<p>Students investigate systems of inequality in this unit. Students are going to:&nbsp;<br>- write inequalities that make up a system that reflects a real-world situation.<br>- graph solutions to individual inequalities.<br>- combine solutions to the inequalities that make up a system to obtain the solution set for the entire system.<br>&nbsp;</p>

<p>- How do you formulate, solve, and evaluate systems of two linear equations and inequalities using graphing and algebraic techniques?</p>

<p>- <strong>Feasible Region:</strong> The set of all solutions that satisfies the constraints of an optimization problem.&nbsp;<br>- <strong>Inequality:</strong> An equation in which one side of the equation is compared to the other side using inequality signs.&nbsp;<br>- <strong>Linear Equation:</strong> An equation whose graph in a coordinate plane is a straight line.&nbsp;<br>- <strong>Linear Inequality:</strong> An inequality that, when graphed, has as its solution half the Cartesian plane; the two regions (solution and non-solution) are divided by a line (as opposed to a curve).&nbsp;<br>- <strong>Linear Optimization:</strong> A technique for determining the best outcome of a given mathematical model.&nbsp;<br>- <strong>Solution:</strong> An ordered pair (<i>x, y</i>) that, when <i>x</i> and <i>y</i> are substituted into an equation or inequality, make the resulting expression true.&nbsp;<br>- <strong>System:</strong> A collection of two or more equations or inequalities; an ordered pair (triple, etc.) is a solution to a system if and only if it is a solution to each equation and inequality in the system.</p>

<p>- colored pencils (several colors for each student)<br>- tracing paper (or light-weight paper that can be seen through) with preprinted Cartesian grids<br>- copies of the Lesson 3 Exit Ticket, one per student (M-A1-4-3_Lesson 3 Exit Ticket and KEY)</p>

<p>- Observe student answers to the introductory activity of graphing and solving a fundamental system of inequalities.&nbsp;<br>- You may verify that students understand that the solution region is the area where the graphs of the inequalities overlap by having them observe the student graphs and overlay the tracing paper in Part 2 of the activity.&nbsp;<br>- You can determine which students need extra practice with a particular ability or an additional learning opportunity by keeping an eye on their review responses, both written and spoken, during the class.&nbsp;<br>- You can determine where students need additional help with identifying constraints, graphing, or connecting the information in the graph to the situation by looking at their performance on the Lesson 3 Exit Ticket, which is distributed at the end of the class (M-A1-4-3_Lesson 3 Exit Ticket and KEY). Students try to graph an inequality system and turn in the outcome on the exit ticket.&nbsp;<br>&nbsp;</p>

<p>Explicit Instruction, Modeling, and Active Engagement&nbsp;<br>W: The lesson begins with <span style="background-color:rgb(255,255,255);color:rgb(34,34,34);">a problem of a type they are familiar with, but with constraints added</span>. This gives students&nbsp;motivation to acquire the new material—to better portray real-world situations—while also assisting them in making connections to prior information.&nbsp;<br>H: Instead of receiving direct instruction, students are prompted to think about the problem in terms of the actual world and are instantly engaged in a question-and-answer session. This will grab students' attention and maintain it throughout the course.&nbsp;<br>E: Students engage with the new material in two ways: first, by graphing the inequalities on a single set of axes, and second, by practically layering the answers on top of each other using tracing paper. This interactive method will engage students who are not primarily auditory learners or those who prefer explicit instruction. Students are better prepared to tackle inequality problems thanks to this approach of investigating alternative approaches to problem-solving.&nbsp;<br>R: There is a review portion of the lesson where students can learn how to graph systems of linear inequalities by working in pairs or as a class. As a result, before moving on to the remainder of the session, students have time to consider and improve the procedure.&nbsp;<br>E: By engaging in the exercises, comparing the solutions to the system in Part 2, and turning in their comprehensive list for solving systems of linear inequalities, students demonstrate their understanding. All linear system components must be included in the evaluation process: Is the solution to the linear inequality accurate? Regarding the solution, are the shaded areas situated correctly? Does a single point in and out of the darkened zone meet the expected conditions? This will allow you to examine students' responses at this point and determine who requires additional guidance.&nbsp;<br>T: Explicit instruction, modeling, and active, hands-on participation are some of the tactics used in the lesson.&nbsp;<br>O: After a teacher-led discussion and analysis of systems, students solve systems in pairs. After delving into a few problems, students work in teams to synthesize their information before returning to the class to reach a resolution.&nbsp;<br>&nbsp;</p>

