Reasoning About Inequality Solution Sets

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SWBAT determine whether a given number in a specified set makes an inequality true and justify their response using substitution.

Big Idea

An inequality is an algebraic statement that a quantity is greater than, less than, or equal to another quantity. A solution to an inequality is any value that makes the inequality true.

Think About It

8 minutes

Students work in partners on the Think About It problem.  The key idea I want to come out during our conversation is that the equation has one solution and the inequality has a solution set.

I plan to ask my students why they chose not to circle 10 as part of the solution to x < 10. My goal is to use this discussion as a lead in to the Introduction of New Material. I hope that my students will articulate that 10 is not less than 10, so it can't be a part of the solution.   


Intro to New Material

20 minutes

An inequality might be a true statement for some values of the variable, and a false statement for others. Solving is a process of reasoning to find the numbers which make an inequality true, which can include checking if a given number is a solution. This is the message that I want to convey to my students today.

At 20 minutes in length, this is one of the longer INM sections that I teach. Many of my students have never seen the 'greater than or equal to' or 'less than or equal to' symbols (the 'equal to' is the new piece).  While they will have compared decimals in 5th grade using the less than and greater than symbols (5.NBT.A.3b), they are extending their knowledge today. 

One habit that I want students to acquire today is 'reading the inequality,' starting with the variable.  During this lesson I'll spend time having my students re-write inequalities so that the variable comes first (rather than 43<y, writing y>43). I find that this helps my students to reason about the solution sets more accurately. It will also make graphing the solution sets easier when we get to that point. 

Partner Practice and Discussion

20 minutes
Students work in partners on the Partner Practice problem set.  As they work, I circulate.
I am looking for:
  • Are students using substitution to solve for the solution set and to justify their answer?
  • Are students precisely describing the solution set?

I'm asking pairs of students:

  • How did you know the solution was x = ?
  • Why can’t x = ?
  • How can you describe the solution of the set? 

After 15 minutes of work time, I will ask my students to complete the Final_Check_for_Understanding problem on their own.  I have students vote with thumbs up/down about whether or not 6 is part of the solution set of the inequality.  I then ask for a volunteer to show his/her work to the class on the document camera. 

Independent Practice

15 minutes

Students work on their own on the Independent Practice problem set.  As I circulate, I am looking for and asking the same questions as in the partner practice section.  

It's important during this lesson that students are practicing using substitution to decide whether a number is part of the solution set.  Some of my more advanced students will use mental math and are able to reason about the solution sets in their heads.  It's important that students are building the habits that will help them be successful in future math classes.  I make sure I am looking to see that students are taking the time to carefully write out the steps needed to substitute and evaluate each expression in the inequalities.  

Closing and Exit Ticket

7 minutes

After student work time, the class comes back together for a conversation about one of the independent work problems.  I like to use problem 2, as it really gets at the key idea that inequalities have a solution set, rather than a solution.

Students work on the Exit Ticket independently.  For the third problem, an exemplar response will show the substitution work, as seen in this Exit Ticket Sample.