6th Grade. Equations & Inequalities.

Transcription

1 1

2 6th Grade Equations & Inequalities

3 Table of Contents Equations and Identities Tables Determining Solutions of Equations Solving an Equation for a Variable Click on a topic to go to that section. Solving One Step Addition & Subtraction Equations Solving One Step Multiplication & Division Equations Writing Equations Writing Simple Inequalities Solutions to Simple Inequalities Graphing Solution Sets to Simple Inequalities Glossary & Standards Teacher Notes 3

4 Equations and Identities Return to Table of Contents 4

5 Equations and Identities What is an equation? How is it different than an identity? Discuss in your groups. Math Practice 5

6 Equations and Identities An equation is created when two expressions are set equal to one another such that they are equal for some values of their variables, but not for all. If they are equal for all values of their variables, then that is an identity, not an equation. So, not all mathematical statements which include an equal sign are equations...some are identities. 6

7 Equations and Identities Here are some identities: = = 7 2x = 2x 9/3 = 3 These are always true...there are no values that can be assigned to the variables for which these would be untrue. 7

8 Equations and Identities Here are some equations: x = 5t v = 7 + x a = x 8 In all these cases, the variables are interdependent. They are only true for certain sets of variables. Changing the value of the variable on the right side of the equation, changes the possible values of the variables on the left side...and vice versa. These are equations (not identities) since knowing the value of one variable changes the possible value(s) of the other(s). 8

9 Tables Return to Table of Contents 9

10 Writing Equations Write an equation in words. Then translate that into a mathematical equation. Tim is eight years older than Kathy. Write an equation for Kathy's age. Math Practice Tim's click age is for equal equation to Kathy's in words plus 8 click for mathematical T = K + 8 equation 10

11 Writing Equations Write an equation in words. Then translate that into a mathematical equation. Tim is eight years older than Kathy. Write an equation for Kathy's age. Math Practice Tim's click age is for equal equation to Kathy's in words plus 8 click for mathematical T = K + 8 equation 11

12 Write an equation in words. Writing Equations Then translate that into a mathematical equation. Bob is 6 inches less than twice the height of Fred. Write an equation for Bob's height. Bob's height click equals for equation double in Fred's words height less 6 click for mathematical B = 2F 6 equation 12

13 Write an equation in words. Writing Equations Then translate that into a mathematical equation. Speed is equal to the distance traveled in an amount of time. Speed click for equals equation distance in words divided by time click for mathematical s = d/t equation 13

14 Tables and Expressions Let's use this table to find some solutions to the equation s = d/t; where s represents speed (in meters/second), d represents distance (in meters) and t represents time (in seconds). We've entered the distance traveled and the time it took to travel that distance in two of the columns. Use the equation (s = d/t) to find the speeds and fill in the blank column. s = d/t d (m) t (s) s (m/s) Math Practice 14

15 Tables and Expressions Note that in the first three sets of answers, the object was moving at a speed of 15 m/s. The final three sets of answers are for an object traveling four times faster, at 60 m/s. s = d/t But, in all cases, knowing the value of two of the three variables determines the values of the third. d (m) t (s) s (m/s)

16 Determining Solutions of Equations Return to Table of Contents 16

17 Determining the Solutions of Equations A solution to an equation is a number that makes the equation true. In order to determine if a number is a solution, replace the variable with the number and evaluate the equation. If the number makes the equation true, it is a solution. If the number makes the equation false, it is not a solution. 17

18 Determining the Solutions of Equations Which of the following is a solution of the equation from the solution set? x + 7 = 9 {2, 3, 4, 5} Write the equation four times. Each time replace x with one of the possible solutions and simplify to see if it is true = = = = 9 9 = 9 10 = 9 11 = 9 12 = 9 Yes No No No Answer: 2 is the solution to x + 7 = 9 18

19 Determining the Solutions of Equations Which of the following is a solution of the equation? y 12 = 8 {17, 18, 19, 20} Write the equation four times. Each time replace y with one of the possible solutions and simplify to see if it is true = = = = 8 5 = 8 6 = 8 7 = 8 8 = 8 No No No Yes Answer: 20 is the solution to y 12 = 8 19

