Working with Inequalities

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Objective

SWBAT: • Explain the difference between an inequality and an equation • Create an inequality for a given situation • Solve an inequality • Graph an inequality on a number line

Big Idea

What’s the difference between s = 40, s > 40, and 4s > 160? Students work on creating, solving, and graphing inequalities for given situations.

Do Now

5 minutes

See my Do Now in my Strategy folder that explains my beginning of class routines.

Often, I create do nows that have problems that connect to the task that students will be working on that day.  Today I want students to quickly review the meaning of these symbols.  Students have previously worked with inequalities in unit 3. I call on students to explain what each symbol means and their example.  

Review and Problem

10 minutes

We review the definitions of an equation and an inequality.  I read over Ashley’s situation.  We work on the first situation together.  What does it mean here if s=40?  I want students to realize that if s=40 then the shoes cost exactly 40 dollars.  The only possible value for s would be 40.  I ask students to generate impossible values for s for this situation.  Students work independently on the other situations.  I walk around and monitor student progress and look for common mistakes.  Students may struggle with the last situation, and this is okay. 

Students participate in a Think Pair Share to talk about their ideas for Ashley’s situations.  Students are engaging in MP2: Reason abstractly and quantitatively, MP3: Construct viable arguments and critique the reasoning of others, and MP7: Look for and make use of structure. I call on students to share their ideas.  I ask students to share all of the possible values for s in the second situation.  I want students to realize that there are an infinite number of possible values to represent s in this situation, so we cannot list them out.   

If I saw a common mistake, I present it and ask students what they think.  If students are stuck on the last situation I ask, “What does s represent?  So what would 4s represent?”  I want students to realize that it means that four pairs of shoes are greater than or equal to $160.  I ask students how this inequality compares to the others in this problem.  I want students to see that the last two situations are equivalent.  I mention that if it was an equation, 4s = 160, what would s equal?  With an inequality we can do the same thing, but instead of using the equal sign we keep the sign that is already there.  This would result in s is greater than or equal to 40 dollars.

Graphing Inequalities

10 minutes

I ask students to share what they remember about graphing inequalities from Unit 3.  I want students to share that if an inequality uses the less than or greater than symbol the circle must be open because the variable cannot equal that value.  If there is a less than or equal to or greater than or equal to sign the circle must be closed because the variable can equal that value. 

I have students work on the problems independently.  Students are engaging in MP2: Reason abstractly and quantitatively and MP7: Look for and make use of structure.  I walk around to monitor student progress.  I am looking to see whether students are able to correctly match 1-3 and what strategies they use for 4-7.  We come together to share ideas.  I ask students, “Why can’t we just plot the number, why do we have to draw the line with the arrow?”  I want students to understand that because there are infinite number of solutions we have to draw and extend the arrow. If students struggle with 4-6 I mention the strategy of solving for x.  If it were 3x = 27, x = 9, therefore 3x < 27 is equivalent with x < 9, which is the same graph for problem 1.

I ask students to share ideas about problem 7.  Some students may immediately say they don’t know because they haven’t seen an inequality like this before.  I encourage them to think about what the numbers and symbols mean.  I cover up the last part of the inequality so students are just looking at 6 < x.  What does this mean?  I want students to be able to explain that 6 is less than x, or x is greater than 6.  Then I cover up the first part so students are just looking at x is less than or equal to 12.  I ask students to explain what this means.  How do we combine these two inequalities?  We draw the inequality with an open circle on 6, a closed circle on 12, and a line connecting them.

Practice and Finding Solutions for Inequalities

15 minutes

Note:

  • Before this lesson I create and Post A Key for this part of the lesson.

I explain that students will be working independently on the next two sections of the lesson.  I explain that problems in “Finding Solutions for Inequalities” are not asking for them to list all of the solutions, but rather to identify which numbers in the box work for that given situation.  We preview the sections and I ask students what strategies they can use if they get stuck.  I want students to realize that they can look back at their notes earlier in the packet to help them.  They can also ask a neighbor a question.  I tell students that when they finish a page, they can raise their hand and show me their work.  Students are engaging in MP2: Reason abstractly and quantitatively, MP4: Model with mathematics, and MP7: Look for and make use of structure.

If they are on track, I send them to check their work with the key.  If they are struggling I ask them one or more of the following questions:

  • What is going on in this situation?
  • How can you represent that with an inequality?
  • What are some possible values for your variable?  Why?  What are some impossible values?  Why?
  • How did we graph inequalities earlier?  Look back at your notes.

If students successfully complete the work they move on to the challenge questions.

Closure and Ticket to Go

10 minutes

I have students turn to the closure graph.  I ask students to come up with a situation that matches the graph using the variable x, in addition to finding as many inequalities as they can that match this graph.  Students participate in a Think Pair Share.  I am looking for students to start with x > 7 and then create equivalent inequalities such as 2x > 14 or x + 3 > 10.  Students share their ideas and I ask the class whether they agree or disagree and why.  Students are engaging in MP4: Model with mathematics and MP7: Look for and make use of structure.

I pass out the Ticket to Go and the HW Working with Inequalities.