Algebra I The Number System & Mathematical Operations

Transcription

1 Slide 1 / 178

2 Slide 2 / 178 Algebra I The Number System & Mathematical Operations

3 Slide 3 / 178 Table of Contents Review of Natural Numbers, Whole Numbers, Integers and Rational Numbers Review of Exponents, Squares, and Square Roots Review of Irrational Numbers Real Numbers Properties of Exponents Future Topics in Algebra II Like Terms Evaluating Expressions Ordering Terms Glossary & Standards Click on a topic to go to that section.

4 Slide 3 (Answer) / 178 Table of Contents Review of Natural Numbers, Whole Numbers, Integers and Rational Numbers Review of Exponents, Squares, and Square Roots Review of Irrational Numbers Real Numbers Properties of Exponents Future Topics in Algebra II Like Terms Teacher Notes Evaluating Expressions Ordering Terms Glossary & Standards Vocabulary Words are bolded in the presentation. The text box the word is in is then linked to the page at the end of the presentation with the word defined on it. [This object is a pull tab] Click on a topic to go to that section.

5 Slide 4 / 178 Review of Natural Numbers, Whole Numbers, Integers & Rational Numbers Return to Table of Contents

6 Slide 5 / 178 Natural Numbers The first numbers developed were the Natural Numbers, also called the Counting Numbers. 1, 2, 3, 4, 5,... The three dots, (...), means that these numbers continue forever: there is no largest counting number.. Think of counting objects as you put them in a container those are the counting numbers.

7 Slide 6 / 178 Natural Numbers Natural numbers were used before there was history. All people use them. This "counting stick" was made more than 35,000 years ago and was found in Lebombo, Swaziland. The cuts in this bone record the number "29."

8 Slide 7 / 178 Natural Numbers and Addition They were, and are, used to count objects > goats, > bales, > bottles, > etc. Drop a stone in a jar, or cut a line in a stick, every time a goat walks past. That jar or stick is a record of the number.

9 Slide 8 / 178 Numbers versus Numerals Numbers exist even without a numeral, such as the number indicated by the cuts on the Lebombo Bone. A numeral is the name we give a number in our culture.

10 Slide 9 / 178 Numbers versus Numerals If asked how many tires my car has, I could hand someone the above marbles. That number is represented by: 4 in our Base 10 numeral system IV in the Roman numeral system 100 in the Base 2 numeral system

11 Slide 10 / 178 Whole Numbers Adding zero to the Counting Numbers gives us the Whole Numbers. Counting numbers were developed more than 35,000 years ago. It took 34,000 more years to invent zero. This the oldest known use of zero (the dot), about 1500 years ago. 0, 1, 2, 3, 4,... It was found in Cambodia and the dot is for the zero in the year

12 Slide 11 / 178 Why zero took so long to Invent Horses versus houses zero horses = zero houses

13 Slide 11 (Answer) / 178 Why zero took so long to Invent Teacher's Note The concept of zero was Horses versus houses. conceptually harder to understand, which is why it took 30,000 years to get 0 to occur. The idea of a number 3 representing nothing was nonexistent back then. People understood 1 horse & 1 house, but had a harder 2 time understanding 0 horses & zero houses. [This object is a pull tab] 1 0 zero horses = zero houses

14 Slide 12 / 178 Why zero took so long Would I tell someone I have a herd of zero goats? Or a garage with zero cars? Or that my zero cars have zero tires? Zero just isn't a natural number, but it is a whole number.

15 Slide 13 / 178 Whole Numbers Each time a marble is dropped in a jar we are doing addition. A number line allowed us to think of addition in a new way

16 Slide 14 / 178 Integers The below number line shows only the integers

17 Slide 15 / 178 Review: Fractions Remember that division led to a new set of numbers: fractions. Fractions are the result when you ask questions like: 1 2 =? 1 3 =? 2 3 =? 1 1,000,000 =? 1 2 =? asks the question: If I divide 1 into 2 equal pieces, what will be the size of each? The answer to this question cannot be found in the integers. New numbers were needed: fractions.

18 Slide 16 / 178 Review: Fractions There are an infinite number of fractions between each pair of integers, since the space between any two integers can be divided by as large a number as you can imagine. On the next slide, a few are shown, but in reality there are as many fractions between any pair of integers as there are integers. Fractions can be written as the ratio of two numbers: -2 / 3, -1 / 4, -1 / 8, 1 / 3, 4 / 5, 7 / 5, 80 / 4, etc. Or in decimal form by dividing the numerator by the denominator: , -0.25, , 0.333, 0.8,1.4, 20, etc. The bar over "666" and "333" means that pattern repeats forever.

19 Slide 17 / 178 Fractions There are an infinite number of fractions between each pair of integers. Using a magnifying glass to look closely between 0 and 1 on this number line, we can locate a few of these fractions Notice that it's easier to find their location when they are in decimal form since it's clear which integers they're between...and closest to. 1 / 5 1 / 4 1 / 3 1 / 2 2 / 3 4 /

20 Slide 18 / 178 Rational Numbers Rational Numbers are numbers that can be expressed as a ratio of two integers. This includes all the fractions, as well as all the integers, since any integer can be written as a ratio of itself and 1. Fractions can be written in "fraction" form or decimal form. When written in decimal form, rational numbers are either: Terminating, such as - 1 = = Repeating, such as 1 = = = =

21 Slide 19 / 178 Review of Exponents, Squares, and Square Roots Return to Table of Contents

22 Slide 20 / 178 Powers of Integers Just as multiplication is repeated addition, exponents are repeated multiplication. For example, 3 5 reads as "3 to the fifth power" = In this case "3" is the base and "5" is the exponent. The base, 3, is multiplied by itself 5 times.