20 1 Which of the following is a solution to the equation? x + 17 = 21 {2, 3, 4, 5} Answer 20

21 2 Which of the following is a solution to the equation? m 13 = 28 {39, 40, 41, 42} Answer 21

22 3 Which of the following is a solution to the equation? 12b = 132 {9, 10, 11, 12} Answer 22

23 4 Which of the following is a solution to the equation? d = 7.5 {2.5, 3, 3.5, 4} Answer 23

24 Solving an Equation for a Variable Return to Table of Contents 24

25 The Rules for Solving Equations Like in any game there are a few rules. There are three rules which will allow you to solve any one step equation. 25

26 The Rules Here are the three rules. Let's examine them, one at a time. 1. To "undo" a mathematical operation, you must perform the inverse operation. 2. You can do anything you want (except divide by zero) to one side of an equation, as long as you do the same thing to the other. 3. You can always switch the left and right sides of an equation. 26

27 The Rules 1. To "undo" a mathematical operation, you must do the opposite. We learned earlier that for every mathematics operation, there is an inverse operation which undoes it: when you do both operations, you get back to where you started. When the variable for which we are solving is connected to something else by a mathematical operation, we can eliminate that connection by using the inverse of that operation. 27

28 The Rules 2. You can do anything you want (except divide by zero) to one side of an equation, as long as you do the same to the other side. If the two expressions on the opposite sides of the equal sign are equal to begin with, they will continue to be equal if you do the same mathematical operation to both of them. This allows you to use an inverse operation on one side, to undo an operation, as long as you also do it on the other side. You can just never divide by zero (or by something which turns out to be zero) since the result of that is always undefined. 28

29 The Rules 3. You can always switch the left and right sides of an equation. Once an equation has been solved for a variable, it'll be a lot easier to use if that variable is moved to the left side. Mathematically, this has no effect since the both sides are equal. But, it's easier to use the equation if the side for which you are solving is on the left and values are substituted on the right. 29

30 Solving Equations When solving equations, the goal is to isolate the variable on one side of the equation in order to determine its value (the value that makes the equation true). x + 7 = 32 The goal: get "x" by itself on one side of the equal sign. 30

31 Inverse Operations For each equation, write the inverse operation needed to solve for the variable. a.) y + 7 = 14 subtract 7 b.) a 21 = 10 add 21 click c.) 5s = 25 click divide by 5 d.) x = 5 multiply click by click Math Practice 31

32 5 What is the inverse operation needed to solve this equation? 7x = 49 A B C D Addition Subtraction Multiplication Division Answer 32

33 6 What is the inverse operation needed to solve this equation? x 3 = 12 A B C D Addition Subtraction Multiplication Division Answer 33

34 7 What is the inverse operation needed to solve this equation? x 5 A B C D = 8 Addition Subtraction Multiplication Division Answer 34

35 8 What is the inverse operation needed to solve this equation? 25 + x = 30 A B C D Addition Subtraction Multiplication Division Answer 35

36 Solving One Step Addition & Subtraction Equations Return to Table of Contents 36

37 One Step Equations HINTS: To solve equations, you must work backwards through the order of operations to find the value of the variable. Remember to use inverse operations in order to isolate the variable on one side of the equation. Whatever you do to one side of an equation, you MUST do to the other side! 37

38 One Step Equations Example: y + 9 = The inverse of adding 9 y = 7 is subtracting 9 Remember whatever you do to one side of an equation, you MUST do to the other!!! 38

39 One Step Equations Try These! Solve each equation. click x 8 = x = 6 click 2 = x = x 39

40 One Step Equations Try These! Solve each equation. click x + 2 = x = 16 click 7 = x = x 40

41 One Step Equations Try These! Solve each equation. click x + 5 = x = 2 15 = x = x click 41

42 9 Solve. x 6 = 11 Answer 42

43 10 Solve. j + 15 = 34 Answer 43

44 11 Solve t = 100 Answer 44

45 12 Solve. w 225 = 300 Answer 45

46 13 Solve x = 57 Answer 46

47 14 Solve. y 17 = 51 Answer 47

48 15 Solve. n 15 = 23 Answer 48

49 Solving One Step Multiplication & Division Equations Return to Table of Contents 49

50 One Step Equations Example: 6m = The inverse of multiplying by 6 m = 12 is dividing by 6 Remember whatever you do to one side of an equation, you MUST do to the other!!! 50

