1. Rational Numbers
Chapter 1, Lesson 1

2. Vocabulary Complete this graphic organizer. Rational Number
Define in your own words.
Percent
Fraction
Decimal
Mixed Number

3. Rational Numbers
All rational numbers are written as a RATIO. Example 1. During a recent regular season, a Texas Ranger baseball player had 126 hits and was at bat 399 times. Write a fraction in the simplest form to represent the ratio of the number of hits to the number of at bats. =
6 19

4. Rational Numbers

5. Repeating vs. Terminating Decimals
Rational
Numbers
Repeating
Decimal
Terminating Decimal
1 2 …
0.5
2 5 ….
0.4
5 6
0.8333…
Does not terminate
(0.83)

6. Write each fraction as a mixed number as a decimal.
Example 1
Write each fraction as a mixed number as a decimal. a
5 ÷ 8 = 0.625 b
−5 3 = -5 ÷ 3 = -1.6

7. Got it? 1 Write each fraction as a decimal. 1. 3 4 2. - 2 9
-0.2
0.75
4.52
3.09

8. Example 2
In a recent season, St. Louis Cardinals first baseman Albert Pujols had 175 hits in 530 at bats. To the nearest thousand, find his batting average. We need to the number of hits, 175, by the number of at bats, ÷ 530 = Round to the nearest thousand

9. Got it?
In a recent season, NASCAR driver Jimmie Johnson won 6 of the 36 total races held. To the nearest thousandth, find the part of races he won. Divide the races he won, 6, out of the races he held, ÷ 36 = … Round to the nearest thousandth

10. Example 3
Write 0.45 as a fraction.
= =

11. Subtract N = 0.55555… to eliminate the repeating part.
Example 4
Write 0.5 as a fraction.
N = …
Multiply each side by 10.
10N = 10( ….)
10N = …..
- N = ….
Subtract N = … to eliminate the repeating part.
9N = 5
N = 5 9
Try 0.7 repeating on the board.

12. Example 5
Write 2.18 as a mixed number. N = … Multiply each side by N = 100( ….) 100N = ….. - N = …. Subtract N = … to eliminate the repeating part. 99N = 216 N = =
Try 0.27 repeating on the board.

13. Homework Independent Practice: 1 – 10, 14 – 15, 17, 19
Try 0.27 repeating on the board.

14. Powers and Exponents
Lesson 2

15. Saving money
Yogi decided to start saving money by putting a penny in his piggy bank, then doubling the amount he saves each week. 1. Complete the table. 2. How many 2’s are multiplied to find his savings in Week 4? Week 5? 3. How much will he save in Week 8?
Week 1 2 3 4 5
Weekly Savings
$0.01
$0.02
Total Savings
$0.03
$0.04
$0.08
$0.16
$0.32
$0.07
$0.15
$0.31
$0.63 4 5
Problem of the Day
$2.56

16. Write and Evaluate Powers
2  2  2  2 = 24
4 factors
Read and Write Powers
Power
Words
Factors 31
3 to the first power 3 32
3 to the second power
3  3 33
3 to the third power
3  3  3 3n
3 to the nth power
3  3  3  3 …3
n factors

17. Example 1 Write each expression using exponents.
(-2)  (-2)  (-2)  3  3  3  3
There are three (-2)’s and four 3’s.
(-2)3  34
a  a  b  b  a
There are three a’s and two b’s.
a3  b2

18. Example 2
Evaluate. ( 2 3 )4 2 3  2 3  2 3  2 3 = 2 𝑥 2 𝑥 2 𝑥 2 3 𝑥 3 𝑥 3 𝑥 3 = Got it? ( 1 5 )3 =

19. Example 3
The deck of a skateboard has an area of about 25  7 square inches. What is the area of the skateboard deck? 25  7 2  2  2  2  2  7 32  square inches

20. Example 4 Evaluate each expression if a = 3 and b = 5. a2 + b4
= 634
(a – b)2
(3 – 5)2
(-2)2
(-2)(-2) = 4

21. Multiply and Divide Monomials
Lesson 3

22. Monomials
Monomial: a number, variable, or a product of a number and variable Examples: a4b8 3x2y g

23. What did you do to the exponents?
Law of Exponents
c  c  c  c  c = c5
c5  c4 = (c  c  c  c  c)  (c  c  c  c)
= c9
What did you do to the exponents?
ADD THE EXPONENTS

