Sample: Do Not Reproduce EE6 STUDENT PACKET. EXPRESSIONS AND EQUATIONS Student Packet 6: Solving Equations 2. Name Period Date

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1 Name Period Date EXPRESSIONS AND EQUATIONS Student Packet 6: EE6.1 Cups and Counters 3 Use a visual model to solve multi-step algebraic equations. Solve equations that require working with variables that are preceded by a minus sign. Use substitution to check solutions. EE6.2 Train Problems Solve and check single-variable word problems involving rate, time, and distance. Use multiple representations for solving word problems, including words, pictures, numbers, and symbols. Use algebraic notation and conventions to solve problems. EE6.3 Solving Equations with Integer Coefficients Solve multi-step algebraic equations. Justify the steps in equation solving. Use substitution to check solutions. EE6 STUDENT PACKET EE6.4 Vocabulary, Skill Builders, and Review Expressions and Equations (Student Pages) EE6 SP

2 equation evaluate expression integers numerical coefficient simplify WORD BANK (EE6) Word Definition Example or Picture solve Expressions and Equations (Student Pages) EE6 SP0

3 6.1 Cups and Markers 3 Ready (Summary) We will solve more linear equations using a visual model and record the steps with pictures and algebraic symbols. For these equations, some of the variables will be preceded by a minus sign. CUPS AND COUNTERS 3 Go (Warmup) Set (Goals) Use a visual model to solve multi-step algebraic equation. Solve equations that require working with variables that are preceded by a minus sign. Use substitution to check solutions. Fill in the missing numbers and name the property of arithmetic illustrated by each equation. Equation 1. (3)(4) ( )(3) (5 + 10) 2(5) + ( )(10) (6) ( )(6) (3 ) 7-14 Property EE6 SP1

4 6.1 Cups and Markers 3 ANOTHER LOOK AT CUPS AND COUNTERS Follow your teacher s instructions to build and record equations. Use a mental strategy to solve and/or to check your answers. 1. Picture Equation Λ x 3 x 2. Picture Equation -2x Picture Equation -(x + 1) 5 4. Picture Equation -2(x + 1) 4 EE6 SP2

5 6.1 Cups and Markers 3 MORE EQUATIONS WITH CUPS AND COUNTERS Build, draw, record, solve, and check each equation. 1. Picture Equation/Steps What did you do? Check your solution using substitution: -x + 4 2x Picture Equation/Steps What did you do? Check your solution using substitution: -2(x + 1) -4x 6 EE6 SP3

6 6.1 Cups and Markers 3 MORE EQUATIONS WITH CUPS AND COUNTERS (continued) Build, draw, record, solve, and check each equation. 3. Picture Equation/Steps What did you do? Check your solution using substitution: 2x -2x 4 4. Picture Equation/Steps What did you do? Check your solution using substitution: -x x + 2 EE6 SP4

7 6.1 Cups and Markers 3 MORE EQUATIONS WITH CUPS AND COUNTERS (continued) Build, draw, record, solve, and check each equation. 5. Picture Equation/Steps What did you do? Check your solution using substitution: -4x 5 -(x 4) 6. Picture Equation/Steps What did you do? Check your solution using substitution: -2(x + 2) -3(x 1) EE6 SP5

8 6.1 Cups and Markers 3 MORE EQUATIONS WITH CUPS AND COUNTERS (continued) Build, draw, record, solve, and check each equation. 7. Picture Equation/Steps What did you do? Check your solution using substitution: -4 x 3x Picture Equation/Steps What did you do? Check your solution using substitution: -x + 4 2x -(2 + x) EE6 SP6

9 6.1 Cups and Markers 3 SOLVING EQUATIONS TEMPLATE Use this page to build, draw, record, solve and check additional equations. Picture Equation/Steps What did you do? Check your solution using substitution: Picture Equation/Steps What did you do? Check your solution using substitution: EE6 SP7

10 6.2 Train Problems Ready (Summary) We will use organized guess-and-check tables, pictures, and algebra to solve rate problems about traveling trains. Use arrows ( numbers. TRAIN PROBLEMS Go (Warmup) Set (Goals) Solve and check single-variable word problems involving rate, time, and distance. Use multiple representations for solving word problems, including words, pictures, numbers, and symbols. Use algebraic notation and conventions to solve problems. ) to sketch each situation. Label with the appropriate colors and Words 1. Two trains start at the same place and travel away from one another. The Blue Train travels 100 miles east and the Green Train travels 150 miles west. 2. Two trains start in the same place and travel in the same direction. The Red Train travels 50 miles south and The Black Train travels 100 miles south. 3. Two trains start 300 miles apart and travel toward one another. The Brown Train travels 100 miles west to east and the Orange Train travels 200 miles east to west to a point at which they meet. Solve: 4. If a train travels 40 miles per hour for 2 hours, how far does it travel? 5. How long does it take a train to travel 150 miles at 50 miles per hour? Arrow Pictures 6. If it takes a train 5 hours to travel 75 miles, what is its rate? EE6 SP8