51 One Step Equations Try These! Solve each equation. 3x = x = 5 4x = x = 3 click click click 25 = 5x = x 51

52 One Step Equations Try These! Solve each equation. click (2) x = 10 (2) x 2 ( 6) = 36 ( 6) 6 x = 20 x = 216 click 52

53 16 Solve. 115 = 5x Answer 53

54 17 Solve. x 9 = 12 Answer 54

55 18 Solve. 51 = 17y Answer 55

56 19 Solve. 3 = x 7 Answer 56

57 20 Solve. 108 = 12r Answer 57

58 21 Solve. 33 = 11m Answer 58

59 22 Solve. 23 = x 5 Answer 59

60 Writing Equations Return to Table of Contents 60

61 Writing Equations We already know how to translate words into expressions. We just need to learn what the equal sign would be represented by. Think of situations where you would use an equals sign. 61

62 Equals List words that indicate equals. Answer 62

63 23 Which equation represents seven minus five is six less than a number. A 7 = 5 + n 6 B 75 = 6 n C 7 5 = n 6 D 7 5 = 6 n Answer 63

64 24 Which equation represents six less than a number comes to the sum of three and seven. A n 6 = B 6 n = C 7 3 = n + 6 Answer D = n 7 64

65 25 Which equation represents ten times a number totals sixty plus twenty. A 10n + 60 = 20 B n10 = C = n Answer D 10n =

66 26 Which equation represents twenty plus four is the same as the product of fourteen and a number. A 24 = 14n B 14n = C = 14n Answer D x 14 = n 66

67 Writing Equations You will now use your knowledge of writing equations to write an equation for real life scenarios. Math Practice George is buying video games online. The cost of the video is $30.00 per game. He spent a total of $ How many games did he buy in all? Lets pull out the pieces of information, and put them in place. 67

68 Writing Equations George is buying video games online. The cost of the video is $30.00 per game. He spent a total of $ How many games did he buy in all? $30.00 per game translates to click 30g Notice that the video games are "per game". We are never told how many games he bought. So we use a variable to represent the number of games. Lets use "g". He spent a total of $ translates to = click 120 We know that total means equals. How many games did he buy in all? means that we are solving for "g". This is the question we need to answer. 68

69 Writing Equations Lets put it all together and solve the equation. 30 g = 120 cost of one video game number of games totals amount he spent Answer 69

70 27 Alice has the 5 newest DVDs, which is 4 less than the amount Jon has. Which equation represents the number of DVDs Jon has. A n + 5 = 4 B 5 = n 4 C 5 4 = n D 4 n = 9 Answer 70

71 28 Now solve the equation... Alice has the 5 newest DVDs, which is 4 less than the amount Jon has. Which equation represents the number of DVDs Jon has. Answer 71

72 29 Jasmine, who bought $5 worth of candy, spent $3 more than Leah spent. Which equation represents the amount that Leah spent? A x 3 = 5 B 5 = x + 3 C 5 + x = 3 Answer D 3x = 5 72

73 30 Now solve the equation... Jasmine, who bought $5 worth of candy, spent $3 more than Leah spent. Which equation represents the amount that Leah spent? Answer 73

74 31 Kate got a 93 on her quiz retake. That is 14 points higher than her original grade. Part A Select an answer from each group to create an equation that can be used to determine g, the original grade. g = A + E 14 G 14 Answer B C x D / F 93 H 93 74

75 32 Kate got a 93 on her quiz retake. That is 14 points higher than her original grade. Part B What is the value of g, the original grade? Answer 75

76 33 Two brothers put their money together to buy a $19 video game. One contributed $8. Part A Select an answer from each group to create an equation that can be used to determine d, the number of dollars the other brother contributed. d = A + E 19 G 19 Answer B C x D / F 8 H 8 76

