In this Warm-up, students interpret the notation , , , and to evaluate whether different values make an inequality true. It is not expected that students will solve the inequalities generally.
Monitor for students who use the following approaches:
Substitute a value in for , evaluate the expression, and think about whether the statement is true. For example, “Is -300 less than 75? Yes, so this is true.”
Draw a number line for each inequality and use it to reason about different values.
Display the task for all to see. Tell students that for each number in the first row, they will decide if it makes the inequality in the first column true.
Activity
None
The table shows four inequalities and four values for . Decide whether each value makes each inequality true, and complete the table with “true” or “false”.
0
100
-100
25
Student Response
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Building on Student Thinking
Students who try to apply what they know about solving equations to solve the inequalities algebraically may come up with incorrect solutions. For instance, may at first glance look equivalent to , since the “less than” sign appears. Students may incorrectly think that is equivalent to . Ask these students, for example, what the solution to means (25 is the value of that makes equal to 100). Then encourage these students to test values like 24 and 26 to see whether they are solutions to . This will be covered in greater detail in a later lesson, so this understanding does not need to be solidified at this time.
Activity Synthesis
The purpose of the discussion is to note the consequences of an inequality using versus , and versus .
Direct students’ attention to and . Substituting 25 for in each of these inequalities gives , which is false, and , which is true. The key distinction is that and inequalities are considered true when both sides are equal, whereas and inequalities are considered false when both sides are equal. Emphasize that substituting a value in for , and thinking about whether the resulting inequality is saying something true, is the most direct way to check whether the value is a solution.
“What was a strategy you used for determining whether a value makes an inequality true?”
“Were you surprised by or initially incorrect about any of the answers?”
14.2
Activity
Standards Alignment
Building On
Addressing
7.EE.B.4.b
Solve word problems leading to inequalities of the form or , where , , and are specific rational numbers. Graph the solution set of the inequality and interpret it in the context of the problem. \$
In this activity, students write an equation to represent a situation, and then they write an associated inequality. They notice that they can express not just that an outcome can be equal to a value, but that an outcome can be at least as much as a value by using the new notation .
When students start with an equation representing a context and use its structure to write a related inequality, they notice and make use of structure (MP7).
This activity uses the Co-Craft Questions math language routine to advance reading and writing as students make sense of a context and practice generating mathematical questions.
Launch
Arrange students in groups of 2. Introduce the context of earning money to save up for sports supplies. Use Co-Craft Questions to orient students to the context and elicit possible mathematical questions.
Display only the first problem stem, without revealing the questions. Give students 1–2 minutes to write a list of mathematical questions that could be asked about the situation before comparing questions with a partner.
Invite several partners to share one question with the class and record responses. Ask the class to make comparisons among the shared questions and their own. Ask, “What do these questions have in common? How are they different?” Listen for and amplify language related to the learning goal, such as “greater than” or “less than.”
Reveal the questions and give students 1–2 minutes to compare them to their own questions and those of their classmates. Invite students to identify similarities and differences before proceeding.
Representation: Access for Perception. Provide appropriate reading accommodations and supports to ensure student access to written directions, word problems, and other text-based content. Supports accessibility for: Language
Activity
None
Andre has a summer job selling magazine subscriptions. He earns \$25 plus \$3 for every subscription he sells. Andre hopes to earn enough money to buy a new pair of soccer cleats.
Write an expression for the amount of money that Andre earns. Use to represent the number of magazine subscriptions he sells.
The cleats that Andre wants cost \$68. Write and solve an equation to find out how many magazine subscriptions Andre needs to sell to buy the cleats.
If Andre sold 16 magazine subscriptions this week, would he reach his goal? Explain your reasoning.
What are some numbers of magazine subscriptions Andre could sell and still reach his goal?
Write an inequality to represent that Andre wants to earn at least \$68.
Write an inequality to represent the number of subscriptions Andre must sell to reach his goal.
Diego has budgeted \$35 to buy shorts and socks for soccer. He needs 5 pairs of identical socks and a pair of shorts. The shorts he wants cost \$19.95.
Write an expression for the total cost of the socks and shorts. Use to represent the price of one pair of socks.
Write and solve an equation that represents Diego spending exactly \$35 on the socks and shorts.
List some other possible prices for the socks that would still allow Diego to stay within his budget.
Write an inequality to represent the amount Diego can spend on a single pair of socks.
Activity Synthesis
The purpose of this discussion is for students to understand that to find the solution to an inequality, it helps to find the solution to the related equation. That value is important to know because it separates numbers that are solutions to the inequality from numbers that are not solutions.
Ask students to share the inequality they wrote to represent the number of subscriptions Andre must sell to reach his goal. For example, . Explain that to find whether the solution to this inequality is or , we can substitute some values of that are greater than and some that are less than to check. Alternatively, we can think about the context: If Andre wants to make more money, he needs to sell more magazines, not fewer.
