<p>Students will learn how to solve a system of equations by graphing, substituting, or eliminating variables. Students will:<br>- determine the intersection of two lines on the same coordinate&nbsp;grid.<br>- solve a system using substitution or elimination method.<br>- determine the most efficient strategy for solving a system of equations.<br>- determine whether a system has a unique solution, no solutions, or an unlimited number of solutions.</p>

<p>- How would you utilize graphical and/or algebraic techniques to solve an equation system, and how would you interpret the results?&nbsp;<br>- How can we demonstrate that algebraic properties and processes are extensions of arithmetic properties and processes, and how can we apply algebraic properties and processes to problems?&nbsp;<br>- What functional representation would you use to simulate a real-world situation, and how would you explain how you solved the problem?&nbsp;<br>- How would you explain the link between quantities represented by linear equations and/or inequalities?&nbsp;<br>&nbsp;</p>

<p><span style="background-color:rgb(255,255,255);color:rgb(8,42,61);"><i>(Vocabulary hasn't been entered into the lesson plan.)</i></span></p>

<p>- mini-whiteboards, whiteboard markers and erasers/paper towels for students&nbsp;<br>- paper and markers (if mini-whiteboards are not available)&nbsp;<br>- graph paper&nbsp;<br>- rulers&nbsp;<br>- Systems of Linear Equations and Solutions Chart for display or one copy per student (M-A1-5-2_Systems of Linear Equations and Solutions Chart)<br>- copies of Exit Ticket (M-A1-5-2_Exit Ticket)</p>

<p>- Teacher observation during class discussion and activities&nbsp;<br>- Work samples on whiteboards&nbsp;<br>- Exit Ticket activity</p>

<p><strong>T:</strong> This lesson is designed to accommodate a variety of learning modalities, mostly visual and auditory. Use the options below to personalize the lesson to your student's specific requirements.&nbsp;<br><br><strong>Routine:</strong> Students are encouraged to assist one another through group and partner work. The emphasis should be on communicating mathematical ideas using vocabulary phrases&nbsp;appropriate for the subject. Encourage students to generate and explain real-world contexts for the many graphs and equations used in the challenges. The lecture involves both precise notes and active participation. Help students with organized and note-taking abilities to improve their learning experience and create a useful resource (notes).&nbsp;<br><br><strong>Small Group:</strong> Struggling students can be assigned to one or more small groups for additional assistance from the instructor. Use the example provided below. Model the cognitive process utilized to begin and solve the problem. Ask students questions geared to assist you in completing the problem's steps, and allow them to ask any questions they may have. Either ask students to finish the problem where you left off or build a similar problem for them to complete entirely on their own.&nbsp;<br><br><strong>Example Problem: </strong>The physical education classes at Century Middle School are organizing a field excursion to the bowling alley. There are two potential bowling alleys. The first bowling alley charges $6.00 per student and $9.25 for an adult chaperone. The second bowling alley charges $7.25 per student and $5.50 for adult chaperones. Assign variables, formulate equations, locate the solution set, and explain what it represents.&nbsp;<br><br><strong>Extension: </strong>Assign pairs of students to compose a real-world problem containing a system of equations. Students should answer the problem using each method (graphing, elimination, and substitution) and create a presentation with a visual display, such as a poster, PowerPoint, transparency, and so on. If time allows, students can:&nbsp;<br><br>- Trade problems with another group and evaluate their answer.&nbsp;<br>- Use the problem to impart a lesson to the class or a small group.&nbsp;<br>- Present their solutions (with visuals) and explain which strategy is the most efficient.&nbsp;</p><p><br><strong>O: </strong>The lesson is structured such that students learn two more approaches to solving problems in addition to the graphic method taught in Lesson 1. They take notes on various strategies for organizing their thoughts and using them&nbsp;as a resource when solving problems. Students learn algebraically solving linear equation systems both individually and in pairs. The class works together to debate techniques and solutions. Students are also taught the ideas of no solution and infinitely many solutions. The activity using mini-whiteboards or sheets of paper allows students to receive fast feedback on their work, allowing them to get back on track quickly.</p>

