Download Lesson 1 - Triumph Learning

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TABLE OF CONTENTS
Introduction .................................................................................................... 1
Test-Taking Tips ................................................................................ 2
Unit 1 – Number Sense .................................................................................. 3
Lesson 1: Number Concepts ............................................................. 4
PASS Objective: 2.2
Lesson 2: Comparing and Ordering Rational Numbers ................. 13
PASS Objective: 2.1
Lesson 3: Computation with Rational Numbers............................. 27
PASS Objective: 2.1
Lesson 4: Ratio, Proportion, and Percent ....................................... 40
PASS Objective: 2.1
Lesson 5: Estimation and Problem Solving.................................... 46
PASS Objectives: 2.1, 2.2
Unit 2 – Algebraic Reasoning...................................................................... 57
Lesson 6: Writing and Solving Equations and Inequalities............ 58
PASS Objectives: 1.1, 1.2
Lesson 7: Graphing Equations and Inequalities ............................. 71
PASS Objectives: 1.1, 1.2
Unit 3 – Geometry........................................................................................ 93
Lesson 8: Plane Figures .................................................................. 94
PASS Objective: 3.2
Maintenance Concept: Geometric Figures
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Lesson 9: Solid Figures................................................................. 112
PASS Objective: 3.1
Unit 4 – Measurement ............................................................................... 121
Lesson 10: Measurement Systems ................................................ 122
PASS Objective: 4.3
Maintenance Concepts: Customary and Metric Measurements
Lesson 11: Geometric Measurement ............................................ 133
PASS Objectives: 4.1, 4.2, 4.3
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Table of Contents
Unit 5 – Data Analysis and Statistics ....................................................... 157
Lesson 12: Data Analysis.............................................................. 158
PASS Objectives: 5.1, 5.2, 5.3
Lesson 13: Probability .................................................................. 184
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Maintenance Concept: Simple Probabilities
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Unit 1 – Number Sense
PASS Objective: 2.2
Lesson 1: Number Concepts
In this lesson, you will write numbers using exponents and scientific notation. You will also
compute with numbers written using exponents and scientific notation.
Exponents
An exponent (or power) shows how many times the base number appears as a factor. To
evaluate 23, multiply three twos together.
exponent
‘
“
23 2 • 2 • 2 8
base
Here are a few reminders for evaluating exponents.
• A base with an exponent of 0 equals 1.
100 1
50 1
25,000,0000 1
• A base with an exponent of 1 equals the base number.
101 10
51 5
25,000,0001 25,000,000
• A positive base with a positive exponent equals a positive number.
42 16
53 125
1
2
2
1
4
• A negative base with an even exponent equals a positive number.
(3)2 (3) • (3) 9
(3)3 (3) • (3) • (3) 27
• A base with a negative sign in front equals a negative number.
33
(3 • 3 • 3) 27
92
(9 • 9) 81
• A base with a negative exponent equals the reciprocal of the base with a
positive exponent.
53 4
1
3
5
1
125
(8)3 1
(8)3
1
512
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• A negative base with an odd exponent equals a negative number.
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Lesson 1: Number Concepts
PASS Objective: 2.2
Rules of exponents
Follow these rules when multiplying or dividing powers with like bases or when simplifying a
power of a power.
Multiplication
To multiply numbers with exponents that have the same bases, add the exponents and keep the
bases the same.
Example
Multiply: 23 • 27
23 • 27 (2 • 2 • 2) • (2 • 2 • 2 • 2 • 2 • 2 • 2)
2(3 7)
210
1,024
The product of 23 and 27 is 1,024.
Division
To divide numbers with exponents that have the same bases, subtract the exponent of the
denominator from the exponent of the numerator and keep the bases the same.
Examples
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Divide:
38
5
3
38
5
3
Divide:
46
9
4
46
9
4
4•4•4•4•4•4
4•4•4•4•4•4•4•4•4
3(8 5)
1
4•4•4
33
1
3
4
27
1
64
3•3•3•3•3•3•3•3
3•3•3•3•3
The quotient of 38 and 35 is 27.
( 4(6 9) 43)
The quotient of 46 and 49 is
1
.
64
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Page 6
Unit 1 – Number Sense
PASS Objective: 2.2
Power of a power
To simplify a number that has a power raised to another power, multiply the exponents and keep
the base the same.
Example
Evaluate: (53)2
(53)2 (5 • 5 • 5) • (5 • 5 • 5)
5(3 • 2)
56
15,625
(53)2 is equal to 15,625.
Practice
Directions: For Numbers 1 through 12, evaluate the expressions.
1. 42 ____________
7. (22)0 ____________
2. (82)2 ____________
8.
109
2
10
3. 32 • 34 ____________
9.
(4)6
(4)3
____________
4. (3)3 ____________
10. (32)3 ____________
5. (25)2 ____________
11. 75 • 72 ____________
6. 64 ____________
12.
6
156
3
15
____________
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____________
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Lesson 1: Number Concepts
PASS Objective: 2.2
Scientific Notation
Scientific notation is used to represent very large or very small numbers. The following table
shows the first five positive and negative powers of 10.
Powers of 10
Negative
Positive
101 10
10 1 0.1
102 100
10 2 0.01
103 1,000
10 3 0.001
104 10,000
10 4 0.0001
105 100,000
10 5 0.00001
and so on . . .
and so on . . .
–
–
–
–
–
Changing from standard form to scientific notation
A number is written in scientific notation as the product of the coefficient (a number greater
than or equal to 1 but less than 10) and some power of 10. Follow these steps to change a
number from standard form to scientific notation.