In the same way, we can think: If Diego wants to spend less than \$35, he needs to spend less for socks, not more. This will help us understand that why is the solution, not .
To promote thinking about a general solving strategy, ask:
“How does solving the related equation help us solve an inequality? What does the solution to the equation tell us about solutions to the inequality?”
“What are some ways we can determine whether the solution to an inequality should use ‘less than’ or ‘greater than’ symbols?”
“How can we check whether a value is a solution to the inequality?”
14.3
Activity
Standards Alignment
Building On
Addressing
7.EE.B.4.b
Solve word problems leading to inequalities of the form or , where , , and are specific rational numbers. Graph the solution set of the inequality and interpret it in the context of the problem. \$
In this activity, students interact with contexts in which the direction of inequality is the opposite of what they might expect if they were to solve it like an equation. For example, in the second problem, the original inequality is , but the solution to the inequality is .
Some students might solve the associated equation and then test values of to determine the direction of inequality. That method will be introduced in more generality in the next lesson. This activity emphasizes thinking about the context in deciding the direction of inequality.
Since the task requires students to interpret the meaning of their answer in a context, they are reasoning abstractly and quantitatively (MP2).
Launch
Keep students in the same groups. Give 5–10 minutes of quiet work time and partner discussion followed by a whole-class discussion.
Representation: Internalize Comprehension. Provide students with a partially completed graphic organizer, such as a two-column table with the columns labeled “amount Kiran withdraws each month” and “account balance one year from now” to help students see the relationship between the variables. Supports accessibility for: Visual-Spatial Processing, Organization
Activity
None
Kiran has \$100. He wants to know how much he could spend each month to still have at least \$25 left one year from now.
To represent this situation, Kiran writes the inequality . What does represent? Why is it negative?
Find some values of that would work for Kiran.
To express all the values that would work, should we use or ? Explain your reasoning.
A teacher wants to buy 9 boxes of granola bars for a school trip. Each box usually costs \$7, but many grocery stores are having a sale on granola bars this week. Different stores are selling boxes of granola bars at different discounts.
If represents the dollar amount of the discount, then the amount the teacher will pay can be expressed as . What does the quantity represent?
The teacher has \$36 to spend on the granola bars. The equation represents her spending all \$36. Solve this equation. What does the solution mean in this situation?
The teacher does not have to spend all \$36. Write an inequality that represents her spending at most \$36.
The solution to this inequality must either look like or . Which one is it? Explain your reasoning.
Activity Synthesis
The purpose of the discussion is to let students voice their reasoning about the direction of the inequality symbol using the context. Ask students to share their reasons for choosing the direction of inequality in their solutions. Some students may notice that the algebra in both problems involves multiplying or dividing by a negative number. Honor this observation, but again, the goal is not to turn this observation into a rule for students to memorize and follow. Interpreting the meaning of the solution in the context should be the focus.
As students model real-world situations, questions about the interpretation of the mathematical solution should continue to come up in the conversation. For instance, the amount of the granola bar discount cannot be \$3.5923, even though this is a solution to the inequality . The value -10 is a solution to Kiran’s inequality , even though he can’t spend a negative number of dollars. Students can argue that negative values for simply don’t make sense in this context. Some may argue that we should interpret to mean that Kiran deposits or earns $10 every month.
MLR8 Discussion Supports. For each observation that is shared, invite students to turn to a partner and restate what they heard, using precise mathematical language. Advances: Listening, Speaking
Lesson Synthesis
Share with students, “Today we solved problems that can be represented with inequalities.”
If desired, use this example to help students differentiate between equations and inequalities. Display and for all to see. Consider asking:
“What is different about these two statements?” (The first is an equation, and the second is an inequality.)
“What is different about the situations that these two statements could represent?” (The equation represents reaching exactly that value, while the inequality represents getting at least that amount. You could also have more.)
“What is different about the solutions to these two statements?” (The solution to the equation is just one number, while there are many possible values that are solutions to the inequality.)
Student Lesson Summary
We can write and solve inequalities to solve problems.
Example: Elena has \$5 and sells pens for \$1.50 each. Her goal is to save \$20. We could solve the equation to find the number of pens, , that Elena needs to sell in order to save exactly \$20. Adding to both sides of the equation gives us , and then dividing both sides by 1.5 gives the solution pens.
What if Elena wants to save more than \$20? The inequality tells us that the amount of money Elena saves needs to be greater than \$20. The solution to the previous equation will help us understand what the solutions to the inequality will be. We know that if Elena sells 10 pens, she will save exactly \$20. Since each pen gives her more money, she needs to sell more than 10 pens to save more than \$20. So, we can represent all the solutions to the inequality with another inequality: . A solution to an inequality is a number that can be used in place of the variable to make the inequality true.
Standards Alignment
Building On
Addressing
Building Toward
7.EE.B.4.b
Solve word problems leading to inequalities of the form or , where , , and are specific rational numbers. Graph the solution set of the inequality and interpret it in the context of the problem. \$