<p>Using an overhead projector or the board, present the following scenario to the class at the start of the lesson. Students should be able to see the problem throughout the first portion of the lecture.&nbsp;Tell the students, <strong>"A hat maker makes two different kinds of hats. He makes berets, which take 40 minutes per beret to make, and he can make top hats, which take 60 minutes per hat to make. He has total 6 hours to make hats."</strong><br><br>Before creating any inequalities to reflect the problem, ask students what the constraints are; in other words, what limits how many or how <i>few</i> hats the hat maker can make. Students should acknowledge time constraints, such as having just 360 minutes to complete tasks.<br><br>Ask students to create an inequality to represent the time constraint, with <i>x </i>being the number of berets and <i>y</i> representing the number of top hats produced. Invite a student to come to the board and write about the inequality. (The equation should be 40<i>x</i> + 60<i>y</i> ≤ 360.)<br><br>Instruct students to concentrate on the minimum quantity of each type of hat that the hat maker is capable of producing:<br><br><strong>"Is there a minimum number of berets he can make?"</strong> <i>(yes)</i><br><strong>"Is there a minimum number of top hats he can make?" </strong><i>(yes)</i><br><br>Ask students to create an inequality for <i>each</i> of the constraints. Students should write <i>x</i> ≥ 0 and <i>y</i> ≥ 0.<br><br>Consider three limitations in this problem: time, inequality direction (&lt;, &gt;, or +), and the fact that no negative number of <i>any</i> kind of hat. Also, note that there is a separate inequality for <i>each</i> constraint: three constraints, three inequalities. <strong>"A </strong><i><strong>system of inequalities</strong></i><strong> (or </strong><i><strong>system of equations</strong></i><strong>) is a system of inequalities (or equations) that all relate to the same problem or circumstance. The good news about finding solutions to systems of inequalities is that it isn't substantially harder than finding answers to each inequality individually."</strong><br><br><strong>Part 1</strong><br><br>Sketch the board with a Cartesian plane. Review briefly the four quadrants' placement and conventional identification. Students could be asked, <strong>"Which quadrant, if any, has the solutions to the inequality </strong><i><strong>x</strong></i><strong> ≥ 0?"</strong><br><br>Remind students that this inequality doesn't care about the value of <i>y</i>, which can range from -1,000,000 to -\(2 \over 3\) to&nbsp;7000. The sole drawback of this inequality is that the <i>x</i>-value is not negative. <strong>"Where are the points on the Cartesian plane that have the </strong><i><strong>x</strong></i><strong>-value equal to zero or have a positive </strong><i><strong>x</strong></i><strong>-value?"</strong><br><br>After students have identified quadrants I and IV, as well as the y-axis, softly shade or crosshatch the relevant area of the Cartesian plane, preferably with colored chalk. Trace along the <i>y</i>-axis with white chalk or colored chalk to indicate its inclusion in the solution and a solid line. Then, ask the same questions about the inequality <i>y</i> ≥ 0. After students have identified quadrants I and II, as well as the <i>x</i>-axis, proceed as follows:<br><br><strong>"Returning to our hat maker problem, we know that he cannot produce a negative number of either type of hat. Where are all the spots that represent non-negative numbers in both </strong><i><strong>x</strong></i><strong> and </strong><i><strong>y</strong></i><strong>?" </strong><i>(Quadrant I and the positive x and y-axes)</i><br><br>Make sure to consistently reinforce the inclusion of the relevant axes on the graph, rather than just the appropriate quadrants; students frequently forget about the axes when focused too much on the four quadrants. Graph and shade the relevant portion of the Cartesian plane, using a different color of chalk or a different direction/type of crosshatch pattern to distinguish it from the first set of shading.