Grade 8 Systems of Linear Equations 8.EE.8a-c

Transcription

1 THE NEWARK PUBLIC SCHOOLS THE OFFICE OF MATHEMATICS Grade 8 Systems of Linear Equations 8.EE.8a-c 2012 COMMON CORE STATE STANDARDS ALIGNED MODULES 2012 COMMON CORE STATE STANDARDS ALIGNED MODULES

2 THE NEWARK PUBLIC SCHOOLS Office of Mathematics MATH TASKS 8.EE.8a-c Expressions and Equations Analyze and solve linear equations and pairs of simultaneous linear equations. Goal: In this module, students will solve systems of two linear equations in two variables and relate the systems to pairs of lines in the plane; these intersect, are parallel, or are the same line. Students will use their understanding of systems of linear equations to analyze situations and solve problems. Essential Question(s): How can you solve a system of equations? Can systems of equations model real-world situations? Explain. Prerequisites: 1.Write linear equations (relationships) 2.Solve linear equations 3.Graph linear equations 4.Understand properties of equality Embedded Mathematical Practices MP.1 Make sense of problems and persevere in solving them MP.2 Reason abstractly and quantitatively MP.3 Construct viable arguments and critique the reasoning of others MP.4 Model with mathematics MP.5 Use appropriate tools strategically MP.6 Attend to precision MP.7 Look for and make use of structure MP.8 Look for and express regularity in repeated reasoning Lesson 2 Solution(s) to Systems of Linear Equations Counting Calories Lesson 1 Introduction to Systems of Linear Equations Kimi and Jordan Lesson 3 Solving Systems of Linear Equations Algebraically Ivan s Furnace Lesson 5 Golden Problem Cell Phone Plans Lesson 4 Using Systems of Linear Equations In Solving Real World Problems Reading Comic Books Lesson Structure Introductory Task Guided Practice Collaborative work Journal Questions Skill Building Homework Page 2 of 23

3 Lesson 1:Introductory Task Kimi and Jordan 8. EE.8a: Understand that solutions to a system of two linear equations in two variables correspond to points of intersection of their graphs, because points of intersection satisfy both equations simultaneously. 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. For example, 3x + 2y = 5 and 3x + 2y = 6 have no solution because 3x + 2y cannot simultaneously be 5 and EE.8c: Solve real-world and mathematical problems leading to two linear equations in two variables. For example, given coordinates for two pairs of points, determine whether the line through the first pair of points intersects the line through the second pair. Prerequisite Competencies 1. Write linear equations (relationships) 2. Solve linear equations 3. Graph linear equations 4. Understanding of properties of equality Page 3 of 23

4 Introductory Task Kimi and Jordan are each working during the summer to earn money in addition to their weekly allowance, and they are saving all their money. Kimi earns $9 an hour at her job, and her allowance is $8 per week. Jordan earns $7.50 an hour, and his allowance is $16 per week. 1. Complete the two tables shown below. Number of hours worked in a week, h Kimi s weekly total savings, K Number of hours worked in a week, h Jordan s weekly total savings, J 2. Write an equation that can be used to calculate the total of Kimi's allowance and job earnings at the end of one week given the number of hours she works 3. Write an equation that can be used to calculate the total of Jordan's allowance and job earnings at the end of one week given the number of hours worked. 4. Sketch the graphs of your two equations on one pair of axes. 5. Jordan wonders who will save more money in a week if they both work the same number of hours. Write an answer for him. Focus Question(s) 1. What is a system of linear equations? 2. How are the solutions to the two equations represented graphically? Page 4 of 23

5 Lesson 1:Guided Practice 1. Graph each equation. Pick 3 values of x. Then find the corresponding values of y. Use a T-chart to show the x and y values. a. b. c. d. e. f. 2. Graph both equations in the same coordinate plane. g. h. i. j. 3. Students Statistics. The number of right-handed students in a mathematics class is nine times the number of left-handed students. The total number of students in the class is 30. Use r to represent the number of right-handed students and use l to represent the number of left-handed students. What two equations can you write to model the situation? Graph the two equations in the same x-y axes to solve for a. the number of right-handed students in the class. b. the number of left-handed students in the class. 4. Graph the equations and on the same coordinate plane. Do you think there is a solution to this system of equation? Explain. Page 5 of 23

