Students sometimes struggle to decide on their own what values to test in a given inequality. In this activity, they are given specific options to consider and get to choose the value they most want to test. This provides an opportunity to talk about why certain values might be chosen. Students are also asked to think about which values will definitely satisfy and not satisfy the inequalities. This encourages them to analyze the structure of the expression and think about how its value will change when testing large numbers, negative numbers, zero, or other types of values. Students look for and express regularity in repeated reasoning when they generalize ideas about what numbers to use to test their solutions (MP8). This activity prepares students for work in the associated Algebra 1 lesson in which they find solutions to inequalities and compare the situations that the inequalities represent.

Monitor for students who use these different strategies:

• Choose to test zero

• Consider values that will result in whole numbers when divided by a whole number or multiplied by a fraction

• Strategically choose large numbers

• Strategically choose negative numbers

• Always choose the same number

The goal of this discussion is to showcase different things to consider when choosing a number to test an inequality.

Display 2–3 approaches from previously selected students for all to see. Use Compare and Connect to help students compare, contrast, and connect the different approaches/representations. Here are some questions for discussion:

• “Why do the different approaches lead to the same or different outcomes?”
• “Are there any benefits or drawbacks to one strategy compared to another?”

If not brought up in students’ explanations, ask students to think about what kinds of numbers they would choose to make each inequality true or false (positive or negative and large or small).

Ask students to share which numbers that they didn’t test that would definitely make a given inequality true or false. This process of reasoning helps students build their intuition about solutions to inequalities. It also helps to check their reasoning, a skill they can use when they try to solve inequalities in the associated Algebra 1 lesson.

The goal is to discuss how to interpret solutions to an inequality. Here are some questions for discussion:

• "How do you know if a specific value is a solution to an inequality?" (The values that are solutions to an inequality are the ones that you substitute for  and the inequality is still true. The values that are not solutions make the inequality false when you substitute them in for .)

• "Why is it important to think about values that satisfy the inequality and values that do not?" (Thinking about this helps to make sense of the solution that you create for an inequality. If I know which values make the inequality true, then I also know if certain values are reasonable answers for an inequality and if they are reasonable for a given context.) 

• "How do you know which verbal descriptions represent the inequality?" (I think about which values are being described in the verbal description, and figure out if some sample values that fit the description also satisfy the inequality.)

• "Which representation (verbal description or symbols) helps you more in understanding the solution?" (The verbal description helps me understand solutions better than symbols do because I can think about more values that make the inequality true rather than thinking more about the meaning of the symbol. Or, the symbols are easier to read than a verbal description.)