23 Slide 21 / 178 Powers of Integers When evaluating exponents of negative numbers, keep in mind the meaning of the exponent and the rules of multiplication. For example, (-3) 2 = (-3)(-3) = 9, is the same as (3) 2 = (3)(3) = 9. However, (-3) 2 = (-3)(-3) = 9 is NOT the same as -3 2 = -(3)(3) = -9, Similarly, (3) 3 = (3)(3)(3) = 27 is NOT the same as (-3) 3 = (-3)(-3)(-3) = -27,

24 Slide 21 (Answer) / 178 Powers of Integers When evaluating MP.6: exponents Attend of to negative precision. numbers, keep in mind the meaning of the exponent and the rules of multiplication. Emphasize the difference between -4 2 For example, and (-4) 2 w/ (-3) even 2 = (-3)(-3) powers. = 9, In -4 2, only the 4 is getting squared and the is the same as negative stays (3) 2 = in (3)(3) front, = 9. whereas in (-4) 2, -4 is being multiplied by itself, making the answer positive. However, (-3) 2 = (-3)(-3) = 9 Ask: What does the exponent (which is NOT the same ever as number -3 2 = it -(3)(3) is) mean? = -9, Math Practice How do you know the sign of your answer? Similarly, (3) [This 3 = object (3)(3)(3) is a pull tab] = 27 is NOT the same as (-3) 3 = (-3)(-3)(-3) = -27,

25 Slide 22 / 178 Special Term: Squares and Cubes A number raised to the second power can be said to be "squared." That's because the area of a square of length x is x 2 : "x squared." A number raised to the third power can be said to be "cubed." That's because the volume of a cube of length x is x 3 : "x cubed." x A = x 2 x V = x 3 x x x

26 Slide 23 / 178 The Root as an Inverse Operation Performing an operation and then the inverse of that operation returns us to where we started. We already saw that if we add 5 to a number and then subtract 5, we get back to the original number. Or, if we multiply a number by 7 and then divide by 7, we get back to the original number.

27 Slide 24 / 178 The Root as an Inverse Operation Inverses of exponents are a little more complicated for two reasons. First, there are two possible inverse operations. The equation 16 = 4 2 provides the answer 16 to the question: what is 4 raised to the power of 2? One inverse operation is shown by: This provides the answer 4 to the question: What number raised to the power of 2 yields 16? This shows that the square root of 16 is 4. It's true since (4)(4) = 16

28 Slide 25 / 178 Logs as an Inverse Operation The other inverse operation will not be addressed until Algebra II. Just for completeness, that inverse operation is 2 = log It provides the answer 2 to the question: To what power must 10 be raised to get 100. You'll learn more about that in Algebra II, but you should realize it's the other possible inverse operation.

29 1 What is? Slide 26 / 178

30 Slide 26 (Answer) / What is? Answer 1 [This object is a pull tab]

31 2 What is? Slide 27 / 178

32 Slide 27 (Answer) / What is? Answer 9 [This object is a pull tab]

33 3 What is the square of 15? Slide 28 / 178

34 Slide 28 (Answer) / What is the square of 15? Answer 225 [This object is a pull tab]

35 4 What is? Slide 29 / 178

36 Slide 29 (Answer) / What is? Answer 16 [This object is a pull tab]

37 5 What is 13 2? Slide 30 / 178

38 Slide 30 (Answer) / What is 13 2? Answer 169 [This object is a pull tab]

39 6 What is? Slide 31 / 178

40 Slide 31 (Answer) / What is? Answer 14 [This object is a pull tab]

41 7 What is the square of 18? Slide 32 / 178

42 Slide 32 (Answer) / What is the square of 18? Answer 324 [This object is a pull tab]

43 8 What is 11 squared? Slide 33 / 178

44 Slide 33 (Answer) / What is 11 squared? Answer 121 [This object is a pull tab]

45 Slide 34 / 178 Review of the Square Root of a Number The following formative assessment questions are review from 8th grade. If further instruction is need, see the presentation at:

46 Slide 35 / A 6 B -6 C is not real

47 Slide 35 (Answer) / A 6 B -6 C is not real Answer A [This object is a pull tab]

48 Slide 36 / A 9 B -9 C is not real

49 Slide 36 (Answer) / A 9 B -9 C is not real Answer C [This object is a pull tab]

50 Slide 37 / A 20 B -20 C is not real

51 Slide 37 (Answer) / A 20 B -20 C is not real Answer A [This object is a pull tab]

52 Slide 38 / (Problem from ) Which student's method is not correct? A Ashley's Method B Brandon's Method On your paper, explain why the method you selected is not correct.

53 Slide 38 (Answer) / (Problem from ) Brandon's method works for the Which student's square method root of is not 4, but correct? it would not A Ashley's work Method for the square root of 36. B Brandon's Half of Method 36 is 18, but the square of On your paper, 36 is explain 6 times why 6 equals the method 36. Ashley you selected is not correct. describes the correct way to find the square root of a number. Answer Brandon's method is not correct. [This object is a pull tab]

54 13 Slide 39 / 178

55 Slide 39 (Answer) / Answer 5 [This object is a pull tab]

56 14 Slide 40 / 178

57 Slide 40 (Answer) / Answer 11 [This object is a pull tab]

58 Slide 41 / A B C D

59 Slide 41 (Answer) / A B C D Answer B [This object is a pull tab]

60 Slide 42 / A 3 B -3 C No real roots

61 Slide 42 (Answer) / A 3 B -3 C No real roots Answer C [This object is a pull tab]

62 Slide 43 / The expression equal to is equivalent to a positive integer when b is A -10 B 64 C 16 D 4 From the New York State Education Department. Office of Assessment Policy, Development and Administration. Internet. Available from accessed 17, June, 2011.

63 Slide 43 (Answer) / The expression equal to is equivalent to a positive integer when b is A -10 B 64 C 16 D 4 Answer A [This object is a pull tab] From the New York State Education Department. Office of Assessment Policy, Development and Administration. Internet. Available from accessed 17, June, 2011.

64 Slide 44 / 178 Review of the Square Root of Fractions The following formative assessment questions are review from 8th grade. If further instruction is need, see the presentation at:

65 Slide 45 / A C B D no real solution

66 Slide 45 (Answer) / A C B D Answer no real solution A [This object is a pull tab]

67 Slide 46 / A C B D no real solution

68 Slide 46 (Answer) / A C B D Answer no real solution C [This object is a pull tab]

69 Slide 47 / 178

70 Slide 47 (Answer) / 178

71 Slide 48 / A C B D no real solution

72 Slide 48 (Answer) / A C B D Answer no real solution D [This object is a pull tab]

73 Slide 49 / A C B D no real solution

74 Slide 49 (Answer) / A C B D Answer no real solution C [This object is a pull tab]

75 Slide 50 / 178 Review of the Square Root of Decimals The following formative assessment questions are review from 8th grade. If further instruction is need, see the presentation at:

