Algebra 1 Unit 3: Systems Of Equations

Course Name: Math I Unit # 2 Unit Title: Systems of Equations & Inequalities Enduring understanding (Big Idea): Students will understand that systems of equations and inequalities can be used to model, analyze and solve real world situations. Essential Questions: 1. How do you choose the best method to solve a system of linear equations or inequalities? and Why? 2. How do you apply the solution to a system of equations or inequalities to a real BY THE END OF THIS UNIT: Students will be able to… A.CED.3: Students will know… I can represent constraints by equations or inequalities. There are no new formulas in this unit. I can represent constraints by systems of equations/inequalities. Vocabulary: I can interpret solutions as viable or nonviable Elimination by multiplication and addition, solutions graphically. substitution A.REI.5: Systems of equations I can understand and can prove that, given a consistent and inconsistent systems system of two equations in two variables, dependent and independent systems replacing one of the equations by the sum of solution set that equation and a multiple of the other objective equation, Constraint, greater than, >, produces a system with the same solutions. less than, <, greater than or equal to, ≥, less A.REI.6: than or equal to, ≤ , inequality, viable, I can solve systems of linear equations in 2 Function, intersection, approximate, linear, variables by graphing. exponential, f(x), g(x), half-plane, solution A.REI.11: region I can explain why the x-coordinate of the point Unit Resources of intersection of the graphs of the equations y1=f(x) and y2=g(x) is a solution to the Learning Task: systems formed by y1 and y2, where y1 and/or  Unit introduction: Rental Cars y2 can be linear, polynomial, rational, absolute Performance Task: value, exponential, and logarithmic.  Best Buy Tickets A.REI.12: Project: I can graph the solution of a linear inequality  A Day at the Food Court in two variables. Unit Review Game: I can graph the solution of a system of linear  Jeopardy: Systems inequalities in two variables. MathForward: Learncheck Question Bank: I can identify the solution of a linear inequality SYSTEMS in two variables. Released Test: 2,7,10,34,35,40 I can identify the solution of a system of linear Suggested Pacing: inequalities in two variables. CCSS-M Included: Mathematical Practices in Focus: 1. Make sense of problems and persevere A.REI.5, A.REI.6, A.REI.11, A.REI.12, in solving them A.CED.3 2. Reason abstractly and quantitatively 6. Attend to precision

Standards are listed in alphabetical /numerical order not suggested teaching order. PLC’s must order the standards to form a reasonable unit for instructional purposes. Course Name: Math I Unit # 2 Unit Title: Systems of Equations & Inequalities

CORE CONTENT Cluster Title: Create equations that describe numbers or relationships Standard: Standard A.CED.3: Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or non-viable options in a modeling context. For example, represent inequalities describing nutritional and const constraints on combinations of different foods. Concepts and Skills to Master:  Determine whether a point is a solution to an equation or inequality  Determine whether a solution has meaning in a real-world context  Write and use a system of equations and/or inequalities to solve a real-world problem  Recognize that the equations and inequalities represent the constraints of the problem. Use the Objective Equations and the Corner Principle to determine the solution to the problem. (Linear Programming) SUPPORTS FOR TEACHERS Critical Background Knowledge  Ability to read and write inequality symbols  Ability to graph equations and inequalities on the coordinate plane Academic Vocabulary Constraint, greater than, >, less than, <, greater than or equal to, ≥, less than or equal to, ≤ , inequality, viable Suggested Instructional Strategies: Resources: •Use a story context such as those found in linear programming problems to write and graph  Textbook Correlation: 6-5, 6-6, CC-7 equations with constraints.  MARS Concept Development Lessons: •Analyze real-world problems connected to student Defining Regions using Inequalities interest Solving Linear Equations in Two Variables DPI Unpacking:  MARS Problem Solving Lesson: When given a problem situation involving limits or Optimization problems restrictions, represent the situation symbolically using an equation or inequality. Interpret the solution(s) in the context of the problem. When given a real world situation involving multiple restrictions, develop a system of equations and/or inequalities that models the situation. In the case of linear programming, use the Objective Equation and the Corner Principle to determine the solution to the problem. Sample Assessment Tasks

