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Rational Numbers
Chapter 1, Lesson 1
Vocabulary
Complete this graphic organizer. Rational Number
Define in your own words.
Percent Fraction
Decimal Mixed Number
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. 126
399 =
6
19
Rational Numbers
Repeating vs. Terminating Decimals
Rational Numbers
Repeating Decimal
Terminating Decimal
1
2 0.50000… 0.5
2
5 0.40000…. 0.4
5
6
0.8333… Does not terminate
(0.83)
Example 1
Write each fraction as a mixed number as a decimal.
a. 5
8
5 ÷ 8 = 0.625
b. -12
3
;5
3 = -5 ÷ 3 = -1.6
Got it? 1
Write each fraction as a decimal.
1. 3
4 2. -
2
9
3. 4 13
25 4. 3
1
11
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.
175 ÷ 530 = 0.3301886792
Round to the nearest thousand.
0.330
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.
Example 3
Write 0.45 as a fraction.
0.45 = 45
100 =
9
20
Example 4
Write 0.5 as a fraction.
N = 0.55555…
Multiply each side by 10.
10N = 10(0.555555….)
10N = 5.5555…..
- N = 0.55555….
Subtract N = 0.55555… to eliminate the repeating part.
9N = 5
N = 5
9
Example 5
Write 2.18 as a mixed number.
N = 2.18181818…
Multiply each side by 100.
100N = 100(2.1818181818….)
100N = 218.181818…..
- N = 2.181818….
Subtract N = 0.181818… to eliminate the repeating part.
99N = 216
N = 216
99 = 2
2
11
Homework
Independent Practice: 1 – 10, 14 – 15, 17, 19
Powers and Exponents
Lesson 2
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 0 1 2 3 4 5
Weekly Savings $0.01 $0.02
Total Savings $0.01 $0.03
$0.04
$0.07
$0.08
$0.15
$0.16
$0.31
$0.32
$0.63
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
Example 1
Write each expression using exponents.
a. (-2) (-2) (-2) 3 3 3 3
There are three (-2)’s and four 3’s.
(-2)3 34
b. a a b b a
There are three a’s and two b’s.
a3 b2
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
= 16
81
Got it?
(1
5)3 =
1
125
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 7
224 square inches
Example 4
Evaluate each expression if a = 3 and b = 5.
a. a2 + b4
32 + 54
9 + 625
= 634
b. (a – b)2
(3 – 5)2
(-2)2
(-2)(-2) = 4
Multiply and
Divide Monomials
Lesson 3
Monomials
Monomial: a number, variable, or a product of a
number and variable
Examples:
32 74 a4b8 3x2y g
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
Product of Powers
Words: To multiply powers with the same base,
add their exponents.
Examples:
24 23 = 24+3 or 27
am an = am+n
Example 1 - Simplify
a. c3 c5
c3 c5
c3 + 5 = c8
b. -3x2 4x5
-3x2 4x5
(-3)(4) x2 x5
-12x7
Law of Exponents
r 4 = r r r r
r2 = r r
= r2
What did you do with the exponents?
SUBTRACT THE EXPONENTS
quotient of Powers
Words: To divide powers with the same base,
subtract their exponents.
Examples:
37
33 = 37 – 3 = 34
𝑎𝑚
𝑎𝑛 = am – n
Example 2 - Simplify
a.48
42
48
42 = 46 = 4,096
b.12𝑥5
2𝑥3
12𝑥5
2𝑥3 = 6x2
Powers of Monomials
Lesson 4
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
Example 1
Simplify.
a. (84)3
84 x 3
812
b. (k7)5
k7 x 5
k35
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
Example 2
Simplify.
a. (4p3)4
44 p3x4
256p12
b. (-2m7n6)5
(-2)5 m75 n65
-32m35n30
Negative Exponents
Lesson 5
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.
Examples:
7-3 = 1
73 = 1
343 k-n =
1
𝑘𝑛
Example 1 - Simplify
a. 6-2
= 1
62 =1
36
b. a-5
= 1
𝑎5
c. 80
= 1
Example 2 Write each fraction using a negative exponent.
a.1
52
= 5-2
b.1
49
= 1
72 = 7-2
Powers of 10
Exponential Form Standard Form How many Zero’s?
103 1,000 3
102 100 2
101 10 1
100 1 0
10-1 𝟏
𝟏𝟎= 𝟎. 𝟏 1
10-2 𝟏
𝟏𝟎𝟎= 𝟎. 𝟎𝟏 2
10-3 𝟏
𝟏𝟎𝟎𝟎= 𝟎. 𝟎𝟎𝟏 3
Example 3
One human hair is about 0.0001 inch in diameter.
Write this decimal as a power of 10.
0.0001 has 4 zeros
0.0001 = 1
10,000=
1
104
= 10-4
Example 4 - Simplify
53 x 5-5
= 53+(-5)
= 5-2
= 1
52 = 1
25
Example 5 - Simplify
𝑏2
𝑏6
=b(2 – 6)
=b-4
= 1
𝑏4
Scientific Notation
Lesson 6
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
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
Two Rules for S.N.
