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## What are we going to learn?

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**Learning Objective**Inverse Operations We will solve two-step equations. + and – What are we going to learn? CFU and Activate Prior Knowledge An equation states1 two expressions are equal. Inverse operations are operations that undo each other. Equation x + 7 = 9 expression expression Solve the one-step equations. 1. x + 7 = 9 2. x + 3 = 7 Students, you already know how to solve one-step equations. Now we will solve two-step equations. Make Connection − 7 − 7 − 3 − 3 x = 2 x = 4 4. 2x = 4 3. 4x = 16 1 says or tells Vocabulary 2 2 4 4 x = 4 x = 2**Concept Development**Inverse Operations A two-step equation contains2 two operations. subtraction + and – addition 2x+ 3 = 7 division and multiplication CFU Solving Two-Step Equations • A two-step equation requires3two inverse operations to solve for the variable. • To keep an equation balanced, inverse operations must be done on both sides of the equation. Which of the following is an example of a two-step equation? How do you know? A 5x= 6 B 5x − 4 = 6 C x − 4 = 6 Which two inverse operations would be used to solve the two-step equation 5x − 4 = 6? How do you know? What is the difference between the solution (x= 2) and non-solution (x= 4)? 2x+ 3 = 7 Animated − 3 − 3 2 1 1 Inverse Operation Inverse Operation Inverse Operation Balance Balance Balance 2 1 1 2x+ 0 = 4 Balance Inverse Operation 2 2 2 2 x= 2 Flashcards: Inverse Operations Addition & Division The solution is the value of the variable that makes the equation true. Checking a Solution NOT a Solution Solution 2x+ 3 = 7 2x+ 3 = 7 2x+ 3 = 7 x= 2 x= 4 2 has within it 3 needs (synonym) Vocabulary ? 2(2) + 3 = 7 ? 2(4) + 3 = 7 7 = 7 11 = 7 True False**Skill Development/Guided Practice**Inverse Operations A two-step equation contains two operations. A two-step equation requires two inverse operations to solve. + and – • To keep an equation balanced, inverse operations must be done on both sides of the equation. The solution is the value of the variable that makes the equation true. and How did I/you isolate the term with the variable? How did I/you solve for the variable? How did I/you check the solution? CFU Solve two-step equations. 2 Read the problem and identify (underline) important information. Connect the problem to the given two-step equation. Isolate4 the term with the variable. Solve for the variable. Check and interpret the solution. Hint: Answer the question. 1 3 a 4 2 Flashcards: Steps to Check a Solution 3 4 2. What is the unknown value of x in the two-step equation? 1. What is the unknown value of x in the two-step equation? Now Check! Now Check! ? ? +6 +6 +3 +3 ? ? 3 •5 5• 1 2• •2 x 15 x 2 “x = 2 is the solution because it makes the equation true.” “x = 15 is the solution because it makes the equation true.” 4 separate Vocabulary**Skill Development/Guided Practice (continued)**Inverse Operations A two-step equation contains two operations. A two-step equation requires two inverse operations to solve. + and – • To keep an equation balanced, inverse operations must be done on both sides of the equation. The solution is the value of the variable that makes the equation true. and How did I/you connect the problem to the given two-step equation. How did I/you isolate the term with the variable? How did I/you solve for the variable? How did I/you check the solution? CFU Solve two-step equations. 1a Read the problem and identify (underline) important information. Connect the problem to the given two-step equation. Isolate the term with the variable. Solve for the variable. Check and interpret the solution. Hint: Answer the question. 1 a 2 2 3 3 4 4 3. A gym membership charges a flat fee5 to sign up, as well as a monthly rate. Jessica paid $50 to sign up and $25 each month. If Jessica has paid $175 to the gym, how long has she been a member? 4. A book club membership charges a flat fee to sign up, as well as an annual6 rate. Maurice paid $15 to sign up and $7 each year. If Maurice has paid $36 to the book club, how long has he been a member? Now Check! 7y + 15 = 36 Now Check! −15 −15 ? ? 7( ) + 15 = 36 7(3) + 15 = 36 25m + 50 = 175 25( ) + 50 = 175 25(5) + 50 = 175 −50 −50 7y = 21 ? 21 + 15 = 36 ? 125 + 50 = 175 7 7 25m = 125 36 = 36 175 = 175 y = 3 25 25 m = 5 If Maurice has paid $36 to the book club, he has been a member for three years. 5 one-time payment 6 yearly Vocabulary If Jessica has paid $175 to the gym, she has been a member for five months.