately. Ask them to share their interpretations later.

Arrange students in groups of 2. Give them a few minutes of quiet think time, followed by some time to share their thinking with their partner.

Some students may relate an inequality to a written description simply based on the letter chosen for the variable (for example, “people” begins with “p”). Push these students to explain how the inequalities express the quantities and constraints in the written descriptions.

For students unfamiliar with the notation \(120 \le p \le 200\), explain that this is a way of stating \(120 \le p\) and \( p \le 200\).

Invite previously identified students to share their responses. Make sure students understand why the symbols accurately represent the constraints in the situation.

For the last two inequalities, make sure students see how the operations represent the constraints on profit and on the number of chaperones.

If needed, use numbers to illustrate the relationship between variables. For example, to help students make sense of \(c\geq \frac{p}{20}\), ask: "How many chaperones are needed if there are 120 students?" (at least \(\frac{120}{20}\) or 6 chaperones, or \(c \ge\frac{120}{20}\)) "180 students?" (at least \(\frac{180}{20}\) or 9 chaperones, or \(c \ge\frac{180}{20}\)).

Previously, students interpreted given inequalities and made sense of them in terms of a situation. In this activity, students write inequalities to represent the constraints in a situation. Students identify key quantities and relationships, and think about ways to represent them. In doing so, they engage in an aspect of mathematical modeling (MP4).

As students work, look for those who represent the same constraint using different inequalities or equations. For example, to represent the total weight that the elevator could carry, some students may write \(w\leq 1,\!500\), some may write \(1,\!500 \geq w\), and others may write \(w<1,\!500\) or \(w=1,500\). Ask students with contrasting statements to share their responses later. 

Keep students in groups of 2.

Students who use a variable for the number of adults and another for the number of children may have trouble accounting for the weight of the gear because it applies to both groups. Calculating the weight of a specific combination of adults and children (with their gear) may help.

Invite students to share their equations and inequalities, starting with those that are more concrete (from the first question) and ending with the ones that are more abstract (from the last question).

Emphasize that the same constraints may be accurately represented by statements of different forms. Consider reading aloud the different inequalities that represent the same constraint. For example, if \(w\) represents the total weight:

• \(w\leq 1,\!500\) can be read: "The total weight is less than or equal to 1,500 kilograms." or "The total weight is at most 1,500 kilograms."
• \(1,\!500 \geq w\) can be read: "Fifteen hundred kilograms is greater than or equal to the total weight."
• \(w<1,\!500\) or \(w=1,500\) can be read: "The total weight is less than 1,500 kilograms, or it is equal to 1,500 kilograms."

For constraints that involve multiple quantities, some students may write, for instance, \(74a + 39c \leq 1,\!500\), while others may write \(70a + 4a + 35c +4c \leq 1,\!500\). Ask students why these expressions are equivalent, encouraging them to use the context in their explanation.

Point out that although a constraint can be written in different ways, writing it using fewer terms may be more convenient and may allow us to gain certain insights about the situation.