76 Slide 51 / Evaluate A B C D no real solution

77 Slide 51 (Answer) / Evaluate A B C D no real solution Answer C [This object is a pull tab]

78 Slide 52 / Evaluate A.06 B.6 C 6 D no real solution

79 Slide 52 (Answer) / Evaluate A.06 B.6 Answer B C 6 D no real solution [This object is a pull tab]

80 Slide 53 / Evaluate A 0.11 B 11 C 1.1 D no real solution

81 Slide 53 (Answer) / Evaluate A 0.11 B 11 C 1.1 D no real solution Answer A [This object is a pull tab]

82 Slide 54 / Evaluate A 0.8 B 0.08 C D no real solution

83 Slide 54 (Answer) / Evaluate A 0.8 B 0.08 C D no real solution Answer B [This object is a pull tab]

84 Slide 55 / 178

85 Slide 55 (Answer) / 178

86 Slide 56 / 178 Review of Approximating Non-Perfect Squares The following formative assessment questions are review from 8th grade. If further instruction is need, see the presentation at:

87 Slide 57 / The square root of 40 falls between which two integers? A 3 and 4 B 5 and 6 C 6 and 7 D 7 and 8

88 Slide 57 (Answer) / The square root of 40 falls between which two integers? A 3 and 4 B 5 and 6 C 6 and 7 D 7 and 8 Answer C [This object is a pull tab]

89 Slide 58 / Which integer is closest to? < < Identify perfect squares closest to 40 < < Take square root Identify nearest integer

90 Slide 58 (Answer) / Which integer is closest to? Answer 6 < < Identify perfect squares closest to 40 < < Take square root [This object is a pull tab] Identify nearest integer

91 Slide 59 / The square root of 110 falls between which two perfect squares? A 36 and 49 B 49 and 64 C 64 and 84 D 100 and 121

92 Slide 59 (Answer) / The square root of 110 falls between which two perfect squares? A 36 and 49 B 49 and 64 C 64 and 84 D 100 and 121 Answer D [This object is a pull tab]

93 Slide 60 / Estimate to the nearest integer.

94 Slide 60 (Answer) / Estimate to the nearest integer. Answer 10 [This object is a pull tab]

95 Slide 61 / Estimate to the nearest integer.

96 Slide 61 (Answer) / Estimate to the nearest integer. Answer 15 [This object is a pull tab]

97 Slide 62 / Estimate to the nearest integer.

98 Slide 62 (Answer) / Estimate to the nearest integer. Answer 9 [This object is a pull tab]

99 Slide 63 / Approximate to the nearest integer.

100 Slide 63 (Answer) / Approximate to the nearest integer. Answer 5 [This object is a pull tab]

101 Slide 64 / Approximate to the nearest integer.

102 Slide 64 (Answer) / Approximate to the nearest integer. Answer 10 [This object is a pull tab]

103 Slide 65 / Approximate to the nearest integer.

104 Slide 65 (Answer) / Approximate to the nearest integer. Answer 13 [This object is a pull tab]

105 Slide 66 / Approximate to the nearest integer.

106 Slide 66 (Answer) / Approximate to the nearest integer. Answer 12 [This object is a pull tab]

107 Slide 67 / Approximate to the nearest integer.

108 Slide 67 (Answer) / Approximate to the nearest integer. Answer 6 [This object is a pull tab]

109 Slide 68 / The expression is a number between: A 3 and 9 B 8 and 9 C 9 and 10 D 46 and 47 From the New York State Education Department. Office of Assessment Policy, Development and Administration. Internet. Available from accessed 17, June, 2011.

110 Slide 68 (Answer) / The expression is a number between: A 3 and 9 B 8 and 9 C 9 and 10 D 46 and 47 Answer C [This object is a pull tab] From the New York State Education Department. Office of Assessment Policy, Development and Administration. Internet. Available from accessed 17, June, 2011.

111 Slide 69 / For what integer x is closest to 6.25? Derived from

112 Slide 69 (Answer) / For what integer x is closest to 6.25? Answer 39 [This object is a pull tab] Derived from

113 Slide 70 / For what integer y is closest to 4.5? Derived from

114 Slide 70 (Answer) / For what integer y is closest to 4.5? Answer 20 [This object is a pull tab] Derived from

115 Slide 71 / Between which two positive integers does lie? A 1 B 2 C 3 D 4 E 5 F 6 G 7 H 8 I 9 J 10 Derived from

116 Slide 71 (Answer) / Between which two positive integers does lie? A 1 B 2 C 3 D 4 E 5 Answer F 6 G 7 H 8 I 9 J 10 G & H [This object is a pull tab] Derived from

117 Slide 72 / Between which two positive integers does lie? A 1 B 2 C 3 D 4 E 5 F 6 G 7 H 8 I 9 J 10 Derived from

118 Slide 72 (Answer) / Between which two positive integers does lie? A 1 B 2 C 3 D 4 E 5 Answer F 6 G 7 H 8 I 9 J 10 H & I [This object is a pull tab] Derived from

119 Slide 73 / Between which two labeled points on the number line would be located? A B C D E F G H I J Derived from

120 Slide 73 (Answer) / Between which two labeled points on the number line would be located? A B C D E F G H I J Answer B & C [This object is a pull tab] Derived from

121 Slide 74 / 178 Review of Irrational Numbers & Real Numbers Return to Table of Contents

122 Slide 75 / 178 Irrational Numbers Just as subtraction led us to zero and negative numbers. And division led us to fractions. Finding the root leads us to irrational numbers. Irrational numbers complete the set of Real Numbers. Real numbers are the numbers that exist on the number line.

123 Slide 76 / 178 Irrational Numbers Irrational Numbers are real numbers that cannot be expressed as a ratio of two integers. In decimal form, they extend forever and never repeat. There are an infinite number of irrational numbers between any two integers (think of all the square roots, cube roots, etc. that don't come out evenly). Then realize you can multiply them by an other number or add any number to them and the result will still be irrational.

124 Slide 77 / 178 Rational & Irrational Numbers is rational. This is because the radicand (number under the radical) is a perfect square. If a radicand is not a perfect square, the root is said to be irrational. Ex:

125 Slide 78 / 178 Irrational Numbers Irrational numbers were first discovered by Hippasus about the about 2500 years ago. He was a Pythagorean, and the discovery was considered a problem since Pythagoreans believed that "All was Number," by which they meant the natural numbers (1, 2, 3...) and their ratios. Hippasus was trying to find the length of the diagonal of a square with sides of length 1. Instead, he proved, that the answer was not a natural or rational number. For a while this discovery was suppressed since it violated the beliefs of the Pythagorean school. Hippasus had proved that 2 is an irrational number.