1. If the number is greater than or equal to 1, the
power of 10 is positive.
2. If the number is between 0 and 1, then power
of ten is negative.
Example 1
Write each number in standard form.
a. 5.34 x 104
5.34 x 10,000
move the decimal point 4 times to the right
= 53,400
b. 3.27 x 10-3
3.27 x 0.001
move the decimal point 3 times to the left
0.00327
Example 2
Write each number in scientific notation.
a. 3,725,000
3.725 x 106
b. 0.000316
3.16 x 10-4
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 107 7.15 x 106
1.03 x 107 1.83 x 106 > STEP 2:
1.06 > 1.03 7.15 > 1.83
CORRECT ORDER:
United Kingdom, Canada, Mexico, India
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.
Compute with
Scientific Notation
Lesson 7
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.
1.152 x 108
Got it?
a. (8.4 x 102)(2.5 x 106)
b. (2.63 x 104)(1.2 x 10-3)
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 109
3 𝑥 108.
(7
3)(101)
2.3 x 10 = 23
The world’s population is about 23 times bigger than the
United States population.
Adding Numbers in Scientific
Notation
a. (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. 6.89 + 92.4 = 99.29
= 99.29 x 104
Put this number in scientific notation.
=9.929 x 105
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. 78.9 + 5.93 = 84.83
= 84.83 x 105
Put this number in scientific notation.
=8.483 x 106
Subtracting Numbers in
Scientific Notation
(7.83 x 108) – 11,610,000
Each number must be in scientific notation.
11,610,000 = 1.161 x 107
Make each number have the same power of ten.
(7.83 x 108) = 78.3 x 107
Subtract the numbers. 78.3 - 1.161 = 77.139
= 77.139 x 107
Put this number in scientific notation.
=7.7139 x 108
Got it?
a. (8.41 x 103) + (9.71 x 104)
b. (1.263 x 109) - (1.525 x 107)
c. (6.3 x 105) + 2,700,000
Roots
Lesson 8
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
Example 1 A. 64 = 8
B. ± 1.21 = ± 1.1 or you could say 1.1, -1.1
C. -25
36 = -
5
6
D. −16 there are not real square roots
Got it? 1
A.9
16
B. ± 0.81
C. - 49
D. −100
Example 2
Solve t2 = 169. Check your solutions.
t2 = 169
𝑡2 = ± 169
t = ± 13
132 = 169 (-13)2 = 169
It checks!!
Got it? 2
A. 289 = a2
B. m2 = 0.09
C. y2 = 4
25
Cube Roots
Cube root: a number is one of three equal factors.
8 = 2 • 2 • 2 = 23
83
= 2
Perfect Cubes: 1, 8, 27, 64, 125, 216…
Also,
-1, -8, -27, -64, -125, -216…
Example 3
A. 1253
= 5
5 • 5 • 5 =125
B. −273
= -3
-3 • -3 • -3 = -27
Got it? 3
A. 7293
B. −643
C. 10003
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
83
= 𝑠33
2 = s
Got it? 4
An aquarium in the shape of a cube that will hold 25
gallons of water has a volume of 3.375 cubic feet.
Solve s3 = 3.375 to find the length of one side of the
aquarium.
Estimating Roots
Lesson 9
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.
Example 1
Estimate 83 to the nearest integer.
81 < 83 < 100
83 is closest to 9.
Got it? 1
A. 35
B. 170
C. 44.8
Example 2
Estimate 3203
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 < 343
3203
is closest to 7
Got it? 2
A. 623
B. 253
C. 129.63
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 4 2.
4 2 is closest to 6 yards.
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?
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 1: 5
2.
Estimate this value.
5 is closest to 2. So, 1:2
2 equals 1.5.
1: 5
2 estimates to 1.5.
“Little Subset” 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, .333333333 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
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, 3.14159265… (Chorus)
Compare Real
Numbers
Lesson 10
Real Numbers
Example 1
Name all sets of numbers in which each real
number belongs.
A. 0.252525… it has a pattern so it’s rational
B. 36 it equals 6, so it’s a natural, whole,
integer and rational
C. - 7 does not repeat, so it is irrational
Got it? 1
A. 10
B. -2 2
5
C. 100
Example 2
Fill in the blank with <, >, and = to make a true
statement.
A. 7 22
3
B. 15.7% 0.02
<
>
Got it? 2
Fill in the blank with <, >, and = to make a true
statement.
A. 11 31
3
B. 17 4.03
C. 6.25 250%
Example 3
Order the set { 30, 6, 5 4
5, 5.3666…} from least to
greatest.
30 ≈ 5.48
6 = 6
5 4
5 = 5.8
5.3666… ≈ 5.36
5.366…, 30, 5 4
5, and 6
Got it? 3
Order -7, - 60, -7.7, and - 66
9 from least to greatest.
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 ≈ 43.05
Kia: 1.23 x 1,362 ≈ 45.51
45.51 – 43.05 = 2.46 miles
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