<p><strong>W:</strong> This lesson teaches two new equation-solving approaches. Students learn how to apply substitution and elimination strategies. In addition, students investigate whether strategy (graphing, substitution, or elimination) is most effective in many situations. Students learn how to determine whether a system has one solution, no solution, or infinitely many solutions, and how this affects efficiency.<br><br><strong>H: "In the previous lesson, we studied how systems of equations are commonly used in real-world situations with varying rates of change. We also learned how to graph solutions and check them. There are numerous approaches to achieving the same solution."</strong><br><br><strong>"How many of you would rather find the sum of 478 + 478 + 478 + 478 + 478 + 478 by hand than with a calculator?"</strong> <i>(Most people will probably say the calculator is easier or more convenient.)</i><br><br><strong>“When we do a task and try to do it quickly and more easily, that is called being efficient.”</strong><br><br><strong>"Has anyone used a dustbuster (a little handheld vacuum)? Have you ever tried a full-sized vacuum? When is it more </strong><i><strong>efficient</strong></i><strong> to use a dustbuster instead of a vacuum cleaner? When is it more </strong><i><strong>efficient</strong></i><strong> to use the vacuum instead of the dustbuster?"</strong><br><br><strong>"Graphs are a very effective means of visualizing a system of equations, and they are frequently utilized when dealing with real-world situations. In some of our cases, graphing proved to be incredibly efficient. Could you please tell me which ones? "</strong><i><strong> </strong>(those with whole numbers or integer values for solutions)</i><br><br><strong>"When you graphed them, did you find any of them to be inaccurate or frustrating?"</strong> <i>(When the answer was nonintegral.)</i><br><br><strong>"You will discover some new, more </strong><i><strong>efficient</strong></i><strong> methods for solving a system of equations today."</strong><br><br><strong>E:</strong> Review the walking competition problem from the previous lesson. To see if students are thinking algebraically, ask them how they would have determined the intersection point without graphing the two equations.<br><br>The notes below should be placed on the board for students to copy.&nbsp;<br><br><strong>Three Ways to Solve a System of Equations</strong><br><br>1. <strong>Graphing:</strong> this method is used to see how the system is solved.<br><br>2. <strong>Substitution Method:</strong> Used when one or both equations are in slope-intercept form (<i>y</i> = m<i>x</i> + b).<br><br><u>Example 1:</u> The walking competition (both equations use slope-intercept form).</p><figure class="image"><img style="aspect-ratio:385/237;" src="https://storage.googleapis.com/worksheetzone/images/Screenshot_5.png" width="385" height="237"></figure><p>Solution to the system: (1, 2.5). The answer is written as an ordered pair.<br><br>What does this means in terms of the walking competition?<br><br>Answer: 1 → 1 hour,&nbsp;<br>&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;2.5 → 2.5 miles<br><br><u>Example 2:</u> One equation is in slope-intercept form.</p><figure class="image"><img style="aspect-ratio:409/72;" src="https://storage.googleapis.com/worksheetzone/images/Screenshot_6.png" width="409" height="72"></figure><p>3<i>x</i> + 4(2<i>x</i> – 3) = -1 ← Distribute 4 and combine like terms to solve for <i>x</i>.<br>3<i>x</i> + 8<i>x</i> – 12 = -1<br>11<i>x</i> – 12 = -1<br>11<i>x</i> = 11<br><i>x</i> = 1<br><br><i>y</i> = 2(1) – 3 &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; Substitute the <i>x</i>-value into one of<br><i>y</i> = 2 – 3 &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;the equations to solve for <i>y</i>.<br><i>y</i> = -1<br><br>Solution to the system: (1, -1)<br><br><br>3.&nbsp;<strong>Elimination Method or Linear Combination Method: </strong>This method is used when both equations are expressed in standard form (A<i>x</i> + B<i>y</i>=C).<br><br>The objective is to eliminate one of the variables to solve for the other.<br>Elimination is done by combining the two equations .<br>The eliminated variable must have the same coefficient but opposite signs.<br>(End of student notes.)<br><br><u>Example 1:</u>&nbsp;</p><figure class="image"><img style="aspect-ratio:82/61;" src="https://storage.googleapis.com/worksheetzone/images/Screenshot_7.png" width="82" height="61"></figure><p>Step 1: Since the <i>y</i>’s have the same coefficient with opposite signs, the y is eliminated by adding the equations together.</p><figure class="image"><img style="aspect-ratio:81/68;" src="https://storage.googleapis.com/worksheetzone/images/Screenshot_8.png" width="81" height="68"></figure><p>Step 2: Solve for <i>x</i>.