126 Slide 79 / 178 Irrational Numbers An infinity of irrational numbers emerge from trying to find the root of a rational number. Hippasus proved that 2 is irrational goes on forever without repeating. Some of its digits are shown on the next slide.

127 Slide 80 / 178 Square Root of 2 Here are the first 1000 digits, but you can find the first 10 million digits on the Internet. The numbers go on forever, and never repeat in a pattern:

128 Slide 81 / 178 Roots of Numbers are Often Irrational Soon, thereafter, it was proved that many numbers have irrational roots. We now know that the roots of most numbers to most powers are irrational. These are called algebraic irrational numbers. In fact, there are many more irrational numbers that rational numbers.

129 Slide 82 / 178 Principal Roots Since you can't write out all the digits of 2, or use a bar to indicate a pattern, the simplest way to write that number is 2. But when solving for the square root of 2, there are two answers: + 2 or - 2. These are in two different places on the number line. To avoid confusion, it was agreed that the positive value would be called the principal root and would be written 2. The negative value would be written as - 2.

130 Slide 83 / 178 Algebraic Irrational Numbers There are an infinite number of irrational numbers. Here are just a few that fall between -10 and

131 Slide 84 / 178 Transcendental Numbers The other set of irrational numbers are the transcendental numbers. These are also irrational. No matter how many decimals you look at, they never repeat. But these are not the result of solving a polynomial equation with rational coefficients, so they are not due to an inverse operation. Some of these numbers are real, and some are complex. But, this year, we will only be working with the real transcendental numbers.

132 Pi Slide 85 / 178 We have learned about Pi in Geometry. It is the ratio of a circle's circumference to its diameter. It is represented by the symbol. Discuss why this is an approximation at your table. Is this number rational or irrational?

133 Slide 85 (Answer) / 178 Pi We have learned about Pi in Geometry. It is the ratio of a circle's circumference to its diameter. It is represented by the symbol Pi is an. irrational number because it has a decimal representation that never ends and never settles into a permanent repeating pattern. Answer Super computers have found over 12 trillion digits in pi. [This object is a pull tab] Discuss why this is an approximation at your table. Is this number rational or irrational?

134 Slide 86 / 178 # is a Transcendental Number The most famous of the transcendental numbers is #. A O B # is the ratio of the circumference to the diameter of a circle. People tried to find the value of that ratio for millennia. Only in the mid 1800's was it proven that there was no rational solution /06/pi-day004.jpg Since # is irrational (it's decimals never repeat) and it is not a solution to a equation...it is transcendental. Some of its digits are on the next page.

135 Slide 87 / 178

136 Slide 88 / 178 Other Transcendental Numbers There are many more transcendental numbers. Another famous one is "e", which you will study in Algebra II as the base of natural logarithms. In the 1800's Georg Cantor proved there are as many transcendental numbers as real numbers. And, there are more algebraic irrational numbers than there are rational numbers. The integers and rational numbers with which we feel more comfortable are like islands in a vast ocean of irrational numbers.

137 Slide 89 / 178 Real Numbers Real Numbers are numbers that exist on a number line.

138 Slide 90 / What type of number is 25? Select all that apply. A Irrational B Rational C Integer D Whole Number E Natural Number

139 Slide 90 (Answer) / What type of number is 25? Select all that apply. A Irrational B Rational C Integer Answer B, C, D, E D Whole Number E Natural Number [This object is a pull tab]

140 Slide 91 / What type of number is - 5? Select all that apply. 3 A Irrational B Rational C Integer D Whole Number E Natural Number

141 Slide 91 (Answer) / What type of number is - 5? Select all that apply. 3 A Irrational B Rational C Integer D Whole Number E Natural Number Answer B [This object is a pull tab]

142 Slide 92 / What type of number is 8? Select all that apply. A Irrational B Rational C Integer D Whole Number E Natural Number

143 Slide 92 (Answer) / What type of number is 8? Select all that apply. A Irrational B Rational C Integer D Whole Number E Natural Number Answer A [This object is a pull tab]

144 Slide 93 / What type of number is - 64? Select all that apply. A Irrational B Rational C Integer D Whole Number E Natural Number

145 Slide 93 (Answer) / What type of number is - 64? Select all that apply. A Irrational B Rational C Integer Answer B, C D Whole Number E Natural Number [This object is a pull tab]

146 Slide 94 / What type of number is 2.25? Select all that apply. A Irrational B Rational C Integer D Whole Number E Natural Number

147 Slide 94 (Answer) / What type of number is 2.25? Select all that apply. A Irrational B Rational C Integer Answer B D Whole Number E Natural Number [This object is a pull tab]

148 Slide 95 / 178 Properties of Exponents Return to Table of Contents

149 Slide 96 / 178 Properties of Exponents The properties of exponents follow directly from expanding them to look at the repeated multiplication they represent. Don't memorize properties, just work to understand the process by which we find these properties and if you can't recall what to do, just repeat these steps to confirm the property. We'll use 3 as the base in our examples, but the properties hold for any base. We show that with base a and powers b and c. We'll use the facts that: (3 2 )= (3 3) (3 3 ) = (3 3 3) (3 5 ) = ( )

150 Slide 97 / 178 Properties of Exponents We need to develop all the properties of exponents so we can discover one of the inverse operations of raising a number to a power...finding the root of a number to that power. That will emerge from the final property we'll explore. But, getting to that property requires understanding the others first.

