Reasoning with Equations and Inequalities


 Martha Manning
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1 Instruction Goal: To provide opportunities for students to develop concepts and skills related to solving linear sstems of equations b graphing Common Core Standards Algebra: Solve sstems of equations. AREI.. Solve sstems of linear equations eactl and approimatel (e.g., with graphs), focusing on pairs of linear equations in two variables. Represent and solve equations and inequalities graphicall. AREI.11. Eplain wh the coordinates of the points where the graphs of the equations = f() and = g() intersect are the solutions of the equation f() = g(); find the solutions approimatel; e.g., using technolog to graph the functions, make tables of values, or find successive approimations. Include cases where f() and/or g() are linear, polnomial, rational, absolute value, eponential, and logarithmic functions. Student Activities Overview and Answer Ke Station 1 Students will be given graph paper and a ruler. Students will graph a sstem of linear equations. The will find the point of intersection of the lines and realize that this is the solution to the linear sstem. The will doublecheck the point of intersection b substituting it into the original equations Walch Education Algebra I Station Activities for Common Core Standards
2 Answers Instruction. Yes, (1, ) 3. The point of intersection is the solution of the sstem of linear equations.. Substitute (1, ) into both equations to verif that it satisfies both equations (3, 1) 1 Algebra I Station Activities for Common Core Standards 11 Walch Education
3 Station Instruction Students will be given four inde cards with the following sstems of linear equations written on them: 1 = 1 ; = = 3 = ; = + = 1 ; = + + = Students will work together to match each sstem of linear equations with the appropriate graph. The will eplain the strateg the used to match the graphs. Answers = = = + = 1 1 = 1 = = +. + =. Answers will var. Possible answer: Finding the point of intersection of the lines. This is the solution to the linear sstem. Substitute this point into the equations to see which sstems of linear equations have this solution. Station 3 Students will be given linear equations and will find points that satisf the equations. Then the will look at the equations as a linear sstem. The will find the solution, or no solution, of the linear sstem. The will describe the strateg the used to determine the solution of the linear sstem. Answers Walch Education Algebra I Station Activities for Common Core Standards
4 3. Yes, (1, 1) Instruction. The had the same ordered pair of (1, 1), which means the lines intersect at that point The table of values ma or ma not contain the solution You can use the slope of each line and intercepts, or ou can graph the equations and see if the intersect. 9. There is no solution. Each equation has the same slope, but different intercepts. Station Students will be given a graphing calculator. Students will be shown how to find the solution of a sstem of linear equations using the graphing calculator. Then the will use the graphing calculator to find the solutions of given sstems of linear equations. Answers 1. ( 1.7, 9.7). (, 1) 3. (1.9, 3.). (.73, 3.7) Materials List/Setup Station 1 graph paper; ruler Station four inde cards with the following sstems of linear equations written on them: 1 = 1 + = 3 ; ; = 7 + = Station 3 none Station graphing calculator = + = 1 ; = + + = 1 Algebra I Station Activities for Common Core Standards 11 Walch Education
5 Discussion Guide Instruction To support students in reflecting on the activities and to gather some formative information about student learning, use the following prompts to facilitate a class discussion to debrief the station activities. Prompts/Questions 1. How do ou solve sstems of linear equations b graphing?. How do ou know if a sstem of linear equations has infinite solutions? 3. How do ou know if a sstem of linear equations has no solutions?. How can ou use a graphing calculator to graph and solve sstems of linear equations?. Give an eample of a graph of a linear sstem that is used in the real world. Think, Pair, Share Have students jot down their own responses to questions, then discuss with a partner (who was not in their station group), and then discuss as a whole class. Suggested Appropriate Responses 1. Graph each line. Their point of intersection is the solution to the linear sstem.. Graph each line. If the lines are the same line then the have infinite solutions. Or, if the lines have the same slope and intercept, then there are infinite solutions. 3. Graph each line. If the lines are parallel or have no point of intersection, the sstem has no solutions.. Tpe the equations into Y1 and Y. Hit GRAPH, nd, Trace, and select : Intersect. Then hit the ENTER ke three times to find the point of intersection. The point of intersection is the solution to the linear sstem.. Answers will var. Possible answer: comparing service plans from two different cell phone companies Possible Misunderstandings/Mistakes Incorrectl graphing the equations b not finding the correct slope and intercepts Not realizing that equations with the same slope and intercept are the same line. If the are a sstem of linear equations then the sstem has infinite solutions. Not realizing that parallel lines have no solutions Not realizing the point of intersection of the two lines is the solution to the sstem of linear equations Walch Education Algebra I Station Activities for Common Core Standards
6 Station 1 At this station, ou will find graph paper and a ruler. You can find the solution to a sstem of linear equations b graphing each equation in the same coordinate plane. 1. As a group, graph each of the following linear equations on the same graph. = 3 + = +. Do the lines intersect? If so, find the point at which the two lines intersect. 3. What does this point of intersection tell ou about the two equations?. How can ou doublecheck our answer to problem 3?. As a group, graph the sstem of linear equations on a new graph. = = 7. Using this graph, what is the solution to the linear sstem? 1 Algebra I Station Activities for Common Core Standards 11 Walch Education
7 Station At this station, ou will find four inde cards with the following linear sstems of equations written on them: 1 = 1 ; = + = 3 ; 7 + = = + = 1 ; = + + = Work together to match each sstem of linear equations with the appropriate graph below. Write the appropriate sstem of linear equations under each graph Answer: Answer: continued 1 11 Walch Education Algebra I Station Activities for Common Core Standards
8 Answer: Answer:. What strateg did ou use to match the sstems of linear equations with the appropriate graph? 1 Algebra I Station Activities for Common Core Standards 11 Walch Education
9 Station 3 Use what ou know about linear equations to solve the following problems. 1. Complete the table below for the linear equation = Complete the table below for the linear equation 3 + = If = 3 + = were a set of linear equations, would it have a solution? If so, what would it be?. What strateg did ou use in problem 3 to find our answer?. Complete the table below for the linear equation = 1. continued Walch Education Algebra I Station Activities for Common Core Standards
10 . Complete the table below for the linear equation = + 1. = 1 7. If were a set of linear equations, wh would ou need more information than = + 1 just the values in the tables ou found for problems and to determine an solutions?. What information would ou need to determine whether or not a sstem of linear equations has a solution, infinite solutions, or no solutions? 9. Does = 1 = + 1 have a solution? Wh or wh not? 1 Algebra I Station Activities for Common Core Standards 11 Walch Education
11 Station At this station, ou will find a graphing calculator. You can graph sstems of linear equations on our calculator to find their point of intersection. Work together to graph sstems of linear equations on our calculator. Given: = 3 = Follow these steps on our graphing calculator to find the point of intersection for this sstem of equations: 1. Hit the Y = ke. Tpe Y1 = 3.. Tpe Y =. 3. Hit the GRAPH ke.. Hit nd and then TRACE. Select : Intersect.. Hit ENTER three times. You will be given the  and values of the intersection. 1. What is the intersection/solution of = 3 =? Using the same steps, find the solution to each sstem of linear equations below on our graphing calculator = = Solution = = + 9 = 3 Solution = + = = 1 Solution = Walch Education Algebra I Station Activities for Common Core Standards
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Instruction Goal: To provide opportunities for students to develop concepts and skills related to solving sstems of linear inequalities, including realworld problems through graphing two and three variables
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