**Skill Development/Guided Practice (continued)**Inverse Operations A two-step equation contains two operations. A two-step equation requires two inverse operations to solve. + and – • To keep an equation balanced, inverse operations must be done on both sides of the equation. The solution is the value of the variable that makes the equation true. and How did I/you connect the problem to the given two-step equation. How did I/you isolate the term with the variable? How did I/you solve for the variable? How did I/you check the solution? CFU Solve two-step equations. 1a Read the problem and identify (underline) important information. Connect the problem to the given two-step equation. Isolate the term with the variable. Solve for the variable. Check and interpret the solution. Hint: Answer the question. 1 a 2 2 3 3 4 4 5. The length of a rectangular computer monitor is 20 cm. The perimeter of the monitor is 72 cm. What is the width of the computer monitor? 6. The length of a rectangular cell phone screen is 2 in. The perimeter of the cell phone screen is 10 in. What is the width of the cell phone screen? 2w + 2l = P 2w + 2l = P 2w + 2(20) = 72 2w + 2(2) = 10 l = 2 in 2w + 40 = 72 2w + 4 = 10 l = 20 cm −40 −40 −4 −4 2w = 32 2w = 6 w w = 3 in 2 2 2 2 w w = 16 cm w = 16 w = 3 Now Check! Now Check! ? ? 2 ( ) + 4 = 10 2(3) + 4 = 10 2(16) + 40 = 72 2( ) + 40 = 72 ? ? 6 + 4 = 10 32 + 40 = 72 10 = 10 72 = 72 The width of the computer monitor is 16 cm. The width of the cell phone screen is 3 in.**Skill Development/Guided Practice (continued)**Inverse Operations A two-step equation contains two operations. A two-step equation requires two inverse operations to solve. + and – • To keep an equation balanced, inverse operations must be done on both sides of the equation. The solution is the value of the variable that makes the equation true. and How did I/you connect the problem to the given two-step equation. How did I/you isolate the term with the variable? How did I/you solve for the variable? How did I/you check the solution? CFU Solve two-step equations. 1a Read the problem and identify (underline) important information. Connect the problem to the given two-step equation. Isolate the term with the variable. Solve for the variable. Check and interpret the solution. Hint: Answer the question. 1 a 2 2 3 3 4 4 7. A middle school hosted a fall festival to raise money for both grades in the school. The seventh grade class used their half of the money to buy a $750 chemistry set. After buying the chemistry set, the 7th graders had $350 left over. How much money did the school earn at the fall festival? 8. Maria’s grandmother gave Maria money to share with her two sisters. The three girls split the money evenly and Maria used her share to buy a computer that cost $550. After buying the computer, she had $25 leftover. How much money did Maria’s grandmother give Maria and her sisters altogether? Now Check! Now Check! ? ? + 550 + 550 + 750 + 750 ? 575 − 550 = 25 ? 1100 − 750 = 350 • 3 3 • • 2 2 • 25 = 25 350 = 350 m = 1725 m = 2200 The fall festival at the middle school earned $2,200. Maria’s grandmother gave Maria and her sisters $1,725 to share.**Relevance**In chemistry, we use equations to show chemical reactions. The equation below shows methane burning with oxygen. CH4 + 2 O2 CO2 + 2 H2O A two-step equation contains two operations. A two-step equation requires two inverse operations to solve. • To keep an equation balanced, inverse operations must be done on both sides of the equation. The solution is the value of the variable that makes the equation true. Solving two-step equations will help you solve problems in other classes. 1 Solving two-step equations will help you solve real-world problems. 2 B Street Bikes rents bicycles for $16 plus $4perhour. Eduardo paid $36 to rent a bicycle. For how many hours did he rent the bicycle? 4h+ 16 = 36 Does anyone else have another reason why it is relevant to solve two-step equations? (Pair-Share) Why is it relevant to solve two-step equations? You may give one of my reasons or one of your own. Which reason is more relevant to you? Why? CFU Solving two-step equations will help you do well on tests. 3 Sample Test Question: 43. Joan needs $60 for a class trip. She has $32. She can earn $4 an hour mowing lawns. If the equation shows this relationship, how many hours must Joan work to have the money she needs? 4h + 32 = 60 A 7 hours B 17 hours C 23 hours D 28 hours**Inverse Operations**A two-step equation contains two operations. A two-step equation requires two inverse operations to solve. + and – • To keep an equation balanced, inverse operations must be done on both sides of the equation. The solution is the value of the variable that makes the equation true. and Skill Closure Solve two-step equations. Read the problem and identify (underline) important information. Connect the problem to the given two-step equation. Isolate the term with the variable. Solve for the variable. Check and interpret the solution. Hint: Answer the question. 