**Unformatted text preview: ** m707_c01_002_003 1/19/06 8:26 PM Page 2 Algebraic
Reasoning
1A Patterns and
Relationships 1-1
1-2
1-3
1-4
LAB Numbers and Patterns
Exponents
Metric Measurements
Applying Exponents
Scientific Notation with
a Calculator
Order of Operations
Explore Order of
Operations
Properties 1-5
LAB
1-6 1B Algebraic Thinking 1-7 Variables and Algebraic
Expressions
1-8 Translate Words
into Math
1-9 Simplifying Algebraic
Expressions
1-10 Equations and Their
Solutions
LAB Model Solving Equations
1-11 Addition and Subtraction
Equations
1-12 Multiplication and
Division Equations KEYWORD: MS7 Ch1 2 Chapter 1 Astronomical
Distances
Object
Mercury
Venus
Earth
Mars
Jupiter
Saturn
Uranus
Neptune
Pluto
Nearest star Distance from
the Sun (km)*
5.80 10 7
1.082 10 8
1.495 10 8
2.279 10 8
7.780 10 8
1.43 10 9
2.90 10 9
4.40 10 9
5.80 10 9
3.973 10 13 *Distances of
planets from
the Sun
are average
distances. Cosmologist
Dr. Stephen Hawking is a cosmologist.
Cosmologists study the universe as a whole.
They are interested in the origins, the structure,
and the interaction of space and time.
The invention of the telescope has extended
the vision of scientists far beyond nearby stars
and planets. It has enabled them to view
distant galaxies and structures that at
one time were only theorized by
astrophysicists such as Dr. Hawking.
Astronomical distances are so
great that we use scientific
notation to represent them. m707_c01_002_003 1/19/06 8:27 PM Page 3 Vocabulary
Choose the best term from the list to complete each sentence.
1. The operation that gives the quotient of two numbers
is __?__. division
multiplication 2. The __?__ of the digit 3 in 4,903,672 is thousands. place value 3. The operation that gives the product of two numbers
is __?__. product
quotient 4. In the equation 15 3 5, the __?__ is 5. Complete these exercises to review skills you will need for this chapter. Find Place Value
Give the place value of the digit 4 in each number.
5. 4,092
9. 3,408,289 6. 608,241
10. 34,506,123 7. 7,040,000 8. 4,556,890,100 11. 500,986,402 12. 3,540,277,009 Use Repeated Multiplication
Find each product.
13. 2 2 2 14. 9 9 9 9 15. 14 14 14 16. 10 10 10 10 17. 3 3 5 5 18. 2 2 5 7 19. 3 3 11 11 20. 5 10 10 10 Division Facts
Find each quotient.
21. 49 7 22. 54 9 23. 96 12 24. 88 8 25. 42 6 26. 65 5 27. 39 3 28. 121 11 Whole Number Operations
Add, subtract, multiply, or divide.
29. 425
12
30. 619
254
31. 33. 62
42
34. 122
15
35. 76
2
3
62
47
32. 373
86
36. 241
4
9
Algebraic Reasoning 3 m707_c01_004_004 12/29/05 11:37 AM Page 4 Previously, you algebraic expression expresión algebraica Associative Property propiedad asociativa Commutative Property used multiplication and
division to solve problems
involving whole numbers. propiedad
conmutativa Distributive Property propiedad distributiva equation ecuación • converted measures within the
same measurement system. exponent exponente • wrote large numbers in
standard form. numerical expression expresión numérica order of operations orden de las
operaciones term término variable variable • Study Guide: Preview • used order of operations
to simplify whole number
expressions without exponents. You will study • simplifying numerical
expressions involving order of
operations and exponents. • using concrete models to solve
equations. • finding solutions to application
problems involving related
measurement units. • writing large numbers in
scientific notation. You can use the skills
learned in this chapter 4 Key
Vocabulary/Vocabulario • to express distances and sizes
of objects in scientific fields
such as astronomy and biology. • to solve problems in math and
science classes such as Algebra
and Physics. Chapter 1 Algebraic Reasoning Vocabulary Connections
To become familiar with some of the
vocabulary terms in the chapter, consider
the following. You may refer to the chapter,
the glossary, or a dictionary if you like.
1. The words equation, equal, and equator all
begin with the Latin root equa-, meaning
“level.” How can the Latin root word help
you define equation ?
2. The word numerical means “of numbers.”
How might a numerical expression differ
from an expression such as “the sum of
two and five”?
3. When something is variable, it has the
ability to change. In mathematics, a
variable is an algebraic symbol. What
special property do you think this type
of symbol has? m707_c01_005_005 12/29/05 11:37 AM Page 5 Reading Strategy: Use Your Book for Success
Understanding how your textbook is organized will help you locate and
use helpful information.
As you read through an example problem, pay attention to the margin notes ,
such as Helpful Hints, Reading Math notes, and Caution notes. These
notes will help you understand concepts and avoid common mistakes. The index is located
at the end of your
textbook. Use it to
find the page where
a particular concept
is taught. Reading and Writing Math The glossary is found
in the back of your
textbook. Use it to
find definitions and
examples of unfamiliar
words or properties. The Skills Bank is
found in the back of
your textbook. These
pages review concepts
from previous math
courses. Try This
Use your textbook for the following problems.
1. Use the index to find the page where exponent is defined.
2. In Lesson 1-9, what does the Remember box, located in the margin of
page 43, remind you about the perimeter of a figure?
