Modeling with Inequalities
17.1: Possible Values (5 minutes)
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.
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 \(x\) represent the number of sandwiches he will order and writes \(5.99x \leq 83\). He solves this to 2 decimal places, getting \(x \leq 13.86\).
Which of these are valid statements about this situation? (Select all that apply.)
- He can call the sandwich shop and order exactly 13.86 sandwiches.
- He can round up and order 14 sandwiches.
- He can order 12 sandwiches.
- He can order 9.5 sandwiches.
- He can order 2 sandwiches.
- He can order -4 sandwiches.
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:
Even though 13.86 makes the inequality true, most sandwich shops would not let you order 13.86 sandwiches.
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.
Might be okay if the shop allows you to order sandwiches
in \(\frac12\)-sandwich increments.
Even though -4 makes the inequality true, that value doesn’t make sense in this context.
17.2: Elevator (15 minutes)
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.
Supports accessibility for: Language; Social-emotional skills
Design Principle(s): Support sense-making
A mover is loading an elevator with many identical 48-pound boxes.
The mover weighs 185 pounds. The elevator can carry at most 2000 pounds.
- Write an inequality that says that the mover will not overload the elevator on a particular ride. Check your inequality with your partner.
- Solve your inequality and explain what the solution means.
Graph the solution to your inequality on a number line.
- If the mover asked, “How many boxes can I load on this elevator at a time?” what would you tell them?
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.)
17.3: Info Gap: Giving Advice (15 minutes)
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.
Supports accessibility for: Memory; Organization
Design Principle(s): Cultivate conversation
Your teacher will give you either a problem card or a data card. Do not show or read your card to your partner.
If your teacher gives you the problem card:
Silently read your card and think about what information you need to be able to answer the question.
Ask your partner for the specific information that you need.
Explain how you are using the information to solve the problem.
Continue to ask questions until you have enough information to solve the problem.
Share the problem card and solve the problem independently.
Read the data card and discuss your reasoning.
If your teacher gives you the data card:
Silently read your card.
Ask your partner “What specific information do you need?” and wait for them to ask for information.
If your partner asks for information that is not on the card, do not do the calculations for them. Tell them you don’t have that information.
Before sharing the information, ask “Why do you need that information?”
Listen to your partner’s reasoning and ask clarifying questions.
Read the problem card and solve the problem independently.
Share the data card and discuss your reasoning.
Pause here so your teacher can review your work. Ask your teacher for a new set of cards and repeat the activity, trading roles with your partner.
Are you ready for more?
In a day care group, nine babies are five months old and 12 babies are seven months old. How many full months from now will the average age of the 21 babies first surpass 20 months old?
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?
In the last few lessons, students have seen a variety of situations in which inequalities described situations. Ask students to think about a career they might be interested in pursuing and have them write a few sentences about the usefulness of inequalities in the work of that profession, including at least one example. Ask them to think about whether inequalities are sometimes more helpful than equations, and if so, why.
17.4: Cool-down - Movies on a Hard Drive (5 minutes)
Student Lesson Summary
We can represent and solve many real-world problems with inequalities. Whenever we write an inequality, it is important to decide what quantity we are representing with a variable. After we make that decision, we can connect the quantities in the situation to write an expression, and finally, the whole inequality.
As we are solving the inequality or equation to answer a question, it is important to keep the meaning of each quantity in mind. This helps us to decide if the final answer makes sense in the context of the situation.
For example: Han has 50 centimeters of wire and wants to make a square picture frame with a loop to hang it that uses 3 centimeters for the loop. This situation can be represented by \(3+4s=50\), where \(s\) is the length of each side (if we want to use all the wire). We can also use \(3+4s\leq50\) if we want to allow for solutions that don’t use all the wire. In this case, any positive number that is less or equal to 11.75 cm is a solution to the inequality. Each solution represents a possible side length for the picture frame since Han can bend the wire at any point. In other situations, the variable may represent a quantity that increases by whole numbers, such as with numbers of magazines, loads of laundry, or students. In those cases, only whole-number solutions make sense.