Lesson 17

The purpose of this warm-up is for students to interpret an inequality in a real-world situation and reason about the quantities in its solution. Some of the statements involve reasoning about the how a sandwich shop sells its sandwiches, however, the focus of the discussion should be on the meaning of the solution to the inequality. Students should reason that they cannot order anything more than 13.86 sandwiches, but can order any amount less than 13.86 sandwiches.

Arrange students in groups of 2. Give students 2 minutes of quiet work time followed by 1 minute to compare their responses with a partner. Follow with a whole-class discussion. 

Some students may think of 13.86 sandwiches as 14 whole sandwiches because it rounds to that number, and 13.86 doesn’t make sense to them in the context of sandwiches. It may be helpful for these students to use a calculator to find the cost of 14 sandwiches to see that is not a solution to the inequality. Tell these students that, although sandwich shops may not sell sandwiches in fractional pieces, the maximum amount that can be ordered is 13.86. 

Poll the class about whether they think each statement is valid. Ask a student to explain why the invalid statements don’t work.Record and display their responses for all to see.

For each statement, students should mention the following ideas:

1. Even though 13.86 makes the inequality true, most sandwich shops would not let you order 13.86 sandwiches.

2. He doesn’t have enough money to order 14 sandwiches. He has to order a number of sandwiches that is less than or equal to 13.86.

3. Works.

4. Might be okay if the shop allows you to order sandwiches
   in \(\frac12\)-sandwich increments.

5. Works.

6. Even though -4 makes the inequality true, that value doesn’t make sense in this context.

This problem is an introduction to the series of modeling problems in the next activity. Here, students read a question and are prompted to think about what extra information they would need to solve it (MP4). Then they write and solve inequalities to answer the question.

The context in this problem provides an opportunity for students to think about aspects of mathematical modeling like discrete versus continuous solutions and rounding. Make sure to touch on these topics in discussion before moving on to the next activity.

Ask students to close their books or devices or to leave them closed. Present this scenario verbally or display for all to see:

“A mover is loading an elevator with identical boxes. He wants to take all the boxes up the elevator at once, but he is worried about overloading the elevator. What are all the possibilities for the number of boxes the mover can take on the elevator at once?”

Give students a few minutes of quiet think time to brainstorm what information they would need to answer this question, followed by 1–2 minutes to discuss with a partner. Ask a few students to share their questions with the class and record them for all to see.

Then, ask students to open their books or devices to this activity and use the given information to help solve the problem.

Many issues will come up in the discussion of this problem that will recur throughout the lesson. Some examples:

• “How can we represent the solution on a number line? Is 5.5 a solution?” (Not in the context of this problem; you can’t have a half a box.)
• “Do we want to change the number line somehow to show this?”
  (We could plot discrete points, or we could simply leave it as is, but just know that for a problem with this context, we’re only going to use integer solutions.)
• “Which type of inequality would you use to describe answers using no more than or no less than?” (\(\leq\) and \(\geq\), respectively.)
• “How did you know which way to round?” (Round down, otherwise you’ve gone over the weight limit.)
• “What other limitations do the contexts place on the solutions?”
  (You must have a positive number of boxes.)

In this activity, students set up and solve inequalities that represent real-life situations. Students will think about how to interpret their mathematical solutions. For example, if they use \(w\) to represent width in centimeters and find \(w<25.5\), does that mean \(w=\text-10\) is a solution to the inequality?

Tell students they will practice using their knowledge of inequalities to think about specific situations and interpret what their solutions mean in those situations. Ask students to represent their solution using words, an inequality, and a graph. Arrange students in groups of 2. If necessary, demonstrate the protocol for an Info Gap activity. In each group, distribute a problem card to one student and a data card to the other student. Give students who finish early a different pair of cards and ask them to switch roles.

If students do not know where to start, suggest that they first identify the quantity that should be variable and choose a letter to represent it.
In Elena’s problem, it may help to remind students that they know how to write a formula for the area of a rectangle.

As the groups report on their work, encourage other students to think and ask questions about whether the answers are plausible. If students do not naturally raise these questions, consider asking:

• In Noah’s problem, is 1.5 loads of laundry a solution to the inequality?
• In Elena’s problem, can the width of the frame be -10 centimeters? Can the width of the frame be 0 centimeters? How about 0.1 centimeters?

Other questions for discussion:

• Which situations are discrete (have only whole-number solutions)?
• In Noah’s problem, should we round up or down?