Algebra 1 Keystone Remediation Packet Module 1 Anchor 2 A.1.1.2.1.1 Write, solve, and/or graph linear equations using various methods. A.1.1.2.1.2 Use and/or identify an algebraic property to justify any step in an equation solving process. A.1.1.2.1.3 Interpret solutions to problems in the context of the situation (linear equations) A.1.1.2.2 Write, solve, a system of linear equations using graphing, substitution, and/or elimination. Interpret solutions in the context of the problem. Name
Keystone Practice Problems A.1.1.2.1.1 Write, solve, and/or graph linear equations using various methods. 1) Jenny has a job that pays her $8 per hour plus tips (B). Jenny worked for 4 hours on Monday and made $65 in all. Which equation could be used to find B the amount Jenny made in tips? a. 65 = 4B + 8 b. 65 = 8B 4 c. 65 = 8B + 4 d. 65 = 8 4 + B 2) In an experiment, a plant grows 0.05 centimeter per day. The plant had a height of 2 centimeters when the experiment started. Which equation describes the relationship between the number of days T the experiment lasts and the height h in centimeters of the plant? a. h = 0.05T + 2 b. h = 0.05 2 + T c. h = 2T + 0.05 d. h = 2.05T 3) Keyshawn buys 100 baseball cards to start a collection. He purchases 25 more baseball cards each week. Which equation could be used to represent the number of baseball cards (b) in Keyshawn s collection after w weeks? a. V = 25W + 100 b. V = 100W + 25 c. W = 25V + 100 d. W = 100V + 25 4) Linda has 50 stamps in her collection. She purchased 10 stamps each week for V weeks. Which equation could be used to represent the number of stamps (s) in Linda s collection after V weeks? a. [ = 10V + 50 b. [ = 50V + 10 c. V = 10[ + 50 d. V = 50[ + 10
A.1.1.2.1.2 Use and/or identify an algebraic property to justify any step in an equation solving process Properties of Real Numbers Commutative Property- Change Order Addition Multiplication 5c + 4 = 4 + 5c 3 8 5c = 5c 3 8 Associative Property- rearrange groups Addition Multiplication 4c + 2c + 7c = 4c + (2c + 7c) (c g 5c) c = c g (5c c) Identity Property Addition 5h + 0 = 5h Multiplication 2c 1 = 2c Distributive Property- The distributive property allows us to simplify an expression by multiplying every term inside the parenthesis by the multiplier. Applying the distributive property to 3 c + 2 gets 3c + 3 2 which equals 3c + 6 Example:
Keystone Practice Problems Eligible Content: A.1.1.2.1.2 Use and/or identify an algebraic property to justify any step in an equation solving process 1) Juanita used the steps shown below to correctly solve an equation. A step is missing. Which step shows the equation that is most likely a missing step and the property that justifies the step? a. 3m 6 + 4m = 10m + 9, associative property b. 3m + 18 + 4m = 10m + 45, associative property c. 3m 6 + 4m = 10m + 9, distributive property d. 3m + 18 + 4m = 10m + 45, distributive property 2)
Keystone Practice Problems A.1.1.2.1.3 Interpret solutions to problems in the context of the problem situation 1) Ms. Bernard monitored the growth of a fish. The fish originally weighed 27 ounces. The fish grew at a rate of 5 ounces per month. The equation below can be used to describe the weight, in ounces, of the fish. 72 = 27 + 5c Ms. Bernard correctly determined that x=9. What does the solution of the equation mean? a. The fish grew at a rate of 9 ounces per month for 72 months. b. The fish grew at a rate of 72 ounces per month for 9 months. c. It took 9 months for the fish to grow 72 ounces. d. It took 72 months for the fish to grow 9 ounces. 2) Sylvia studied a new language. The equation below describes how many words she knew (y) after studying the language for x days. h = 5c + 18 The ordered pair (6,48) is a solution of the equation. What does the solution represent? a. Sylvia knew 6 words after 6 days. b. Sylvia knew 6 words after 48 days. c. Sylvia knew 48 words after 6 days. d. Sylvia knew 48 words after 48 days. 3) Jake bought x pairs of thermal socks and y shirts at the department store. He spent a total of $80. The following equation describes the relationship between the number of pairs of thermal socks and the number of shirts purchased. 11x + 18y = 80 The ordered pair (4, 2) is a solution of the equation. What does (4, 2) represent? a) Shirts cost 2 times as much as socks b) Jake purchased 4 pairs of socks and 2 shirts c) Socks cost $4 each and shirts cost $2 each d) Jake spent $4 on socks and $2 on shirts.