151 Slide 98 / 178 Multiplying with Exponents When multiplying numbers with the same base, add the exponents. (a b )(a c ) = a (b+c) (3 2 )(3 3 ) = 3 5 (3 3 )(3 3 3) = ( ) (3 2 )(3 3 ) = 3 5 (3 2 )(3 3 ) = 3 (2+3)

152 Slide 98 (Answer) / 178 Multiplying with Exponents When multiplying numbers with the same base, add the exponents. MP3 Construct viable arguments and critique (a b )(a c ) = the a (b+c) reasoning of others is address on this slide and the (3 2 )(3 next 3 ) = slide 3 5 Math Practice (3 We 3 )(3 are 3 using 3) = examples (3 3 3 to 3 prove 3) that the properties are true. (3 2 )(3 3 ) = 3 5 (3 2 )(3 3 ) = 3 (2+3) [This object is a pull tab]

153 Slide 99 / 178 Dividing with Exponents When dividing numbers with the same base, subtract the exponent of the denominator from that of the numerator. (a b ) (a c ) = a (b-c) (3 3 ) (3 2 ) = 31 (3 3 3) (3 3 ) = 3 (3 3 ) (3 2 ) = 3 (3-2) (3 3 ) (3 2 ) = 3 1

154 Slide 100 / Simplify: A 5 2 B 5 3 C 5 6 D 5 8

155 Slide 100 (Answer) / Simplify: A 5 2 B 5 3 C 5 6 D 5 8 Answer C [This object is a pull tab]

156 Slide 101 / Simplify: A B 7 2 C 7 10 D 7 24

157 Slide 101 (Answer) / Simplify: A B 7 2 C 7 10 D 7 24 Answer B [This object is a pull tab]

158 Slide 102 / 178

159 Slide 102 (Answer) / 178

160 Slide 103 / Simplify: A 8 2 B 8 3 C 8 9 D 8 18

161 Slide 103 (Answer) / Simplify: A 8 2 B 8 3 C 8 9 D 8 18 Answer C [This object is a pull tab]

162 Slide 104 / Simplify: A 5 2 B 5 3 C 5 6 D 5 8

163 Slide 104 (Answer) / Simplify: A 5 2 B 5 3 C 5 6 D 5 8 Answer A [This object is a pull tab]

164 Slide 105 / Simplify: 4 3 (4 5 ) A 4 15 B 4 8 C D

165 Slide 105 (Answer) / Simplify: 4 3 (4 5 ) A 4 15 B C Answer B D 4 7 [This object is a pull tab]

166 Slide 106 / Simplify: A 5 2 B C D

167 Slide 106 (Answer) / Simplify: A 5 2 B C Answer D D 5 4 [This object is a pull tab]

168 Slide 107 / 178 An Exponent of Zero Any base raised to the power of zero is equal to 1. a (0) = 1 Based on the multiplication rule: (3 (0) )(3 (3) ) = (3 (3+0) ) (3 (0) )(3 (3) ) = (3 (3) ) Any number times 1 is equal to itself (1)(3 (3) ) = (3 (3) ) (3 (0) )(3 (3) ) = (3 (3) ) Comparing these two equations, we see that (3 (0) )= 1 This is not just true for base 3, it's true for all bases.

169 57 Simplify: 5 0 = Slide 108 / 178

170 Slide 108 (Answer) / Simplify: 5 0 = Answer 1 [This object is a pull tab]

171 58 Simplify: = Slide 109 / 178

172 Slide 109 (Answer) / Simplify: = Answer 2 [This object is a pull tab]

173 59 Simplify: (7)(30 0 ) = Slide 110 / 178

174 Slide 110 (Answer) / Simplify: (7)(30 0 ) = Answer 7 [This object is a pull tab]

175 Slide 111 / 178 Negative Exponents A negative exponent moves the number from the numerator to denominato and vice versa. (a (-b) )= 1 a b Based on the multiplication rule and zero exponent rules: (3 (-1) )(3 (1) ) = (3 (-1+1) ) (3 (-1) )(3 (1) ) = (3 (0) ) (3 (-1) )(3 (1) ) = 1 But, any number multiplied by its inverse is 1, so 1 (3 = (1) ) (3 (-1) )(3 (1) ) = 1 1 a b = (a (-b) ) Comparing these two equations, we see that 1 (3 (-1) )= 3 1

176 Slide 112 / 178 Negative Exponents By definition: x -1 =, x 0

177 Slide 113 / Simplify the expression to make the exponents positive. 4-2 A 4 2 B 1 4 2

178 Slide 113 (Answer) / Simplify the expression to make the exponents positive. 4-2 A 4 2 B Answer B [This object is a pull tab]

179 Slide 114 / Simplify the expression to make the exponents positive. A 4 2 B

180 Slide 114 (Answer) / Simplify the expression to make the exponents positive. A B Answer A [This object is a pull tab]

181 Slide 115 / Simplify the expression to make the exponents positive. x 3 y -4 A x 3 y 4 B y 4 x 3 x 3 C D y 4 1 x 3 y 4

182 Slide 115 (Answer) / Simplify the expression to make the exponents positive. x 3 y -4 A x 3 y 4 B y 4 x 3 x 3 C D y 4 1 x 3 y 4 Answer C [This object is a pull tab]

183 Slide 116 / Simplify the expression to make the exponents positive. a -5 b -2 A a 5 b 2 B C D b 2 a 5 a 5 b 2 1 a 5 b 2

184 Slide 116 (Answer) / Simplify the expression to make the exponents positive. a -5 b -2 A a 5 b 2 B b 2 a 5 C D a 5 b 2 1 a 5 b 2 Answer D [This object is a pull tab]

185 Slide 117 / Which expression is equivalent to x -4? A B x 4 C -4x D 0 From the New York State Education Department. Office of Assessment Policy, Development and Administration. Internet. Available from accessed 17, June, 2011.

186 Slide 117 (Answer) / Which expression is equivalent to x -4? A B x 4 C -4x D 0 Answer A [This object is a pull tab] From the New York State Education Department. Office of Assessment Policy, Development and Administration. Internet. Available from accessed 17, June, 2011.

187 Slide 118 / What is the value of 2-3? A B C -6 D -8 From the New York State Education Department. Office of Assessment Policy, Development and Administration. Internet. Available from accessed 17, June, 2011.

188 Slide 118 (Answer) / What is the value of 2-3? A B C -6 D -8 Answer B [This object is a pull tab] From the New York State Education Department. Office of Assessment Policy, Development and Administration. Internet. Available from accessed 17, June, 2011.

189 Slide 119 / Which expression is equivalent to x -1 y 2? A B xy 2 C D xy -2 From the New York State Education Department. Office of Assessment Policy, Development and Administration. Internet. Available from accessed 17, June, 2011.

190 Slide 119 (Answer) / Which expression is equivalent to x -1 y 2? A B xy 2 C D xy -2 Answer C [This object is a pull tab] From the New York State Education Department. Office of Assessment Policy, Development and Administration. Internet. Available from accessed 17, June, 2011.