24. Product of Powers 24  23 = 24+3 or 27 am  an = am+n
Words: To multiply powers with the same base, add their exponents.
Examples:
24  23 = 24+3 or 27
am  an = am+n

25. Example 1 - Simplify c3  c5 c3 + 5 = c8 -3x2  4x5 (-3)(4)  x2  x5

26. SUBTRACT THE EXPONENTS
Law of Exponents
r 4 = r  r  r  r
r = r  r
= r2
What did you do with the exponents?
SUBTRACT THE EXPONENTS

27. quotient of Powers 37 33 = 37 – 3 = 34 𝑎𝑚 𝑎𝑛 = am – n
Words: To divide powers with the same base, subtract their exponents.
Examples:
= 37 – 3 = 34
𝑎𝑚 𝑎𝑛 = am – n

28. Example 2 - Simplify
48 42
= 46 = 4,096
12𝑥5 2𝑥3
12𝑥5 2𝑥3 = 6x2

29. Powers of Monomials
Lesson 4

30. Power of a Power
Words: To find the power of a power, multiply the exponents. Examples: (52)3 = 52 x 3 = 56 (am)n = am  n (64)5 = (64)(64)(64)(64)(64)
5 factors

31. Example 1
Simplify.
(84)3
84 x 3
812
(k7)5
k7 x 5
k35

32. Power of a Product
Words: To find the power of a product, find the power of each factor and multiply. Examples: (6x2)3 = 63  (x2)3 = 216x6 (6x2)3 = (6x2)(6x2)(6x2)
3 factors

33. Example 2 Simplify. (4p3)4 44  p3x4 256p12 (-2m7n6)5

34. Negative Exponents
Lesson 5

35. Zero and Negative Exponents
Words: Any number to the zero power is 1. Examples: 40 = 1 b0 = 1 Words: Any number to the negative power is the multiplicative inverse of its nth power. 7-3 = 1 73 = k-n = 1 𝑘𝑛

36. Example 1 - Simplify
6-2
= = 1 36
a-5
= 1 𝑎5 80
= 1

37. Example 2 Write each fraction using a negative exponent. 1 52 = 5-2
1 49
= = 7-2

38. Powers of 10 Exponential Form Standard Form How many Zero’s? 103 1,000
102
100 2
101 10 1
10-1
𝟏 𝟏𝟎 =𝟎.𝟏
10-2
𝟏 𝟏𝟎𝟎 =𝟎.𝟎𝟏
10-3
𝟏 𝟏𝟎𝟎𝟎 =𝟎.𝟎𝟎𝟏

39. Example 3
One human hair is about inch in diameter. Write this decimal as a power of has 4 zeros = 1 10,000 = = 10-4

40. Example 4 - Simplify
53 x 5-5
= 53+(-5)
= 5-2
= = 1 25

41. Example 5 - Simplify
𝑏2 𝑏6 =b(2 – 6) =b-4 = 1 𝑏4

42. Scientific Notation
Lesson 6

43. Scientific Notation Table
Expression
Product
4.7 x 103 = 4.7 x 1000
4,700
4.7 x 102 = 4.7 x 100
470
4.7 x 101 = 4.7 x 10 47
4.7 x 10-1 = 4.7 x 0.1
0.47
4.7 x 10-2 = 4.7 x 0.01
0.047
4.7 x 10-3 = 4 x 0.001
0.0047

44. Scientific Notation
Words: when a number is written as the product of a factor and an integer power of 10. The number must be between 1 and 10. Symbols: a x 10n, where a is between 1 and 10 Example: 435,000,000 = 4.35 x 108

45. Two Rules for S.N.
If the number is greater than or equal to 1, the power of 10 is positive.
If the number is between 0 and 1, then power of ten is negative.

