<p>In this lesson, students will learn how to graph solutions of linear inequality. Students are going to:&nbsp;<br>- understand the logic behind each step.<br>- realize there are typically an infinite number of solutions to linear inequalities.</p>

<p>- How do you create, solve, graph, and interpret linear equations and inequalities to represent relationship between quantities?&nbsp;<br>&nbsp;</p>

<p>- Inequality: An equation in which one side of the equation is compared to the other side using inequality signs.<br>- Linear Equation: An equation whose graph in a coordinate plane is a straight line.<br>- Linear Inequality: An inequality that, when graphed, has as its solution half the Cartesian plane; the two regions (solution and non-solution) are divided by a solid or dashed boundary line (as opposed to a curve).<br>- Solution: An ordered pair (x, y) that, when x and y are substituted into an equation or inequality, make the resulting expression true.</p>

<p>- copies of Inequality Worksheet, one per student (M-A1-4-2_Inequality Worksheet and KEY)<br>- graph paper for each student (enough for students to make three complete graphs)<br>- copies of the Lesson 2 Exit Ticket, one per student (M-A1-4-2_Lesson 2 Exit Ticket and KEY)</p>

<p>- Throughout the exercises, keep an eye on the students' comments as they offer ideas for solutions to the more difficult conceptual problems. This will enable you to help students with the concepts and point them in the direction they need to go to&nbsp;solve the challenge.&nbsp;<br>- While helping students solve linear inequalities, keep an eye on their progress on the inequality worksheet.&nbsp;<br>- The Lesson 2 Exit Ticket (M-A1-4-2_Lesson 2 Exit Ticket&nbsp;and KEY) is a tool that can be used to assess students' understanding of both setting up and graphing inequalities. This enables you to help students who might require more practice and determine whether a concept review is required before moving on to the next lesson.&nbsp;<br>&nbsp;</p>

<p>Scaffolding, Explicit Instruction<br>W: Students can see where they are heading and how it connects to their prior knowledge by giving them a real-world problem that is somewhat similar to one they have seen before, such as an inequality instead of an equation. Additionally, students can have a preview of their assessment process by having the Inequality Worksheet available early in the course.&nbsp;<br>H: An engaging real-world example of athletic training that involves exercise and relates to a topic not usually related to mathematics captures the attention of the students. This merging of seemingly unrelated topics of interest can help to maintain student attention. Maintaining student interest in this problem is facilitated by the ongoing examination of it throughout the lesson.&nbsp;<br>E: Instead of having the full process demonstrated for them and then attempting to duplicate it, students are allowed&nbsp;to engage with and practice their new information in short, defined steps when they are introduced to new content.&nbsp;<br>R: Students use the Inequality Worksheet to reflect on the lesson's material and solve the three problems. Students can use this as an opportunity to reconsider and improve their ideas on this topic as needed. Students are also encouraged to complete their exit ticket with the instructions needed to graph linear inequalities as a way to reflect on the entire class.&nbsp;<br>E: Students use the Inequality Worksheet to demonstrate their learning, and they assess their own performance on the graphing of linear inequalities by contrasting their work with that of their peers to make sure they understand.&nbsp;<br>T: Explicit instruction and independent or pairs practice are alternated throughout the lesson. Students are also taught both the more logical, conceptual approach and reasoning, as well as the "straightforward," or non-conceptual, method of graphing linear inequalities.&nbsp;<br>O: The purpose of both exercises is to gradually advance from teacher-led discussion and analysis of the original topic to partnered work using the Inequality Worksheet on individual problems.</p>

