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Julia Jung KimBook Title: Big Ideas Math Chapter 5: Systems of Linear EquationsTitle of Lesson: 5.3 Solving Systems of Linear Equations by EliminationLearning Goals/Objectives?In groups, students will discover how and when they can add or subtract the equations in a system to eliminate a variable.?Students will be able to multiply, if necessary, one or both equations by a constant so at least 1 pair of like terms has the same or opposite coefficients.?Students will be able to write and solve systems of linear equations by elimination. Common Core State Standards for Mathematics (CCSSM)8.EE.8b: Solve systems of two linear equations in two variables algebraically, and estimate solutions by graphing the equations. Solve simple cases by inspection.8.EE.8c: Solve real-world and mathematical problems leading to two linear equations in two variables. Grade Level8th gradeDuration3 days (60 minutes each)Class InformationThree Regular Math classes, including students with IEPs and 504 plansCommon Core Mathematical Practice FocusMP1, MP1a, MP3a, MP7Prior Knowledge?Students should know how to solve linear equations and evaluate expressions at a specified value of the variable. Materials?For teacher: Smart Board, PowerPoint, Blackboard, Dynamic classroom?For students: Calculator, Straightedge, Scissors, Construction Paper, MarkersAccommodations Provide students with every material they need for the lesson such as calculators, straightedges, scissors, and construction paper. Smart Board and PowerPoint will be used at the same time due to the fact that I can demonstrate and point out key points to prevent any students from being left behind. This lesson focuses on a variety of levels of students, especially students with IEPs and ELL’s. All activities and exercises encourage students’ critical thinking skills, and they are designed for step-by-step procedures, detailed explanations, and hands-on learning. Additionally, I aim to apply whole group, small group, and individual work into exercises and activities. Hence, all different types of learners can participate in the lesson without any hindrance or obstacle.AssessmentFormative: Observation, creating a model, class discussions, using repeated reasoning, exit ticket, and class work/homework assignmentsSummative: Timed, in-class quiz (after complete Section 5.3 and review) and a chapter 5 test at the end of three sectionsLesson ProcedureReal-Life Application Project (25 minutes) –Posters, Markers, Straight Edges, Grid Papers, Laptops ?Students will complete the project from the previous section (5.2 substitution).?Assign students twenty minutes to complete and collect the project at once (Students are permitted to come during or after school for a help period to finish it for full credit).?Let students clear the desks, put the materials back, and be ready to start a lesson.Motivating Introduction (10 minutes) –Smart Board + PowerPoint Essential Question: How can you use elimination to solve a system of linear equations?Play the Opposites Game?With the group members, students will create a two column list of words that are opposites (example: hot and cold, fast and slow). I will give students two minutes to complete on a piece of paper.?At the end of the game, find a group that has the longest list and have them read their list of words. I will ask if any other groups want to share any new words from their lists.?Relate this activity back to our topic, solving systems of linear equations, ask students to write down the opposite of 5, -2.3, and 4x. (-5, 2.3, -4x)Exercise (20 minutes) –Smart Board + PowerPoint Using Elimination to Solve a SystemIntroduce how to solve a system of linear equations by using two methods.Add or subtract the equations to eliminate one of the variablesMethod 1: Add.Add the two equations.Method 2: Subtract.Subtract Equation 2 from Equation 1.?The teacher demonstrates the first question using two methods. Then, let students try questions 2 to 4 in their notebooks individually. 4x + y = 8 2) 3x – y = 4 3) x + 2y = 7 4) 2x + y = 164x – y = 0 3x + y = 2 x – 2y = -5 2x – y = 0 (1,4) (1,-1) (1,3) (4,8)?After students find the value of one variable, I will ask, “How can you find the value of the other variable?” ?Discuss the solutions as a whole class. Ask, “Is the solution the same using both methods?” (YES!)Exit Ticket (5 minutes) – Smart Board, A Piece of Paper for Individual Student?As closure, check if students have realized that one pair of like variable terms in a system must have the same or opposite coefficients before you can eliminate a variable by adding or subtracting the equations. I will provide an exit ticket question on the Smart Board: Solve the system by elimination. Find the solution using the addition or subtraction method.x + 3y = -2 x – 3y = 10 (4,-2)Lesson ProcedureMotivating Introduction (10 minutes) – Smart Board + PowerPointEssential Question: How can you use elimination to solve a system of linear equations??Yesterday students discovered how and when they can add the equations in a system to eliminate a variable. ?Start with the attention-getter question to remind them of previous lesson (using addition method).?In order to guide students to see the basic concept of the lesson, I will present the following equations: 2x + y = 17 x – y = 4?Demonstrate how to combine two equations in order to eliminate the y variable.?Give students 2 minutes to figure out the following system using the same method: 2x + y = 17x + y = 4?Then, introduce how to distribute a negative sign to the second equation to eliminate the y variable.?Make sure students can solve a system of linear equations by using two methods.Add or subtract the equations to eliminate one of the variables.Add the equations.Subtract the second equation from the first.?Explain how important it is for students to understand that the approach they use usually depends on the coefficients of the variables. ? Students are using the Distributive Property. Remind students that when distributing a negative sign, make sure that they distribute every term by the negative sign.Exercise (15 minutes) –Smart Board + PowerPoint ?Since students learned from the previous lesson how to solve a system of linear equations by using the addition method, we will focus on subtraction method.?Ask the question “Why do we have to use subtraction method?” and prove with the question 1. 