Skill-based task Problem Task 1.Given y ≤ 2x + 1 and y > x – 3 find a point that: 1.Iced tea costs $1.50 a glass and lemonade a. Satisfies both costs $2.00. If you have $12, what can you buy? b. Satisfies one, but not the other Justify your answer using multiple representation. c. Satisfies neither 2.Imagine that you are a production manager at 2.All American Bats, produces two different quality wooden a calculator company. Your company makes two baseball bats, the Aaron Bat and the DiMaggio Bat. The types of calculators, a scientific calculator and a Aaron Bat takes 8 hours to trim and 2 hours to finish it. It graphing calculator. has a profit of $16. The DiMaggio Bat takes 5 hours to trim a. Each model uses the same plastic case and the same circuits. However, the graphing calculator and 5 hours to finish it, but it has a profit of $27. The total requires time available per day for trimming is 80 hours and 50 20 circuits and the scientific calculator requires hours for finishing. What is the maximum profit the All only 10. The company has 240 plastic cases and American Bats can make each day? 3200 circuits in stock. Graph the system of inequalities that represents these constraints. 3.Anthonie sells meringue cookies and butter cookies. Baking a batch of meringue cookies takes 3 egg whites and b. The profit on a scientific calculator is $8.00, while the profit on a graphing calculator is 1 cup of sugar. Baking a batch of butter cookies takes 1 egg $16.00. Write an white and 2 cups of sugar. Anthonie has 7 egg whites and 9 equation that describes the company’s profit cups of sugar. He makes $2 profit per batch of meringue from calculator sales. cookies and $1 profit per batch of butter cookies. How many c. How many of each type of calculator should batches of butter cookies should Anthonie make to the company produce to maximize profit using the stock on hand? maximize his profit? 4.Gigi sells whole wheat bread and banana bread. Her recipe for whole wheat bread calls for 4 cups of milk and 3 cups of wheat flour. Her recipe for banana bread calls for 2 cups of milk and 3 cups of wheat flour. Gigi has 16 cups of milk and 15 cups of wheat flour. She makes $3.50 profit on her wheat bread and $1.55 profit on her banana bread. What is the maximum profit Gigi can make? CORE CONTENT Cluster Title: Solve Systems of Equations Standard A.REI.5: Prove that, given a system of two equations in two variables, replacing one equation by the sum of that equation and a multiple of the other produces a system with the same solutions. Concepts and Skills to Master:  Explain the use of the multiplication property of equality to solve a system of equations.  Explain why the sum of two equations is justifiable in the solving of a system of equations (property of equality)  Relate the process of linear combinations with the process of substitution for solving a system of linear equations. SUPPORTS FOR TEACHERS Critical Background Knowledge  Graph a line  Solve systems of equations

Academic Vocabulary Elimination by multiplication and addition, substitutio Suggested Instructional Strategies: Resources:  Modeling the property of equations with a balance scale to make the point you  Textbook Correlation: 6-3 can add the same quantity to both sides of an equation  khanacademy.org – instructional video: i.e. given 6x – 2y = 22 and -6x + 5y = Solving Systems of Equations by -37, we’re given that -6x + 5y is equal to Elimination -37 and the property of equality tells us that we can add the same quantity to both sides, we could add -37 to both sides of the equal sign in the first equation or we could add -6x + 5y to both sides of the first equation or we could add -6x + 5y to the left side of the first equation and -37 to the right side of the first equation.

DPI Unpacking: Solve systems of equations using the elimination method (sometimes called linear combinations). Sample Assessment Tasks Skill-based task Problem Task Skill-based task Given the system shown below, create a new a. Verify that (-4, 13) is the solution to the system using both the addition and system. multiplication properties of equality. Then verify 2x  y  5 that the new system has the same solution as  the original.  5x  2y  6 2x  y  5 b. Justify that the following is an equivalent  system  5x  2y  6  3x  y  1   5x  2y  6 Verify that (-4, 13) is the solution to this new system.