<p>To begin the class, provide students with the following situation: <strong>"Assume an athlete trains by running and jogging. The athlete burns 8 calories per minute while running and 5 calories per minute while jogging. The athlete intends to burn a total of 240 calories during her training. The question is: What are all the possible running and jogging combinations that will allow the athlete to meet her goal of burning 240 calories?"</strong><br><br>Guide students through the process of creating a linear equation that represents the problem solutions. Students must identify the two variables (time running, <i>x</i>, and time jogging, <i>y</i>), as well as the constraint(burning 240 total calories). Ask the students,&nbsp;<br><br><strong>"What equation can we represent the answers to this situation with?"</strong><br><br>(8<i>x</i> + 5<i>y</i> = 240)<br><br>Once students have correctly identified the linear equation, investigate the problem in greater detail.<br><br><strong>"Is it likely that you want to burn </strong><i><strong>exactly</strong></i><strong> 240 calories when working out as an athlete?" </strong><i>(no)</i><br><br><strong>"What is more likely to be your goal—less than 240 calories, exactly 240 calories, or more than 240 calories?"</strong> <i>(more than 240 calories)</i><br><br><strong>"If we want to show that the athlete wants to burn </strong><i><strong>more</strong></i><strong> than 240 calories, we need to use an inequality symbol instead of an equal sign or an equation."</strong><br>Remove the equal sign from the equation and replace it with an inequality sign so that it reads<br><br>8<i>x</i> + 5<i>y</i> ≥ 240.<br><br>Provide them with this information: <strong>"In many situations, an inequality represents the reality of the situation better than an equal sign."</strong><br><br><strong>Activity 1</strong><br><br>Tell the students,<strong> "We'll continue to look at inequality signs with some problems. In the case of the athlete, how would our inequality be affected if she desired to expend</strong><i><strong> more</strong></i><strong> than 240 calories rather than a minimum of 240 calories?"</strong> Students should respond by stating that the inequality sign will be changed to strictly <i>greater than</i>. Explain to students that the line beneath the inequality sign is similar to the equal sign and represents both equality and inequality. Ask the question: <strong>"How would our inequality sign change if our athlete's goal was to&nbsp;burn 240 calories </strong><i><strong>or less</strong></i><strong>—a strange goal?"</strong> (<i>The inequality sign changes to ≤.</i>)<br><br>Repeat this question with the goal being burning strictly <i>less than</i> 240 calories.<br><br>Give each student a copy of the Inequality Worksheet (M-A1-4-2_Inequality Worksheet and KEY). Allow students to work in pairs, but each should finish their own worksheet. Tell students that the class will work together to complete the entire worksheet in steps. The first step is to put down the inequality that reflects each scenario.<br><br>Encourage students to think of each problem in terms of a linear equation. Students should identify the variables as well as the constraints (such as time) and use them to write the equation.&nbsp;Then, students should identify the part of the problem that causes an inequality and replace the equal sign in their equation with the appropriate inequality sign, paying special attention to whether the inequality sign is strictly less than (or greater than) or less than <i>or equal to </i>(or greater than <i>or equal to</i>).<br><br>Students should just complete the inequalities to represent each case at this point in the worksheet. After each pair has completed this section of the worksheet, have them compare their results to another pair who has completed the worksheet to ensure they have the same inequalities; if not, the foursomes should collaborate to address any problems.<br><br><strong>Activity 2</strong><br><br>On the board, draw the first quadrant of a Cartesian plane. Tell them, <strong>"Now, let's continue to look at our athlete problem; we'll come back and finish the next section of the worksheet once we finish our sample problem. </strong><i><strong>What if</strong></i><strong> our athlete is determined to achieve her specific goal? In other words, what if we look at all of the solutions to the </strong><i><strong>equation</strong></i><strong> we started with? What would our graph look like if we only considered the equation?"</strong><br><br>Students should work in pairs to identify the graph of 8<i>x</i> + 5<i>y</i> = 240. Remind them that they can plot points or convert the equation to slope-intercept form to determine how it appears. The slope-intercept equation is as follows:<br><br>y = -\(8 \over 5\)x + 48.<br><br><strong>"The first step in graphing a linear inequality is to graph the </strong><i><strong>line</strong></i><strong>—consider it like an equation."</strong><br><br>Sketch the line on the Cartesian plane. Ask students, <strong>“Remember that she wants to burn </strong><i><strong>at least</strong></i><strong> 240 calories, so it's okay if she burns exactly 240 calories. This signifies that all of the options in our line are acceptable to the runner. Because the line reflects some of the solutions to our inequality, we will make the line solid.”