Modeling Inequalities on Number Lines

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Objective

SWBAT represent a context using a simple inequality. SWBAT model the solution using a number line diagram and describe the solution set.

Big Idea

Inequalities can be graphed on a number line to represent all possible numbers in the solution set.

Think About It

7 minutes

Students work on the Think About It problem in partners.

I ask the class these questions, allowing each speaker to 'roll it' to the student who will answer my next question:

  • why do we use an inequality to represent this scenario?
  • which inequality did you write?
  • why did you use the 'greater than' symbol, and not the 'greater than or equal to' symbol?
  • what are all of the possible values that are in this solution set?
  • what's one number that's not in this solution set?
  • what are two reasons that -5 could not be in this solution set? (with this question, not only am I looking for the answer that -5 is less than 6, but also that -5 does not make sense, given the context of this problem)

 

 

Intro to New Material

10 minutes

In the Intro to New Material, students are introduced to the concepts of open and closed circles on the number line.  

When students are given number lines that do not have integers already labeled, I teach students to label 3 integers:

  • The integer in the statement of inequality
  • One number greater than the integer in the inequality
  • One number lower than the integer in the inequality

I find that this process gives students a way to check that they are 'shading' the number line in the correct direction: Students can use the labeled integers to reason about the solution set.  

In the final problem in the new material section, together we create an inequality to represent the real-world scenario and then graph the solution set on the number line.

Partner Practice

23 minutes

Students work in partners on the Partner Practice problems.  As they work, I am circulating around the room.

I am looking for:

  • Are students correctly using open or closed circles on the number lines?
  • Are students shading in the correct direction on the number lines?
  • Are students correctly representing real world problems with inequalities?
  • Are students defining their variables?
  • Are students discussing and debating possible solutions using substitution to justify their solution?

 I'm asking pairs:

  • How did you know x = ___ is a possible value?
  • Can x = ___?
  • Why does this inequality represent the given situation?
  • Why does this inequality have a closed/open circle? What does that represent?
  • Why did you make your graph go in this direction?
  • Is ___ a number included in the solution? Why or why not?

 

After 10 minutes of partner work time, I call on a student to share what would happen to graph #8, if we needed to include the number 4 in our solution set.  

Students complete the final Check for Understanding problems on their own.  I call on a student to share the inequality that they've written for the first problem.  I then ask students to show me if, on the number line, we need to use an open circle or a closed circle.  I go through the same process for the remaining two problems.

 

Independent Practice

10 minutes

Students work on their own on the Independent Practice problems.  As they are working, I circulate and look for and ask the same questions as I did during the partner practice.

Some students will look to make generalizations about graphing, and think that the inequality sign essentially 'points' in the direction that they'll need to shade.   I want students to really understand inequalities, rather than rely on short cuts, so I do not share this observation with the entire class. When students share this with me, I point to problem #1 in this problem set (7 < x) and ask why this isn't the case here.  For students who are ready to notice things like this, though, I want them to test their hypotheses out and decide for themselves when this pattern holds true.

Closing and Exit Ticket

10 minutes

After independent work time, we discuss problem 13b as a class.  I ask for someone who feels confident about his/her work to share his/her paper on the document camera.  This problem requires students to graph the starting point in the solution set between the integers that are marked off on the number line.  

I ask students to give feedback about the work that has been shared out.

Students then work on the Exit Ticket independently.