Solving and Proving Linear Inequalities in One Variable

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SWBAT... 1. Solve linear inequalities in one variable, including inequalities with letters as coefficients.

Big Idea

Students learn that being close enough counts in horseshoes and inequalities!

Entry Ticket: Solving Inequalities

15 minutes

To begin class, students complete an Entry Ticket that asks them to translate different number sentences into inequalities and also solve the inequalities. 

I like to have students translate because it activates their prior knowledge and is something most students will probably be able to do to open the class. I ask students to solve the inequalities to give them a clue about what we will be working on in class and to assess where students are at in the process. The entry ticket combined with the exit ticket and homework provides a nice opportunity for a pre/post comparison of student learning and understanding.

Active Note Taking: Solving Inequalities

20 minutes

In this section, I have students take Two-column notes as a way to be active participants in direct instruction. I play a video from Khan Academy that goes in depth to solving a challenging example of solving an inequality. I like this particular problem because it requires some prerequisite skills (adding fractions with unlike denominators, for example).

I stress the same Five Star Steps used to solving equations for each problem:

Five-Star Solutions to Equations and Inequalities 

Step 1 - rewrite

Step 2 - Simplify each side (distribute and/or combine like terms)

Step 3 - Get variables on one side, constants on the other (addition and/or subtraction)

Step 4 - Isolate the variable (divide or multiply by the reciprocal)

Step 5 - Prove (Check)

Reminder: For inequalities we have to remember to change the direction of the inequality sign if we multiply or divide by a negative number!

During the video, students write notes about the problem, with the main idea on the left third of their paper and details, notes, and steps on the right two-thirds of their notes.

I also pause the video at various steps along the way and ask students what they think the next logical step in the process should be. I like to do this because it engages students to use expressive language as a way to process the new information and skills. 

The video portion of the example takes about 5-7 minutes (including the pausing and class discussion). 

I conclude this portion of the class by solving 2-3 other inequalities with students as a class.

For example,

  • -7x < 42                     to focus on dividing by a negative number
  • 10y + 25 > 75            is a two step problem
  • 5/4(Z) - 13 > 7/6        requires students to use fractions


Collaborative Group Work: Practice Solving Inequalities

20 minutes

Now that students have had a chance to see how to solve inequalities and compound inequalities I have them work in pairs on practicing these types of problems. I like to use Kuta Software to generate worksheets for students. This is a nice differentiation tool as I tend to give one step worksheets to students who need more practice with the basic, and multi-step and/or compound problems for more advanced students. Another strategy s to print out a number of each worksheet and let students choose which worksheet they want to practice on. 

While students are working, I am navigating the room and checking in with students to ensure that they are starting and giving them cues as to next steps, questions that can help them better frame their approach to the problem they are working on. 

For the final 5 minutes of this section I like to reconvene the class and have a discussion about the different strategies they used. One strategy that is useful is to encourage students to talk about strategies for problems that were difficult for them or even problems that they did not solve. This helps students articulate their approach to a problem and also encourages a whole class problem solving environment where the perspective of other students comes into play to help the whole class have more tools and strategies to solve more difficult problems. 

Active Note-Taking: Solving Inequalities with letters as coefficients

20 minutes

In this section, I have students take Two-column notes as a way to be active participants in direct instruction. I go through notes for examples of solving inequalities with letters as coefficients (see resource for a copy of teacher notes). 

After walking through the first example and proving the solution for situations where a>0, I have students complete a Think-Pair-Share with the prompt:

What happens when a<0?

I suggest that the class use a = -12 as an example. We also review what happens when a = 0 (no solution - dividing by 0). 

I want students to make the connection that the value of the variable, a, may affect the solution to the inequality. I conclude the note-taking with a practice problem. Depending on the class, teachers can go through the problem with the class, ask students to work in pairs, or assign  the problem as a quick formative assessment to gauge student understanding.

There is a sample set of Class Notes as a resource in this section. Teachers can use the notes as a guide to write on the board. I find that students tend to like taking notes off of the white board and one problem at a time as opposed to having notes already written and projected on a SmartBoard. I also use the typed notes as a resource for students who are absent (more more info, see the strategy video on Student Absences and Missing Work). The notes present all the steps for the first problem and gradually decrease the amount of support. 

Exit Ticket: Solving an Inequality with a Letter as a Coefficient

15 minutes

To conclude the class I have students complete an Idea Organizer to give them the time to write out their thoughts as they solve a problem involving inequalities. I have found students tend to have difficulty with the problem, so I use this activity as a short Formative Assessment. I review their work with respect to their mathematical understanding and their writing.