<br><br><strong>"So all the points that we’ve identified―do they represent all the possible solutions to our hat maker problem?"</strong> Students should understand that they do not, since we have not yet considered the time constraints. Ask students whether the hat manufacturer can create 200 top hats and 100 berets if they do not immediately realize this. At this stage, students should be informed of the time constraints. <strong>"So far, we've eliminated a large portion of the Cartesian plane simply by applying the two inequalities that represent the impossibility of producing a negative number of either type of hat. But now we must remove the area of the Cartesian plane that contains all of the values that indicate combinations of hats that take </strong><i><strong>too long to make</strong></i><strong>."</strong><br><br>Ask students to graph the inequality in slope-intercept form, which illustrates the time constraint (40<i>x</i> + 60<i>y</i> ≤ 360). Students can review their work with their classmates. (Using slope-intercept form, the inequality is <i>y</i> ≤ -\(2 \over 3\)<i>x</i> + 6.)<br><br>Ask a student to come to the board and graph the inequality. Remind the student to consider whether to represent the inequality with a solid line or dashed line. Ask the class how they choose which side of the line to shade as the student graphs the line. Students should reply by using a test point, such as (0, 0), which is a solution to the linear inequality, and shading the side of the plane containing (0, 0). Give the learner colored chalk or have him/her draw a different crosshatch pattern to show the answer to the inequality. Ask students, <strong>"Do </strong><i><strong>all</strong></i><strong> the points below&nbsp;y ≤ -\(2 \over 3\)x + 6 represent solutions to the </strong><i><strong>system</strong></i><strong> of inequalities?"</strong> <i>(no)</i><br><br>Help students understand that the only points that are acceptable combinations for the system are those that fulfill each of the three system inequalities. <strong>"How can we determine which part of the Cartesian plane has the solutions to each of the three inequality problems?"</strong> (identify the section with all three shading colors or all three styles of crosshatching.)<br><br>Outline the relevant region to make it apparent where the solution is.<br><br>Ask, <strong>"How did the process used to arrive at this solution differ from the simple graphing of a single inequality?"</strong><br><br>Identify a feasible area. Students should understand that they only wrote and graphed each inequality separately before identifying the overlapping region. The only new step is identifying the overlapping zone; otherwise, the techniques for graphing linear inequalities are the same as before.<br><br>Before proceeding, underline that finding solutions to inequalities is simply a repetition of previously mastered abilities. Also, emphasize the necessity of keeping work neat, as graphs with three, four, or more inequalities can become messy. To help students distinguish between areas where different colors overlap, encourage them to use colored pencils and shade very lightly.<br><br>Another strategy that students can use is to put an arrow on the graph of each line to indicate which side of the line to shade, then shade the region where all of the arrows point. This has the advantage of keeping the graphic neater, but it makes it harder to see where the shading goes when there are a lot of inequalities to graph at once.<br><br><strong>"Let's look at another example,"</strong> says&nbsp;the class. <strong>“Consider the inequality </strong><i><strong>y</strong></i><strong> &gt; \(1 \over 3\)</strong><i><strong>x</strong></i><strong> - 2 and </strong><i><strong>y</strong></i><strong> ≥ \(4 \over 3\)</strong><i><strong>x</strong></i><strong> + 4. What would the solution look like?”</strong></p><figure class="image"><img style="aspect-ratio:314/282;" src="https://storage.googleapis.com/worksheetzone/images/Screenshot_95.png" width="314" height="282"></figure><p><strong>Part 2</strong><br><br>Assign students to work in pairs. Give each pair five sheets of tracing paper with Cartesian graphs on them. Create the system shown below on the board:<br><br><i>y</i> &gt; 0.5<i>x</i> - 1<br><br><i>y</i> &gt; –2<i>x</i> – 2<br><br><i>y</i> ≤ –<i>x</i> + 5<br><br><i>y</i> ≤ 2<i>x</i> + 2<br><br>Each pair should divide the four inequalities so that each member has two of them. Each student should next graph his or her two inequalities on separate sheets of tracing paper. Each student should shade lightly but dark enough so that the shading and line can be <i>clearly</i> seen through the tracing paper. (It must be visible through four pieces of tracing paper by the end of the task.)