Lesson 20

The purpose of this Math Talk is to elicit strategies and understandings students have for solving inequalities. These understandings help students develop fluency and will be helpful later in this lesson when students will need to be able to solve inequalities in order to match them with the appropriate graph and situation.

Display one problem at a time. Give students quiet think time for each problem and ask them to give a signal when they have an answer and a strategy. Keep all problems displayed throughout the talk. Follow with a whole-class discussion.

Ask students to share their strategies for each problem. Record and display their responses for all to see. To involve more students in the conversation, consider asking:

• “Who can restate \(\underline{\hspace{.5in}}\)’s reasoning in a different way?”

• “Did anyone have the same strategy but would explain it differently?”

• “Did anyone solve the problem in a different way?”

• “Does anyone want to add on to \(\underline{\hspace{.5in}}\)’s strategy?”

• “Do you agree or disagree? Why?”

The purpose of this activity is for students to practice interpreting the solution of an inequality, especially those involving negative numbers. Students also practice representing inequalities written in natural language with symbols and number lines, and connecting the solution of an inequality with a number line. The work in this activity will be useful when students solve more complex inequalities in the associated Algebra 1 lesson. 

The goal of this discussion is to help students make sense of their solutions. Here are questions for discussion:

• "After you checked your answers using a value from your solution, did any of your answers change?" (Yes. Initially I thought the solution was \(x <\text-2\), but I used \(\text-3\) to check my answer and realized it was incorrect. When I substituted \(\text-3\) for \(x\) in the original inequality, I got \(6<4\) which is not true. This reminds me to change the inequality symbol when I multiply or divide by a negative number.)
• "Could the elevator hold 1,300 pounds?" (No. It holds up to 1,200, so this is the largest amount the elevator can hold.)
• "Could the senator’s approval rating be 54%?" (Yes. The situation says the approval rating can be 3 points above and below 53%, and 54% fits that requirement.)
• "How can the number line be useful in writing the solution to an inequality?" (It helps to see which values are included in the solution and which are not. It can also be used to check your answer after you solve an inequality.)
• "On the graph of the minimum years of experience for a candidate, does it make sense to have an upper bound? What examples can you think of that would make this situation have an upper bound?" (Yes, because realistically a person cannot have infinity years of experience and that is what the current graph of the inequality reprsents. For example, many people retire after a certain age, so it would be hard for someone to have more than 50 years of experience. Also, the average lifespan is about 110 and people ususally start working as teens or young adults, so this creates another limit on how many years of experience someone can actually have.)

A sorting task gives students opportunities to analyze representations, statements, and structures closely and make connections (MP2, MP7). In this activity, students practice connecting verbal, symbolic, and graphical descriptions of inequalities, including activities involving negative numbers.

The practice will pay off when they write inequalities and interpret their solutions in the associated Algebra 1 lesson. Monitor for different ways groups choose to categorize the representations. Look for groups that categorize each representation of the same situation (graph, symbolic, and verbal) together, and for groups that sort strictly greater than and less than statements together and less than or equal and greater than or equal to statements together.

As students work, encourage them to refine their descriptions using more precise language and mathematical terms (MP6).

Arrange students in groups of 2. Tell them that in this activity, they will sort some cards into categories of their choosing. When they sort the representations, they should work with their partner to come up with categories. Encourage students to find more than one way to sort the cards.

Ask previously identified students to share a way of categorizing the cards that distinguishes between greater/less than and equal situations and strictly greater/less than situations. Ask previously identified students to share a way of categorizing the cards that groups together representations that describe the same situation. If time permits, invite students to share other ways of categorizing the cards.

Possible questions for discussion:

• "Describe why you grouped these cards together." (All the cards in each group share a characteristic such as; they all represent a situation in which the variable is less than (or greater than) the given quantity, they all represent a situation in which the given quantity is included in the solution, they are all graphs of inequalities, or they are all expressions that had to be simplified to find the solution to the inequality.)

• "On card 13, why does the graph stop at 0?" (Although the mattresses are less than 93.99 and that technically includes negative numbers, it is not reasonable to pay a negative amount to purchase something.)

• “Why might someone have put cards 1, 3, 5, 8, and 16 together? How are they different from cards 2, 4, 6, 7, 9, 10, 11, 13, 12, 14, and 15?” (The first list of cards are examples of inequalities in which the given quantity is included in the solution, and the second list of cards is not.)

• "Are there any other ways to group the cards that have not been discussed?"