Math 803
Unit 2 – Rational Numbers and Variable
Expressions
Textbook (Lessons 1.1, 1.2, 1.10, and 8.1-8.3)
Name ________________________________________
Period ____________
Teacher Name __________________
1
Lesson 2.1 – Rational Numbers (8.NS.1) Textbook: 1.1
THE REAL NUMBER SYSTEM
Natural Numbers: The set of counting numbers not containing zero.
{𝟏, 𝟐, 𝟑, … }
Whole Numbers: The set of Natural numbers plus zero
{𝟎, 𝟏, 𝟐, 𝟑, … . . }
Integers: The set of all whole numbers and their opposites. These include
all the positive and negative “whole” numbers on the number line.
Rational Numbers: Numbers that can be written as a comparison of two
integers and expressed as a fraction where the denominator is not zero.
Examples of Rational Numbers
Fraction:
3
4
1
Mixed Number: 3 11 =
= 0.75
Whole Numbers: 15 =
15
34
11
3
Ratio: 15 dogs to 10 cats = 2
1
8
̅̅̅̅ = 1
Repeating Decimals: .33
3
Terminating Decimals: .32 = 25
19
5
Perfect squares: √25 = 5 = 1
Percent: 19% = 100
2
Converting Rational Numbers to decimal form
The fraction bar is a division bar. To write a rational number in fraction
form as a decimal, divide the numerator by the denominator.
Examples:
5
8
2
means 5 ÷ 8
3
means 2 ÷ 3
Practice
Write each fraction as a decimal
1)
3)
3
2)
8
5
4)
12
4
5
15
25
Writing terminating decimals as rational numbers in fraction form
0.25 =
25
100
=
1
4
1.4 = 1
(25 hundredths)
4
10
(1 and 4 tenths)
2
= 1 5 (reduce)
(reduce)
Practice
Write each decimal as a fraction
1) 0.35
2) 2.6
3) 3.26
4) 0.39
3
Writing repeating decimals as rational numbers in fraction form
Example 1:
0.555…
N = 0.555…
10(N) = (0.555…)10
(assign a variable to the repeating decimal)
(multiply each side by 10 to move decimal point one right)
10N = 5.555…
-N = -0.555…
9N = 5
9 = 9
5
N=9
(subtract the original equation to eliminate the repeating part)
(divide each side by 9)
Example 2:
0.2323…
N = 0.2323…
(assign a variable to the repeating decimal)
100(N) = (0.2323…)100
(multiply each side by 100 to move decimal point one right)
100N = 23.2323…
-N = -0.2323…
(subtract the original equation to eliminate the repeating part)
99N = 23
99 = 99
23
N = 99
(divide each side by 99)
Example 2:
0.125125…
N = 0.125125…
(assign a variable to the repeating decimal)
1000(N) = (0.125125…)1000
(multiply each side by 100 to move decimal point one right)
1000N = 125.125125…
-N = -0.125125…
999N = 125
999 = 999
125
N = 999
(subtract the original equation to eliminate the repeating part)
(divide each side by 99)
Notice any patterns?
4
In your own words, write a shortcut for changing repeating decimals to
fractions. Be specific!
Use the shortcut to change each decimal into a fraction.
̅̅̅̅
1) 0.12
2) 0.6̅
3) 0.2525….
4) 0.453453…
̅̅̅̅̅
5) 0.682
6) 2.7̅
5
Lesson 2.2 – Powers and Exponents (8.EE.1) Textbook: 1.2
Powers and Exponents
Power: A product of repeated factors expressed using an exponent and
a base.
Base:
the common factor
Exponent:
the ‘raised’ number, tells how many times the base is used as
a factor
Exponent
2 ⋅ 2 ⋅ 2 ⋅ 2 = 24
Base
Factors
Power
Words
Factors
51
3 to the second power
or
3 squared
4⋅4⋅4
64
5𝑛
6
Write an expression using exponents
The product of repeated factors can be expressed as a power. A power consists of a base and an
exponent. The exponent tells how many times the base is used as a factor.
Write each expression using exponents.
2 ⋅ 2 ⋅ 2 ⋅ 2 ⋅ 2 = 𝟐𝟓
The base 2 is a factor 5 times
(-5) ⋅ (-5) ⋅ (-5) = (−5)3
The base 5 is a factor 3 times and the base 2 is a factor 2 times
x ⋅ x ⋅ y ⋅ x ⋅ y ⋅ y ⋅ x = 𝒙 𝟒 ⋅ 𝒚𝟑
The base x is a factor 4 times and the base y is a factor 3 times
𝟏
𝟏
𝟏
𝟏 𝟑
⋅ ⋅ = (𝟑 )
𝟑 𝟑 𝟑
1
The base 3 is a factor 3 times
PRACTICE:
Write each expression using exponents
1) 4 ⋅ 4 ⋅ 4 ⋅ 4 =
2) (-11)(-11)(-11) =
3) r ⋅ s ⋅ r ⋅ r ⋅ s⋅ s ⋅ r =
4) 2 ⋅ 2 ⋅ 7 ⋅ 7 ⋅ 7 =
7
Evaluating Powers
To evaluate a power, perform the repeated multiplication to find the product.
