Mathematics Success Grade 8 T429 [OBJECTIVE] The student will solve systems of equations by graphing. [PREREQUISITE SKILLS] solving equations [MATERIALS] Student pages S207 S220 Rulers [ESSENTIAL QUESTIONS] 1. Explain the three possible outcomes when solving systems of linear equations. 2. Describe the graph of a system of linear equations when there is only one solution. 3. How do the graphs differ when comparing a system of equations with no solution and a system of equations with infinite solutions? [WORDS FOR WORD WALL] system of linear equations, graphs, solution, intersection, no solution, parallel lines, one solution, infinite solutions, y-intercept, slope, slope-intercept form [GROUPING] Cooperative Pairs (CP), Whole Group (WG), Individual (I) *For Cooperative Pairs (CP) activities, assign the roles of Partner A or Partner B to students. This allows each student to be responsible for designated tasks within the lesson. [LEVELS OF TEACHER SUPPORT] Modeling (M), Guided Practice (GP), Independent Practice (IP) [MULTIPLE REPRESENTATIONS] SOLVE, Verbal Description, Pictorial Representation, Concrete Representation, Graphic Organizer, Graph, Algebraic Formula [WARM-UP] (IP, WG) S207 (Answers on T438.) Have students turn to S207 in their books to begin the Warm-Up. Students will solve equations. Monitor students to see if any of them need help during the Warm- Up. After students have completed the Warm-Up, review the solutions as a group. {Graphic Organizer, Algebraic Formula} [HOMEWORK] Take time to go over the homework from the previous night. [LESSON] [1 2 Days (1 day = 80 minutes) M, GP, WG, CP, IP]
T430 Mathematics Success Grade 8 SOLVE Problem (WG, GP) S208 (Answers on T439.) Have students turn to S208 in their books. The first problem is a SOLVE problem. You are only going to complete the S step with students at this point. Tell students that during the lesson they will learn how to solve systems of linear equations by graphing. They will use this knowledge to complete this SOLVE problem at the end of the lesson. {SOLVE, Verbal Description, Graphic Organizer} Solving Systems by Graphing One Solution (M, GP, IP, CP, WG) S208, S209, S210 (Answers on T439, T440, and T441.) M, GP, CP, WG: Students will work together to graph equations and find the solution to the equations. Be sure to assign the roles of Partner A and Partner B. {Concrete Representation, Graph, Verbal Description, Graphic Organizer} MODELING Solving Systems by Graphing One Solution Step 1: Direct students attention to the bottom of S208. Partner A, what are the two equations that are given for Question 1? (y = - x + 5 and y = 2x 4) Together, let s graph these two lines on the coordinate plane to the right. Partner B, what form are the lines given in? (slope-intercept form) Partner A, explain how we graph a linear equation when it s given in slopeintercept form. (We graph the y-intercept and then apply the slope.) Partner B, what is the y-intercept of the first equation? (5) Record the y-intercept. Have students plot the y-intercept at (0, 5). Step 2: Partner B, what is the slope of the first equation? ( - 1 or - 1 1 ) Partner A, what should we do after we plot the y-intercept? (Apply the slope by moving down 1 unit and to the right 1 unit to plot more points.) Have students continue with this pattern to plot two or three points, then draw the line through the points. Have students label the line with the equation. Partner B, what is the y-intercept of the second equation? ( - 4) Have students plot the y-intercept at (0, - 4) on the same coordinate plane to the right. Partner A, what is the slope of the second equation? (2 or 2 1 ) Partner B, what should we do after we plot the y-intercept? Justify your thinking. (Apply the slope by moving up 2 units and to the right 1 unit to plot more points.) Have students continue with this pattern to plot two or three points, then draw the line through the points. Have students label the line with the equation.
Mathematics Success Grade 8 T431 Step 3: Direct students attention to the top of S209. Partner A, when we are solving an equation, what does it mean to find a solution? (The solution is the value that when substituted back into the equation will make the statement true.) Using the equations we began with, let s substitute some values on the lines into the equations to see if they are solutions. Partner B, for y = - x + 5, is (0, 5) a solution? (Yes) Explain your thinking. (If we substitute 0 into the equation for x and substitute 5 into the equation for y, the left side of the equation simplifies to 5 and the right side of the equation also simplifies to 5. Therefore, a true statement is created and the point is a solution.) (5) = - (0) + 5 5 = 0 + 5 5 = 5 YES Partner A, for y = 2x 4, is (0, 5) a solution? (No) Explain your thinking. (If we substitute 0 into the equation for x and substitute 5 into the equation for y, the left side of the equation simplifies to 5 and the right side of the equation simplifies to - 4. Therefore, a false statement is created and the point is not a solution.) (5) = 2(0) 4 5 = 0 4 5-4 NO Step 4: Direct students to Question 3. Is (0, 5) a solution for both equations? (No, it is only a solution to the first equation.) With a partner, substitute the other two points provided in the table into the equations to see if they are solutions to either or both of the equations. Give students a few minutes to accomplish this. Suggest that they each take a column and then share their answers, discussing their solutions. Partner B, is (2,0) a solution for both equations? (No, it is only a solution to the second equation.) Partner A, is (3, 2) a solution for both equations? (Yes, it creates true statements for both equations.) Step 5: Direct students attention to Question 6. Look back at the graph. What do you notice about these three points in regard to the lines we plotted? Record your observations in the table. Partner B, what do you notice about (0, 5)? (This point lies on the first line, but not the second.) Record in the chart.
