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1 A Park Ranger s Work Is Never Done Solving Problems Using Equations Learning Goal In this lesson, you will: Write and solve two-step equations to represent problem situations. Solve two-step equations. Key Terms inverse operations two-step equation solution coefficient constant Properties of Equality The first equations to ever be written down using symbolic notation are much like the equations you will study in this lesson. They first appeared in a book called the Whetstone of Witte by Robert Recorde in 17. This book is notable because it contained the first recorded use of the equals sign. Recorde got tired of writing the words is equal to in all his equations so he decided that a pair of parallel lines of the same length sitting sideways would be the perfect symbol because, as he said, no two things can be more equal. Here are the first two equations ever written: 14x x Can you solve the world s oldest symbolic equations? 1.1 Solving Problems Using Equations 3

2 Problem 1 An existing 10 foot walkway is in the process of being extended. The rate at which the additional walkway can be built is feet per hour. Students define variables, write equations, and use the equations to solve for unknown values. Have students complete Question 1 with a partner. Then share the responses as a class. Problem 1 Building a Walkway Many situations can be modeled by equations that need more than one operation to solve them. 1. At a local national park, the park rangers decide that they want to extend a wooden walkway through the forest to encourage people to stay on the path. The existing walkway is 10 feet long. The park rangers believe that they can build the additional walkway at a rate of about feet per hour. a. How many total feet of walkway will there be after the park rangers work for hours? After hours of work, there will be 17 feet of total walkway. b. How many total feet of walkway will there be after the park rangers work for 7 hours? After 7 hours of work, there will be 18 feet of total walkway. Share Phase, Question 1 What length of walkway already exists? How long is 10 feet? Compare it to something you are familiar with. How fast is the new walkway being built? Why isn t there feet of walkway completed after the rangers worked for hours? Why isn t there 3 feet of walkway completed after the rangers worked for 7 hours? How is the number 10 used in this problem situation? How can the expression you wrote in part (c) help you? Why didn t you need to divide 00 by to determine the time it will take the rangers to complete 00 feet of walkway? Can you imagine how long 00 feet is? It's about 1 _ 3 football fields. c. Define a variable for the amount of time that the rangers will work. Then, use the variable to write an expression that represents the total number of feet of walkway built, given the amount of time that the rangers will work. Let t represent time in hours. t 1 10 d. How many hours will the rangers need to work to have a total of 00 feet of walkway completed? The rangers will need to work 70 hours to have 00 feet of walkway completed. e. Explain the reasoning you used to solve part (d). First, I subtracted the existing amount from 00, then I divided that number by. f. What mathematical operations did you perform to calculate your answer to part (d)? I used subtraction and division to determine my answer. How can the expression you wrote in part (c) help you? Why do you need to subtract 10 from 00? How long is 00 feet? Compare it to something you are familiar with. 4 Chapter 1 Linear Equations

3 g. Write an equation that you can use to determine the amount of time the rangers need to build a total of 00 feet of walkway. Then, determine the value of the variable that will make this equation true. t t 30 t 30 t 70 h. Interpret your answer in terms of this problem situation. It will take the rangers 70 hours to complete 00 feet of walkway. Have students complete Questions and 3 with a partner. Then share the responses as a class. Share Phase, Question How long does it take for the rangers to build one foot of walkway? To determine the time needed for the rangers to have a total of 70 feet of walkway completed, do you need to subtract or divide first? Explain. How is the expression you wrote in the last question used to write the equation to determine the amount of time it will take to have a total of 70 feet of walkway completed? How many values of the variable will make the equation true?. How many hours will the rangers need to work to build a total of 70 feet of walkway? Explain your reasoning. The rangers will need to work 4 hours to build 70 feet of walkway. To determine my answer, I subtracted 10 from 70, then divided by. a. What mathematical operations did you perform to determine your answer? I used subtraction and division to determine my answer. b. Explain why using these mathematical operations gives you the correct answer. Part of the path was already there, so I need to determine the difference between the total walkway and the part of the walkway that already exists. This takes subtraction. Then, I need to divide to determine the time. c. Write an equation that you can use to determine the amount of time it will take to have a total of 70 feet of walkway completed. t Don't forget to take a minute to estimate your answer before starting to work. 1.1 Solving Problems Using Equations

