Grade 8 Linear Equations in One Variable 8.EE.7a-b

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1 THE NEWARK PUBLIC SCHOOLS THE OFFICE OF MATHEMATICS Grade 8 Linear Equations in One Variable 8.EE.7a-b 2012 COMMON CORE STATE STANDARDS ALIGNED MODULES

2 THE NEWARK PUBLIC SCHOOLS Office of Mathematics MATH TASKS 8.EE.7a-b Expressions and Equations Analyze and solve linear equations and pairs of simultaneous linear equations. Goal: In Module 1, students will strategically choose and efficiently implement procedures to solve linear equations in one variable, understanding that when they use the properties of equality and the concept of logical equivalence, they maintain the solution of the original equation. By being able to transform linear equations in one variable into simpler forms, students will discover that equations can have one solution, infinitely many solutions, or no solutions. Essential Question(s): How are properties related to algebra? How can you represent quantities, patterns, and relationships? Can equations that appear to be different be equivalent? Explain. Prerequisites: Simplify algebraic expressions by collecting like terms and using the distributive property Fluency with operations involving integers and rational numbers Understanding of 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 Solving Simple and Multi Steps Equations in One Variable Lesson 1 Simplifying Algebraic Expressions Using Distributive Property Lesson 3 Finding the Number of Solutions of Linear Equations in One Variable Lesson 5 Golden Problem Lesson 4 Writing and Solving Multi Steps Equations in One Variable Lesson Structure Introductory Task Guided Practice Collaborative work Journal Questions Skill Building Homework Page 2 of 22

3 Lesson 1: Introductory Task - Coupon vs. Discount 8.EE.7a: Give examples of linear equations in one variable with one solution, infinitely many solutions, or no solutions. Show which of these possibilities is the case by successively transforming the given equation into simpler forms, until the equivalent equation of the form x = a, a=a, or a =b results (where a and b are different numbers. 8.EE.7b: Solve linear equations with rational number coefficients, including equations whose solutions require expanding expressions using the distributive property and collecting like terms. Prerequisite Competencies 1. Simplifying algebraic expressions by collecting like terms 2. Performing operations with rational numbers 3. Understanding of distributive property 4. Understanding of properties of equality Introductory Task You have a coupon worth $20 off the purchase of a scientific calculator. At the same time the calculator is offered with a discount of 20%, but no further discounts may be applied. For what tag price on the calculator do you pay the same amount for each discount? Focus Question(s) How can you simplify an algebraic expression with terms that are alike? Page 3 of 22

4 Lesson 1: Guided Practice 1. Simplify each expression. a. 6k + 7k g. 11r 12r b. 12r 8 12 h. v + 12v c. n n i. 8x 11x d. 4x 10x j. 4p + 2p e. r 10 k. 5n + 11n f. 2x x l. n n 2. A reef explorer dives 25 ft to photograph brain coral and then rises 16 ft to ravel over a ridge before diving 47 ft to survey the base of the reef. Then the diver rises 29 ft to see an underwater cavern. What is the location of the cavern in relation to sea level? 3. Evaluate the variable expression 5(6 + 12x) + 3(20x +10) when x = Restate each expression using the Distributive Property. a. g. b. h. ( 10)p c. i. d. j. e k. f. l. Page 4 of 22

5 Lesson 1: Collaborative Work Collaborative Work 1. Without solving them, say whether these equations have a positive solution, a negative solution, a zero solution, or no solution. a. b. c. d. e. 2. Simplify the variable expression 3. Maria opened a savings account with $50. She deposited $35.50 into the account every month until she had a total of $ a. Write the equation that could be used to find m, the number of months she made deposits to the account. b. Proceed to solve for m. Journal Question(s) 1. Solve the equation. 2. Simplify the expression 3. Solve the equation 4. Solve the equation Page 5 of 22

6 Lesson 1: Homework 1. Anna used the distributive property to rewrite the variable expression. Decide whether Anna rewrote the expression correctly. If not, correct her work. 2. Simplify each variable expression. a. f. b. g. c. h. d. 1. i. e. j. 3. Write an equivalent equation to. Page 6 of 22

7 Lesson 2: Introductory Task The Oregon Trail Introductory Task Covered wagons traveled about 15 miles per day on the Oregon Trail. Write an equation that you can use to find the days d if the journey took 2,025 miles. Solve the equation. Show your work at how you arrived at your solution. Explain why you think your solution makes sense. About how many miles would a covered wagon travel in 162 days? Show your work. Focus Question(s) 1. What are the differences between an expression and an equation? 2. Does a mathematical expression have any solution? Explain. Page 7 of 22

8 Lesson 2: Guided Practice Guided Practice 1. The recommended heart rate for exercise, in beats per minute, is given by the expression where y is a person s age in years. Rewrite this expression using the distributive property. What is the recommended heart rate for a 20 year old? What is the recommended heart rate for a 50 year old? Use mental math. 2. Write an expression in simplified form for the area of each rectangle? a) b) c) d) Page 8 of 22

9 Lesson 2: Guided Practice Continued 3. Bushels of apples cost $7.95 at a farmer s market. How much would 8 bushels of apples cost? Use mental math. 4. Solve the equation. 5. Is a solution of the equation? Show your work to justify your answer. Page 9 of 22

