Solution Sets to Equations/Inequalities

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SWBAT represent the solution to an equation or inequality using both set builder and interval notation.

Big Idea

Mathematically literate students understand the language of mathematics and how to represent information symbolically.

Warm Up

5 minutes

At the start of class, I will write the following on the board: 

2x - 6 = 10        and       2x - 6 > 10

Explain what you would think of as the "solution set" for each of these sentences.

Are these two sentences always, sometimes, or never true? 

I will allow students to work on these two questions with their partner.  I want them to go through the statements one at a time. As they work I will circulate and review student ideas about whether the inequality is always, sometimes, or never true.  This case was not covered in previous lessons. If necessary, however, I will guide students to the understanding that the statement x > 8 indicates that there are an infinite set of numbers that make the statement true, but, the inequality statement itself is not always true. I want students to recognize the similarities and differences between equations and inequalities in this case regard. 

Teaching Point: This is a brainstorming session and you want to get as many ideas on the table as possible.  That being said, you don't want to sacrifice quantity for quality.  Let a student propose an idea and then set the rule that the next person called on must comment on the first idea before giving their own idea.  This technique allows for many responses in a short period of time but also forces students to listen to each other first before sharing an idea. For more about this technique, please watch this video_narrative

Independent Study-Set Notation

15 minutes

For this part of the lesson I will either have laptops or a computer lab available for students. Everyone will begin their work by going to the link below:

Once the students load this page, I will explain to students that they will be doing some Check for Understanding questions after they review the information on the webpage. 

Teaching Point: Oftentimes my students are unfamiliar with reading mathematics. So, at the start I pull up the webpage on a projector and model for students how to read mathematical text.  I focus on demonstrating to my students that reading mathematical text is a non-linear process.  By this I mean that in a math text, you may read a few words and then look at a diagram, then go back and read the sentence again, then back to the diagram.  As a result, it is often important to read slowly and persistently.  It is important to pay attention to notation and make notes as you read.

The keys to student success in this lesson are perseverance (MP1) and precision (MP6). The lesson introduces students to many symbols and terms. As they practice explaining solutions to themselves, they should try to use the correct terms and symbols. 

After reading they should answer Questions #1-3 (at the bottom of the webpage) to check their understanding of the content.  If time allows, I may use Questions 4-8 as an extension. I will certainly encourage students to challenge themselves to complete them if the opportunity presents itself. 

Discovery-Interval Notation

15 minutes

The second piece of content for this lesson is for students to understand and be able to use interval notation when writing solution_sets. I run this discovery activity as follows:

  • In each pair, students will use the front side of the sheet to determine the meaning about rows 1-3 (Each row expresses the same solution set but using different notation).  
  • I tell students that they may not go onto the back of the sheet until they have verbalized the meaning of all three lines to you. If I do not have a second adult in the room, I will check a few groups who get done first and then have those students check the answers of other groups. 

When I check with students, I will see if they can successfully verbalize the meaning (MP3) of an open and a closed bracket. I want to make sure that students are relating these symbols to the representation of a solution on the number line.  Once students have a satisfactory understanding they can check that understanding by completing the fill in the blank on the back of the sheet. 

To bring this part of the lesson to a close, I will go over the answers with the class. I will call on pairs of students and ask them to explain their answer. Depending on where the class is along a continuum of understanding, I may want to use a chart on the board to summarize the symbols. For example, to show how "<" and "(" and "an open circle" all can be used to represent the same idea.


5 minutes

Today, I will ask students to work on this Closure Task individually as a Check for Understanding.  Question #1 will help me to determine if students are able to use important vocabulary to describe the given set.  Question #2 gives the students an opportunity to check their understanding of set builder notation.  If students struggle, it will most likely be with question #2.  Again, precision of language is key.  I will try to help students understand what they are trying to say rather than simply memorizing the symbols that are used in set builder notation.  I expect that Question #3 may yield some interesting responses.  My goal is to see how students write the left hand side of the interval (should be negative infinity).  Use the student responses to craft a revisit of this concept if necessary during a future lesson.  Question #4 is open-ended and will yield many different results.  Students can be as creative as they want in making an inequality that yields the appropriate set of solutions.