77 34 Two brothers put their money together to buy a $19 video game. One contributed $8. Part B What is the value of d, the number of dollars the other brother contributed? Answer 77

78 35 James is 3 times as old as Thomas, who is 8 years old. Part A Select an answer from each group to create an equation that can be used to determine j, James' age. = Answer A 3 B 8 C j D + E x F 8 G j H 3 I 8 J j 78

79 36 James is 3 times as old as Thomas, who is 8 years old. Part B What is the the age of James, j? Answer 79

80 37 Kellie bought 8 towels and spend $ Each towel costs the same amount. Part A Select an answer from each group to create an equations that can be used to determine t, the price, in dollars, of 1 towel. t = A + E 8 G 8 Answer B C x F H D / From PARCC EOY sample test calculator #1 80

81 38 Kellie bought 8 towels and spend $ Each towel costs the same amount. Part B What is the price, in dollars, of 1 towel? Answer From PARCC EOY sample test calculator #1 81

82 Writing Simple Inequalities Return to Table of Contents 82

83 Symbols What do these symbols mean? Less Than click to reveal Less Than or Equal To Greater Than Greater Than or Equal To 83

84 Inequality An inequality is a statement that two quantities are not equal. The quantities are compared by using one of the following signs: Symbol Expression Words < A < B A is less than B > A > B A is greater than B < A < B > A > B A is less than or equal to B A is greater than or equal to B 84

85 Inequality When am I ever going to use it? Your parents and grandparents want you to start eating a healthy breakfast. The table shows the nutritional requirements for a healthy breakfast cereal with milk. Healthy Breakfast Cereals (per serving) Fat Less than 3 grams Protein More than 5 grams Answer & Math Practice Fiber Sugar At least 3 grams At most 5 grams Suppose your favorite cereal has 2 grams of fat, 7 grams of protein, 3 grams of fiber and 4 grams of sugar. Is it a healthy cereal? 85

86 Inequality Healthy Breakfast Cereals (per serving) Fat Protein Fiber Sugar Less than 3 grams More than 5 grams At least 3 grams At most 5 grams Answer Is a cereal with 3 grams of fiber considered healthy? 86

87 Inequality Healthy Breakfast Cereals (per serving) Fat Protein Fiber Sugar Less than 3 grams More than 5 grams At least 3 grams At most 5 grams Answer Is a cereal with 5 grams of sugar considered healthy? 87

88 Inequality When you need to use an inequality to solve a word problem, you may encounter one of the phrases below. Important Words Sample Sentence Equivalent Translation is more than is greater than must exceed 88

89 Inequality Here are some more expressions you may encounter: Important Words Sample Sentence Equivalent Translation cannot exceed is at most is at least 89

90 Read Inequalities How are these inequalities read? > 3 Two plus two is greater than > 3 Two plus two is greater than or equal to Two plus two is greater than or equal to < 5 Two plus two is less than Two plus two is less than or equal to Two plus two is less than or equal to 4 90

91 Writing Inequalities Let's translate each statement into an inequality. x is less than 10 words translate to x < 10 inequality statement 20 is greater than or equal to y 20 > y 91

92 Try These is greater than a 2. b is less than or equal to is less than the product of f and 20 Answer 4. The sum of t and 9 is greater than or equal to 36 92

93 Try These 5. 7 more than w is less than or equal to decreased by p is greater than or equal to 2 7. Fewer than 12 items Answer 8. No more than 50 students 9. At least 275 people attended the play 93

94 Writing Inequalities A store's employees earn at least $7.50 per hour. Define a variable and write an inequality for the amount the employees may earn per hour. Let e represent an employee's wages. An employee earns e at least > $

95 39 Write an inequality for the sentence: m is greater than 9 A m < 9 B m < 9 C m > 9 D m > 9 Answer 95

96 40 Write an inequality for the sentence: 12 is less than or equal to y A 12 < y B 12 < y C 12 > y D 12 > y Answer 96

97 41 Write an inequality for the sentence: The grade, g, on your test must exceed 80% A g < 80 B g < 80 C g > 80 D g > 80 Answer 97