</strong></p><figure class="image"><img style="aspect-ratio:399/388;" src="https://storage.googleapis.com/worksheetzone/images/Screenshot_92.png" width="399" height="388"></figure><p><strong>"What if her objective was to burn </strong><i><strong>more</strong></i><strong> calories than 240? What if our athlete couldn't burn precisely 240 calories? Will the solutions that comprise the line genuinely be solutions?</strong>" <i>(no)</i><br><br>" <strong>If she is required to burn a calorie intake that is strictly more than 240, how would that alter our inequality?"</strong> <i>(Instead of greater-than-or-equal-to, the inequality sign would just be greater-than.)</i><br><br><strong>“In that case, we would continue to graph the line; however, we would make it dashed rather than solid.”</strong></p><figure class="image"><img style="aspect-ratio:510/407;" src="https://storage.googleapis.com/worksheetzone/images/Screenshot_93.png" width="510" height="407"></figure><p>Here, it is important to remind students how to graph linear inequality such as <i>x</i> &gt; 5 (or <i>x</i> ≥ 5),&nbsp;and how to choose an open or closed circle depending on the inequality sign. Show students the analogy between creating a closed circle and creating a solid line.<br><br>Instruct students to return to their pairs and complete the Inequality Worksheet. Explain,<strong> "Now, for each of the three issues, you will perform the following step in graphing inequalities: Graph the line that describes the situation </strong><i><strong>as if it were an equation</strong></i><strong>. You've already written the equations; all you need to do now is graph them." </strong>Remind students that they can graph equations by plotting points or converting them to slope-intercept form.<br><br><strong>"At this point, it makes a difference if the line is dashed or solid."</strong> Students should be aware that whether an inequality indicator is "or equal to," which determines whether the line should be solid or dashed, determines the importance of the inequality.&nbsp;<br><br>After each pair has finished this section of the worksheet, compare them to another pair who has completed the worksheet to ensure they have the same lines; if not, the two pairs should collaborate to resolve any misunderstandings or disagreements.<br><br><strong>Activity 3</strong><br><br>To start this task, remind students: <strong>"So far, we've only dealt with graphing the answers as if they were equations rather than inequality. We will now tackle the </strong><i><strong>Inequality </strong></i><strong>part. What are some of the different running and jogging times that the athlete could use to reach her goal of burning </strong><i><strong>at least</strong></i><strong> 240 calories? Remember, we've already discussed all of the ways she can burn exactly 240 calories."</strong><br><br>As students offer time estimates, plot them on the <i>x-y</i> coordinate (Cartesian) plane. Remind the students that the competitor can burn the most calories. For this scenario, ignore the reality that she may get tired, run out of time, and so on. Encourage students to consider both "extreme" answers (e.g., running for 1000 minutes) and "borderline" solutions where the athlete barely accomplishes her goal.<br><br>When students have offered a dozen or more solutions, ask them what they have in common when plotted on a graph. They should realize that they are all above the line. Ask:<strong> "Are there any points </strong><i><strong>above</strong></i><strong> the line that represent times for which our athlete would </strong><i><strong>not</strong></i><strong> meet her goal?" </strong>If students think there are, remember that any point above the line has a point on the line directly "below" it, and that point signifies burning exactly 240 calories. The point above the line represents exercising <i>more</i>, which has to represent burning <i>more than</i> 240 calories.<br><br>Say it again: <strong>"Are there any points </strong><i><strong>below</strong></i><strong> the line that represent times for which our athlete </strong><i><strong>would</strong></i><strong> achieve her goal?"</strong> Use the same rationale to help students understand that any position below the line represents <i>less</i> exercise than a point right above it on the line, and hence must indicate burning <i>less than</i> 240 calories.<br><br><strong>“So our whole answer consists of all the spots on the line where she burns exactly 240 calories, as well as </strong><i><strong>all</strong></i><strong> the points above the line. To demonstrate this, because we can't draw dots for every single point, we simply shade over the line to indicate that all of it represents answers to our problem.”</strong></p><figure class="image"><img style="aspect-ratio:267/267;" src="https://storage.googleapis.com/worksheetzone/images/Screenshot_94.png" width="267" height="267"></figure><p>Make it clear to students that when graphing the solution to this problem, they must consider some real-world constraints. For instance, the athlete cannot run or jog for 1,000 minutes or for a negative amount of time. Inform students that learning to graph linear inequalities without regard for extra limitations is the first step toward analyzing more complicated, realistic scenarios. In the future, the class will explore scenarios with many real-world constraints.