CORE CONTENT Cluster Title: Solve Systems of Equations Standard A.REI.6: Solve systems of linear equations exactly and approximately (e.g. with

graphs), focusing on pairs of linear equations in two variables. Concepts and Skills to Master:  Solve a system of equations exactly (with algebra) and approximately (with graphs)  Test a solution to the system in both original equations (both graphically and algebraically)  Analyze a system of equations using slope to predict one, infinitely many or no solutions SUPPORTS FOR TEACHERS Critical Background Knowledge  Graph a line  Find the slope of a line  Solve systems of equations Academic Vocabulary Systems of equations, consistent and inconsistent systems, dependent and independent systems, solution set Suggested Instructional Strategies: Resources:  Solve contextual problems using systems of  Textbook Correlation: 6-1, 6-2, 6-3, equations 6-4, CC-7  Have students generate scenarios which  MARS lesson: Solving Linear Equations in might yield one, many, or no solutions Two Variables(students choose a method  Have students create systems of equations of solving) to model various contextual situations such  MathForward activities: as cell phone costs Use graphing calculators to estimate All on the Line solutions to systems DVD Purchases NCDPI Unpacking: How many solutions Solve systems of equations using graphs The Dirty Dawg Salon Understanding Solutions of Systems  TI84 App: Alg1Chap5 (the app can be downloaded from HERE Sample Assessment Tasks Skill-based task Problem Task

1.Approximate the solution to the system of 1.The high school is putting on the musical equations graphically. Verify the solution Footloose. The auditorium has 300 seats. algebraically. Student tickets are $3 and adult tickets are $5. 3x – 5y = -15 The royalty for the musical is $1300. What 2x – y = 2 combination of student and adult tickets do you need to fill the house and pay the royalty? How 2.Solve the system algebraically using either could you change the price of tickets so more elimination or substitution. Explain why you students can go? 3y  4x  1 chose the method you used.  4x  2y 10 2. The graphs of lines m and n are parallel and have different y-intercepts. 3.The equations y = 18 + .4m and y = 11.2 + . a. How many solutions exist for a system 54m Give the lengths or two different springs of equations containing lines m and n? Explain in centimeters, as mass is added in grams, m, your answer. to each separately. b. A new line, p, is added to the graph. Lines p and n have infinitely many solutions.

a. Graph each equation. What, if anything does this tell you about lines p b. When are the springs the same length? and n? c. When is one spring at least 10cm longer than the other? d. Write a sentence comparing the two springs CORE CONTENT Cluster Title: Represent and solve equations and inequalities graphically Standard: A.REI.11: Explain why the x-coordinates of the points where the graphs of the equations y = f(x) and y= g(x) intersect are the solutions of the equation f(x) = g(x); find the solutions approximately, e.g., using technology to graph the functions, make tables or values, or find successive approximations. Include cases where f(x) and/or g(x) are linear, polynomial, rational, absolute value, or exponential, and logarithmic functions. Concepts and Skills to Master:  Approximate solutions to systems of two equations using graphing technology  Approximate solutions to systems of two equations using tables of values  Explain why the x-coordinates of the points where the graphs of the equations y = f(x) and y = g(x) intersect are the solutions of the equation f(x) = g(x)  Be able to express that when f(x) = g(x), the two equations have the same solution(s). SUPPORTS FOR TEACHERS Critical Background Knowledge  Evaluate expressions  Construct a table of values for a given function  Graph functions using graphing technology Academic Vocabulary Function, intersection, approximate, linear, exponential, f(x), g(x) Suggested Instructional Strategies: Resources:  Use graphing technology to approximate the point(s) of intersection of two graphs  Textbook Correlation: CB4-4, Make comparisons between tables of values 9-8, CC-18 NCDPI Unpacking: Construct an argument to demonstrate understanding that the solution to every equation can be found by treating each side of the equation as separate functions that are set equal to each other, f(x) = g(x). Allow y1=f (x) and y2= g(x) and find their intersection(s). The x-coordinate of the point of intersection is the value at which these two functions are equivalent, therefore the solution(s) to the original equation. Students should understand that this can be treated as a system of equations and should also include the use of technology to justify their argument using graphs, tables of values, or successive approximations. Sample Assessment Tasks