4 d. Determine the value of the variable that will make this equation true. t t 10 t 10 t 4 Share Phase, Question 3 How many feet of the walkway are already completed? Does a negative answer make sense? What does a negative answer tell you with respect to the problem situation? If the time it took for the rangers to build a portion of the walkway was 1, what would that mean? If the time it took for the rangers to build a portion of the walkway was, what would that mean? What do negative numbers representing time mean with respect to the problem situation? e. Interpret your answer in terms of this problem situation. It will take 4 hours to complete 70 feet of walkway. 3. How many hours will the rangers need to work to build a total of 100 feet of walkway? Explain your reasoning. The rangers do not need to work because the existing walkway is already 10 feet long. a. What mathematical operations did you perform to determine your answer? I subtracted 10 from 100 which is 0. The length of the existing walkway, 10 feet, exceeds 100 feet. b. Write an equation that you can use to determine the amount of time it will take to have a total of 100 feet of walkway completed. t c. Determine the value of the variable that will make this equation true. t t 0 t 10 Does your answer make sense? 6 Chapter 1 Linear Equations

5 d. Interpret your answers in terms of this problem situation. I can represent 10 hours ago as feet of walkway were already completed when this project started. At the current rate the park rangers are working, it would have taken 10 hours to complete the first 100 feet of walkway. Problem A bear cub has fallen into an abandoned mine shaft 77 feet below the surface of the ground. The rate at which the cub is being pulled out of the shaft is 7 feet per minute. Students define variables, write equations, and use the equations to solve for unknown values. Have students complete Questions 1 and with a partner. Then share the responses as a class. Share Phase, Question 1 How deep is the mine shaft? How long is 77 feet? Compare it to something you are familiar with. How fast is the basket with the cub being pulled up? How did you determine how many feet below the surface of the ground the cub will be in 6 minutes? How did you determine how many feet below the surface of the ground the cub will be in 11 minutes? Problem Rescuing a Bear Cub 1. Part of a park ranger s job is to perform rescue missions for people and animals. Suppose that a bear cub has fallen into a steep ravine on the park grounds. The cub is 77 feet below the surface of the ground in the ravine. A ranger coaxed the cub to climb into a basket attached to a rope and is pulling up the cub at a rate of 7 feet per minute. a. How many feet below the surface of the ground will the cub be in 6 minutes? After 6 minutes, the cub will be 3 feet below the surface of the ground. b. How many feet below the surface of the ground will the cub be in 11 minutes? After 11 minutes, the cub will be at 0 feet, or at the surface. c. Define a variable for the amount of time spent pulling the cub up the ravine. Then use the variable to write an expression that represents the number of feet below the surface of the ground the cub is, given the number of minutes that the ranger has spent pulling up the cub. Let t represent time in minutes. 77 7t d. In how many minutes will the cub be 14 feet from the surface? In 9 minutes, the cub will be 14 feet from the surface. How is the number 77 used in this problem situation? How can the expression you wrote in part (c) help you? What does the expression 7t 77, where t is the time in minutes, tell you about the problem situation? What does the expression 77 t, where t is the time in minutes, tell you about the problem situation? 1.1 Solving Problems Using Equations 7

6 Share Phase, Question How long does it take for the cub to be lifted one foot? If the cub is 8 feet from the surface is it also 8 feet from the bottom of the mine shaft? Explain. To determine the time needed for the cub to be 8 feet from the surface, do you need to subtract or divide first? Explain. How is the expression you wrote in the last question used to write the equation to determine the amount of time it will take to lift the cub 8 feet from the surface? How many values of the variable will make the equation true? How many feet does the cub have to be lifted to reach the surface? How much time will it take for the cub to be lifted half-way to the surface? Explain. How much time will it take for the cub to be lifted to the surface? Explain. What are some values of the variable that result in answers that don t make sense in this problem situation? e. Explain your reasoning you used to solve part (d). First I subtracted 14 from 77, and then I divided by 7. f. What mathematical operations did you perform to determine your answer to part (d)? I used subtraction and division to determine my answer. g. Explain why using these mathematical operations gives you the correct answer. I needed to use subtraction because I had to calculate the difference between where the cub started and where the cub ended up (14 feet from the surface). Then, I had to divide by the rate to determine the time. h. Write an equation that you can use to determine the number of minutes it takes for the cub to be 14 feet below the surface of the ground by setting the expression you wrote in part (c) equal to 14. Then determine the value of the variable that will make the equation true. 77 7t 14 7t 63 7t t 9. In how many minutes will the cub be 8 feet from the surface? In 7 minutes, the cub will be 8 feet from the surface. a. Explain your reasoning. I subtracted 8 from 77, and then divided by 7. b. What mathematical operations did you perform to determine your answer? I used subtraction and division to determine my answer. 8 Chapter 1 Linear Equations

