Unit 4, Lesson 9
Testing Points to Solve Inequalities
Extra Support Materials for Algebra 1
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The purpose of this activity is to examine how to solve an inequality using a test point.
This activity uses the Worked Example routine. In this routine, students analyze steps used to solve a problem and discuss specific steps with a partner and the whole class. During the discussions, students construct viable arguments to justify the steps taken by the problem solver (MP3). Later, students can apply these strategies to solve similar problems.
Arrange students in groups of 2. Tell them to close their books or devices (or to keep them closed). Reveal the prompt and the first step. Give students a brief quiet think time, and ask them to give a signal when they have read the step and are ready to see the next. Once most students have given the signal, reveal the next step. Repeat until all steps have been revealed.
If previous work indicates that most students will struggle to make sense of a step, pause and give students 1 minute to discuss with their partner and formulate a question about the step. Ask 1–3 groups to share their questions, inviting other groups to answer while recording responses next to the step for all to see. Once questions have been answered, continue to the next reveal.
Solve the inequality and use a test point to check your answer.
Step 1:
Step 2:
When .
, so 0 is not a solution.
Step 3:
, so all solutions are .
The purpose of this discussion is to analyze the steps taken by the solver to solve an inequality and use a test point.
Tell students to open their books or devices. Display the following questions, and give students 1–2 minutes of quiet think time to consider their responses.
Ask partners to compare their responses to each question and decide if either or both answers are correct. During this partner discussion, monitor to select groups who reason about the questions in different ways. Follow with a whole-class discussion, inviting 1–2 selected groups to share their answers for each question.
If time allows, consider displaying a second worked example next to the first that either:
The mathematical purpose of this lesson is to help students reason about why testing points is an important step in solving inequalities, and how to do so purposefully and efficiently. This helps make explicit the reasoning behind some of the work students are doing in the associated Algebra 1 lesson. Students construct viable arguments and critique the reasoning of others when they explain how to be more efficient and identify and describe an error (MP3).
Display Andre’s work. Ask students what they notice, and what they wonder about Andre’s work. Give students an opportunity to work on the questions about Andre’s work.
If students are struggling to make sense of Mai’s work, encourage students to notice and wonder about it in small groups, or pause the class when most students are ready to move on to Mai’s work. Display Mai’s work. Ask students what they notice, and what they wonder about Mai’s work. Explain that Mai has made an error, and ask them to notice and wonder again. Give students an opportunity to work on the questions about Mai’s work.
Andre is working on . He figured out that when . He tested all these points:
Based on these results, Andre decides that solutions for should include -1 and 0, but not 2 or 3.
The purpose of this discussion is to help students continue to practice reasoning about what is going on mathematically when they test points to help them solve inequalities. Students should be beginning to recognize the value of solving the related equation and testing a point to the left or right, and picking those points strategically.
Display a number line.
Here are some questions for discussion:
Ask students to explain to a partner how Andre could have figured out the solutions to the inequality without having to test all the points.
Instruct students to tell their partner how they could use Mai’s experience to convince someone that it is a good idea to test more than one point.