Skill-based task Problem Task 1.Use technology to graph and compare a beginning 1.Explain why a company has to sell 100 soccer salary of $30 per day increased by $5 each day and a balls before they will make a profit. The cost of beginning salary of $0.01 per day, which doubles each producing a soccer ball is modeled by C = 10x + day. When are the salaries equal? How do you know? 1000. The sales price of a soccer ball is $20. 2.If you’re getting an iPhone 4S, you have three 2.John and Jerry both have jobs working at the options for wireless service: AT&T, Verizon and Sprint. town carnival. They have different employers, so Unfortunately, there is no APP for choosing the right their daily wages are calculated differently. plan. What are the plans? Sprint offers an unlimited John’s earnings are represented by the equation, data and texting plan for $80 a month. AT&T charges p(x) = 2x and Jerry’s by g(x) = 10 + 0.25x. 20 cents per text and $55 for a 200 MB data plan. a. What does the variable x represent? Verizon charges 10 cents per text and $70 for a 200 b. If they begin work next Monday, Michelle told MB data plan. (source: money.cnn.com) them that Friday would be the only day they  a.Write equations to represent AT&T’s plan and made the same amount of money. Is she correct Verizon’s plan. in her assumption? Explain your reasoning.  b.Graph the equations for AT&T and Verizon c. When will Jerry earn more money than John?  c.When are the plans the same cost? When will John earn more money than Jerry?  d.When does the plan for AT&T cost $5 more than During what day will their earnings be the same? Verizon? Justify your conclusions.  e.Write a short paragraph comparing the price plans for AT&T and Verizon. CORE CONTENT Cluster Title: Represent and solve equations and inequalities graphically Standard A.REI.12: Graph the solutions to a linear inequality in two variables in a half-plane (excluding the boundary in the case of a strict inequality), and graph the solution set to a system of linear inequalities in two variables as the intersection of the corresponding half-planes. Concepts and Skills to Master:  Graph the solution to linear inequalities in two variables  Graph the solution to systems of linear inequalities in two variables  Identify the solutions as a region of the plane SUPPORTS FOR TEACHERS Critical Background Knowledge  Graph linear equations  Graph systems of linear equations  Simplify inequalities to represent them in a format that is easy to graph Academic Vocabulary Inequality, solution, half-plane, solution region

Suggested Instructional Strategies: Resources:  Use technology to model examples of intersections of inequalities  Textbook Correlation: 6-5, 6-6, CB 6- Use colored pencils to find the region of 6 solutions  MARS lesson: Defining Areas using NCDPI Unpacking: Inequalities By graphing a two variable inequality, students  MathForward activities: understand that the solutions to this inequality are all Testing for Truth the ordered pairs located on a portion or side of the coordinate plane that, when substituted into the The Impossible Task inequality, make the equation true. Students should be Tri This able to graph the inequality, specifying whether the Winning Inequalities points on the boundary line are also solutions by using a dotted or solid line. Using a variety of methods, which include selecting and substituting test points into the inequality, students should be able to determine which portion or side of the graph contains the ordered pairs that are the solutions to the original inequality. Sample Assessment Tasks Skill-based task Problem Task

Graph the solution set of x + 2y > 12 and 3x – y < 9

Create a context that represents the shaded area. Write the system of inequalities that models the meaning of the context. Describe the connections between the context, inequalities, and graph.

Standards are listed in alphabetical /numerical order not suggested teaching order. PLC’s must order the standards to form a reasonable unit for instructional purposes.