Step 1: Move the decimal point to the left or right until you have a number greater than
or equal to 1 but less than 10.
Step 2: Count the number of places you moved the decimal point to the left or right and
use that number as the positive or negative power of 10.
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Step 3: Multiply the decimal (in Step 1) by the power of 10 (in Step 2).
Example
Write 3,945,600 in scientific notation.
Move the decimal point 6 places to the left.
3.945600.
Since the decimal point moved 6 places to the left, multiply by 106.
3.9456 • 106
3,945,600 3.9456 • 106
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Unit 1 – Number Sense
PASS Objective: 2.2
Example
Write 0.0000451 in scientific notation.
Move the decimal point 5 places to the right.
0.00004.51
Since the decimal point moved 5 places to the right, multiply by 105.
4.51 • 105
0.0000451 4.51 • 105
Changing from scientific notation to standard form
To change a number from scientific notation to standard form, either multiply or divide by a
power of 10. A number written in scientific notation with a positive power of 10 shows that
you are multiplying by a power of 10. To change a number written in scientific notation with
a positive power of 10 to standard form, move the decimal point to the right. The exponent
shows the number of places the decimal point will be moved.
Examples
2.001 • 105 2.00100. 200,100
A number written in scientific notation with a negative power of 10 shows that you are
dividing by a power of 10. To change a number written in scientific notation with a negative
power of 10 to standard form, move the decimal point to the left. The exponent shows the
number of places the decimal point will be moved.
Examples
9.232 • 104 0.0009.2932 0.0009232
5.30704 • 103 0.005.30704 0.00530704
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6.173 • 107 6.1730000. 61,730,000
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Lesson 1: Number Concepts
PASS Objective: 2.2
Practice
Directions: For Numbers 1 through 5, write
each number in scientific notation.
Directions: For Numbers 6 through 10, write
each number in standard form.
1. 0.0000821 _________________
6. 8.7 • 108 _________________
2. 5,033 _________________
7. 9.162 • 105 _________________
3. 1,743.55 _________________
8. 6.524 • 105 _________________
4. 0.0005894 _________________
9. 5.00961 • 104 _________________
5. 34,008 _________________
10. 3.00006 • 104 _________________
Directions: For Numbers 11 through 14, write each number in standard form or scientific
notation.
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11. The speed of light is approximately 3 • 108 meters per second. How is 3 • 108 written in
standard form?
____________________
12. Sound travels 10 feet through water in about 0.00206 seconds. How is 0.00206 written in
scientific notation?
____________________
13. Jake’s computer can do one calculation in 6 • 105 seconds. How is 6 • 105 written in
standard form?
____________________
14. In 2004, Oklahoma produced about 164,500,000 bushels of wheat. How is 164,500,000
written in scientific notation?
____________________
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Unit 1 – Number Sense
PASS Objective: 2.2
Multiplying and dividing numbers in scientific notation
When multiplying or dividing numbers written in scientific notation, multiply or divide the
coefficients and use the rules of exponents to determine the power of 10. Make sure the product
or quotient is written in correct scientific notation.
Example
Multiply: (4.5 • 105) • (6 • 108)
Multiply the coefficients.
4.5 • 6 27
Add the powers of 10.
5 8 13
Make sure the quotient is written in correct scientific notation.
(4.5 • 105) • (6 • 108) 27 • 1013
2.7 • 1014
Therefore, (4.5 • 105) • (6 • 108) 2.7 • 1014.
Example
Divide: (2.8 • 106) (5.6 • 103)
Divide the coefficients.
2.8 5.6 0.5
6 (3) 9
Make sure the quotient is written in correct scientific notation.
(2.8 • 106) (5.6 • 103) 0.5 • 109
5.0 • 108
Therefore, (2.8 • 106) (5.6 • 103) 5 • 108.
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Subtract the powers of 10.
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Lesson 1: Number Concepts
PASS Objective: 2.2
Practice
Directions: For Numbers 1 through 8, multiply or divide the numbers in scientific notation.
1. (9.23 • 104) • (7.85 • 107) ____________________
2. (6.3 • 103) (2.5 • 108) ____________________
3. (2.25 • 105) • (4 • 105) ____________________
4. (5.3 • 102) • (2.06 • 109) ____________________
5. (8 • 107) (6.25 • 102) ____________________
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6. (9.1 • 103) • (4.5 • 108) ____________________
7. (9 • 107) (2.4 • 102) ____________________
8. (1.9 • 1010) (8 • 103) ____________________
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Unit 1 – Number Sense
OCCT Practice
1
5
9
Multiply: 12 • 12
3
What is the value of (2)4?
A 123
A 16
6
B 12
B 8
C 1212
C 8
D 1227
D 16
6
2
The distance from the Sun to
the Earth is approximately
1.5 • 108 kilometers. How is this
number written in standard form?
How is 0.00001578 written in
scientific notation?
A 1.578 • 105
B 1.578 • 106
A 15,000,000 km
5
C 15.78 • 10
B 150,000,000 km
D 15.78 • 105
C 1,500,000,000 km
D 15,000,000,000 km
3
4
7
A 9 • 10
Divide:
B 9 • 105
2515
255
A 253
6
C 9 • 10
B 2510
7
D 9 • 10
C 2520
D 2575
4
How is 5,200,000,000 written in
scientific notation?
A 5.2 • 109
B 5.2 • 108
C 52 • 108
D 0.52 • 1010
12
8
What is the value of 252?
A 5
B 50
C 225
D 625
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Divide: (7.2 • 105) (8 • 1010)