𝟐𝟔 = 𝟐 ∙ 𝟐 ∙ 𝟐 ∙ 𝟐 ∙ 𝟐 ∙ 𝟐 = 𝟔𝟒
(−4)3 =
(6)5 =
𝟖𝟑 ⋅ 𝟐𝟓 = 𝟖 ∙ 𝟖 ∙ 𝟖 ∙ 𝟐 ∙ 𝟐 ∙ 𝟐 ∙ 𝟐 ∙ 𝟐 = 𝟓𝟏𝟐 ∙ 𝟑𝟐 = 𝟏𝟔, 𝟑𝟖𝟒
42 ⋅ 94 =
123 ⋅ 23 =
55 ⋅ 86 =
𝟏 𝟑
(− 𝟑)
𝟏
𝟏
𝟏
𝟏
= − 𝟑 ∙ − 𝟑 ∙ − 𝟑 = − 𝟐𝟕
𝟐 𝟒
(𝟑) =
3 5
( ) =
4
1) The deck of a skateboard has an area of about 25 ∙ 7 square inches. What is the
area of the skateboard deck?
2) As of January 2012 there are 140 million active Twitter users. There
are 23 ∙ 31 ∙ 52 ∙ 17 million tweets a month. About how many
tweets is that?
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Evaluating expressions
The order of operations states that exponents are evaluated before multiplication, division, addition, and
subtraction.
Evaluate m2 + (n – m)3 if m = –3 and n = 2.
m2 + (n – m)3 = (–3)2 + (2 – (–3))3
Replace m with –3 and n with 2.
= (–3)2 + (5)3
Perform operations inside parentheses.
= (–3 • –3) + (5 • 5 • 5)
Write the powers as products.
= 9 + 125 or 134
Add.
Evaluate each expression if a = 5 and b = –4.
1. 𝑎2 + 𝑏 2
2. (𝑎 + 𝑏)2
3. 𝑎 + 𝑏 2
4.
𝑥 3 + 𝑦 2 ; x = 2 and y = 12
5. 2𝑛 - 𝑛3 ; n = 5
9
Lesson 2.3 – Compare real Numbers (8.NS. 1, 8.EE.1, 8.EE.2)
Textbook: 1.10
Compare Real Numbers
Numbers that are not rational are called irrational numbers.
Irrational Numbers: A number that cannot be expressed as a ratio of two
integers.
Examples of Irrational Numbers
Square roots of Non-perfect squares: √3
Non-repeating, non- terminating Decimals: 2.645751311….
pi: π = 3.141592…
THE REAL NUMBER SYSTEM
Name all sets of numbers to which each real number belongs.
1. 5
whole number, integer, rational number
2. 0.666… Decimals that terminate or repeat are rational numbers, since they can be
expressed
2
as fractions. 0.666 … 3
3. −√25
Since −√25 = −5, it is an integer and a rational number.
4. √11
√11 ≈ 3.31662479 … Since the decimal does not terminate or repeat, it
10
is an irrational number.
Name all sets of number to which each real number belongs.
Exercises
1. 30
3. 5
2. – 11
4
4. √21
7
6. −√9
5. 0
7.
6
8. −√101
3
11
Compare and Order Real Numbers
To compare and order real numbers, write them in the same notation.
To compare real numbers write each number as a decimal and then compare the decimal values.
Example 5
Replace
with <, >, or = to make 𝟐
𝟏
𝟒
√𝟓 a true statement.
Write each number as a decimal.
1
2 = 2.25
4
√5 ≈ 2.236067 …
1
Since 2.25 is greater than 2.236067 … , 2 > √5.
4
Compare the numbers using <, >, or = to make a true statement
1) √7
2
3) √11
3
2
3
1
3
2) 15.7%
4)
√6.25
√0.02
250%
Order the set of number from least to greatest
4
5) {√30, 6, 5 , 5.36}
5
7
6) {√5, √6, 2.5, 2.55, }
3
12
Area of Circles REVIEW
The area A of a circle equals the product of pi (π) and the square of its radius r.
A = πr2
Example
Find the area of the circle. Use 3.14 for π.
A = πr2
Area of circle
A ≈ 3.14 • 5
Replace π with 3.14 and r with 5.
A ≈ 3.14 • 25
52 = 5 • 5 = 25
2
A ≈ 78.5
The area of the circle is approximately 78.5 square centimeters.
Find the area of each circle. Round to the nearest tenth. Use 3.14 or
1.
2.
4.
5.
7. diameter = 9.4 mm
8. radius = 3 2 ft
𝟐𝟐
for
𝟕
π.
3.
6.
1
13
9. radius = 8 in.