4) Francisco purchased x hot dogs and y hamburgers at a baseball game. He spent a total of $10. The equation below describes the relationship between the number of hot dogs and the number of hamburgers purchased. 3x + 4y = 10 The ordered pair (2, 1) is a solution of the equation. What does the solution (2, 1) represent? a) Hamburgers cost 2 times as much as hot dogs. b) Francisco purchased 2 hot dogs and 1 hamburger. c) Hot dogs cost $2 each and hamburgers cost $1 each. d) Francisco spent $2 on hot dogs and $1 on hamburgers. 5) At the theater, Fred purchased x popcorns and y drinks. He spent a total of $26. The equation below describes the relationship between the number of popcorns and the number of drinks he purchased. 4c + 3h = 26 The ordered pair (5, 2) represents the solution of the equation. What does the solution (5,2) represent? a) Popcorn costs $5 each and drinks cost $2 each b) Fred purchased 5 popcorns and 2 drinks. c) Fred spent $5 on popcorn and $2 on drinks. d) Fred purchased 2 popcorns and 5 drinks. 6) Juan answered all 50 questions on a test. He earned 3 points for each question he answered correctly. He lost 1 point for each question he answered incorrectly. His final test score was 102 points. The system of equations below describes the relationship between the number of questions he answered correctly (x) and the number he answered incorrectly (y). Part of the solution of the system is x=38. What does this value represent? a) The number of questions he answered correctly b) The number of questions he answered incorrectly c) The number of points he lost from questions answered incorrectly d) The number of points earned from questions he answered correctly
Systems of Equations A.1.1.2.2.1, A.1.1.2.2.2 Solve and/or graph a system of linear equations using various methods including graphing, substitution, and/or elimination. Interpret the results in the context of the situation. A system of equations is a set of two or more linear equations in the same variables. There are three possible answers: one solution, no solutions, or infinite solutions. Looking at the left graph, you will notice that the two lines intersect at exactly one point. Only one value of (x, y) makes both equations true. A solution of a system of linear equations in two variables is an ordered pair that is a solution of each equation in the system. Example 1: Determine whether the ordered pair 2, 5 is a solution of the system of linear equations. c + h = 7 2c 3h = 11 Example 2: A system of equations is shown below. What is the solution of the system of equations? 2c + 2h = 10 5c 2h = 4 a) 2, 7 b) 2, 7 c) (2, 3) d) (3, 2)
Keystone Practice Problems A.1.1.2.2.1 Write and/or solve a system of linear equations 1) Jenna and her friend Jada are having a contest to see who can save the most money. Jenna has already saved $120 and every week she saves an additional $15. Jada has already saved $80 and every week she saves an additional $25. Write a system of linear equations that represents the situation. In how many weeks will Jenna and Jada have the same amount of money? 2) At a baseball game, James purchased 3 large soft drinks and 5 nachos. He spent $42. Later in the game, his friend Laura purchased 2 large soft drinks and 4 nachos. She spent $32. Write a system of linear equations that represents the situation. Solve the system to determine how much large soft drinks and nachos cost. 3) Raul has $640 saved, and Jaime has $320 saved. They each begin a new job on the same day and save all of their money. Raul earns $180 per day and Jaime earns $200 per day. In how many days will they have an equal amount of money? a) 8 b) 16 c) 24 d) 32