1 a 2 3 4 1. The length of the basketball backboard is 2 m. The perimeter of the backboard is 6 m. What is the width of the basketball backboard? l = 2 m 2w + 2l = P Now Check! 2w + 2(2) = 6 ? 2w + 4 = 6 2(1) + 4 = 6 2w + 4 = 6 − 4 − 4 w = 1 m w ? 2 + 4 = 6 2w = 2 6 = 6 2 2 w = 1 The width of the basketball backboard is 1 m. Constructed Response Closure Write an example of a two-step equation. How do you know it is an example of a two-step equation? Summary Closure What did you learn today about solving two-step equations? (Pair-Share)**Independent Practice**Inverse Operations A two-step equation contains two operations. A two-step equation requires two inverse operations to solve. + and – • To keep an equation balanced, inverse operations must be done on both sides of the equation. The solution is the value of the variable that makes the equation true. and Solve two-step equations. Read the problem and identify (underline) important information. Connect the problem to the given two-step equation. Isolate the term with the variable. Solve for the variable. Check and interpret the solution. Hint: Answer the question. 1 a 2 3 4 1. What is the unknown value of x in the two-step equation? 2. What is the unknown value of x in the two-step equation? Now Check! Now Check! -3x – 12 = -3 +12 +12 -8 -8 -3x 9 -3 -3 -3 4• •4 x x -12 -3 “x = -12 is the solution because it makes the equation true.” “x = -3 is the solution because it makes the equation true.” 3. A cell phone service charges a flat fee for activation1, as well as a monthly rate. Petra paid $145 to sign up and $50 each month. If Petra has paid $395, how long has she had her cell phone service? Now Check! If Petra has paid $395, she has had her cell phone service for five months. Vocabulary 1 start ? 50m + 145 = 395 50(5) + 145 = 395 − 145 − 145 ? 250 + 145 = 395 50m = 250 395 = 395 50 50 m = 5**Independent Practice (continued)**Inverse Operations A two-step equation contains two operations. A two-step equation requires two inverse operations to solve. + and – • To keep an equation balanced, inverse operations must be done on both sides of the equation. The solution is the value of the variable that makes the equation true. and Solve two-step equations. Read the problem and identify (underline) important information. Connect the problem to the given two-step equation. Isolate the term with the variable. Solve for the variable. Check and interpret the solution. Hint: Answer the question. 1 a 2 3 4 4. The length of a football field is 360 ft. The perimeter of the football field is 1040 ft. What is the width of the football field? 5. Bill and his four friends earned money doing chores for their neighbors. Bill spent $15 of his share on a new book and had $20 left. How much money did Bill and his friends earn doing chores? 2w + 2l = P l = 360 ft 2w + 2(360) = 1040 2w + 720 = 1040 Now Check! − 720 − 720 ? w + 15 + 15 2w = 320 w = 160 ft Home Visitor 2 2 ? • 5 35 − 15 = 20 5 • w = 160 20 = 20 m = 175 Now Check! ? 2(160) + 720 = 1040 ? 320 + 720 = 1040 1040 = 1040 Bill and his friends earned $175 doing chores. The width of the football field is 160 ft.**Periodic Review 1**Inverse Operations A two-step equation contains two operations. A two-step equation requires two inverse operations to solve. + and – • To keep an equation balanced, inverse operations must be done on both sides of the equation. The solution is the value of the variable that makes the equation true. and Solve two-step equations. Read the problem and identify (underline) important information. Connect the problem to the given two-step equation. Isolate the term with the variable. Solve for the variable. Check and interpret the solution. Hint: Answer the question. 1 a 2 3 4 1. To ensure a swimming pool is safe, a fence is being built to surround it. To surround the pool, the perimeter of the fence must be 170 m and the length 55 m. How wide does the fence need to be to surround the entire pool? 2. In basketball, each basket scores two points and free throws one point. Chandra made seven free throws. If she scored a total of 23 points, how many baskets did she make? l = 55ft 2w + 2l = P 2w + 2(55) = 170 2b + f = 23 2w + 110 = 170 ? 2b + 7 = 23 2(8) + 7 = 23 − 110 − 110 − 7 − 7 w = 30ft ? Home Visitor 16 + 7 = 23 2w = 60 2b = 16 2 2 23 = 23 2 2 w = 30 b = 8 Now Check! ? 2(30) + 110 = 170 ? 60+ 110 = 170 170 = 170 If Chandra made seven free throws and scored 23 points, she made eight baskets. The width of the fence must be 30 m.