6 Lesson 1:Collaborative Work Collaborative Work 1. Plants. A plant nursery is growing a tree that is 3-ft tall and grows at an average rate of 1 foot per year. Another tree at the nursery is 4-ft tall and grows at an average rate of 0.5 ft. per year. a. Write two equations that will model the situation. b. Graph the two equations in the same coordinate plane. c. After how many years will the trees be the same height? 2. Fitness. At a local fitness center, members pay a $20 membership fee and $3 for each aerobics class. Nonmembers pay $5 for each aerobics class. 3. a. Write two equations that will model the situation. b. Graph the two equations in the same coordinate plane. c. For what number of aerobics classes will the cost for members and nonmembers be the same? d. Would you rather be a member or nonmember to this fitness center? Explain. 4. Graph each pair of equations on the same coordinate plane. (For each equation in the system, pick 3 values of x. Then find the corresponding values of y. Use a T-chart to show the x and y values.) Then write the solution of each system. a. b. c. 5. Which ordered pair is a solution of the linear system and? A. ( 3,3) B. ( 3,6) C. (3,3) D. (3,6) Journal Question(s) What is the solution of each system? Graph each system in the same coordinate plane. 1. and 2. and 3. and Page 6 of 23

7 Lesson 1: Homework Tell whether the ordered pair is a solution of the linear system. 1. (-3,1) ; and 2. (5,2) ; and 3. (-2,1) ; and 4. (-3,6) ; and 5. Vocabulary Complete the statement. A(n) of a system of linear equation in two variables is an ordered pair that satisfies each equation in the system. Solve the linear system by graphing. Check your solution Solve the equation given The cost to join an art museum is $60. If you are a member, you can take lessons at the museum for $2 each. If you are not a member, lessons cost $6 each. Write a system of equations that can be used to find the number of x lessons after which the total cost of y lessons with the membership is the same as the total cost of lessons without a membership. Page 7 of 23

8 Lesson 2:Introductory Task Counting Calories Introductory Task Focus Question(s) 1. How do you know if the solution to a linear system is correct? 2. Explain how to use the graph-and-check method to solve a linear system of two equations in two variables. Page 8 of 23

9 Lesson 2: Guided Practice Guided Practice 1. The parks and recreation department in your town offers a season pass for $90. As a season pass holder, you pay $4 per session to use the town s tennis courts. Without the season pass, you pay $13 per session to use the tennis courts. a) Write two equations that will model the situation. b) Graph the two equations in the same coordinate plane. c) Find the number of sessions after which the total cost with a season pass, including the cost of the pass, is the same as the total cost without a season pass? 2. Solve the linear system by graphing and algebraically (by substitution). a. b. c. (1, 2) (4, 0) (-2,-2) 3. Use the graph to solve the system. Check your solution. (a) (b) (c) 4. RENTAL BUSINESS A business rents in-line skates and bicycles. During one day, the business has a total of 25 rentals and collects $450 for the rentals. Find the number of skates rented and the number of bicycles rented. Page 9 of 23

10 Lesson 2: Collaborative Work 1. State whether or not the given ordered pair is a solution of the given systems of equations. a. b. c. 2. Find the solution set of each system. Use any method. a. b. c. d. e. 3. The sum of two numbers is 40, and their difference is 6. Find the numbers. (23 and 17) 4. The sum of two numbers is five time their difference. If one number exceeds the other by 7, find the numbers. (14 and 21 Journal Question(s) Learning to be a watchmaker, a student spent a total of 1430 hours in two courses: Watchmaking and Watch Repair. If she spent 328 more hours in Watchmaking than she did in Watch Repair, how many hours did she spend in each course? Show the steps. (879 and 551) Page 10 of 23

11 Lesson 2: Homework Homework 1. Find the solution set of each system. Use any method. a. b. c. d. e. f. g. 2. I am thinking of two numbers whose difference is 20. Twice one number equals three times the other. Find the numbers. 3. You burned 8 calories per minute on a treadmill and 10 calories per minute on an elliptical trainer for a total of 560 calories in 60 minutes. How many minutes did you spend on each machine?. Page 11 of 23