11 6.2 Train Problems Two trains that are 360 miles apart are approaching one another. The Blue Train is traveling at a rate of 50 miles per hour. The Green Train is traveling at a rate of 40 miles per hour. How long will it take for them to meet? How far does each train travel? Guess Number of hours (x) x Distance The Blue Train travels TRAIN PROBLEM 1 Distance The Green Train travels Make a sketch (if needed) Total distance traveled Check Total distance 360 miles? Equation and Solution Write an equation and solve How long will it take for the Verify the solution using the it if you can. trains to meet? equation. How far does each train travel? EE6 SP9

12 6.2 Train Problems Two trains start traveling in the same direction, from the same place and at the same time. The Green Train travels at a rate of 65 miles per hour. The Red Train travels at a rate of 90 miles per hour. After some time has passed, one train is ahead by 150 miles. Which train is it? How many hours have passed? Guess Number of hours (x) x Distance The Green Train travels TRAIN PROBLEM 2 Distance The Red Train travels Make a sketch (if needed) Distance between them Check? Equation and Solution Write an equation and solve it Which train is ahead? Check your answer using the if you can. equation. How much time has passed? EE6 SP10

13 6.2 Train Problems Two trains start traveling from the same place and at the same time. The Black Train travels west at a rate of 75 miles per hour and the Orange Train travels east at a rate of 80 miles per hour. How long will it take for the trains to be 1,085 miles apart? Guess (x) x TRAIN PROBLEM 3 Make a sketch (if needed) Check? Equation and Solution Write an equation and solve it if you can. How long will it take for the trains to be 1,085 miles apart? Check your answer using the equation. EE6 SP11

14 6.2 Train Problems Two trains start traveling in the same direction from the same place and at the same time. The Purple Train travels at a rate of 110 miles per hour, and the Red Train travels at a rate of 85 miles per hour. After some time has passed, one train is ahead by 125 miles. Which train is it, and how much time has passed? Guess (x) x TRAIN PROBLEM 4 Make a sketch (if needed) Check? Equation and Solution Write an equation and solve Answer the questions. Check your answer using the it if you can. equation. EE6 SP12

15 6.2 Train Problems Two trains that are 50 miles apart are traveling in opposite directions. The Black Train travels at a rate of 48 miles per hour, and the Red Train travels at a rate of 52 miles per hour. How many hours will it be until the trains are 650 miles apart? Guess (x) x TRAIN PROBLEM 5 Make a sketch (if needed) Check? Equation and Solution Write an equation and solve Answer the questions. Check your answer using the it if you can. equation. EE6 SP13

16 6.2 Train Problems Two trains start traveling in the same direction at the same time, with the Green Train starting out 100 miles east of the Blue Train. The Blue Train travels at a rate of 65 miles per hour. The Green Train travels at a rate of 45 miles per hour. How many hours will it take for the Blue Train to catch up to the Green Train? Guess (x) x TRAIN PROBLEM 6 Make a sketch (if needed) Check? Equation and Solution Write an equation and solve Answer the questions. Check your answer using the it if you can. equation. EE6 SP14

17 6.3 Solving Equations with Integer Coefficients SOLVING EQUATIONS WITH INTEGER COEFFICIENTS Ready (Summary) We will begin to transition from solving equations with a model to using more algebraic notation. Solve the equation. Fill in the table completely. Go (Warmup) Set (Goals) Solve multi-step algebraic equations. Justify the steps in equation solving. Use substitution to check solutions. Picture Equation/Steps What did you do? Check solution by substitution: -3(x 1) x 5 EE6 SP15

18 6.3 Solving Equations with Integer Coefficients PROPERTIES OF ARITHMETIC AND EQUALITY Property Associative property of addition Commutative property of addition Identity property of addition Inverse property of addition Associative property of multiplication Commutative property of multiplication Identity property of multiplication Inverse property of multiplication Distributive property of multiplication over addition Properties of Arithmetic Symbols Example with numbers (for any number a, b, c) a + (b + c) (a + b) + c a + b b + a a + 0 a a + (-a) 0 a (b c) (a b) c a b b a a 1 1 a a a 1 a 1 a 0 a(b + c) ab + ac Abbreviation Properties of Equality Property Description Example with numbers Abbreviation Addition property of equality Multiplication property of equality If a b and c d, then a + c b + d If a b and c d, then ac bd EE6 SP16

