Modeling with Inequalities

10 min

Teacher Prep
Setup
Students in groups of 2. Give students 2 minutes of quiet work time followed by a 1-minute partner discussion, then whole-class discussion.

Narrative

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

The context in this problem provides an opportunity for students to think about aspects of mathematical modeling like discrete versus continuous solutions and rounding (MP4).

Launch

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.

Student Task

The stage manager of the school musical is trying to figure out how many sandwiches he can order with the $83 he collected from the cast and crew. Sandwiches cost $5.99 each, so he lets xx represent the number of sandwiches he will order and writes 5.99x≤835.99x \leq 83. He solves this to 2 decimal places, getting x≤13.86x \leq 13.86.

Determine whether each statement about this situation is true. Be prepared to explain your reasoning.

  1. He can call the sandwich shop and order exactly 13.86 sandwiches.
  2. He can round up and order 14 sandwiches.
  3. He can order 12 sandwiches.
  4. He can order 9.5 sandwiches.
  5. He can order 2 sandwiches.
  6. He can order -4 sandwiches.

Sample Response

  1. Probably not. Sample reasoning: It is unlikely a sandwich shop would sell precisely .86 of a sandwich.
  2. No. Sample reasoning: The solution of x≤13.86x \leq 13.86 means that 14 sandwiches would cost more than the $83 the group can spend.
  3. Yes. Sample reasoning: The solution of x≤13.86x \leq 13.86 means that 12 sandwiches will cost less than (or equal to) $83.
  4. Possibly. Sample reasoning: The sandwich shop may sell half sandwiches for half the price of a whole sandwich.
  5. Yes. Sample reasoning: At a glance, 2 sandwiches will cost much less than $83.
  6. No. Sample reasoning: Though -4≤13.86\text-4 \leq 13.86 is true and -4 is a numerical solution to the inequality, it does not make sense to order -4 sandwiches.
Activity Synthesis (Teacher Notes)

The purpose of this discussion is to highlight how the situation represented by a solution might further constrain the solution.

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. He can order 12 sandwiches.
  4. He may be able to order 9.5 sandwiches if the shop allows orders in 12\frac12-sandwich increments.
  5. He can order 2 sandwiches.
  6. Even though -4 makes the inequality true, that value doesn’t make sense in this context.
Anticipated Misconceptions

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. Consider asking these students to use a calculator to find the cost of 14 sandwiches to show that it is not a solution to the inequality. Explain that although sandwich shops may not sell sandwiches in fractional pieces, the maximum amount that can be ordered is 13.86. 

Standards
Building Toward
  • 7.EE.4.b·Solve word problems leading to inequalities of the form px + q &gt; r or px + q &lt; r, where p, q, and r are specific rational numbers. Graph the solution set of the inequality and interpret it in the context of the problem. <em>For example: As a salesperson, you are paid $50 per week plus $3 per sale. This week you want your pay to be at least $100. Write an inequality for the number of sales you need to make, and describe the solutions.</em>
  • 7.EE.B.4.b·Solve word problems leading to inequalities of the form <span class="math">\(px + q &gt; r\)</span> or <span class="math">\(px + q &lt; r\)</span>, where <span class="math">\(p\)</span>, <span class="math">\(q\)</span>, and <span class="math">\(r\)</span> are specific rational numbers. Graph the solution set of the inequality and interpret it in the context of the problem. \$

20 min