<p>The lesson relates prior experience and knowledge of linear functions to the concept of linear systems. Students are going to:&nbsp;<br>- find the point of intersection of two lines on a coordinate grid.<br>- use substitution to verify a solution's validity.<br>- find the intersection of two lines to model the answer to a real-world problem with varying rates of change.<br>- define system of equations and solutions.</p>

<p>- How can we demonstrate that algebraic processes and properties are arithmetic properties and processes extended, and what are some applications of algebraic properties and processes in problem-solving?&nbsp;<br>- Which functional representation would you use to simulate a real-world scenario, and how would you justify your choice of action?&nbsp;<br>- How would you characterize the relationship between quantities represented by linear equations and/or inequalities?&nbsp;<br>- How would you utilize graphical and/or algebraic approaches to solve a system of equations, and how would you interpret the results?&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>- two 8-foot ropes per group<br>- several pieces of string cut to a variety of lengths from 1 to 8 feet<br>- graph paper<br>- rulers<br>- poster-sized graph paper, overhead projector or smart board<br>- colored sticky dots<br>- tape<br>- copies of Exit Ticket (M-A1-5-1_Exit Ticket)</p>

<p>- Think-Pair-Share activity&nbsp;<br>- Teacher observations during group activities and class discussion&nbsp;<br>- Exit Ticket activity</p>

<p><strong>T:</strong> Use the following methodologies to customize the lesson to accommodate the requirements of your students throughout the academic year.<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 that are appropriate for the subject. Encourage students to construct and explain real-world contexts for the many graphs and equations used in the challenges.<br><br><strong><u>Partner and Small-Group Activities (Visual):</u></strong><br><br>Pair up the students. Give each pair a poster-sized piece of graph paper with the x and y axes printed and two equations written at the top. In addition, give students sticky dots (five in two different colors and one in a third color), two pieces of string, and tape.</p><figure class="image"><img style="aspect-ratio:92/73;" src="https://storage.googleapis.com/worksheetzone/images/Screenshot_1.png" width="92" height="73"></figure><p>1. Students A and B plot five points for the first and second equations, respectively.<br>2. Students should position the third colored dot on the intersection point if there are two sticky dots on the same point.<br>3. Students should tape a string to the graph to depict the lines.<br>4. If students plot points that do not overlap, they must determine the coordinates of the location where the two strings intersect.<br>5. Students substitute the <i>x</i>- and <i>y</i>-values of the shared point to demonstrate that point satisfies both equations and thus represents the genuine intersection point of the graphs.</p><p><strong>Extension:</strong></p><p>1. Give students&nbsp;four equations to graph, solve, and check. The graphs for these equations form a quadrilateral. Students are to identify the quadrilateral using its most exact name.&nbsp;<br><br>Equation 1: <i>y</i> = \(1 \over 2\)<i>x</i> – 2&nbsp;<br><br>Equation 2: <i>y</i> = \(1 \over 2\)<i>x</i> – 5&nbsp;<br><br>Equation 3: <i>y</i> = -\(1 \over 4\)<i>x</i> – 2&nbsp;<br><br>Equation 4: <i>y</i> = -\(1 \over 4\)<i>x</i> – \(1 \over 2\)&nbsp;<br><br>Solution: parallelogram;&nbsp;vertices (0, -2), (2, -1), (4, -3),&nbsp;(6, -2).&nbsp;<br><br>2. On a property map of a state forest preserve, the <i>y</i>-axis goes north-south, while the <i>x</i>-axis runs east-west. A pipeline from northwest to southwest intersects the west boundary (<i>y</i>-axis) at (0, 6), with a slope of -\(2 \over 5\)<i>x</i>. At which ordered pair does the pipeline intersect the south boundary (<i>x</i>-axis)?&nbsp;<br><br>Solution: (15, 0)<br><br><strong>Technology Connection: </strong>Have students use graphing calculators or software to investigate systems of equations with nonintegral solutions.<br><br><strong>O:</strong> To keep students interested in exploring equation systems, this lesson uses a competitive context. Students go over how to graph linear functions. The emphasis is on interpreting the significance of the sites of intersection as a solution. Students learn how to connect mathematical systems to real-world issues</p>