Keystone Practice Problems A.1.1.2.2.2 Interpret solutions to problems in the context of the problem situation 1) A bakery charges x per cupcake and y per icing color used to decorate the cupcakes. Jan purchase 8 cupcakes, each decorated with 4 colors. Tony purchase 5 cupcakes, each with 6 colors. Jan spent $26 and Tony spent $18. The equations below represent the situation. How much does the bakery charge for one cupcake and for each color used to decorate the cupcakes? 8c + 4h = 26 5c + 6h = 18 a) $2 per cupcake, $1 per color b) $3 per cupcake, $0.50 per color c) $4 per cupcake, $1.50 per color d) $6 per cupcake, $2 per color 2) A poster printing company charges x per square foot of paper and y per color used on a poster. Tim has two posters printed. The first poster has an area of 2 square feet and uses 3 colors. It costs $13. The second posted has an area of 3 square feet and uses 5 colors. It costs $21. The equations below represent the situation. How much does the company charge per square foot of paper and for each color used on a poster? 2c + 3h = 13 3c + 5h = 21 a) $3 per square foot, $2 per color b) $3 per square foot, $3 per color c) $2 per square foot, $5 per color d) $2 per square foot, $3 per color 3) Christine sells a total of 50 tickets to a school play. Student tickets sell for $1.50 each, and adult tickets sell for $4.00 each. Christine made a total of $112.50 in ticket sales. Christine writes a system of equations to represent this information. Which statement best describes the solution to the system of equations? a) Christine sells 35 student tickets and 15 adult tickets. b) Christine would earn $200 by selling 50 adult tickets. c) Christine sells a pair of tickets for between $3.00 and $8.00 d) Christine could not earn $112.50 by selling 50 student tickets.
4) Mary measured the heights of two different plants every day. Plant A was 1 inch tall when Mary began her measuring, and it grew 0.5 inch per day. Plant B was 3 inches tall, and it grew 0.25 inch per day. On what day were plant A and plant B the same height? a) Day 5 b) Day 8 c) Day 12 d) Day 16 5) Anna burned 15 calories per minute running for x minutes and 10 calories per minute hiking for y minutes. She spent a total of 60 minutes running and hiking and burned 700 calories. The system of equations shown below can be used to determine how much time Anna spent on each exercise. What is the value of x, the minutes Anna spent running? 15c + 10h = 700 c + h = 60 a) 10 b) 20 c) 30 d) 40 6) Samantha and Maria purchased flowers. Samantha purchased 5 roses for x dollars each and 4 daisies for y dollars each. She spent $32 on the flowers. Maria purchased 1 rose for x dollars and 6 daises for y dollars each. She spent $22. The system of equations shown below represents the situation. Which statement is true? 5c + 4h = 32 c + 6h = 22 a) A rose cost $1 more than a daisy. b) Samantha spent $4 on each daisy. c) Samantha spent more on daisies than roses. d) Maria spent 6 times as much on daisies as she did on roses. 7) Henry purchased x candy bars and y soft drinks for his friends at a cost of $16. The next day, at the same store, Henry s friend purchased twice as many candy bars and three times as many soft drinks and spent $38. Which of the following statements is true? a) Henry spent $4 more on drinks than candy b) Henry spent more on candy than Henry s friend spent on drinks c) Henry spent less on candy that Henry s friend spent on drinks d) Henry spent more on drinks than his friend spent on candy
CONSTRUCTED-RESPONSE ITEM Algebra I 1 MODULE 1 13. Small baskets of tomatoes are sold at a vegetable stand for $3 per basket. Large baskets of tomatoes are sold at the stand for $5 per basket. Only whole numbers of baskets may be purchased. A customer purchases a total of 8 baskets of tomatoes and pays $36. A. Write and solve a system of equations that models the number of small baskets (x) and the number of large baskets (y) that the customer purchases. Show or explain all your work. Go to the next page to finish question 13. GO ON Keystone Algebra I Item and Scoring Sampler September 2017 22
Algebra I 1 MODULE 1 13. Continued. Please refer to the previous page for task explanation. Another customer claims that he can purchase a total of 10 baskets of tomatoes and pay $45. B. Use a system of equations that describes this other customer s purchase to explain why the claim is incorrect. AFTER YOU HAVE CHECKED YOUR WORK, CLOSE YOUR ANSWER BOOKLET AND TEST BOOKLET SO YOUR TEACHER WILL KNOW YOU ARE FINISHED. STOP Keystone Algebra I Item and Scoring Sampler September 2017 23