**Periodic Review 2**Inverse Operations A two-step equation contains two operations. A two-step equation requires two inverse operations to solve. + and – • To keep an equation balanced, inverse operations must be done on both sides of the equation. The solution is the value of the variable that makes the equation true. and Solve two-step equations. Read the problem and identify (underline) important information. Connect the problem to the given two-step equation. Isolate the term with the variable. Solve for the variable. Check and interpret the solution. Hint: Answer the question. 1 a 2 3 4 1. A rental car charges an initial1 payment of $125, plus $34 every day the car is rented. Charles rents a car and pays $363. How long did Charles rent the car? 2. In basketball, each basket scores two points and free throws one point. Troy made eleven free throws. If he scored a total of 19 points, how many baskets did he make? 1 beginning ? 34d + 125 = 363 34(7) + 125 = 363 ? 2b + f = 19 2(4) + 11 = 19 − 125 − 125 ? 2b + 11 = 19 238 + 125 = 363 ? 8 + 11 = 19 34d = 238 − 11 − 11 363 = 363 34 34 19 = 19 2b = 8 d = 7 2 2 b = 4 Charles rented the car for seven days. If Troy made eleven free throws and scored 19 points, he made four baskets.**Periodic Review 3**Inverse Operations A two-step equation contains two operations. A two-step equation requires two inverse operations to solve. + and – • To keep an equation balanced, inverse operations must be done on both sides of the equation. The solution is the value of the variable that makes the equation true. and Solve two-step equations. Read the problem and identify (underline) important information. Connect the problem to the given two-step equation. Isolate the term with the variable. Solve for the variable. Check and interpret the solution. Hint: Answer the question. 1 a 2 3 4 1. Janine and her three brothers are given spending money to share. Janine sets $75 aside to save, and spends the other $5 on lunch that afternoon. How much money were Janine and her brothers given? 2. An oversized America flag is created to hang above the fence of a softball field. The flag must have a width of 10 ft. and a perimeter of 58 ft. What must the length of the American flag be? 2w + 2l = P l = 19 ft 2(10) + 2l = 58 ? 20 + 2l = 58 − 20 − 20 + 75 + 75 ? 80 − 75 = 5 2l = 38 w = 10 ft 5 = 5 2 2 ● 4 4 ● l = 19 m = 320 ? 20 + 2(19) = 58 ? 20 + 38 = 58 58 = 58 Janine and her brothers were given $320. The length of the flag must be 19 ft.**Access Common Core**Inverse Operations A two-step equation contains two operations. A two-step equation requires two inverse operations to solve. + and – • To keep an equation balanced, inverse operations must be done on both sides of the equation. The solution is the value of the variable that makes the equation true. and 1. A rental car charges an initial1 payment of $125, plus $34 every day the car is rented. Charles rents a car and pays $363. How long did Charles rent the car? Which of the following statements is true? How do you know? The inverse operation subtraction (-) will be used to solve this problem. Subtraction (–) will be the first inverse operation to solve this problem. The equation used to solve this problem is The meaning of the solution to this problem is the amount of money he earned. Yes/No Yes/No Yes/No 34d + 125 = 363. Yes/No 2. In basketball, each basket scores two points and free throws one point. Troy made eleven free throws. If he scored a total of 19 points, how many baskets did he make? Which of the following statements is true? How do you know? The inverse operation addition (+) will be used to solve this problem. The equation used to solve this problem is The equation used to solve this problem is The meaning of the solution to this problem is the total number of points that Troy scored. Yes/No Yes/No 2b + f = 19. Yes/No 2b + 11 = 19. Vocabulary 1 beginning Yes/No**Access Common Core**Inverse Operations A two-step equation contains two operations. A two-step equation requires two inverse operations to solve. + and – • To keep an equation balanced, inverse operations must be done on both sides of the equation. The solution is the value of the variable that makes the equation true. and 1. 4x+5 =19 2. 