19 6.3 Solving Equations with Integer Coefficients RIGHT OR WRONG Check each solution. If it is correct, write in what was done for each step. If it is incorrect, find the mistake and correct it. Use pictures if needed. 1. Equation/Steps What was done? 16x 9-20x+ 19 given equation 16x 9-20x x 0 16x -9 4x x x 4-7 x Check solution by substitution: subtract 16x from both sides; addition (subtraction) property of equality 2. Equation/Steps What was done? -x + 7-3(x 5) given equation -x + 7-3x x + 7-3x x x 2x x x 4 Check solution by substitution: EE6 SP17

20 6.3 Solving Equations with Integer Coefficients RIGHT OR WRONG (continued) Check each solution. If it is correct, write in what was done for each step. If it is incorrect, find the mistake and correct it. Use pictures if needed. 3. Equation/Steps What was done? -2x 1 -(x 1) given equation -2x 1 -x 1-2x 1 -x 1 +x 0 +x -x x 0 x 0 Check solution by substitution: 4. Equation/Steps What was done? -2(x + 3) 12-2x x x x 9 given equation Check solution by substitution: EE6 SP18

21 6.3 Solving Equations with Integer Coefficients SOLVING MORE EQUATIONS Write all the steps used to solve the equations. Provide justifications/explanations for each step. Use pictures as needed. 1. Equation/Steps Why can you do that? -3x + 7 3x + 19 given equation -3x + 7 3x + 19 subtraction (addition) -7-7 property of equality Check solution by substitution: 2. Equation/Steps Why can you do that? -4x 20 6x Check solution by substitution: EE6 SP19

22 6.3 Solving Equations with Integer Coefficients SOLVING MORE EQUATIONS (continued) Write all the steps used to solve the equations. Provide justifications/explanations for each step. Use pictures as needed. 3. Equation/Steps Why can you do that? -x 7-5x + 5 Check solution by substitution: 4. Equation/Steps Why can you do that? 3(x + 1) 2(x 1) 6 given equation Check solution by substitution: EE6 SP20

23 6.4 Vocabulary, Skill Builders, and Review FOCUS ON VOCABULARY (EEQ6) Fill in the crossword puzzle using the word list and vocabulary from this lesson. 4 ACROSS 1. To a variable expression, substitute a number for each variable. DOWN 1. A mathematical statement that asserts the equality of two expressions. 2. Whole numbers and their opposites. 3. A combination of numbers, variables, and operation symbols. 4. A number which is multiplied by 5 one or 5. To find a solution for a variable that more variables or powers of variables. makes an equation true. 5. Converting an expression to a simpler form EE6 SP21

24 6.4 Vocabulary, Skill Builders, and Review SKILL BUILDER 1 Use each of the numbers 1, 3, 5, and 8 exactly once to write an expression for each target value. Use any operation symbols or grouping symbols necessary. # Target Value Expression Evaluate each expression for x -3 and y x +5y x + y x+ y 6. -(x + y) 2 7. (-x + y) x + y 2 Compute mentally EE6 SP22

25 6.4 Vocabulary, Skill Builders, and Review Compute SKILL BUILDER Simplify x x x x What is the result for problems #7-10? Why is the result always the same in the problems above? EE6 SP23

26 6.4 Vocabulary, Skill Builders, and Review Compute SKILL BUILDER Simplify x x What is the result for problems #10-11? Why is the result always the same in the problems above? EE6 SP24

27 6.4 Vocabulary, Skill Builders, and Review SKILL BUILDER 4 Draw the next step suggested by this pattern. Then complete the table and find a rule for the number of toothpicks at step n. step1 step 2 step 3 step 4 Step # n Arithmetic Number of toothpicks 1. Label the horizontal and vertical axes and graph the data points. 2. Recursive Rule: Start with toothpicks, and then each time. 3. Explicit Rule: Explain what to do to the input number at each step to get the corresponding output number. 4. In which step number are there exactly 84 toothpicks? EE6 SP25

28 6.4 Vocabulary, Skill Builders, and Review SKILL BUILDER 5 Without plotting the ordered pairs, match each input-output table with a graph below. Write one or two sentences to justify each choice. 1. Graph: Explain: Input (x) Output (y) Graph: Explain: A. y B. y C. Input (x) Output (y) Make up reasonable x-y values for the graph. Could this graph represent a function? Explain. x y x 10 y x 10 x y x EE6 SP26