1) 6x – 2y = 18 2) 7x + 2y = 24 3) 7x + 8y = 20 6x – 7y = 3 8x + 2y = 30 7x – y = 29 (4 , 3) (6 , -9 ) (4 , -1)?Demonstrate the question 1 how to solve the system by subtracting the second equation from the first.?First of all, write Equation 1 and Equation 2.Ask the question “What are the coefficients of the x-terms?”Ask the question “What are the coefficients of the y-terms?”(Remind students to line up like terms vertically so that you can add the terms in each column).?Show students question 1, how to distribute every term of the second equation by the negative sign. Then, combine two equations in order to find two variables, x and y. - (6x – 7y = 3) 6x – 2y = 18 y = 3 -6x + 7y = -3 -6x + 7y = -3 6x – 2(3) = 18 --------------------- x = 4 5y = 15 (4 , 3) ? Have students work the questions 2 to 3 independently and go over as a whole class. Ask volunteers to share their work at the Smart Board.Activity 1 (35 minutes) –Construction Paper, Scissors, Markers, Smart Board + PowerPoint Pick a linear equations card from a pocket!?Up to second day of the lesson, students have learned how to solve a system of linear equations using elimination by addition and subtraction. Starting with this activity, I will group students into 3 or 4 and each group will be provided a pink paper that has total 8 linear equations to solve and 2 different pockets that I have already created for them.?Each color of pocket represents a different method, and students will cut their pink papers and sort it into two piles: addition and subtraction. ?Inform students that each individual person needs to complete at least 3 cards from each pocket. Therefore, everyone has to solve at least 6 cards in total. At the end, I will tell them to pick one card from each pocket that they did not have chance to finish and assign that as homework. ? As closure, recommend to students that if they have not finished or have any questions, to come after school for help. Make sure students put the materials back where they belong.Lesson ProcedureMotivating Introduction (10 minutes) – Smart Board + PowerPointEssential Question: How can you use elimination to solve a system of linear equations??Start with a question that requires assumption and critical thinking. Have students write down the question and answer in their notebook: “What if this system had been written this way?” 5y ? 6x = 25?2x ? 4y = 14 (Students should recognize that the like terms are not lined up in the same columns, so one of the equations should be rewritten)?After students recognize that one of the equations should be rewritten, ask next question:“What do you notice about this system?”(None of the coefficients of like variable terms are the same or opposite)?Encourage the students to make a mathematical guess on the last question, which is related to what they are going to learn today: “Can you think of a way to rewrite the equations so that either the x-terms or the y-terms have coefficients that are the same or opposite?”(Multiply through one equation by a constant)Exercise 1 (15 minutes) –Smart Board + PowerPoint ?Start with the questions that students already learned in previous lessons. Give each student two minutes to work on these questions to fresh their memory. Addition/Subtraction: -5x + 2y = 13 x + 3y = -2 5x + y = -1 x – 3y = 16?Clarify once more: When the coefficients of one variable are opposites you add the equations to eliminate a variable; and when the coefficients of one variable are equal, you subtract the equations to eliminate a variable.?Introduce how to solve systems of equations using the elimination with multiplication method. From the previous lesson, students should have knowledge of how to multiply one of the equations with -1. This time, the students will learn multiply one, or both of the equations with a constant to obtain an equivalent linear system, in order to eliminate one of the variables by addition or subtraction.?Demonstrate: 3x – 5y = 9 6x – 6y = 6?Show the two options: 1. Multiplying the first equation by 2 and subtracting 2. Multiplying the first equation by -2 and adding?Prove how both approaches give the same result. (Explain that sometimes fewer computation errors occur when equations are added rather than subtracted.)?Remind students that when multiplying through by a constant, make sure that they multiply every term by the constant. Students are using the Distributive Property.Activity 1 (15 minutes) –Smart Board + PowerPoint ?I will ask two questions before students get into the activity:“Can you think of a way to rewrite the equations so that either the x-terms or the y-terms have coefficients that are the same or opposite?”“If you wanted to eliminate the x-term, what could you do?”-6x + 5y = 25-2x -4y = 14?Divide a class into half and assign the both groups different options to solve the same question.Group 1: Multiplying the second equation by -3 and addingGroup 2: Multiplying the second equation by 3 and subtracting?One person from each group will come up and explain their work on the board.?Both groups will switch the options and solve in different way. Therefore, everyone tried both ways to solve the question.?One person from each group will come up and explain their work on the board.?Make sure students realize that both approaches give them the same solution.Exercise 2/Exit Ticket (15 minutes) –Smart Board + PowerPoint?I will display 3 conditions on the Smart Board:Adding or subtracting equations will eliminate one of the variables.You need to multiply one of the equations by an integer before you add or subtract.You need to multiply both of the equations by a constant before you add or subtract.?Each group member picks a different condition and writes an example of a system for the condition relative to the solution by elimination.?Share and make sure each member of the group has correct example for each condition. Then, gather all the group members’ papers and staple them.?I will collect and grade them for the participation grade.Closure (5 minutes) – Ask students if they have any questions about the lesson.Make sure everyone understands the homework assignment for Chapter 5 Lesson 3 by next class: pg.221-3 #1-3, 9-21 odd, 22-32 even + be prepared for the 5.3 quiz on Friday. ................
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