Unit 4, Lesson 1
Representing Situations with Inequalities
Algebra 1
1.1
Warm-up
Standards Alignment
Building On
6.EE.B.5
Understand solving an equation or inequality as a process of answering a question: which values from a specified set, if any, make the equation or inequality true? Use substitution to determine whether a given number in a specified set makes an equation or inequality true.
Addressing
Building Toward
HSA-CED.A.3
Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or nonviable options in a modeling context. For example, represent inequalities describing nutritional and cost constraints on combinations of different foods.
Instructional Routines
NoneMaterials
NoneActivity Narrative
In this activity, students recall the meaning of inequality symbols (, , , and ) 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.
Launch
Give students 1–2 minutes of quiet work time. Follow with a whole-class discussion.
Activity
None-
Match each inequality to the meaning of a symbol within it. Read the inequality from left to right.
- less than or equal to
- greater than
- greater than or equal to
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Is 25 a solution to any of the inequalities? Which one(s)?
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Is 40 a solution to any of the inequalities? Which one(s)?
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Is 30 a solution to any of the inequalities? Which one(s)?
Activity Synthesis
Draw students' attention to the last inequality (). 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 ," which means that 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: , , and . 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.
1.2
Activity
Materials
NoneActivity Narrative
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.
This activity uses the Three Reads math language routine to support reading comprehension and sense making.
1.3
Activity
Materials
NoneActivity Narrative
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 , some may write , and others may write or . Ask students with contrasting statements to share their responses later.
Lesson Synthesis
Solicit from students any advantages and disadvantages for representing constraints as inequalities. Some advantages students might bring up:
- Compared to written words, inequalities are a simpler and quicker way to describe what is happening in a situation.
- It is easier to see what values a certain quantity could or could not take when the constraint is written with symbols and numbers.
Possible disadvantages:
- Unless we know what the variables stand for, we can't be sure about the meaning of an inequality.
- If we don't recall what the symbols mean or how to read them, we can't access the information.
Next, invite students to share some advice for students who might just be learning to write inequalities to represent constraints. What should they pay attention to? What are some potential sources of confusion or error they should look out for? Students might mention that the following are important things to attend to:
- Specifying the meaning of each variable
- Not using the same variable to represent different quantities
- Making sure that the correct symbols are used to represent relationships
- Using words to read each inequality to make sure that it fully represents a constraint
Student Lesson Summary
We have used equations and the equal sign to represent relationships and constraints in various situations. Not all relationships and constraints involve equality, however.
In some situations, one quantity is, or needs to be, greater than or less than another. To describe these situations, we can use inequalities and symbols such as , or .
When working with inequalities, it helps to remember what the symbol means, in words. For example:
- means “100 is less than .”
- means “ is less than or equal to 55,” or " is not more than 55."
- means “20 is greater than 18.”
- means “ is greater than or equal to 40,” or " is at least 40."
These inequalities are fairly straightforward. Each inequality states the relationship between two numbers (), or it describes the limit or boundary of a quantity in terms of a number ().
Inequalities can also express relationships or constraints that are more complex. Here are some examples:
- The area of a rectangle, , with a length of 4 meters and a width of meters is no more than 100 square meters.
- To cover all the expenses of a musical production each week, the number of weekday tickets sold, , and the number of weekend tickets sold, , must be greater than 4,000.
- Elena would like the number of hours she works in a week, , to be more than 5 but no more than 20.
- The total cost, , of buying adult shirts and child shirts must be less than 150. Adult shirts are \$12 each, and child shirts are \$7 each.
In upcoming lessons, we’ll use inequalities to help us solve problems.