<p><strong>W:</strong> After this lesson, students will learn that the solution to a system of equations is the place where the graphs of two lines intersect. Due to the prevalence of linear equations that represent real-world scenarios and involve different rates of change, students are developing the ability to solve a system of equations. Plotting the lines of the two equations will allow students to solve a system of equations. They will be able to double-check their work by substituting the intersection point's coordinates into both equations to demonstrate that it satisfies both equations.<br><br><strong>H:</strong> <strong><u>Think-Pair-Share</u></strong><br><br>Use the scenario below to create a real-world scenario with a system of linear equations. <strong>"Consider a school recycling competition with your partner. Two classes are competing in a recycling competition, with one class collecting three-quarters of a pound of cans per day. The other class began with two pounds and gathered one quarter-pound each day after being informed about the competition in advance. Determine when the first class will pass the second class in terms of total pounds collected."</strong> Students may use a variety of strategies, most will likely use a guess-and-check method. Ask pairs to present their responses (<i>4 days is the correct answer</i>). Allow students to share the various methods they used to tackle the challenge.<br><br><strong>“Could we write equations that model the class can-collection scenarios?”</strong><br><br>Review the process of formulating a linear equation.&nbsp;<br>Define the variables: <i>x</i> represents the number of days, and <i>y</i> represents the weight of the cans in pounds.</p><figure class="image"><img style="aspect-ratio:543/508;" src="https://storage.googleapis.com/worksheetzone/images/Screenshot_2.png" width="543" height="508"></figure><p><br><strong>Step for Graphing a Line:</strong><br>1. To graph a line, first rewrite the equation into slope-intercept form.&nbsp;<br><br>2. Plot the <i>y</i>-intercept at (0, b).&nbsp;<br><br>3. Draw two or three additional points by counting the rise and run from the <i>y</i>-intercept.&nbsp;<br>a. If the slope is positive, count up for the rise and right for the run (also down and left).&nbsp;<br><br>For example, <i>y</i> = \(2 \over 3\)<i>x</i> + 1 &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; Count up 2 and to the right 3.&nbsp;<br><br>b. If the slope is negative, count down for the rise and right for the run (also up and left).&nbsp;<br><br>For example, <i>y</i> = -\(2 \over 3\)<i>x</i> + 1 &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;Count down 2 and to the right 3.&nbsp;<br><br>4. Draw a line through the points and add arrows to the ends. Extend the lines to cover the whole grid (rather than just connecting two points).&nbsp;<br><br><br>Create small groups by putting students in pairs. Distribute graph paper and markers to each group. Have each group agree on two equations that they identified during the preceding activity. On the same grid, request that they plot both lines. (Have any pairs of students who used this method to solve the previous equation assisted other students who were experiencing difficulty with the equation or the graphing by walking around.)&nbsp;<br><br>After students have completed their graphs, present the two graphed equations on a transparency overhead, poster-sized graph paper, or smart board.<br><br><strong>"What do you notice about the two graphs?"</strong> The students are expected to respond that the two graphs intersect at the same place as the answer.<br><br><strong>"What does the intersection point (4, 3) mean?"</strong> Students should be able to explain how the coordinate pair symbolizes the fact that both classes will get three pounds of cans in four days. It also signifies a critical point of change in the data, where the team that was previously behind has caught up and will continue to surpass the previous leader.<br><br><strong>"</strong><i><strong>A system of equations</strong></i><strong> is defined as having more than one equation on a grid. The </strong><i><strong>solution</strong></i><strong> to the system is where the lines intersect."