<br><br>Shade the appropriate area of the graph. Explain to students: <strong>"This is the second stage in graphing linear inequalities: shading the appropriate side of the graph. Notice that the graphed line divides the </strong><i><strong>entire</strong></i><strong> </strong><i><strong>x-y</strong></i><strong> coordinate plane into two halves: one half represents the solutions to the problem, and the other half represents all the points that are </strong><i><strong>not</strong></i><strong> solutions. The trick is deciding which part is which. How did we determine which side to shade in this problem?"&nbsp;</strong><br><br>Students solved the problem by finding spots that satisfied the inequality and shading the corresponding area of the <i>x-y</i> coordinate plane.<br><br><strong>"Is it necessary to identify a dozen different solutions to decide which side of the plane to shade if all the solutions are consistently found on one half of the </strong><i><strong>x-y</strong></i><strong> coordinate plane?"</strong><i><strong> </strong>(no)</i><br><br><strong>"In actuality, how many solutions must we find before we can decide which side to shade?"</strong><i><strong> </strong>(one)</i><br><br><strong>"We can choose any point as long as it is not </strong><i><strong>on</strong></i><strong> the line to determine which side to shade and if it is a solution. Since we can choose any point, does it make sense to choose </strong><i><strong>x</strong></i><strong> = 8.25 and </strong><i><strong>y</strong></i><strong> = 3.14?"</strong> <i>(no)</i><br><br><strong>"Why not?"</strong> <i>(Students should understand that selecting a "complicated" point makes the job far more difficult.)</i><br><br><strong>"What would be an easy point to pick and use for a test?" </strong>[<i>(0, 0)</i>]<br><br>Direct students to choose (0, 0) to answer the question. Have students work in pairs to substitute (0, 0) for <i>x</i> and <i>y</i> in the first issue on the Inequality Worksheet. If students are stuck, have them go through the first problem on the board. Begin with the inequality: Each student should have 5<i>x</i> + 12<i>y</i> ≤ 240.<br><br>Then: 5(0) + 12(0) ≤ 240;<br><br>0 + 0 ≤ 240<br><br>0 ≤ 240<br><br><strong>"Is this a true statement? Is 0 less than or equal to 240?" </strong><i>(yes)</i><br><br><strong>"So, is (0, 0) a solution to our inequality―does it make our inequality true?"</strong> <i>(yes)</i><br><br>Explain, <strong>"Because (0, 0) </strong><i><strong>is</strong></i><strong> a solution, we know all of the solutions to our inequality for problem #1 are on the same side of the line as (0, 0). So, to complete Problem #1, shade the side of the line that includes (0, 0)."</strong><br><br>After students have finished shading, ask them what they would have done differently if they had entered (0, 0) and it did <i>not</i> make the inequaility true. Remind students that the line they graphed splits the plane into two regions: one with all the solutions and one that has all points that are <i>not</i> solutions. A solution cannot exist for any point on that side of the line if (0, 0) is not a solution. What does this mean about the <i>other</i> side of the line? (<i>The other side represents the solution.</i>)<br><br>Students should work in pairs to finish the final two challenges. After each pair has completed the worksheet, have them compare it to another couple who has also completed it to ensure they have the same inequalities and graphs; if not, the two pairs should collaborate to address any misunderstandings.<br><br>Students should write the steps for graphing a linear equation on the back of their worksheets. The steps should be:<br><br>1. Graph the associated line.<br>2. Select and replace a test point.<br>3. Shade the proper side of the line.<br><br>Students can submit their completed Inequality Worksheet (with steps on the back) before leaving class. In addition, distribute the Lesson 2 Exit Ticket (M-A1-4-2_Lesson 2 Exit Ticket and KEY) for students to complete before the next class.<br><br><strong>Extension:</strong><br><br>Students begin to modify their solutions based on the real-world restrictions of the challenges. Students should not only modify their solutions to reflect "real-world" solutions&nbsp;but also create inequalities that indicate these limits (e.g., <i>x</i> ≥ 0, <i>y</i> ≥ 0).<br>Giving students linear inequalities to graph through (0, 0) will help them recall that, although (0, 0) is nearly always the best test point, there are situations when it is not an option.</p>