191 Slide 120 / a) Write an exponential expression for the area of a rectangle with a length of 7-2 meters and a width of 7-5 meters. b) Evaluate the expression to find the area of the rectangle. When you finish answering both parts, enter your answer to Part b) in your responder.

192 Slide 120 (Answer) / a) Write an exponential expression for the area of a rectangle with a length of 7-2 meters and a width of 7-5 meters. Answer a) 7-2 x 7-5 = 7-7 m 2 b) Evaluate the expression to find the area of the rectangle. b) When you finish answering both parts, enter your answer to Part b) in your responder. [This object is a pull tab]

193 Slide 121 / Which expressions are equivalent to? Select all that apply. A 3-12 B 3-4 C 3 2 D E F From PARCC EOY sample test non-calculator #13

194 Slide 121 (Answer) / Which expressions are equivalent to? Select all that apply. A 3-12 B 3-4 C 3 2 D E F Answer B & E [This object is a pull tab] From PARCC EOY sample test non-calculator #13

195 Slide 122 / Which expressions are equivalent to? Select all that apply. A 3 3 B 3-3 C 3-10 D E F

196 Slide 122 (Answer) / Which expressions are equivalent to? Select all that apply. A 3 3 B 3-3 C 3-10 D E F [This object is a pull tab] Answer B, D & E

197 Slide 123 / 178 Raising Exponents to Higher Powers When raising a number with an exponent to a power, multiply the exponents. (a b ) c = a (bc) (3 2 ) 3 = 3 6 (3 2 )(3 2 )(3 2 ) = 3 (2+2+2) (3 3 )(3 3)(3 3) = ( ) (3 2 ) 3 = 3 6

198 Slide 124 / Simplify: (2 4 ) 7 A B 2 3 C 2 11 D 2 28

199 Slide 124 (Answer) / Simplify: (2 4 ) 7 A B 2 3 C 2 11 D 2 28 Answer D [This object is a pull tab]

200 Slide 125 / Simplify: (g 3 ) 9 A g 27 B g 12 C g 6 D g 3

201 Slide 125 (Answer) / Simplify: (g 3 ) 9 A g 27 B g 12 C g 6 D g 3 Answer A [This object is a pull tab]

202 Slide 126 / Simplify: (x 4 y 2 ) 7 A x 3 y 5 B x 1.75 y 3.5 C x 11 y 9 D x 28 y 14

203 Slide 126 (Answer) / Simplify: (x 4 y 2 ) 7 A x 3 y 5 B x 1.75 y 3.5 C x 11 y 9 D x 28 y 14 Answer D [This object is a pull tab]

204 Slide 127 / 178

205 Slide 127 (Answer) / 178

206 Slide 128 / The expression (x 2 z 3 )(xy 2 z) is equivalent to: A x 2 y 2 z 3 B x 3 y 2 z 4 C x 3 y 3 z 4 D x 4 y 2 z 5 From the New York State Education Department. Office of Assessment Policy, Development and Administration. Internet. Available from accessed 17, June, 2011.

207 Slide 128 (Answer) / The expression (x 2 z 3 )(xy 2 z) is equivalent to: A x 2 y 2 z 3 B x 3 y 2 z 4 C x 3 y 3 z 4 D x 4 y 2 z 5 Answer B [This object is a pull tab] From the New York State Education Department. Office of Assessment Policy, Development and Administration. Internet. Available from accessed 17, June, 2011.

208 Slide 129 / The expression is equivalent to: A 2w 5 B 2w 8 C 20w 8 D 20w 5 From the New York State Education Department. Office of Assessment Policy, Development and Administration. Internet. Available from accessed 17, June, 2011.

209 Slide 129 (Answer) / The expression is equivalent to: A 2w 5 B 2w 8 C 20w 8 D 20w 5 Answer D [This object is a pull tab] From the New York State Education Department. Office of Assessment Policy, Development and Administration. Internet. Available from accessed 17, June, 2011.

210 Slide 130 / When -9 x 5 is divided by -3x 3, x 0, the quotient is A 3x 2 B -3x 2 C 27x 15 D 27x 8 From the New York State Education Department. Office of Assessment Policy, Development and Administration. Internet. Available from accessed 17, June, 2011.

211 Slide 130 (Answer) / When -9 x 5 is divided by -3x 3, x 0, the quotient is A 3x 2 B -3x 2 C 27x 15 D 27x 8 Answer A [This object is a pull tab] From the New York State Education Department. Office of Assessment Policy, Development and Administration. Internet. Available from accessed 17, June, 2011.

212 Slide 131 / Lenny the Lizard's tank has the dimensions b 5 by 3c 2 by 2c 3. What is the volume of Larry's tank? A 6b 7 c 3 B 6b 5 c 5 C 5b 5 c 5 D 5b 5 c 6

213 Slide 131 (Answer) / Lenny the Lizard's tank has the dimensions b 5 by 3c 2 by 2c 3. What is the volume of Larry's tank? A 6b 7 c 3 B 6b 5 c 5 Answer & Math Practice C 5b 5 c 5 D 5b 5 c 6 B b 5 3c 2 2c 3 = 6b 5 c 5 MP.2: Reasoning quantitatively and abstractly. MP.4: Model with mathematics. Ask: How can you represent the problems with symbols and numbers? What connections do you see? Write a number sentence to describe this [This object is a pull tab] situation.

214 Slide 132 / A food company which sells beverages likes to use exponents to show the sales of the beverage in a 2 days. If the daily sales of the beverage is 5a 4, what is the total sales in a 2 days? A a 6 B 5a 8 C 5a 6 D 5a 3

215 Slide 132 (Answer) / A food company which sells beverages likes to use exponents to show the sales of the beverage in a 2 days. If the daily sales of the beverage C is 5a 4, what is the total sales in a 2 days? 5a 4 a 2 = 5a 6 A a 6 Answer & Math Practice B 5a 8 C 5a 6 D 5a 3 MP.2: Reasoning quantitatively and abstractly. MP.4: Model with mathematics. Ask: How can you represent the problems with symbols and numbers? What connections do you see? Write a number sentence to describe this [This object is a pull tab] situation.