&nbsp;<br>7<i>x</i> = 14&nbsp;<br><i>x</i> = 2<br><br>Step 3: Substitute the x-value into one of the original equations to solve for <i>y</i>.<br>3(2) + <i>y</i> = 3&nbsp;<br>6 + <i>y</i> = 3&nbsp;<br><i>y</i> = -3&nbsp;<br>Solution: (2, -3)<br><br><br><u>Example 2:</u>&nbsp;</p><figure class="image"><img style="aspect-ratio:92/52;" src="https://storage.googleapis.com/worksheetzone/images/Screenshot_9.png" width="92" height="52"></figure><p>Step 1: Since neither variable has the same coefficient with opposite signs, one of the equations needs to be multiplied by a constant. Because <i>x</i> is the only variable in the second equation and <i>x</i> has a coefficient of 2 in the first equation, multiply the bottom equation by -2.&nbsp;</p><figure class="image"><img style="aspect-ratio:272/51;" src="https://storage.googleapis.com/worksheetzone/images/Screenshot_14.png" width="272" height="51"></figure><p>Step 2: Since the <i>x</i>'s have the same coefficient but different signs, we can combine the two equations to eliminate the <i>x</i>'s.</p><figure class="image"><img style="aspect-ratio:104/69;" src="https://storage.googleapis.com/worksheetzone/images/Screenshot_15.png" width="104" height="69"></figure><p>Step 3: Solve for <i>y</i>.&nbsp;<br>-10<i>y</i> = 10<br><i>y</i> = -1&nbsp;<br><br>Step 4: Substitute the <i>y</i>-value into one of the original equations to solve for <i>x</i>.<br>2<i>x</i> – 4(-1) = 14&nbsp;<br>2<i>x</i> + 4 = 14&nbsp;<br>2<i>x</i> = 10&nbsp;<br><i>x</i> = 5&nbsp;<br>Solution: (5, -1)<br><br><br><u>Example 3:</u>&nbsp;</p><figure class="image"><img style="aspect-ratio:102/53;" src="https://storage.googleapis.com/worksheetzone/images/Screenshot_16.png" width="102" height="53"></figure><p>Step 1: In this system, neither variable can be eliminated by simply multiplying one of the equations. Both equations need to be multiplied by a constant. We can decide to eliminate <i>x</i>. Because 3 does not divide evenly into 4, we will multiply the top and bottom equations by whatever constant produces the least common multiple of 3 and 4 (12). We will multiply the top equations by 4 and the bottom equation by -3, resulting in coefficients of 12 and -12.</p><figure class="image"><img style="aspect-ratio:291/49;" src="https://storage.googleapis.com/worksheetzone/images/Screenshot_10.png" width="291" height="49"></figure><p>Step 2: Since the <i>x</i>'s now have the same coefficient but opposite signs, we can add the two equations to eliminate the <i>x</i>'s.</p><figure class="image"><img style="aspect-ratio:118/69;" src="https://storage.googleapis.com/worksheetzone/images/Screenshot_11.png" width="118" height="69"></figure><p>Step 3: Solve for <i>y</i>.&nbsp;<br>23<i>y</i> = -92<br><i>y</i> = -4<br><br>Step 4: Substitute the <i>y</i>-value into one of the original equations to solve for <i>x</i>.<br>3<i>x</i> + 2(-4) = -17&nbsp;<br>3<i>x</i> -8 = -17&nbsp;<br>3<i>x</i> = -9&nbsp;<br><i>x</i> = -3<br>Solution: (-3, -4)<br><br><br>Give students the six problems below to work on independently for about 10 to 15 minutes. Ask students to write the method they used next to the problem. After the independent work period, have students work in pair to complete and discuss the problems.</p><figure class="image"><img style="aspect-ratio:476/120;" src="https://storage.googleapis.com/worksheetzone/images/Screenshot_12.png" width="476" height="120"></figure><p>*5 and 6 are more challenging.</p><p>Solutions:&nbsp; 1. (4, 5); 2. (2, 4); 3. (-2, -8); 4. (-5, -2); 5. (-8, 0); 6. (0, 3)<br><br>When most students have completed their task, have them write it on the board. Discuss the problems as a class.<br><br>Following the discussion, ask students to solve the following two systems as a think-pair-share task.</p><figure class="image"><img style="aspect-ratio:497/212;" src="https://storage.googleapis.com/worksheetzone/images/Screenshot_13.png" width="497" height="212"></figure><p>&nbsp;</p><p>Since 0 does not equal -5, there are <strong>no solutions</strong>.&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<br>Since 0 is equal to 0, there are <strong>infinitely many solutions</strong>.<br><br>Ask students why there are no solutions to the first problem. Tell students to look closely at the equations. To assist students in answering this and subsequent questions, display the Systems of Linear Equations and Solutions Chart (M-A1-5-2_Systems of Linear Equations and Solutions Chart). Ask students questions like:<br><br><strong>"Are there any similarities between the equations?"</strong><br><strong>"Why do you think there are no solutions&nbsp;for the first problem?"</strong><br><strong>"What do you think the graph will look like? Graph the system of equations."</strong><br><strong>"Why are there an infinite number of solutions to the second problem?"