4x+11 =13 3. In your own words, describe why equations #1 and #2 have the same solution? ____________________________________________________________________________________________________________________________ ____________________________________________________________________________________________________________________________ ______________________________________________________________**Access Common Core**Inverse Operations A two-step equation contains two operations. A two-step equation requires two inverse operations to solve. + and – • To keep an equation balanced, inverse operations must be done on both sides of the equation. The solution is the value of the variable that makes the equation true. and 1. An oversized American flag is created to hang above the fence of a softball field. The flag must have a width of 10 ft. and a perimeter of 58 ft. What must the length of the American flag be? Which of the following statements is true? How do you know? The inverse operation subtraction (-) will be used to solve this problem. The length of the American flag will be 19 ft. The equation used to solve this problem is The meaning of the solution to this problem is the length in feet of the flag. Yes/No Yes/No 20 + 2l = 58. Yes/No Yes/No 2. To ensure a swimming pool is safe, a fence is being built to surround it. To surround the pool, the fence must be 170 m long and have a length of 55 m. How wide does the fence need to be to surround the entire pool? Which of the following statements is true? How do you know? The meaning of the solution to the problem is the distance around the fence. The width of the fence will be 60 m. The inverse operation addition (+) will be used to solve the problem. The width of the fence will be 110 m. Yes/No Yes/No Yes/No Yes/No**Flash Cards (Inverse Operations)**Return to Concept A two-step equation contains two operations. A two-step equation requires two inverse operations to solve. What are the two operations of the equation? What two inverse operations would be used to solve the equation? How do you know? Subtraction Division Addition Multiplication**Flash Cards (Inverse Operations)**Return to Concept A two-step equation contains two operations. A two-step equation requires two inverse operations to solve. 3x- 4 =-13 What are the two operations of the equation? What two inverse operations would be used to solve the equation? How do you know? Subtraction Multiplication Addition Division**Flash Cards (Inverse Operations)**Return to Concept A two-step equation contains two operations. A two-step equation requires two inverse operations to solve. 6 + 4x= 14 What are the two operations of the equation? What two inverse operations would be used to solve the equation? How do you know? Addition Multiplication Subtraction Division**Flash Cards (Inverse Operations)**Return to Concept A two-step equation contains two operations. A two-step equation requires two inverse operations to solve. What are the two operations of the equation? What two inverse operations would be used to solve the equation? How do you know? Addition Division Subtraction Multiplication**Flash Cards (Checking Solution)**The solution is the value of the variable that makes the equation true. Check the solution to the two-step equations. Solve two-step equations Substitue1 the given solution for the variable. Simplify the equation. Interpret2 the given solution. “___ is/is not the solution because __________.” 1 2 3 Is 20 the value of x that makes the equation true? Whiteboard “20 _____ the value of x that makes the equation true because _________.” 1 replace 2 explain Vocabulary**Flash Cards (Checking Solution)**The solution is the value of the variable that makes the equation true. Check the solution to the two-step equations. Solve two-step equations Substitue1 the given solution for the variable. Simplify the equation. Interpret2 the given solution. “___ is/is not the solution because __________.” 1 2 3 Is 12 the value of x that makes the equation true? Whiteboard “12 _____ the value of x that makes the equation true because _________.” 1 replace 2 explain Vocabulary**Flash Cards (Checking Solution)**The solution is the value of the variable that makes the equation true. Check the solution to the two-step equations. Solve two-step equations Substitue1 the given solution for the variable. Simplify the equation. Interpret2 the given solution. “___ is/is not the solution because __________.” 1 2 3 Is 5 the value of x that makes the equation true? Whiteboard 6 + 4x= 14 “5 _____ the value of x that makes the equation true because _________.” 1 replace 2 explain Vocabulary**Flash Cards (Checking Solution)**The solution is the value of the variable that makes the equation true. Check the solution to the two-step equations. Solve two-step equations Substitue1 the given solution for the variable. Simplify the equation. Interpret2 the given solution. “___ is/is not the solution because __________.” 1 2 3 Is 4 the value of x that makes the equation true? Whiteboard 7x- 6 = 24 “4 _____ the value of x that makes the equation true because _________.” 1 replace 2 explain Vocabulary