Lesson 2.4 – Volume of Cylinders (8.G.9) Textbook: 8.1
Volume: ____________________________________________________
____________________________________________________________
Cylinder: ___________________________________________________
___________________________________________________________
FORMULA
V = Bh
B = area of the base (circle) = π𝑟 2
h = height
As with prisms, the area of the base of a cylinder tells the number of cubic units in one
layer. The height tells how many layers there are in the cylinder. The volume V of a
cylinder with radius r is the area of the base B times the height h.
V = Bh, where B = 𝜋𝑟 2 , or V = 𝜋𝑟 2 h
The area of the base of a cylinder tells the number of cubic units in one
layer.
The height of the cylinder tells how many layers there are.
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Example
Find the volume of the cylinder. Round to the nearest tenth.
V ≈ 𝜋𝑟 2 h
Volume of a cylinder
V ≈ π(2)2 (5)
Replace r with 2 and h with 5.
V ≈ 62.8318
Use a calculator
The volume is about 62.8 cubic inches.
Find the Volume of the Cylinder. Round to the nearest tenth.
Given a cylinder with the radius 5cm and height 8cm. Find the volume of
this cylinder. Take π as 3.14.
1) Find the volume of a cylinder with a height of 5 mm and a diameter of
24 mm. Take π as 3.14.
2) Given a cylinder with a volume of 502.4 in3 and radius 4 in. Find the
height of the cylinder. Take π as 3.14.
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3) Find the volume of a cylinder with height 8 yd. and radius 10.5 yd.
4) Find the volume of the cylinder.
5) Find the height of a cylinder whose volume is 167.1cm3 and radius of 2
cm.
6) A metal paperweight is in the shape of a cylinder. The paperweight
has a height of 1.5 inches and a diameter of 2 inches. How much does
the paperweight weigh if 1 square inch weights 1.8 ouches? Round to
the nearest tenth.
16
Lesson 2.5 – Volume of Cone (8.G.9) Textbook: 8.2
A cone is a three-dimensional shape with one circular base.
The volume V of a cone with radius r is one third the area of the base B times the height h.
1
1
V = 3Bh or V = 3πr2h
FORMULA
1
V = Bh
3
B = area of the base (circle) = π𝑟 2
h = height
Example
Find the volume of the cone. Round to the nearest tenth.
1
V = 3πr2h
Volume of a cone
1
V = 3(π • 62 • 12)
r = 6 and h = 12
V ≈ 452.4
Simplify.
The volume is about 452.4 cubic feet.
17
1) Find the volume of the cone.
2) Find the volume of the cone. Round to the nearest tenth.
3) Find the volume of the cone. Round to the nearest tenth.
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4) Find the volume of the cone. Round to the nearest tenth.
5) cone-shaped paper cup is filled with water. The height of the cup is
10 centimeters and the diameter is 8 centimeters. What is the
volume of the paper cup?
6) April is filling six identical cones for her piñata. Each cone has a
radius of 1.5 inches and a height of 9 inches. What is the total
volume of the cones? Round to the nearest tenth.
19
Lesson 2.6 – Volume of Sphere (8.G.9) Textbook: 8.3
A sphere is a set of all points in space that are a given distance from a given point.
The volume V of a sphere with radius r is four thirds the product of π and the cube of the
radius r.
4
V = 3πr3.
Hemisphere: ______________________________________________
__________________________________________________________
FORMULAS
Sphere
4
V = π𝑟 3
3
Hemisphere
1 4
V = ( π𝑟 3 )
2 3
20
Example
Find the volume of the sphere. Round to the nearest tenth.
4
V = πr3
3
4
Volume of a sphere
V = (π • 43)
3
r=4
V ≈ 268.1 Simplify. Use a calculator.
The volume is about 268.1 cubic feet.
1) Find the volume of the sphere. Round to the nearest tenth.
2) Find the volume of a sphere with radius 5mm.
3) Find the volume of the hemisphere with radius 8 in.
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4) Find the volume of the hemisphere with diameter 2 cm.
5) Sarah is blowing up spherical balloons for her brother’s birthday
party. One of the balloons has a radius of 3 inches. What is the
volume of the balloon?
6) Find the volume of the sphere. Round to the nearest tenth.
7) Find the volume of a sphere with diameter 22 cm.
8) A spherical stone in the courtyard of the National Museum of Costa
Rica has a diameter of about 8 feet. Find the volume of the spherical
stone. Round to the nearest tenth.
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9) A basketball has a diameter of about 9.5 inches. Find the volume of
the basketball to the nearest tenth.
10)
An ice cream cone is packed full of ice cream and a generous
hemisphere of ice cream is placed on top. If the volume of ice cream
inside the cone is the same as the volume of ice cream outside the
cone, find the height of the cone.
11)
Jackie bought a game that contained a ball and 10 jacks. The
ball had a radius of 2 inches. What is the volume of the ball? Round
to the nearest tenth.
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