46. Example 1 Write each number in standard form. 5.34 x 104 5.34 x 10,000
move the decimal point 4 times to the right
= 53,4000
3.27 x 10-3
3.27 x 0.001
move the decimal point 3 times to the left

47. Example 2 Write each number in scientific notation. 3,725,000
3.16 x 10-4
show work on the board for these problems.

48. Dollars Spent by International Visitors in the U.S.
Example 3 - comparing
Refer to the table at the right. Order the countries according to the amount of money visitors spent in the US from greatest to least.
Dollars Spent by International Visitors in the U.S.
Country
Dollars Spent
Canada
1.03 x 107
India
1.83 x 106
Mexico
7.15 x 106
United Kingdom
1.06 x 107
STEP 1:
1.06 x x 106
1.03 x x 106
STEP 2:
1.06 > > 1.83
>
Students will need to fill in the answer
CORRECT ORDER:
United Kingdom, Canada, Mexico, India

49. Example 4
If you could walk to the moon at a rate of 2 meters per second, it would take you 1.92 x 108 seconds to walk to the moon. Is it more appropriate to report this time as 1.92 x 108 seconds, or 6.09 years? Explain. The measure 6.09 years is more appropriate. The number 1.92 x 108 seconds is too large of a number to describe a walk to the moon.

50. Compute with Scientific Notation
Lesson 7

51. Example 1
Evaluate (7.2 x 103)(1.6 x 104). Express in scientific notation. Rearrange the numbers (7.2 x 1.6)(103 x 104) (11.52)(107) Move the decimal over to that the number is in scientific notation x 108

52. Got it?
(8.4 x 102)(2.5 x 106)
2.1 x 109
(2.63 x 104)(1.2 x 10-3)
3.156 x 101
Evaluate on the board.

53. Example 2
In 2010, the world population was about 6,860,000,000. The population of the United States was about 3 x 108. About how many times larger is the world population than the population of the United States? Estimate 6,860,000,000 ≈ 7 x 109 Find 7 𝑥 𝑥 ( 7 3 )(101) 2.3 x 10 = 23 The world’s population is about 23 times bigger than the United States population.

54. Adding Numbers in Scientific Notation
(6.89 x 104) + (9.24 x 105)
Make each number have the same power of ten.
(9.24 x 105) = 92.4 x 104
Add the numbers = 99.29
= x 104
Put this number in scientific notation.
=9.929 x 105
Method 2: simply each number into standard form, then add old school.

55. Adding Numbers in Scientific Notation
b. 593,000 + (7.89 x 106)
Each number must be in scientific notation.
593,000 = 5.93 x 105
Make each number have the same power of ten.
(7.89 x 106) = 78.9 x 105
Add the numbers = 84.83
= x 105
Put this number in scientific notation.
=8.483 x 106
Method 2: simply each number into standard form, then add old school.

56. Subtracting Numbers in Scientific Notation
(7.83 x 108) – 11,610,000
Each number must be in scientific notation.
11,610,000 = x 107
Make each number have the same power of ten.
(7.83 x 108) = 78.3 x 107
Subtract the numbers =
= x 107
Put this number in scientific notation.
= x 108
Method 2: simply each number into standard form, then subtract old school.

57. Got it?
(8.41 x 103) + (9.71 x 104)
x 105
(1.263 x 109) - (1.525 x 107)
x 109
(6.3 x 105) + 2,700,000
3.33 x 106

58. Roots
Lesson 8

59. Vocabulary
Square Root: A number is one of its two equal factors. 121 = 11 Perfect Square: Squares of integers: 1, 4, 9, 16, 25, 36, 49, 64… Also, (-1)2 = 1 (-2)2 = 4 (-6)2 = 36

60. Example 1
A. 64 = 8 B. ± 1.21 = ± 1.1 or you could say 1.1, -1.1 C = D. −16 there are not real square roots

61. Got it? 1
9 16 ±
- 49
−100
3 4
± 0.9
- 7
No real solution

62. Example 2
Solve t2 = 169. Check your solutions. t2 = 169 𝑡2 = ± 169 t = ± = 169 (-13)2 = 169 It checks!!

63. Got it? 2
289 = a2
m2 = 0.09
y2 = 4 25
a = ± 17
m = ± 0.3
y = ± 2 5

64. Cube Roots
Cube root: a number is one of three equal factors. 8 = 2 • 2 • 2 = = 2 Perfect Cubes: 1, 8, 27, 64, 125, 216… Also, -1, -8, -27, -64, -125, -216…