216 Slide 133 / A rectangular backyard is 5 5 inches long and 5 3 inches wide. Write an expression for the area of the backyard as a power of 5. A 5 15 in 2 B 8 5 in 2 C 25 8 in 2 D 5 8 in 2

217 Slide 133 (Answer) / A rectangular backyard is 5 5 inches long and 5 3 inches wide. Write an expression for the area of the backyard as a power of 5. A 5 15 in 2 Answer & Math Practice B 8 5 in 2 C 25 8 in 2 D 5 8 in 2 D MP.2: Reasoning quantitatively and abstractly. MP.4: Model with mathematics. Ask: How can you represent the problems with symbols and numbers? What connections do you see? Write a number sentence to describe this [This object is a pull tab] situation.

218 Slide 134 / Express the volume of a cube with a length of 4 3 units as a power of 4. A 4 9 units 3 B 4 6 units 3 C 12 6 units 3 D 12 9 units 3

219 Slide 134 (Answer) / Express the volume of a cube with a length of 4 3 units as a power of 4. A 4 9 units 3 Answer & Math Practice B 4 6 units 3 C 12 6 units 3 D 12 9 units 3 A MP.2: Reasoning quantitatively and abstractly. MP.4: Model with mathematics. Ask: How can you represent the problems with symbols and numbers? What connections do you see? Write a number sentence to describe this [This object is a pull tab] situation.

220 Slide 135 / 178 Future Topics for Algebra II Return to Table of Contents

221 Slide 135 (Answer) / 178 Teacher's Note These topics are discussed in Future this lesson Topics briefly as an introduction of future concepts. This lesson should only take a few minutes to complete. for Algebra II [This object is a pull tab] Return to Table of Contents

222 Slide 136 / 178 The operation of taking the root combined with negative number allows us to ask for the square root of a negative number. The letter i represents -1. Imaginary Numbers An infinite set of square roots of negative numbers emerge from the use of i. These numbers are no more or less real than any other numbers, and they have very practical uses. Understanding their meaning and how to use them is best after studying both Algebra I and Geometry.

223 Slide 137 / 178 Complex Numbers Combining a real number and an imaginary number results in a Complex Number. Complex numbers are written as a + bi, where a is the real part and bi is the imaginary part. All numbers are included in the complex numbers since setting b to zero results in a real number and setting a to zero results in a pure imaginary number.

224 Slide 138 / 178 Like Terms Return to Table of Contents

225 Slide 138 (Answer) / 178 This lesson addresses MP1, MP2 & MP5 Math Practice Like Terms Additional Q's to address MP standards: What is the problem asking? (MP1) How could you start the problem? (MP1) How could you represent the problem with symbols & numbers? (MP2) How could you use drawings or manipulatives to show your thinking? (MP5) [This object is a pull tab] Return to Table of Contents

226 Slide 139 / 178 Like Terms Like Terms: Terms in an expression that have the same variable(s) raised to the same power Like Terms 6x and 2x 5y and 8y 4x 2 and 7x 2 NOT Like Terms 6x and x 2 5y and 8 4x 2 and x 4

227 Slide 140 / Identify all of the terms like 14x 2. A 5x B 2x 2 C 3y 2 D 2x E -10x 2

228 Slide 140 (Answer) / Identify all of the terms like 14x 2. A 5x B 2x 2 C 3y 2 D 2x E -10x 2 Answer B, E [This object is a pull tab]

229 Slide 141 / Identify all of the terms like 0.75w 5. A 75w B 75w 5 C 3v 2 D 2w E -10w 5

230 Slide 141 (Answer) / Identify all of the terms like 0.75w 5. A 75w B 75w 5 C 3v 2 D 2w E -10w 5 Answer B, E [This object is a pull tab]

231 Slide 142 / Identify all of the terms like A 5u B 2u 1 4 u2 C 3u 2 D 2u 2 E -10u

232 Slide 142 (Answer) / Identify all of the terms like A 5u B 2u 1 4 u2 C 3u 2 D 2u 2 E -10u Answer C, D [This object is a pull tab]

233 Slide 143 / 178 Combining Like Terms Simplify by combining like terms 6x + 3x (6 + 3)x 9x Notice when combining like terms you add/subtract the coefficients but the variable remains the same.

234 Slide 143 (Answer) / 178 Combining Like Terms Simplify by combining like terms 6x + 3x Math Practice MP6: Attend to precision. (6 + 3)x Emphasize the addition/subtraction of the coefficients w/ the 9x degree of the variable remaining the same. Notice when combining like terms you add/subtract the coefficients but the variable remains the same. [This object is a pull tab]

235 Slide 144 / ) 4x + 4x is equivalent to 8x 2. True False

236 Slide 144 (Answer) / ) 4x + 4x is equivalent to 8x 2. True False Answer False [This object is a pull tab]

237 Slide 145 / ) 3z 2 + 7z + 5(z + 3) + z 2 is equivalent to 3z z True False

238 Slide 145 (Answer) / ) 3z 2 + 7z + 5(z + 3) + z 2 is equivalent to 3z z True False Answer True [This object is a pull tab]

239 Slide 146 / ) 9r 3 + 2r 2 + 3(r 2 + r) + 5r is equivalent to 9r 3 + 5r 2 + 6r. True False

240 Slide 146 (Answer) / ) 9r 3 + 2r 2 + 3(r 2 + r) + 5r is equivalent to 9r 3 + 5r 2 + 6r. True False Answer False [This object is a pull tab]

241 Slide 147 / ) 10p 3-8p 2 + 9p - 13p(p 2 + 2p) - 12 is equivalent to 10p 3 + 7p 2-2p - 12 True False

242 Slide 147 (Answer) / ) 10p 3-8p 2 + 9p - 13p(p 2 + 2p) - 12 is equivalent to 10p 3 + 7p 2-2p - 12 True False Answer False [This object is a pull tab]

243 Slide 148 / ) 12m 3 + 6m 2-15m - 3m(2m 2 + 5m) + 20 is equivalent to 6m 3-9m 2-15m + 20 True False

244 Slide 148 (Answer) / ) 12m 3 + 6m 2-15m - 3m(2m 2 + 5m) + 20 is equivalent to 6m 3-9m 2-15m + 20 True False Answer True [This object is a pull tab]

245 Slide 149 / ) 8x 2 y + 9x 2-15xy - 3xy(2x + 3y) + 14 is equivalent to 8x 2 y + 9x 2-18xy + 2x + 3y + 14 True False

246 Slide 149 (Answer) / ) 8x 2 y + 9x 2-15xy - 3xy(2x + 3y) + 14 is equivalent to 8x 2 y + 9x 2-18xy + 2x + 3y + 14 True False Answer False [This object is a pull tab]