98 42 Write an inequality for the sentence: y is not more than 25 A y < 25 B y < 25 C y > 25 D y > 25 Answer 98

99 43 Write an inequality for the sentence: The total, t, is fewer than 15 items. A t < 15 B t < 15 C t > 15 D t > 15 Answer 99

100 44 Write an inequality for the sentence: k is greater than or equal to twenty A k < 20 B k < 20 C k > 20 D k > 20 Answer 100

101 45 Cirrus clouds form more than 6,000 meters above Earth. Choose an inequality to represent h, the height, in meters, of cirrus clouds. A h < 6000 B h < 6000 C h > 6000 D h > 6000 Answer From PARCC EOY sample test non calculator #20 101

102 46 Let x represent any number in the set of even integers greater than 1. Which inequality is true for all values of x? A x < 0 B x > 0 Answer C x < 4 D x > 4 From PARCC PBA sample test calculator #3 102

103 Solutions to Simple Inequalities Return to Table of Contents 103

104 Solution Sets Remember: Equations have one solution. Solutions to inequalities are NOT single numbers. Instead, inequalities will have more than one value for a solution. Math Practice This would be read as, "The solution set is all numbers greater than or equal to negative 5." 104

105 Solution Sets Let's name the numbers that are solutions to the given inequality. r > 10 Which of the following are solutions? {5, 10, 15, 20} 5 > 10 is not true So, 5 is not a solution 10 > 10 is not true So, 10 is not a solution 15 > 10 is true So, 15 is a solution 20 > 10 is true So, 20 is a solution Answer: {15, 20} are solutions to the inequality r >

106 Solution Sets Which of the following numbers are solutions to the given inequality. 30 5d; {4,5,6,7,8} 30 5d 30 5(4) d 30 5(5) d 30 5(6) Answer 30 5d 30 5d 30 5(7) 30 5(8)

107 47 Which of the following are solutions to the inequality: Select all that apply. x > 11{9, 10, 11, 12} A 9 B 10 C 11 D 12 Answer 107

108 48 Which of the following are solutions to the inequality: m < 15{13, 14, 15, 16} Select all that apply. A 13 B 14 C 15 D 16 Answer 108

109 49 Which of the following are solutions to the inequality: Select all that apply. x > 34{32, 33, 34, 35} A 32 B 33 C 34 D 35 Answer 109

110 50 Which of the following are solutions to the inequality: Select all that apply. 3x > 15{4, 5, 6, 7} A 4 B 5 C 6 D 7 Answer 110

111 51 Which of the following are solutions to the inequality: Select all that apply 6y < 42{6, 7, 8, 9} A 6 B 7 C 8 D 9 Answer 111

112 Graphing Solution Sets to Simple Inequalities Return to Table of Contents 112

113 Graphing Inequalities Since inequalities have more than one solution, we show the solution two ways. The first is to write the inequality. The second is to graph the inequality on a number line. In order to graph an inequality, you need to do two things: 1. Draw a circle (open or closed) on the number that is your boundary. 2. Extend the line in the proper direction. 113

114 Graphing Inequalities The Circle Determining Whether to Use an Open or Closed Circle An open circle on a number shows that the number is not part of the solution. It serves as a boundary only. It is used with "greater than" and "less than". The word equal is not included. < > A closed circle on a number shows that the number is part of the solution. Teacher Notes It is used with "greater than or equal to" and "less than or equal to". < > 114

115 Determining Which Direction to Extend the Line Extend Line to the Left: If the variable is smaller than the number, then you extend your line to the left (since smaller numbers are on the left). Extend the line to the left in these situations: # > variable variable < # Extend Line to the Right: If the variable is larger than the number, then you extend your line to the right (since bigger numbers are on the right). Teacher Notes Extend the line to the right in these situations: # < variable variable > # 115

116 Graphing Inequalities When graphing inequalities, ask yourselves each question below. What is the number in the inequality? Math Practice What kind of circle should be used? In what direction does the line go? 116

