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1 Name Period Date EQUATIONS AND INEQUALITIES Student Packet 2: Solving Equations 2 EQ2.1 Cups and Counters 3 Use a visual model to solve multi-step algebraic equations. Solve equations that require working with integer coefficients.. Use substitution to check solutions. EQ2.2 Train Problems Solve single-variable word problems involving rate, time, and distance, and check the solutions. Use multiple representations for solving word problems, including words, pictures, numbers, and symbols. Use algebraic notation and conventions to solve problems. EQ2.3 Solving Equations with Integer Coefficients Solve multi-step algebraic equations. Justify the steps in equation solving. Use substitution to check solutions. EQ2 STUDENT PACKET EQ2.4 Vocabulary, Skill Builders, and Review Equations and Inequalities (Student Pages) EQ2 SP

2 WORD BANK (EQ2) equation evaluate expression integers numerical coefficient simplify Word Definition Example or Picture solve Equations and Inequalities (Student Pages) EQ2 SP0

3 6.1 Cups and Markers 3 CUPS AND COUNTERS 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. Go (Warmup) Set (Goals) Use a visual model to solve multi-step algebraic equations. Solve equations that require working with variables with integer coefficients. Use substitution to check solutions. Fill in the missing numbers and name the property of arithmetic illustrated by each equation. Equation Property 1. (3)(4) ( )(3) (5 + 10) 2(5) + ( )(10) (6) ( )(6) (3 ) 7-14 EQ2 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 EQ2 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 EQ2 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 EQ2 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) EQ2 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) EQ2 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: EQ2 SP7

10 6.2 Train Problems TRAIN PROBLEMS Ready (Summary) We will use organized guess-and-check tables, pictures, algebra, and words to solve rate problems about traveling trains. Use arrows ( numbers. 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 east and The Black Train travels 100 miles east. 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? Arrow Pictures 5. How long does it take a train to travel 150 miles at 50 miles per hour? 6. If it takes a train 5 hours to travel 75 miles, what is its rate? EQ2 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 Check your answer by it if you can. trains to meet? substituting into the equation. How far does each train travel? EQ2 SP9

12 6.2 Train Problems TRAIN PROBLEM 2 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 Distance The Red Train travels Make a sketch (if needed) Distance between them Equation and Solution Which train is ahead? Write an equation and solve it if you can. Check? How much time has passed? Check your answer by substituting into the equation. EQ2 SP10

13 6.2 Train Problems TRAIN PROBLEM 3 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 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 by substituting into the equation. EQ2 SP11

14 6.2 Train Problems TRAIN PROBLEM 4 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 Make a sketch (if needed) Check? Equation and Solution Write an equation and solve Answer the questions. Check your answer by it if you can. substituting into the equation. EQ2 SP12

15 6.2 Train Problems TRAIN PROBLEM 5 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 Make a sketch (if needed) Check? Equation and Solution Write an equation and solve Answer the questions. Check your answer by it if you can. substituting into the equation. EQ2 SP13

16 6.2 Train Problems TRAIN PROBLEM 6 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 Make a sketch (if needed) Check? Equation and Solution Write an equation and solve Answer the questions. Check your answer by it if you can. substituting into the equation. EQ2 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. Go (Warmup) Solve the equation. Fill in the table completely. 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 EQ2 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 when a 0 a(b + c) ab + ac Abbreviation Properties of Equality Property Description Example with numbers Abbreviation Addition property of equality If a b and c d, then a + c b + d Multiplication property of equality If a b and c d, then ac bd EQ2 SP16

19 6.3 Solving Equations with Integer Coefficients RIGHT OR WRONG Check each solution. If any steps are not correct, find the mistake and correct it. If all steps are correct, write what was done for each step. 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: EQ2 SP17

20 6.3 Solving Equations with Integer Coefficients RIGHT OR WRONG (continued) Check each solution. If any steps are not correct, find the mistake and correct it. If all steps are correct, write what was done for each step. 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 Check solution by substitution: given equation EQ2 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: EQ2 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: EQ2 SP20

23 6.4 Vocabulary, Skill Builders, and Review FOCUS ON VOCABULARY (EQ2) 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. The numerical factor of a term 5. To find a value for a variable that makes an equation true. 5. To convert an expression to a simpler form EQ2 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 EQ2 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? EQ2 SP23

26 6.4 Vocabulary, Skill Builders, and Review Compute SKILL BUILDER Simplify. 2 x x What is the result for problems #10-11? Why is the result always the same in the problems above? EQ2 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? EQ2 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 EQ2 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: EQ2 SP27

30 6.4 Vocabulary, Skill Builders, and Review SKILL BUILDER 7 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 Make a sketch (if needed) Check Equation and Solution Write an equation and solve How long will it take for the Check your answer by it if you can. trains to meet? substituting into the equation. How far does each train travel? EQ2 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: EQ2 SP29

32 6.4 Vocabulary, Skill Builders, and Review SKILL BUILDER 9 Solve the inequalities. Use mental math. Remember: to show solutions are less than (<) or greater than (>) on the graph, use an open circle ( ) and an arrow (called a ray). To show solutions are less than or equal to ( ) or greater than or equal to ( ) on the graph, use a closed circle ( ) and a ray. Check a point on the ray to verify that it makes the inequality true. 1. 4x < -8 Graph: Check a point on the ray. 3. x 5 > -3 Graph: Check a point on the ray. 5. x Graph: Check a point on the ray. 2. x Graph: Check a point on the ray. x Graph: Check a point on the ray. x 6. > 4 2 Graph: Check a point on the ray. EQ2 SP30

33 6.4 Vocabulary, Skill Builders, and Review TEST PREPARATION (EQ2) 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 EQ2 SP31

34 6.4 Vocabulary, Skill Builders, and Review This page is left blank intentionally. EQ2 SP32

35 6.4 Vocabulary, Skill Builders, and Review KNOWLEDGE CHECK (EQ2) 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 far does each train travel? 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 EQ2 SP33

36 Home-School Connection (EQ2) 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 2012 EQ2 SP34