</strong><br><br><strong>“A system of equations is a set or collection of equations with the same variables. Systems may have braces to indicate them: ”</strong></p><figure class="image"><img style="aspect-ratio:226/68;" src="https://storage.googleapis.com/worksheetzone/images/Screenshot_3.png" width="226" height="68"></figure><p><strong>The point where the two lines intersect is a representation of the x and y values that satisfy both equations and are known as the </strong><i><strong>solution to the system</strong></i><strong>."</strong><br><br><strong><u>Activity 1 (Kinesthetic): The Human Graph</u></strong>&nbsp;<br><br>Gather students in a spacious area (e.g., foyer, cafeteria, gym, large classroom, outdoor area) and divide them into groups. The optimal group size is ten students. Eight students will serve as human coordinates (four for each equation), while the remaining two will use paper to sketch the graphs to document the group's achievements. Each group needs two long ropes and two long strings. Tape markers on the long cables should be placed at one-foot intervals. Set the ropes for the x- and y-axes on the ground.<br><br>1. Provide the groups with two equations. Example: <i>y</i> = –4<i>x</i> – 2 and <i>y</i> = –3<i>x</i> – 1.<br><br>2. Four students represent four points on the graph of the first equation, and four students&nbsp;represent four points on the graph of the second equation. Instruct them to utilize identical <i>x</i>-coordinates.<br><br>3. Each group of four will use the string to connect themselves to create two straight lines when they are "graphed."<br><br>4. Recorders will plot the graph of the equation for their respective groups. One student constructs a graph of the line <i>y</i> = –4<i>x</i> – 2, while the other student constructs a graph of <i>y</i> = –3<i>x</i> –1 on the same sheet of paper using the same coordinate grid.<br><br>5. Ask the group to write down their observations of the two lines. Students representing the line stay in their graph formation while the two student recorders who created the graphs write down the responses of their group members to the following questions:<br><strong>"Are all the lines moving in the same way?"</strong><br><strong>"What are the slope and </strong><i><strong>y</strong></i><strong>-intercept of each equation?"</strong><br><strong>"Are there two students who are standing in the same place?"</strong><br><strong>"If so, what do you think this might indicate?"</strong><br><strong>"What coordinates does that shared point have?"</strong><br><br>6. The group should switch roles and complete an additional system of equations if there is sufficient time, to establish a new recorder.<br><br>7. Discuss the significance of two students standing on the same point when they represent the same point on two different equations in the classroom after all of the groups have completed.<br><br>8. Instruct students to substitute the <i>x</i>-value of the shared point into both equations following the discussion and ask about the result.<br><strong>"What does that point represent in relation to the graphs of the equations?"</strong><br><br><strong><u>Activity 2 (Auditory/Visual)</u></strong><br><br>This is an individual activity, and each student should have access to graph paper and a ruler. You will encounter two equations. Example: <i>y</i> = 2<i>x</i> – 5 and <i>y</i> = -<i>x</i> + 7 Instruct students to plot both lines on the same coordinate grid and determine the point at which they intersect. Upon completion of the graph, instruct students to demonstrate their work algebraically by substituting the <i>x</i>- and <i>y</i>-values into both equations to verify that the coordinate pair satisfies both equations. Perform this exercise with a variety of linear systems. Emphasize to students that the intersection of the two lines is the location of the shared point's coordinates or the solution to the system.<br><br><strong><u>Activity 3: Parallel and Concurrent Lines</u></strong><br><br>Craft a large Post-It or chart that is display-size and includes a coordinate grid that extends at least 15 units in both horizontal and vertical directions from the origin. Construct three right triangles of varying colors that are similar in shape and have integer bases and altitudes that correspond to the units on the image grid.