12 Lesson 3:Introductory Task Ivan s Furnace Introductory Task Ivan s furnace has quit working during the coldest part of the year, and he is eager to get it fixed. He decides to call some mechanics and furnace specialists to see what it might cost him to have the furnace fixed. Since he is unsure of the parts he needs, he decides to compare the costs based only on service fees and labor costs. Shown below are the price estimates for labor that were given to him by three different companies. Each company has also given him an estimate of the time it will take to fix the furnace. Company A charges $35 per hour to its customers. Company B charges a $20 service fee for coming out to the house and then $25 per hour for each additional hour. Company C charges a $45 service fee for coming out to the house and then $20 per hour for each additional hour. For which time intervals should Ivan choose Company A, Company B, Company C? Support your decision with sound reasoning and representations. Consider including equations, tables, and graphs. Focus Question(s) Explain why and have no solution. Page 12 of 23

13 Lesson 3:Guided Practice Guided Practice A. When solved graphically, which system of equations will have exactly one point of intersection? No point of intersection? Infinitely many points of intersection? B. Find the slope of the lines for each equation Solve C. A bottle of cola costs $0.25. The deposit on the bottle is $0.15 less than the price of the cola. How much is the deposit? D. The length of a rectangle is three times the width. The perimeter is 48. Find the length and the width. E. Use any method to solve the linear system: F. Use any method to solve the linear system: G. Last year, Albert planted a rectangular garden with a perimeter of 34 feet. This year, he made his garden half as long and twice as wide. The perimeter of the new garden is 26 feet. What are the dimensions of each garden? H. Yvette is 3 years younger than Lauren. Their ages total 33. How old is each person? Show the steps. Page 13 of 23

14 Lesson 3:CollaborativeWork Collaborative Work Use two variables and two equations. 1. Bill buys 2 hot dogs and soda for $1.20. Rosie buys 3 hot dogs and 2 sodas for $1.95. Use any method to find the cost of each item. 2. Laura has $4.50 in dimes and quarters. She has 3 more dimes than quarters. Use any method to find the number of quarters Laura has. 3. Solve the linear system by graphing:. 4. A taxi company charges $2.80 for the first mile and $1.60 for each additional mile. Another taxi company charges $3.20 for the first mile and $1.50 for each additional mile. After how many miles will each taxi cost the same? Use a table to solve the problem. 5. Use algebra to solve the linear system :. Kaitlin is making a quilt out of fabric that has alternating stripes of regular quilting fabric and satin fabric. She spends $76 on a total of the two fabrics ar a fabric store. Write and solve a system of equations to find the amount x (in yards) of regular quilting and the amount of y (in yards) of satin fabric she purchased. Satin fabric costs $6 per yard. Quilting fabric costs $4 per yard. Journal Question(s) Use two variables and two equations. At the first meeting of the Chess Club, 12 students were present. After efforts were made to increase interest in the club, twice as many girls and 3 times as many boys attended the second meeting as those that attended the first. If there were 29 students at the second meeting, Write two equations that will model the situation. Use any method to determine how many boys and how many girls attended each meeting. Show the steps Page 14 of 23

15 . Lesson 3:Homework 1. When solved graphically, which system of equations will have exactly one point of intersection? no point of intersection? or infinitely many points of intersection? a. b. c. d. 2. Use two variables and 2 equations to solve each of the problems. a. Lea paid $5 for two adult tickets and 1 student ticket. Paul paid $6 for 1 adult ticket and 4 student tickets. Find the price for of each kind of ticket. b. Two adult tickets and 3 student tickets cost $12. One adult ticket and two student tickets cost $7. How much does each kind of ticket cost? c. Two hot dogs and 1 cola cost $ hot dog and 2 colas cost $1.30. How much does each item cost? d. One number is 6 less than another. The sum of the two numbers is zero. Find the numbers. e. One pizza and 4 colas cost $3.40. Two pizzas and 6 root beers cost $6.10. How much does each item cost? 3. Solve using any method. a. b. c. d. Page 15 of 23