65. Example 3
A = 5 5 • 5 • 5 =125 B. 3 −27 = • -3 • -3 = -27

66. Got it? 3
A B. 3 −64 C 9 -4 10

67. Example 4
Dylan has a planter in the shape of a cube that holds 8 cubic feet of potting soil. Solve the equation 8 = s3 to find the side length s of the container. 8 = s3 3 8 = 3 𝑠3 2 = s

68. Got it? 4
An aquarium in the shape of a cube that will hold 25 gallons of water has a volume of cubic feet. Solve s3 = to find the length of one side of the aquarium. s = 15

69. Estimating Roots
Lesson 9

70. Estimating Square and cube Roots
Non-perfect squares can be estimated. 8 is between 4 and 9 8 is closest to 3 since 8 is closest to 9 . Let’s make a number line.

71. Example 1
Estimate 83 to the nearest integer. 81 < 83 < is closest to 9.

72. Got it? 1 35
170
44.8 6 13 7

73. Example 2
Estimate to the nearest integer. Find all the cube roots 1, 8, 27, 64, 125, 216, 343, 512, 729, 1000, … Where does 320 fall? 216 < 320 < is closest to 7

74. Got it? 2
3 62
3 25 4 3 5

75. Example 3
Wyatt wants to fence in a square portion of the yard to make a play area for his new puppy. The area covered is 2 square meters. Approximately, how much fencing should Wyatt buy? Each side is 2 , so the perimeter is is closest to 6 yards.

76. Got it? 3
Sue wants to fence in a square portion of the yard to make a play area for her new puppy. The area covered is 3 square meters. Approximately, how much fencing should Sue buy? Each side is 3 , so the perimeter is is closest to 7 yards.

77. Example 4
The golden rectangle is found frequently in the nautilus shell. The length of the longer side divided by the length of the shorter side is equal to Estimate this value. 5 is closest to 2. So, equals estimates to 1.5.

78. Compare Real Numbers
Lesson 10

79. Real Numbers

80. “Little Subset”
A rational subset are integers
They walk this number line
Go both directions from zero
They go left, they go right
Now, take the positive integers
And let’s give them a name
zero, 1,2,3,4,5 etc…
That’s the whole number game
(Chorus)
Bridge:
Bummed irrational numbers
Feel such heavy shame
They’re real, but that’s just not the same
They envy subsets that complain
So they complain
blah blah blah blah blah
Verse 3:
We can’t be written as fractions
Else we’d be rational
We don’t repeat and/or terminate
Like Pi, …
Verse: Give me a number that’s rational Like any fraction that hurts Accepting positive or negative Are you ready…for two thirds? Or I’ll take the terminating decimal .15, it will be If it’s repeating, it’s sensible So How about, Chorus: Hey little subset, I’m a real number The big super-set, rational and irrational Hey smaller subset You call this place an integer? It’s bigger than the whole numbers and counting without the zeros

81. Example 1 Name all sets of numbers in which each real number belongs.
… it has a pattern so it’s rational
36 it equals 6, so it’s a natural, whole, integer and rational
does not repeat, so it is irrational

82. Got it? 1 10 irrational -2 2 5 100 rational
100
irrational
rational
natural, whole, integers, and rational

83. Example 2
Fill in the blank with <, >, and = to make a true statement.
15.7%
<
>

84. Got it? 2 %
<
> =

85. Example 3
Order the set { 30 , 6, , …} from least to greatest. 30 ≈ = = … ≈ 5.36
5.366…, 30 , , and 6

86. Got it? 3
Order -7, - 60 , -7.7, and from least to greatest. -7 = = = = -7.33
- 60 , -7.7, , and -7

87. Example 4
On a clear day, the number of miles a person can see to the horizon is about 1.23 times the square root of his or her distance from the ground in feet. Suppose Frida is at the Empire Building observation deck at 1,250 feet and Kia is at the Freedom Tower observation deck at 1,362 feet. How much farther can Kia see then Frida? Frida: 1.23 x 1,250 ≈ Kia: 1.23 x 1,362 ≈ – = 1.90 miles

88. vocabulary
base cube root exponent irrational number monomial perfect cube perfect square power radical sign
real number
rational number
repeating decimal
scientific notation
square root
terminating decimal