117 Step 1: Rewrite this as x < 1. Graphing Inequalities x is less than 1 Step 2: What kind of circle? Because it is less than, it does not include the number 1 and so it is an open circle

118 Graphing Inequalities x < 1 Step 3: Draw an arrow on the number line showing all possible solutions. Numbers greater than the variable, go to the right. Numbers less than the variable, go to the left Step 4: Draw a line, thicker than the horizontal line, from the dot to the arrow. This represents all of the numbers that fulfill the inequality

119 Graphing Inequalities You try Graph the inequality x > click 2 on the number line for answer Extra Note Graph the inequality 3 > x click 3 on the number line for answer

120 Graphing Inequalities Try these. Graph the inequalities. 1. x > Answer 2. x <

121 Graphing Inequalities Try these. State the inequality shown Answer

122 52 This solution set would be x 4. True False Answer 122

123 53 State the inequality shown A x > 3 B x < 3 Answer C x < 3 D x > 3 123

124 54 State the inequality shown. A B C D < x 11 > x 11 > x 11 < x Answer 124

125 55 State the inequality shown A x > 1 B x < 1 Answer C x 1 D x 1 125

126 56 State the inequality shown A B C D 4 < x 4 > x 4 x 4 x Answer 126

127 57 State the inequality shown A x > 0 B x < 0 C x 0 Answer D x 0 127

128 A store's employees earn at least $7.50 per hour. Define a variable and write an inequality for the amount the employees may earn per hour. Graph the solutions. Let e represent an employee's wages. Answer & Math Practice

129 58 The sign shown below is posted in front of a roller coaster ride at the Wadsworth County Fairgrounds. If h represents the height of a rider in inches, what is a correct translation of the statement on this sign? A h < 48 B h > 48 C h 48 D h 48 All riders MUST be at least 48 inches tall. Answer From the New York State Education Department. Office of Assessment Policy, Development and Administration. Internet. Available from accessed 17, June,

130 Glossary & Standards Teacher Notes Return to Table of Contents 130

131 Equation A mathematical statement, in symbols, that two things are exactly the same (or equivalent). 4x+2 = 14 7x = 21 (where x = 3) 3y + 2 = 11 (where y = 3) 11 1 = 3z + 1 (where z = 3) a.k.a. function d = rt Back to Instruction 131

132 Identity An equation that has infinitely many solutions. 3(x 1) = 3x 3 3x 3 = 3x 3 3x 3x 3 = 3 7(2x + 1) = 14x x + 7 = 14x x 14x 7 = 7 3x 1 = 3x + 1 3x 3x 1 = +1 Back to Instruction 132

133 Inequality A comparison of two numbers that are not, or may not, be equal. larger Greater than smaller smaller larger Less than Greater than or equal to Less than or equal to Back to Instruction 133

134 Inverse Operation The operation that reverses the effect of another operation. Addition_ + Subtraction Multiplication x Division 5 + x = x = = 3y = 3y = y Back to Instruction 134

135 Solution A value you can put in place of a variable that would make the statement true. x + 4 = 9 Solution: x = 5 3y = 6 Solution: y = 2 The answer to a math problem. Back to Instruction 135

136 Solution Set A set of values that can make a statement true. The #s in a solution set are written in curly brackets. { } 2y = 16 y = {8} 3 < y < 7 y= {4,5,6,7} Back to Instruction 136

137 Variable A letter or symbol that represents a changeable or unknown value. 4x + 2 variable x x 2x = 6 x =? Back to Instruction 137

138 Throughout this unit, the Standards for Mathematical Practice are used. MP1: Making sense of problems & persevere in solving them. MP2: Reason abstractly & quantitatively. MP3: Construct viable arguments and critique the reasoning of others. MP4: Model with mathematics. MP5: Use appropriate tools strategically. MP6: Attend to precision. MP7: Look for & make use of structure. MP8: Look for & express regularity in repeated reasoning. Math Practice Additional questions are included on the slides using the "Math Practice" Pull tabs (e.g. a blank one is shown to the right on this slide) with a reference to the standards used. If questions already exist on a slide, then the specific MPs that the questions address are listed in the Pull tab. 138