16 Lesson 4:Introductory Task Reading Comic Books Introductory Task Bill and John, his brother collect comic books. Bill currently has 15 books and adds 2 books to his collection every month. His brother currently has 7 books and adds 4 books to his collection every month. Find the number of x of months after which Bill and his brother will have the same number of comic books in their collection? Let let number of months that the brothers are collecting comic books and number of comic books in the collection 1. Write two equations that will model the number of comic books in Bill s collection and John s collection. 2. Find the number of books in Bill s collection for Find the number of books in Bill s collection for Use the table below to show your answers. x Number of books in Bill s collection (y) Number of books in John s collection (y) 3. Based on the table, when Bill and John have the same number of books in their collections, how many books will each of them have? 4. Graph the equations above on the same coordinate plane. Do you yield the same solution as in number 3 above? Explain 5. Solve to find the number of months after which Bill and John will have the same number of comic books in their collections algebraically. Page 16 of 23

17 Lesson 4: Guided Practice Guided Practice 1. A rectangle has a perimeter of 18 inches. A new rectangle is formed by doubling the width w and tripling the length l, as shown. The rectangle has a perimeter P of 46 inches. P 2w 3l Write and solve a system of linear equations to find the length and width of the original rectangle. Find the length and width of the new rectangle. 2. Find the slope of a line that is perpendicular to the graph of each equation. a. b. c. 3. Find the slope of a line that is parallel to the graph of each equation. a. b. c. 4. Graph the linear equation. Find another equation of a line that will intersect at exactly one point. Find another equation of a line that will never intersect. Find another equation of a line that will intersect at exactly two points. 5. What is the value of the x-coordinate in the solution for the given system? Focus Question(s) 1. Is it possible for two lines to have exactly two, three, or four points in common? Explain. 2. What does a solution to both equations look like? 3. How many solutions does each equation have? Page 17 of 23

18 Lesson 4:Collaborative Work Show the steps at how you arrived at your answer GEOMETRY. A rectangle has a perimeter of 34 cm and an area of 52 cm². Its length is 5 more than twice its width. Write and solve a system of equations to find the dimensions of the rectangle. 2. ART. An artist is going to sell two sizes of prints at an art fair. The artist will charge $20 for a small print and $45 for a large print. The artist would like to sell twice as many small prints as large prints. The booth the artist is renting for the day costs $510. Write a system of linear equations to find the number each size print must the artist sell in order to break even at the fair. How many of each size print must the artist sell in order to break even at the fair? 3. Graph the linear equation. Find another equation of a line that will intersect at exactly one point. Find another equation of a line that will never intersect. Find another equation of a line that will intersect at exactly two points. 4. CHALLENGE PROBLEM AGRICULTURE. A farmer grows corn, tomatoes and sunflowers on a 320-acre farm. This year, the farmer wants to plant twice as many acres of tomatoes as acres of sunflowers. The farmer also wants to plan 40 more acres of corn than of tomatoes. How many acres of each crop should the farmer plant? Journal Question(s) 1. Using a graph, how can you tell when a system of linear equations has no solution? 2. Solving algebraically, how can you tell when a system of linear equations has no solution? 3. How is the number of solution(s) related to the number of point(s) of intersection when a system of equations is graphed? 1 Source: Problem(s) adapted from Algebra 1 Common Core ( 2012 by Pearson Education, Inc.) Page 18 of 23

19 Lesson 4: Homework Homework Solve each system by graphing. Tell whether the system has one solution, infinitely many solutions, or no solution Solve each system algebraically. Check your answer Solve each of the problems. Show your steps. 1. THEATER TICKETS. Adult tickets to a play cost $22. Tickets for children cost $15. Tickets for a group of 11 people cost a total of $228. Write and solve a system of equations to find how many children and how many adults were in the group. 2. TRANSPORTATION. A school is planning a field trip for 142 people. The trip will use 6 drivers and two types of vehicles: buses and vans. A bus can seat 51 passengers. A van can seat 10 passengers. Write and solve a system of equations to find how many buses and how many vans will be needed? 3. GEOMETRY. The measure of one acute angle in a right triangle is four times the measure of the other acute angle. Write and solve a system of equations to find the measures of the acute angles. 4. Graph the linear equation. Find another equation of a line that will intersect at exactly one point. Find another equation of a line that will never intersect. Find another equation of a line that will intersect at exactly two points. Page 19 of 23