<br><br><strong>Note: </strong>If standard paper is used instead of tracing paper, have each student graph both of his/her inequalities on the <i>same</i> sheet rather than two separate sheets to make the final step of sketching the answer easier.<br><br>While students graph their inequalities, stroll around,&nbsp;and examine their work, As students work, remind them to pay close attention to the line style (dashed or solid) as well as the direction of their shading. Remind students to shade in the appropriate direction using a test point (0, 0) in this assignment.<br><br>After each pair has graphed the system's four inequalities, they should trade with their partner to ensure that the inequalities are appropriately graphed.<br><br>For the final step of the tracing assignment, students should stack all four pieces of tracing paper on top of one another. (Holding the papers up to a window may help you see the four graphs through the layers of tracing paper.) Then, for each pair, take a fifth sheet of tracing paper with a Cartesian plane printed on it and set it on top of their stack. They should be able to identify the region that contributes to the solution of <i>all four</i> inequalities.<br><br>Have students select the four points that define the solution region. (The four points are&nbsp;(-1, 0), (0, -2), (4, 1), and (1, 4). Students should plot those points on their fifth sheet of tracing paper and then connect them with the appropriate form of line (dashed&nbsp;or solid). Then, ask students to shade in the quadrilateral that illustrates the solution.<br><br>Have each pair of students compare their solutions to those of the other pairs. The solutions can be easily compared by stacking tracing paper with the solution quadrilaterals on top of each other and ensuring that they overlap properly.<br><br><strong>Review</strong><br><br>Assign students to continue working in pairs from Part 2: <strong>"Imagine you have a system of inequalities similar to the one you recently worked on with your partner, but instead of using tracing paper, you will graph all of the inequalities on a </strong><i><strong>single</strong></i><strong> Cartesian plane. From the time you are presented with the four inequalities, contemplate the steps you must take. With your partner, create a set of clear, easy actions that anyone can take to solve the problem."</strong><br><br>Allow students around 5 minutes to brainstorm a list of steps before bringing the class back together.<br><br>Ask one group for their initial step, and then ask the other groups if they agree or disagree. If groups disagree, ask them which step(s) should precede the stated first step. Continue until the class has compiled a list of steps for solving a system of inequalities. A possible sequence of steps might be:<br><br>1. Convert each inequality to a slope-intercept form.<br>2. Graph a single inequality by first graphing its corresponding line. When graphing a line, use solid ( for ≤ or ≥) or dashed lines (for &lt; or &gt;).<br>3. Select a test point for the inequality and enter the coordinates into the inequality. (If the line does not pass through the origin, select (0, 0).<br>4. If the test point proves the inequality, shade&nbsp;the side of the line containing the test point; otherwise, shade the opposite side.<br>5. Repeat steps 2 to 4 for each inequality in the system.<br>6. Identify the region that contributes to the solution of each inequality in the system.<br><br>Each pair should write down the agreed-upon sequence of steps. Also, have each pair put their names on their original set of steps and submit them. Remind the pairs that their original set of steps will not be used to determine their score or grade. However, reviewing the initial stages that students devised will assist in anticipating potential places of uncertainty for both individual students and the class as a whole. Also, once the class ends, give them the Lesson 3 Exit Ticket (M-A1-4-3_Lesson 3 Exit Ticket and KEY) to hand in for the next class.<br><br><strong>Extension:</strong><br><br>Introduce students to linear optimization problems (for example, profit maximization while bound by several inequalities). Explain the corner-point principle, which states that the solution region's corners are where maximum and minimum values are reached.</p>