29 6.4 Vocabulary, Skill Builders, and Review SKILL BUILDER 6 Check each solution. If it is correct, write in what was done for each step. If it is incorrect, find the mistake and correct it. Use pictures if needed. 1. Equation/Steps What was done? -6 6 x +12 given equation -6 6x x 6-3 x Check solution by substitution: subtract 12 from both sides; addition (subtraction) property of equality 2. Equation/Steps What was done? 2( x 4) x+1 given equation (nothing done) 2x 8 x+1 2x 8 x+1 - x -x x x 9 Check solution by substitution: EE6 SP27

30 6.4 Vocabulary, Skill Builders, and Review 1. Two trains that are 720 miles apart are approaching one another. The Red Train is traveling at a rate of 80 miles per hour and the Blue Train is traveling at a rate of 100 miles per hour. How long will it take for them to meet? How far does each train travel? Guess (x) x SKILL BUILDER 7 Make a sketch (if needed) Check Equation and Solution Write an equation and solve How long will it take for the Check your answer using the it if you can. trains to meet? equation. How far does each train travel? EE6 SP28

31 6.4 Vocabulary, Skill Builders, and Review SKILL BUILDER 8 Write all the steps used to solve the equations. Provide justifications/explanations for each step. Use pictures as needed. 1. Equation/Steps Why can you do that? (include the property used) 2(x + 3) 4 10 Given equation Check solution by substitution: 2. Equation/Steps Why can you do that? 2x 3-6 x Check solution by substitution: EE6 SP29

32 6.4 Vocabulary, Skill Builders, and Review TEST PREPARATION (EE6) Show your work on a separate sheet of paper and choose the best answer. 1. Solve for x: 9 x -3(x + 1) A. x -6 B. x -4 C. x -2 D. x 6 2. Solve for x: -5x -7x 10 A. x -10 B. x -5 C. x 5 D. x Two trains start traveling from the same place and at the same time. The Orange Train travels north at a rate of 50 miles per hour, while the Green Train travels south at a rate of 70 miles per hour. How long will it take for them to be 600 miles apart? A. 4 hours B. 4.5 hours C. 5 hours D. 5.5 hours 4. Two trains start at the same place and at the same time. The Blue Train travels east at a rate of 60 miles per hour and the Yellow Train travels west at a rate of 65 miles per hour. How long will it take for them to be 500 miles apart? A. 4 hours B. 4.5 hours C. 5 hours D. 5.5 hours 5. Solve: -3(x + 5) 2x + 35 Step 1: -3x 15 2x + 35 Step 2: -3x 2x + 50 Step 3: x 50 Which is the first incorrect step in the solution shown above? A. Step 1 B. Step 2 C. Step 3 D. There are no mistakes EE6 SP30

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35 6.4 Vocabulary, Skill Builders, and Review KNOWLEDGE CHECK (EE6) Show your work on a separate sheet of paper and write your answers on this page. 6.1 Cups and Markers 3 Solve each equation. Show all steps and check your solution using substitution. Draw pictures if needed. 1. 4x -4x (x + 1) Train Problems Two trains that are 520 miles apart are approaching one another. The Brown Train is traveling at a rate of 60 miles per hour and the Orange Train is traveling at a rate of 70 miles per hour. 3. How long will it take for the trains to meet? 4. How much of that distance does each train cover? 6.3 Solving Equations with Integer Coefficients Write all steps in solving the equations. Provide justification/explanations for each step. Use pictures as needed x 10 7x ( x 2) 3( x+ 1) 0 EE6 SP33

36 Home-School Connection (EE6) Here are some questions to review with your young mathematician. 1. Solve -10 x 7x 6 for x and check your solution using substitution. 2. Two trains that are 750 miles apart are approaching one another. The Black Train is traveling at a rate of 70 miles per hour and the Red Train is traveling at a rate of 55 miles per hour. How long will it take for the trains to meet? How much of that distance does each train cover? 3. Solve for x. 2(-7x 6) 16 : Provide justifications for each step. Use pictures if necessary. Parent (or Guardian) signature 7.EE.1 7.EE.4a 8.EE.7b A-SSE-1a SELECTED COMMON CORE STATE STANDARDS FOR MATHEMATICS Apply properties of operations as strategies to add, subtract, factor, and expand linear expressions with rational coefficients. Solve word problems leading to equations of the form px + q r and p(x + q) r, where p, q, and r are specific rational numbers. Solve equations of these forms fluently. Compare an algebraic solution to an arithmetic solution, identifying the sequence of the operations used in each approach. For example, the perimeter of a rectangle is 54 cm. Its length is 6 cm. What is its width? Solve linear equations with rational number coefficients, including equations whose solutions require expanding expressions using the distributive property and collecting like terms. Interpret parts of an expression, such as terms, factors, and coefficients. FIRST PRINTING DO NOT DUPLICATE 2011 EE6 SP34