10 Lesson 2: Collaborative Work 1. How can you express 499 to find the product using mental math. Explain. 2. Solve the equation. 3. Solve the equation 4. The manager of a restaurant earns $2.25 more each hour than the host of the restaurant. Write an equation that relates the amount h that the host earns each hour when the manager earns $11.50 each hour. 5. For each diagram given, find the value of if the perimeter of figure A is 136 units and B is 134 units. Give the unknown dimension. (A) (B) Journal Question(s) 1. When solving a one-step equation, how do you know which operation to use in order to isolate the variable? Show an example. 2. Solve the equation Page 10 of 22

11 Lesson 2: Homework Homework Solve each of the problems given below About 100 acres of pizza are eaten each day in the United States. San Francisco, California, has an area of about of about 29,887 acres. Write an expression that will calculate the number of days d it would take people in the United States to eat a pizza the size of San Francisco. Use that expression to solve for the value of d. 14. Is a solution of the equation? Explain. Page 11 of 22

12 Lesson 3: Introductory Task Error Analysis Introductory Task 1. Hoover checked whether is a solution for the equation, as shown: Describe and correct Hoover s error. 2. Identify and correct the error shown. Focus Question(s) What are equivalent equations? Page 12 of 22

13 Lesson 3: Guided Practice Guided Practice Solve each of the problems given below Solve each equation. Then state the number of solution(s) as one solution, no solutions or infinitely many solutions. a. b. c. d. e. f. 2. You have already read 120 pages of a book. You are one third of the way through the book. Write and solve an equation to find the number of pages in the book. 3. You have $16 and a coupon for a $5 discount at a local supermarket. A bottle of olive oil costs $7. How many bottles of olive oil can you buy? Page 13 of 22

14 Lesson 3: Collaborative Work Collaborative Work Solve each of the problems given. Show the steps. Check if your solution is correct Write an equation. Then solve. Show the steps. 1. Three times a number plus five times the same number is 40. What is the number? 2. University High School s basketball team played 35 games. They won 7 more games than they lost. How many games did they win and lose? 3. Al has twice as much money as Chip. Together they have $78. How much money does each have? 4. There are 15 more girls than boys in a summer camp. There are 135 campers in all. How many boys were there? Journal Question(s) Solve 1. The three sides a triangle are. Draw and label the triangle. Find the value of if the perimeter of the triangle is 48. Identifying solution(s) of an equation. 2. Is a solution to the equation Show your work to justify your answer. Page 14 of 22

15 . Lesson 3: Homework 1. Solve each of the equations given below: a. b. c. d. e. What pattern did you observe in solving equations a through e above? Discuss and/or explain. 2. Solve each of the equations given below: i. What pattern did you observe in solving equations f through j above? Discuss and/or explain. Page 15 of 22

16 Lesson 4: Introductory Task Food Calories Introductory Task A piece of pizza has 50 more calories than a can of cola. Three pieces of pizza have as many calories as four cans of cola. Let number of calories in a can of soda. Write an algebraic equation expression that will calculate information given. based on the Use the algebraic equation to solve for x. Show the steps. How many calories does a piece of pizza have? Show your steps. Focus Question(s) How can you tell the number of solutions when solving equations? Page 16 of 22

17 Lesson 4: Guided Practice Guided Practice A. Solve each of the equations given below. Then, for each of the equation tell which of the following is true a. No number is a solution b. One number is a solution. c. There are infinitely many solutions. B. Al has three times as much money as Peter. Together they have $40. Let amount of money that Peter has. Write an algebraic equation that will calculate based on the information given. Use the algebraic equation to solve form. Show the steps. How much money does Al have? Do you think your solution makes sense? Explain. Page 17 of 22

18 Lesson 4: Collaborative Work Show the steps at how you arrived at your answer. 1., what is the product of x and m? 2. At the beginning of her mathematics class, Mrs. Reno gives a warm-up problem. She says, "I am thinking of a number such that 6 less than the product of 7 and this number is 85." Which number is she thinking of? 3. Robin spent $17 at an amusement park for admission and rides. If she paid $5 for admission, and rides cost $3 each, what is the total number of rides that she went on? 4. How many solution(s) does the equation have? Explain. 5. What is the value of p in the equation? 6. The sum of the ages of the three Romano brothers is 63. If their ages can be represented as consecutive integers, what is the age of the middle brother? 7. How many solution(s) does the equation have? Explain. 8. How many solution(s) does have? Journal Question(s) 1. The junior class is selling granola bars to raise money. They purchased 1250 granola bars and paid a delivery fee of $25. The total cost including the delivery fee, was $800. What was the cost of each granola bar? Show your work. 2. Describe the steps that you would use to solve the equation. Page 18 of 22

19 Lesson 4: Homework Homework Solve all of the problems given below. 1. Lou has twice as much money as Sherlene. Sherlene has $11 more than Sherry. Together they have $89. How much does each have? Show your work. 2. Rich is 3 years older than Carla. Ruth is twice as old as Rich. Their ages total 33 years. How old is each person? Show your work 3. Solve the equation. 4. Solve the equation. 5. Solve the equation. 6. Solve the equation. Page 19 of 22

20 Lesson 5: Golden Problem Golden Problem When your car breaks down, you may have to call a tow truck. Towing company usually charge one fee to hook up the car. They also may charge for each mile the car is towed. The table shows towing charges for two companies. Company Charge to hook up car Charge per mile A $34 $2 B $28 $3 Suppose Company A charges you $70. About how far was your car towed? Show your work. A friend tells you that Company B would have been a better bargain. Do you agree or disagree? Justify your answer by using mathematical work. Under what circumstances would you recommend using Company B? What other information would you like to know before hiring a towing company? Page 20 of 22

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 22

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 22