T432 Mathematics Success Grade 8 Partner B, what did we find out about (0, 5) when we substituted it into the equations? (It was a solution for the first equation, but not the second.) Partner A, what do you notice about (2, 0)? (This point lies on the second line, but not the first.) Record in the chart. Partner A, what did we find out about (2, 0) when we substituted it into the equations? (It was a solution for the second equation, but not the first.) Partner B, what do you notice about (3, 2)? (This point is found on both lines because it is the intersection point of both lines.) Record in the chart. Partner A, what did we find out about (3, 2) when we substituted it into the equations? (It was a solution for both equations.) Step 6: What conclusions can be drawn from this experiment? Justify your thinking. [A solution, in terms of a graph, is a point that will lie on the graph of a line. (0, 5) was a solution to the first equation because the point fell on its line, just as (2, 0) was a solution to the second equation. If a point is going to be a solution to more than one equation, it must fall on both lines, in which case there will be only one solution.] Therefore, what is the intersection point of two lines? (the solution to a system of linear equations) Have students return to the graph on S208. Looking back at the graph, plot a point where the two lines intersect on the coordinate plane. (3, 2) Plot the point. What is the solution to this system of equations? (3, 2) Record at the bottom of the box for Question 1 on S208. IP, CP, WG: Have students complete Questions 1 3 on S210. Have students work together to plot the two lines on the coordinate plane provided. Then have students identify the point of intersection on the graphs so that the solution of the system of equations is found. Review the solutions as a whole group. {Concrete Representation, Graph, Verbal Description, Graphic Organizer} Solving Systems by Graphing No Solution (M, GP, CP, WG) S211 (Answers on T442.) M, GP, CP, WG: Have students turn to S211 in their books. Students will complete the same type of activity as S210, but they will be introduced to a new scenario. Students will see a situation where they will not have solutions because the lines that are graphed will be parallel. {Concrete Representation, Verbal Description, Graph, Graphic Organizer}
Mathematics Success Grade 8 T433 MODELING Solving Systems by Graphing No Solution Step 1: Direct students attention to the top of S211. Have students take a look at Question 1. Partner A, is the first equation in slope-intercept form? (No) Partner B, what can we do to put it into slope-intercept form, for easy graphing? Explain your thinking. (We can subtract x from each side of the equation so that y is isolated.) Have students subtract x from each side of the equation. Partner A, what is the equation in slope-intercept form? (y = - x + 6) Partner B, is the second equation in slope-intercept form? (No) Partner A, what can we do to put it into slope-intercept form? Explain your thinking. (We can subtract x from each side of the equation, so that y is isolated.) Have students subtract x from each side of the equation. Partner B, what is the equation in slope-intercept form? (y = - x 1) Have partners look at Question 2. Partner A, is the first equation for Question 2 in slope-intercept form? (Yes) Partner B, is the second equation in slope-intercept form? (No) Partner B, what can we do to change it to slope-intercept form? (We can add 3x to each side of the equation so that y is isolated.) Have students add 3x to each side. Partner A, what is the equation in slope-intercept form? (y = 3x 1) Step 2: Now that we have changed all of our equations to slope-intercept form for easy graphing, Partner A, graph the lines for Problem 1, while Partner B graphs the lines for Problem 2. When finished, have students discuss the questions at the bottom of the page. Give students a moment to graph the lines of the equations for Problems 1 and 2. Allow partners to share their work and discuss what they are finding with their graphs. Partner A, what does the graph of the equations look like for Problem 1? (parallel lines) Partner B, what does the graph of the equations look like for Problem 2? (parallel lines) Partner A, what do you notice about the slope of the line? Discuss this with your partner. (The slope in each problem is the same value.) Partner B, what do you notice about the y-intercept of the lines? (The y-intercept of each line is different.) What conclusion can you make from these equations and graphs? Justify your thinking. (If the slopes of the lines are the same and the y-intercepts are different, the lines are parallel. Parallel lines never intersect; therefore there will be no solution to this system.)