247 Slide 150 / ) 5jk 2-2k 2 + 6jk - 4jk(j + 5k) is equivalent to 4j 2 k - 15jk 2-2k 2 + 6jk True False

248 Slide 150 (Answer) / ) 5jk 2-2k 2 + 6jk - 4jk(j + 5k) is equivalent to 4j 2 k - 15jk 2-2k 2 + 6jk True False Answer True [This object is a pull tab]

249 Slide 151 / 178 Evaluating Expressions Return to Table of Contents

250 Slide 151 (Answer) / 178 Math Practice This lesson addresses MP1 Additional Q's to address MP standards: What is the problem asking? (MP1) How could you start the problem? (MP1) Evaluating Expressions [This object is a pull tab] Return to Table of Contents

251 Slide 152 / 178 Evaluating Expressions When evaluating algebraic expressions, the process is fairly straight forward. 1. Write the expression. 2. Substitute in the value of the variable (in parentheses). 3. Simplify/Evaluate the expression.

252 Slide 153 / Evaluate 3x + 17 when x = -13

253 Slide 153 (Answer) / Evaluate 3x + 17 when x = -13 Answer 3(-13) [This object is a pull tab]

254 Slide 154 / Evaluate 4u 2-11u + 5 when u = -5

255 Slide 154 (Answer) / Evaluate 4u 2-11u + 5 when u = -5 Answer 4(-5) 2-11(-5) + 5 4(25) - 11(-5) [This object is a pull tab]

256 Slide 155 / Evaluate 8v 2 + 9vw - 6w 2 when v = 4 and w = -3

257 Slide 155 (Answer) / Evaluate 8v 2 + 9vw - 6w 2 when v = 4 and w = -3 Answer 8(4) 2 + 9(4)(-3) - 6(-3) 2 8(16) + 9(-12) - 6(9) [This object is a pull tab]

258 Slide 156 / Evaluate -10p pq - 64q 2 when p = -2 and q = 1 4

259 Slide 156 (Answer) / Evaluate -10p pq - 64q 2 when p = -2 and q = 1 4 Answer -10(-2) (-2)( 1 ) - 64( 1 ) (4) - 32( 1 ) - 64( 1 ) [This object is a pull tab]

260 Slide 157 / Evaluate x 3-2x 2 y + 64xy y 3 when x = -3 and y = 3 2

261 Slide 157 (Answer) / Evaluate x 3-2x 2 y + 64xy y 3 when x = -3 and y = 3 2 Answer (-3) 3-2(-3) 2 ( 3 ) + 64(-3)( 3 3 ) ( ) (9)( 3 ) + 64(-3)( 9 ) + 16( 27 ) [This object is a pull tab]

262 Slide 158 / 178 Ordering Terms Return to Table of Contents

263 Slide 159 / 178 Putting Terms in Order A mathematical expression will often contain a number of terms. As you'll recall, the terms of an expression are separated by either addition or subtraction. While the value of an expression is independent of the order of the terms, it'll help a lot to put them in a specific order.

264 Slide 160 / 178 Degree of a Variable It's good practice to order the terms by the degree of one of the variables. The degree of a variable is the value of the exponent in a term. 5x 3 - x x The degree is: constant; degree = 0

265 Slide 161 / 178 Order Terms by Degree of a Variable For instance, the expression below is not ordered: 5x 3 - x x This is mathematically correct, but will lead to some mistakes later in this course if you're not careful.

266 Slide 162 / 178 Order Terms by Degree of a Variable 5x 3 - x x It's better to write the final expression with the highest degree terms on the left and with each term to the right being next highest. In this case, the above becomes: - x 4 + 5x 3-3x + 2 This will make it easier to keep track of these terms when we do polynomials and algebraic fractions.

267 Slide 163 / 178 Order Terms by Degree of a Variable If there are two or more variables, we have to choose one of them for this purpose. - 9xy 2 + x 3 + 7x 2 y - y 3 We can choose to put the terms in order using either x or y. They would both work. Since x comes before y in the alphabet, I'll order the terms using x. x 3 + 7x 2 y - 9xy 2 - y 3 Notice that the degrees of the variable y ascend from left to right. This won't always happen, but sometimes it will.

268 Slide 164 / 178 Order Terms by Degree of a Variable - 9xy 2 + x 3 + 7x 2 y - y 3 Let's rearrange the terms in descending order using the powers of y. - y 3-9xy 2 + 7x 2 y + x 3 After this rearrangement, the degrees of the variable x ascend from left to right, too. Both these answers work and are correct.

269 Slide 165 / 178 Order Terms by Degree of a Variable Let's try one more example. 3x 3 - x 4 y xy 2 Since there are more terms in x than y, I'll order by using x. -x 4 y 2 + 3x 3-3xy This will make it easier to keep track of these terms when we do polynomials and algebraic fractions.

270 Slide 166 / Order the terms by the degree of the variable in the expression 7x x + 4x 3. A 4x 3-7x 2-8x + 23 B x 2 + 4x 3-8x C 7x 2-8x + 4x D 4x 3 + 7x 2-8x + 23

271 Slide 166 (Answer) / Order the terms by the degree of the variable in the expression 7x x + 4x 3. A 4x 3-7x 2-8x + 23 B x 2 + 4x 3-8x Answer C 7x 2-8x + 4x D D 4x 3 + 7x 2-8x + 23 [This object is a pull tab]

272 Slide 167 / Order the terms by the degree of the variable in the expression 12 - x 3 + 7x 5-8x x. A x + 7x 5 - x 3-8x 2 B 7x 5 - x 3-8x x - 12 C 7x 5 - x 3-8x x + 12 D - x 3 + 7x x x

273 Slide 167 (Answer) / Order the terms by the degree of the variable in the expression 12 - x 3 + 7x 5-8x x. A x + 7x 5 - x 3-8x 2 B 7x 5 - x 3-8x x - 12 Answer C 7x 5 - x 3-8x x + 12 C D - x 3 + 7x x x [This object is a pull tab]

274 Slide 168 / Order the terms by the degree of the variable in the expression 41 + x xy + y 2. A x xy + y B xy + x 2 + y 2 C xy + y 2 + x 2 D x 2 + y y + 41