</strong><br><strong>"What do you think the graph will look like? Graph the system of equations."</strong><br><strong>"In each problem, how does the first equation relate to the second equation?"</strong><br><br><strong>R:</strong> Give students small whiteboards or paper and markers. Display the six challenges listed below on a board or above. Instruct students to use a whiteboard or paper to demonstrate their efforts in solving equation systems. When they've finished a problem, ask them to hold their whiteboard or paper up in the air, facing you. Tell them you'll say yes or no. If you respond yes, they either erase the board and go on to the next challenge or use another sheet of paper to move on to the next problem. If you reply no, they will need to try the problem again. Students cannot move on to the next problem until you have approved their effort and solution.<br><br>Remind students that this is not a race. It may be beneficial to keep track of how many times you say yes and no to each problem, as well as to identify students who are suffering. When everyone has finished or after a set period of time, go over the problems with the students to clarify any misconceptions or areas of concern.</p><figure class="image"><img style="aspect-ratio:427/134;" src="https://storage.googleapis.com/worksheetzone/images/Screenshot_17.png" width="427" height="134"></figure><p>Solutions:&nbsp; 1. (5, -3); 2. (-1, 3); 3. no solution; 4. infinitely many solutions; 5. (-7, 2); 6. (5, 2)<br><br><strong>E:</strong> Hand out exit tickets (M-A1-5-2_Exit Ticket) to assess students comprehension. <strong>"The local movie theater was offering ticket discounts in honor of its 25th anniversary. You had no idea what the ticket prices were, but you heard two of your classmates discussing how much it cost their families to go to the movies last weekend. One student stated that the cost of two adults and two children was $12. Another student stated that the fee for three adults and four youngsters was $20."</strong><br><br>It would be beneficial to display the scenario on the overhead or&nbsp;the board.<br><br><strong>"In this situation, what are the unknowns or variables?"</strong><br><br><strong>"What information do we know?"</strong><br><br><strong>"Whenever you are using your variables in a real-world scenario, make sure you define them."</strong><br><br><strong>"Work independently to write the system of equations for this situation, select the most effective way to solve it, and then describe the situation in terms of the solution."</strong><br><br>Allow students to work independently to complete the problem. They will turn in their work after class. If there is time, we can discuss the problem in class.<br><br>Answers:<br><br>1. Create an equation system that reflects the scenario for the movie ticket.</p><figure class="image"><img style="aspect-ratio:148/61;" src="https://storage.googleapis.com/worksheetzone/images/Screenshot_18.png" width="148" height="61"></figure><p>2. Which solution method will you use to solve the system? Explain why. (<i>Elimination</i>)<br><br>3. Solve the system of equations. (<i>4, 2</i>)<br><br>4. What does the solution represent in terms of the movie tickets? (<i>Adult tickets are $4 and children tickets are $2.)</i><br><br><br><strong>Second problem:</strong> The first student in line at the snack shop bought 2 pears and 1 apple for $2.10. The second student bought 3 apples and 1 pear for $2.30. What is the cost for one pear, one apple?<br><br>It would be beneficial to display the scenario on the overhead or&nbsp;the board.<br><br><strong>"In this situation, what are the unknowns or variables?"</strong><br><br><strong>"What information do we know?"</strong><br><br><strong>"Whenever you are using your variables in a real-world scenario, make sure you define them."</strong><br><br><strong>"Work independently to write the system of equations for this situation, select the most effective way to solve it, and then describe the situation in terms of the solution."</strong><br><br>Allow students to work independently to complete the challenge. They will turn in their work after class. If there is time, we can discuss the problem in class.<br><br>Answers:<br><br>1.&nbsp;Create an equation system that reflects the scenario for the apple and pear cost.</p><figure class="image"><img style="aspect-ratio:145/73;" src="https://storage.googleapis.com/worksheetzone/images/Screenshot_19.png" width="145" height="73"></figure><p>2. Which solution method will you use to solve the system? Explain why. (<i>Elimination</i>)<br><br>3. Solve the system of equations. (<i>a = 0.80, p = 0.50</i>)<br><br>4. What does the solution represent in terms of the cost of one apple and one pear? (<i>One apple costs $0.50 and one pear costs $0.80.</i>)</p>