T434 Mathematics Success Grade 8 Solving Systems by Graphing Infinite Solutions (M, GP, CP, WG) S212 (Answers on T443.) M, GP, CP, WG: Have students turn to S212 in their books. Students will continue graphing equations and identifying solutions. In the case of the problems on S212, students will be working with equations that have infinite solutions because they can be manipulated to be the same line in slope-intercept form. {Concrete Representation, Verbal Description, Graph, Graphic Organizer} MODELING Solving Systems by Graphing Infinite Solutions Step 1: Direct students attention to the first equation, 2x + 2y = 2. Partner B, is the first equation in slope-intercept form? (No) Partner A, what can we do to put it into slope-intercept form, for easy graphing? Explain your thinking. (We can divide each term in the equation by 2, to begin isolating y.) Have students divide each term, on both sides of the equation, by 2. Partner B, what is the simplified form of the equation? (x + y = 1) Partner A, is the equation in slope-intercept form now? (No) Partner B, what can we do to put it into slope-intercept form? (We can subtract x from each side of the equation so that y is isolated.) Have students subtract x from each side of the equation. Partner A, what is the equation in slope-intercept form? (y = - x + 1) Partner B, is the second equation in slope-intercept form? (Yes) Have students take a look at the two equations for Question 2. Partner A, are the equations in slope-intercept form? (No) Ask students to take a moment to work through these two equations and find the slope-intercept form of each. (For the first equation, students can divide each term by 2 and then subtract 2x from each side. For the second equation, they only need to subtract 2x from each side.) Step 2: Now that we have changed all of our equations to slope-intercept form for easy graphing, Partner B, graph the lines for Problem 1, while Partner A graphs the lines for Problem 2. When finished, have students discuss the questions on the bottom of page S212. Give students a moment to graph the lines of the equations for Problems 1 and 2. Allow partners to share their work and discuss what they are finding with their graphs.
Mathematics Success Grade 8 T435 Partner B, what does the graph of the equations look like for Problem 1? (identical lines) Partner A, what does the graph of the equations look like for Problem 2? (identical lines) Partner B, what do you notice about the slope of the line? Discuss this with your partner. (The slope in each problem is the same value.) Partner A, what do you notice about the y-intercept of the lines? (The y-intercept in each problem is the same value.) What conclusion can you make from these equations and graphs? (If the slope of each line is the same and the y-intercepts are the same, the lines are identical. Identical lines intersect at every point on the line, with every intersection being a solution. Therefore, identical lines have infinite solutions.) Summarize Results Analyze Equations (M, GP, IP, CP, WG) S213 (Answers on T444.) M, GP, CP, WG: Have students turn to S213 in their books. Now that students have seen the three different types of results for solving systems of linear equations, they will create a chart to summarize. After summarizing, students will analyze equations and decide the result without ever graphing them. Be sure students know their designation as Partner A or Partner B, {Verbal Description,Table, Graphic Organizer, Algebraic Formula} MODELING Summarize Results Analyze Equations Step 1: Direct students attention to the top of S213. What three outcomes have we found when solving systems of linear equations? (We can have one solution, no solutions, or infinite solutions.) Have students list them in the far left column of the table. Partner A, is the slope the same or different when the equations only have one solution? (The slopes are different.) Record in the chart. Partner B, are the y-intercepts the same or different when the equations only have one solution? Explain your thinking. (The y-intercepts can be the same or different. As long as the slopes are different, we can be sure there is only one solution.) Have students record N/A or any notation that the y-intercept has no effect on the outcome if the slopes are different.
T436 Mathematics Success Grade 8 Partner A, how would you describe the lines of the equations that have only one solution? (intersecting) Partner B, is the slope the same or different when the equations have no solutions? (The slopes are the same.) Record in the chart. Partner A, are the y-intercepts the same or different when the equations have no solutions? (The y-intercepts are different.) Partner B, how would you describe the lines of the equations that have no solutions? (parallel) Partner A, is the slope the same or different when the equations have infinite solutions? (The slopes are the same.) Record in the chart. Partner B, are the y-intercepts the same or different when the equations have infinite solutions? (The y-intercepts are the same.) Partner A, how would you describe the lines of the equations that have infinite solutions? Explain your thinking. (identical) Step 2: Direct students to Question 1 on S213. Identify if the following systems of equations have one solution, no solutions, or infinite solutions. Partner B, is the first equation in slope-intercept form? (No) Partner A, how can we manipulate the equation so that we can change it to slope-intercept form to quickly identify the slope and y-intercept? Justify your thinking. (We can subtract 4x from each side of the equation.) Partner B, what is the equation written in slope-intercept form? (y = - 4x + 9) Partner A, what is the slope of the first equation? ( - 4) Partner B, what is the y-intercept of the first equation? (9) Partner A, is the second equation in slope-intercept form. (Yes) Partner B, what is the slope of the second equation? ( - 1) Partner A, what is the y-intercept of the second equation? ( - 7) Looking at the chart, what do you think the outcome of the system of equations will be? (one solution) Explain your thinking. (If we start with the slope, the only scenario where the slopes are different is the result where there is one solution.) IP, CP, WG: Have students solve Questions 2 4. Encourage students to change all equations to slope-intercept form and then use the table to identify the result. Also explain to students that they want to begin with the slope when comparing, then move to the y-intercept, if necessary. Take a moment to review student solutions as a group. {Verbal Description, Table, Graphic Organizer, Algebraic Formula}