20 Lesson 5: Golden Problem 2 Golden Problem Cell Phone Plans You are a representative for a cell phone company and it is your job to promote different cell phone plans. 1. Your boss asks you to use graphs to visually display three plans and compare them so you can point out the advantages of each plan to your customers. o Plan A has a basic fee of $30.00 per month and 10 cents per text message o Plan B has a basic fee of $90.00 per month and has unlimited text messages o Plan C has a basic fee of $50.00 per month and 5 cents per text message o All plans offer unlimited calling o Calling on nights and weekends are free o Long distance calls are included 2. A customer wants to know how to decide which plan will save her the most money. Determine which plan has the lowest cost given the number of text messages a customer is likely to send. 2 Source: Problem(s) adapted from The Illustrative Mathematics Project: Page 20 of 23

21 *Grade 8 Required Fluency: Solve simple 2 x 2 systems by inspection Systems of linear equations can also have one solution, infinitely many solutions or no solutions. Students will discover these cases as they graph systems of linear equations and solve them algebraically. A system of linear equations whose graphs meet at one point (intersecting lines) has only one solution, the ordered pair representing the point of intersection. A system of linear equations whose graphs do not meet (parallel lines) has no solutions and the slopes of these lines are the same. A system of linear equations whose graphs are coincident (the same line) has infinitely many solutions, the set of ordered pairs representing all the points on the line. By making connections between algebraic and graphical solutions and the context of the system of linear equations, students are able to make sense of their solutions. Students need opportunities to work with equations and context that include whole number and/or decimals/fractions. Examples: Find x and y using elimination and then using substitution. 3x + 4y = 7-2x + 8y = 10 Plant A and Plant B are on different watering schedules. This affects their rate of growth. Compare the growth of the two plants to determine when their heights will be the same. Let W= number of weeks Let H= height of the plant after W weeks Plant A Plant B W H W H 0 4 (0,4) 0 2 (0,2) 1 6 (1,6) 1 6 (1,6) 2 8 (2,8) 2 10 (2,10) 3 10 (3,10) 3 14 (3,14) Page 21 of 23

22 Given each set of coordinates, graph their corresponding lines. Solution: Write an equation that represent the growth rate of Plant A and Plant B. Solution: Plant A H = 2W + 4 Plant B H = 4W + 2 At which week will the plants have the same height? Solution: The plants have the same height after one week. Plant A: H = 2W + 4 Plant B: H = 4W + 2 Plant A: H = 2(1) + 4 Plant B: H = 4(1) + 2 Plant A: H = 6 Plant B: H = 6 After one week, the height of Plant A and Plant B are both 6 inches. *Fluent in the Standards means fast and accurate. It might also help to think of fluency as meaning the same thing as when we say, that somebody is fluent in foreign language; when you re fluent, you flow. Fluent isn t halting, stumbling, or reversing oneself. Assessing fluency requires attending to issues of time (and even perhaps rhythm, which could be achieved with technology). Source: Page 22 of 23

23 Grade 8 Fluency Problems 3 1. Tanisha and Rachel had lunch at the mall. Tanisha ordered three slices of pizza and two colas. Rachel ordered two slices of pizza and three colas. Tanisha's bill was $6.00, and Rachel's bill was $5.25. What was the price of one slice of pizza? What was the price of one cola? 2. When Tony received his weekly allowance. He decided to purchase candy bars for all his friends. Tony bought three Milk Chocolate bars and four Creamy Nougat bars, which cost a total of $4.25 without tax. Then he realized this candy would not be enough for all his friends, so he returned to the store and bought an additional six Milk Chocolate bars and four Creamy Nougat bars, which cost a total of $6.50 without tax. How much did each type of candy bar cost? 3. Sal keeps quarters, nickels, and dimes in his change jar. He has a total of 52 coins. He has three more quarters than dimes and five fewer nickels than dimes. How many dimes does Sal have? 4. Ramón rented a sprayer and a generator on his first job. He used each piece of equipment for 6 hours at a total cost of $90. On his second job, he used the sprayer for 4 hours and the generator for 8 hours at a total cost of $100. What was the hourly cost of each piece of equipment? 5. At a concert, $720 was collected for hot dogs, hamburgers, and soft drinks. All three items sold for $1.00 each. Twice as many hot dogs were sold as hamburgers. Three times as many soft drinks were sold as hamburgers. Find the number of soft drinks sold. 6. Solve the linear system: 7. Solve the linear system: 8. Solve the linear system: 9. Solve the linear system: 10. Solve the linear system: 3 Source: Problem(s) adapted from: Page 23 of 23