<br><br>Cut several shapes of right triangles out of construction paper. For instance, right triangles with bases of 10, 15, 20,... and altitudes of 4, 6, 8,... resemble those with bases of 5 units and altitudes of 2. Because the hypotenuses of all of these comparable right triangles have the same slope (when oriented the same way), each is parallel to the other. Also, the hypotenuses are parallel when the hypotenuse of any similar right triangle crosses the <i>y</i>-axis at (0, 9), with the height being parallel to the <i>y</i>-axis and the base being parallel to the <i>x</i>-axis.&nbsp;<br><br>1. Cut out three right triangles: one with a base of 10 units and a height of 4 units, a second with a base of 5 units and an altitude of 2 units, and a third with a base of 15 units and an altitude of 6 units.<br><br>2. Mark the grid with the ordered pairs (0, 5), (0, 9), and (10, 5). Position the initial right triangle on the grid with the vertex of its right angle at (0, 5) and the vertex of its smallest angle at (10, 5). Apply a thin layer of tape to the grid to secure the triangle in place. Emphasize to the students that the hypotenuse's slope is negative in its current orientation, which is from the upper left to the lower right.<br><br>3. Instruct one student to place the larger right triangle on the grid with the altitude parallel to the <i>y</i>-axis, the base parallel to the <i>x</i>-axis, and the vertex of its smallest angle pointing to the right. The student should move the triangle to other locations on the grid in the same orientation and emphasize the importance of the hypotenuse remaining parallel to the first triangle as long as the <i>x</i>- and <i>y</i>-axis orientations remain consistent.<br><br>4. Instruct another student to place the smaller right triangle on the grid such that its hypotenuse matches the hypotenuse of the first triangle. Show how, by moving the smaller triangle left and right with the two hypotenuses aligned, the lines representing the hypotenuses are parallel. Equations for lines with slope -\(2 \over 5\) and y-intercept (0, 9) include 5<i>y</i> + 2<i>x</i> = 45, \(2 {1 \over 2} \)<i>y</i> + <i>x</i> = \(22 {1 \over 2} \), and 10<i>y</i> + 4<i>x</i> = 90. All three of these equations are equivalent, and they all have the same graph of&nbsp;the line with slope -\(2 \over 5\) and <i>y</i>-intercept (0, 9). The typical form of this equation is <i>y</i> = -\(2 \over 5\)<i>x</i> + 9, which is generalized to <i>y = mx + b</i>, where <i>m</i> is the slope and <i>b</i> is the <i>y</i>-intercept.&nbsp;<br><br><strong>R:</strong> Activity 4<br>You can also use Activity 2 to graph lines that meet at a point using rational, nonintegral coordinates. Tell them that we are substituting the intersection point coordinate back into the equation to ensure that our estimates are accurate.&nbsp;</p><figure class="image"><img style="aspect-ratio:88/52;" src="https://storage.googleapis.com/worksheetzone/images/Screenshot_4.png" width="88" height="52"></figure><p>1. In Activity 2, students generate graphs for the above&nbsp;system.&nbsp;<br><br>2. Students record their estimated point&nbsp;of intersection next to the graph.&nbsp;<br><br>3. At the bottom of their papers, students substitute the <i>x</i>- and <i>y</i>-values into both equations to determine whether their guesses were correct.<br><br><strong>E:</strong> Exit tickets (M-A1-5-1_Exit Ticket) can quickly assess student comprehension of the concepts. Return to the recycling competition discussion that was initiated at the outset of the class. Set up an new scenario: <strong>"Our class will participate in a fundraising event that involves a walking competition with the principal around the school's track. We will provide the principal with a two-mile lead, and she will walk at a pace of three-quarters of a mile per hour. Each hour, the student who represents our class walks at a pace of 2.5 miles."</strong> On a graph paper, have students do the following:&nbsp;Formulate the two equations for the scenario that has been provided.<br><br>1. Write the two equations for the given scenario.<br><br>2. Graph both lines, indicating the axes, the <i>y</i>-intercepts, and the site of intersection.<br><br>3. Indicate that the intersection point satisfies both equations by substituting the <i>x</i>- and <i>y</i>-values into the equations.<br><br>4. Explain the significance of the intersection point to the walking competition.</p>