7 c. Explain why using these mathematical operations gives you the correct answer. I needed to use subtraction because I had to determine the difference between where the cub started and where the cub ended up (8 feet from the surface). Then I had to divide by the rate to determine the time. d. Write an equation that you can use to determine the number of feet below the surface of the ground that the cub will be in 8 minutes. Then, determine the value of the variable that will make the equation true. 77 7t 8 7t 49 7t t 7 Problem 3 Students analyze two methods to solve two-step equations. They will solve a number of two-step equations. Ask a student to read the introduction to Problem 3 aloud. Discuss the worked example and complete Questions 1 through 3 as a class. Problem 3 Solving Two-Step Equations In Problems 1 and, you were solving two-step equations. To solve two-step equations you need to perform inverse operations. Inverse operations are operations that undo each other. For example, adding 3 and subtracting 3 are inverse operations. Two-step equations are equations that require two inverse operations to solve. A solution to an equation is any value for a variable that makes the equation true. Let s consider the equation: m 6 The left side of this equation has two terms separated by the subtraction operation. The in the first term of the left side of the equation is called the coefficient. A coefficient is the number that is multiplied by a variable. The terms 6 and are called constants. A constant is a term that does not change in value. When solving any equation, you want to get the variable by itself on one side of the equals sign. Remember, when you are solving equations you must maintain balance. 1.1 Solving Problems Using Equations 9

8 Discuss Phase, Worked Example What is the first step in each method? Is it the same step for each method? What is the second step in each method? Is it the same step for each method? Why would someone prefer method 1? Why would someone prefer Method? Which method do you prefer? Why? Two different examples of ways to solve the same two-step equation are shown. Method 1 Method m 6 m 6 Step 1: m m m 8 Step : m 8 m 8 m 14 m Describe the inverse operations used in each step. Step 1: A 6 was added to both sides of the equation to undo the subtraction. Step : A was divided by both sides to undo the multiplication.. What is the difference between the strategies used to solve the equation? In method 1, the first inverse operation is written horizontally. In method, the first inverse operation is written vertically. 3. Verify the solution is m 14. (14) What are the general strategies to solve any twostep equation? Have students complete Question 4 with a partner. Then share the responses as a class. 4. Solve each two-step equation. Show your work. a. v 34 6 b. 3x v x v 60 3x 30 v 60 3x v 1 x Chapter 1 Linear Equations

9 Share Phase, Question 4 What inverse operations will you have to use in this problem? Explain What was you first step in this problem? How did you decide to do that first? What was you second step in this problem? How did you decide to do that second? How did you check that your solution is correct? c x 83 d..c x 83 3.c x 60.c 4x 4 60.c 4.. x 1 c 10 e. 3 4 x f. 3 b x b x 3 3 b ( 3 4 x ) ( 3 3 ( 3 b ) ( 6 x 3 9 3) 4 ) 3 b 9 3 Don't forget to check your solution. Substitute your answer back into the original equation and make sure it is true. g. t 9 1 h..7 s 4 t t s 4 t 1 ( ) (30) ( 1 ) 4 1 (0.7) s ( 4 ) 4 1 t 10 s 1.08 i. 1m j d 1m d 1m d 1m d 4 m 13 d Solving Problems Using Equations 11

10 k. 3z l d 3z d 3z d 3z d z 3 d 1468 Talk the Talk The Properties of Equality are defined. Students describe the strategies used to solve two-step equations. Ask a student to read the definition aloud. Discuss this information and analyze the table as a class. Have students complete Questions 1 and with a partner. Then share the responses as a class. Talk the Talk The Properties of Equality allow you to balance and solve equations involving any number. Properties of Equality For all numbers a, b, and c, Addition Property of Equality If a b, then a 1 c b 1 c. Subtraction Property of Equality If a b, then a c b c. Multiplication Property of Equality If a b, then ac bc. Division Property of Equality If a b and c fi 0, then a c b c. 1. Describe the strategies you can use to solve any two-step equation. When solving two-step equations, I need to first isolate the term with the variable on one side of the equation by adding or subtracting the constant. Then isolate the variable by multiplying or dividing by the coefficient.. Describe a solution to any equation. A solution makes the equation true. Be prepared to share your solutions and methods. 1 Chapter 1 Linear Equations

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