Solving Absolute Value Equations and Inequalities (Day 1 of 2)

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SWBAT solve Absolute Value Equations and Inequalities

Big Idea

To develop a deeper understanding of Absolute Value Equations and Inequalities from the distance on a number line and writing sentences about that distance.


20 minutes

The purpose of this lesson is for my students to develop a conceptual understanding of what absolute value means, and apply it to solving Absolute Value Equations and Inequalities.  So in the Introduction Activity, I have students complete a hands on activity with a table partner.  This activity should help students think about the possible solutions to Absolute Value Equations and Inequalities in terms of "distance from zero."  

When I introduce the activity to my students and I go over the example:  the absolute value of x is equal to three.  I ask, "What points on the number line are a distance of three units from zero?" Most of my students conclude three, several include negative three.  I tell my students to:

  • Ask themselves this question when working through the activity
  • Remember what a solution means
  • Substitute your solutions in for x  in the original equation to determine if they are solutions or not.

For the Activity, table partners need a set of index cards numbered negative five through five, including zero.  The cards are to be placed face up to form a number line.  Each partner takes turns turning over what they believe to be the solutions to the equation or inequality.  The other partner either agrees or disagrees.  Partners should discuss if the solutions make the Absolute Value Equation (or Inequality) true.  If not, the partners should discuss the corrections that need to be made and the reasons for those corrections.

There are three parts to the Introduction Activity.  

  • Part One is solving Absolute Value Equations
  • Part Two is to solve Absolute Value Inequalities of the form less than or equal to
  • Part Three is to solve Absolute Value Inequalities of the form greater than or equal to

After completing all three parts, students are to answer the three concluding questions about the activity.  If I observe students are struggling, I will have them complete only three problems from each section before moving on.  I want to make sure that we get to the class discussion. By having students complete at least three problems in each section, they have a better understanding of the different types of problems that we are comparing and they will be able to benefit from the discussion.

When we discuss the activity, table partners will share their responses from the activity, explaining why they chose what the answers that they did.

  • Question 1 allows us to bring up the possibility of no solution if the equation is set equal to a negative number, and the possibility of only one solution for the absolute value of zero.
  • Questions 2 and 3 allow students to think about when to include the boundary number(s) and when they are not part of a solution. 


Guided Practice

20 minutes

After reviewing the Introduction Activity, I hand my students today's Guided Practice.  It is helpful to to post a number line on the board, and for students to have a number line to share among partners. I use a number line that I found on  Students may also use individual white boards and markers if preferred.  It is better to get the number lines laminated, so that the students may mark on them, and erase the marker.  

The purpose of this Guided Practice is to lead students from thinking about the distance from zero, to questioning on their own, and eventually setting up the problems without the use of a manipulative. As we complete the activity, I alternate between working some examples and then letting students work some problems on their own, discussing and checking their work as we move forward. Again, going from the concrete to the abstract of solving these Absolute Value Equations and Inequalities helps the students remember based on reasoning, not memorization of the procedure.  I demonstrate some of the problems from the Guided Practice in the video below.

Exit Slip

10 minutes

After completing the Guided Practice with the students, I hand each student an Exit Slip.  I try to hand the students the Exit Slip with about 10 minutes remaining in class.

My expectation is for my students to hand in the Slip before they leave.  However, this is a time- consuming lesson, so I may assign the Exit Slip as homework if the Guided Practice ran long.

I will use the Exit Slip as a quick formative assessment, and we will use it during the Warm Up of tomorrow's lesson.