Solving Linear Inequalities: Multiplication and Division
Lesson 14 of 15
Objective: SWBAT solve an inequality with a focus on inequalities with negative coefficients.
I will start class by sharing the First Slide from my presentation with the class. Students are faced with a more complex inequality which requires them to see the structure in the expression in order to find 4 viable solutions. I give students a few minutes to work on finding 4 solutions with their partners.
In a similar way to the previous lesson, the next slide asks them to determine why the series of inequalities are equivalent to the original. I want students to be able to cite the commutative and distributive properties for #1 and #2 respectively. Question #3 deals with adding a term to each side of the inequality. Question #4 deals with multiplying each side of the inequality by a constant.** For each of these questions, I encourage students to verify that the four solutions that they determined would make the original inequality true also make each of these new inequalities true as well.
** This is only true if the constant is positive. Multiplying by a constant that is negative will be the focus of the next portion of the lesson.
I have my students do a Think-Pair-Share to discuss the Third Slide. I ask them to determine the moves that do not change the solution set to an inequality. This gives students time to think and process both by themselves and with a partner before summarizing as a class. When students are sharing their ideas I ask them to validate them based on what they had seen in the warm up activity. I can then guide students to the following:
- Using the properties: commutative, associative, and distributive
- Adding the same term to each side of the inequality (subtraction is also implied).
- Multiplying by a constant* If students bring up the concept of negative vs. positive it will serve as a perfect lead into the next portion of the lesson. If not, explain to students that we will be exploring this third move more in depth.
A main focus of this lesson will be a development of the idea that multiplying or dividing each side of an inequality by a negative number does not maintain the solution set of the original inequality. Take your time in this portion of the lesson and use think-pair-share and turn and talk where you feel they may be appropriate to help students process and reach valuable conclusions about this concept (see Collaboration: Building Communication and Student Ownership to learn more about why this is important in my classroom). I will add points of emphasis in the narrative below, but understand that each of my classes is different and may have unique needs. Here is the set of slides that I use for this section of the class: Solving Inequalities Multiply Launch.
Slide 5: I allow students to explore this "potential problem" with their partner. Before having students begin, I make sure that they realize why they would multiply by -1 (in order to have the coefficient of x be positive). This allows me to ensure that when students are multiplying through by -1 they distribute the value correctly on the left hand side of the inequality. The students should come to the conclusion that the solutions that they found to the initial equation do not make the new equation true.
Slide 6: This question requires students to think critically and understand and make sense of the question being asked (MP1). I have students discuss this problem with a partner to gain awareness of the relevant concepts. Once students have an opportunity to work on this I have students offer their solutions.
Teaching point: Students often get confused with the statment A + c > B + c to mean that c must be a constant not a variable term. Adding x to both sides of this inequality is the crucial move to understand how to make the variable positive without resorting to multiplying through by a negative value.
Lastly, I have students show how their original solutions to the inequality also make the new inequality (1 < x) true.
Slide 7: I want to give students an opportunity to practice the following with a partner. In each case students should be focusing on moving the variable with a negative coefficient to the other side of the inequality in order to ensure the solution set of the inequality remains the same.
Slide 8: Slide 8 allows me to summarize as a class what has been seen so far in solving inequalities. In the next slide the goal is to explore the case of multiplying (or dividing) and formalizing the results.
Slide 9: I use the first question, "What do you notice?" as an attempt to get students discussing the fact that the inequality symbol has reversed directions. They why is based on the relative size of the variables A and B. The animation on this slide brings up the four variables on the number line one at a time. While showing this animation I allow students to think through the placement of each of the variables (You could also choose not to use the animation and just use the number line as a diagram, letting the conversation flow more easily with student ideas)
- If A < B, for the sake of argument let's say they are both positive, what could their values be on this number line?
- Get two values from students:
- So then what do we know about -A and -B? Where would they be located on the number line?
- Remember that "greater than" means that a number is farther to the right than another number on the number line. So why is -A > -B?
Slide 10: This slide will serve as a summary for our two inequalities lessons. I have students take time to process #2 and #3 and ensure they make sense before moving on.
The solving_inequalities_multiply_practice lets students see that of the two ways to solve an inequality with a negative coefficient (moving the negative variable to the opposite side or multiplying by a negative) both will result in the same solution set. The tables allow students to formalize this understanding through practice and to reason both abstractly and quantitatively about the solutions obtained (MP2). There is also a group of practice questions on the back of the worksheet for students to practice all that they have learned dealing with inequalities. I encourage students to take their time with the solutions and justifications. It is more important to have quality work with justifications that make sense.
Solving_inequalities closure will assess students at both the procedural and conceptual level. The first two exercises require students to apply what they have learned about solving and graphing inequalities. The third question allows students an opportunity to summarize what they have learned about multiplying/dividing and inequality by a negative number in a more conceptual way. Results for this question may vary but most students will show how the problem can be rewritten by putting the variable on the opposite side, this shows why the inequality is "reversed" in the solution to the inequality.