3. Use the glossary to find the definition of each term: order of operations,
numerical expression, equation.
4. Where can you review how to multiply whole numbers?
Algebraic Reasoning 5 m707_c01_006_009 12/29/05 11:37 AM 1-1
Learn to identify and extend patterns. Page 6 Numbers
and Patterns
Each year, football teams battle for
the state championship. The table
shows the number of teams in
each round of a division’s football
playoffs. You can look for a pattern
to find out how many teams are in
rounds 5 and 6.
Football Playoffs EXAMPLE 1 Round 1 2 3 4 Number of Teams 64 32 16 8 5 6 Identifying and Extending Number Patterns
Identify a possible pattern. Use the pattern to write the next
three numbers.
64, 32, 16, 8, ,
64 , ,... 32 16 8 2 2 2 2 2 2
A pattern is to divide each number by 2 to get the next number.
824
422
221
The next three numbers will be 4, 2, and 1.
51, 44, 37, 30, , 51 ,
44 ,...
37 30 7 7 7 7 7 7
A pattern is to subtract 7 from each number to get the next
number.
30 7 23
23 7 16
16 7 9
The next three numbers will be 23, 16, and 9.
2, 3, 5, 8, 12,
2 , ,
3 ,...
5 8 12 1 2 3 4 5 6 7
A pattern is to add one more than you did the time before.
12 5 17
17 6 23
23 7 30
The next three numbers will be 17, 23, and 30. 6 Chapter 1 Algebraic Reasoning m707_c01_006_009 12/29/05 11:37 AM EXAMPLE Page 7 2 Identifying and Extending Geometric Patterns
Identify a possible pattern. Use the pattern to draw the next three
figures. The pattern is alternating squares and circles with triangles
between them.
The next three figures will be
. The pattern is to shade every other triangle in a clockwise direction.
The next three figures will be EXAMPLE 3 . Using Tables to Identify and Extend Patterns
Make a table that shows
the number of triangles in
each figure. Then tell how
Figure 1
Figure 2
Figure 3
many triangles are in the
fifth figure of the pattern. Use drawings to justify your answer.
The table shows the number of triangles in each figure.
Figure 1 2 3 4 5 Number of
Triangles 2 4 6 8 10 2 2 2
Figure 4 has 6 2 8 triangles. Figure 4 The pattern is to add
2 triangles each time. 2
Figure 5 has 8 2 10 triangles. Figure 5 Think and Discuss
1. Describe two different number patterns that begin with 3, 6, . . .
2. Tell when it would be useful to make a table to help you identify
and extend a pattern. 1-1 Numbers and Patterns 7 m707_c01_006_009 12/29/05
1-1 11:37 AM Page 8 Exercises KEYWORD: MS7 1-1
KEYWORD: MS7 Parent GUIDED PRACTICE
See Example 1 Identify a possible pattern. Use the pattern to write the next three numbers.
1. 6, 14, 22, 30,
3. 59, 50, 41, 32, See Example 2 , ,
, 2. 1, 3, 9, 27, ,...
, 4. 8, 9, 11, 14, ,... ,
, ,...
, ,... Identify a possible pattern. Use the pattern to draw the next three figures.
5. See Example 3 , 6. 7. Make a table that shows the number of green triangles in each figure.
Then tell how many green triangles are in the fifth figure of the pattern.
Use drawings to justify your answer. Figure 1 Figure 2 Figure 3 INDEPENDENT PRACTICE
See Example 1 See Example 2 Identify a possible pattern. Use the pattern to write the next three numbers.
8. 27, 24, 21, 18, , , ,... 10. 1, 3, 7, 13, 21, , , ,... 11. 14, 37, 60, 83, , ,
, ,
,... 13. 14. Make a table that shows the number of dots in each figure. Then tell how many
dots are in the sixth figure of the pattern. Use drawings to justify your answer. Figure 1 Figure 2 Figure 3 Figure 4 PRACTICE AND PROBLEM SOLVING
Extra Practice
See page 724. Use the rule to write the first five numbers in each pattern.
15. Start with 7; add 16 to each number to get the next number.
16. Start with 96; divide each number by 2 to get the next number.
17. Start with 50; subtract 2, then 4, then 6, and so on, to get the next number.
18. Critical Thinking Suppose the pattern 3, 6, 9, 12, 15 . . . is continued
forever. Will the number 100 appear in the pattern? Why or why not? 8 ,... Identify a possible pattern. Use the pattern to draw the next three figures.
12. See Example 3 9. 4,096, 1,024, 256, 64, Chapter 1 Algebraic Reasoning m707_c01_006_009 12/29/05 11:37 AM Page 9 Identify a possible pattern. Use the pattern to find the missing numbers.
19. 3, 12,
21. , , 192, 768, , , 19, 27, 35, 20. 61, 55, , ... 22. 2, , 51, . . . , 43, , 8, , 32, 64, 23. Health The table shows the target heart rate
during exercise for athletes of different ages.
Assuming the pattern continues, what is the
target heart rate for a 40-year-old athlete? a
65-year-old athlete? 25. 1
,
4 , ,
5 13 9 5 ,
7 , , 21 17
, 10 , 14 , , 19 , , . . .
25 , 25, . . .
, ... Target Heart Rate Draw the next three figures in each pattern.
24. , Age Heart Rate
(beats per minute) 20 150 25 146 30 142 35 138 , . . . 26. Social Studies In the ancient Mayan civilization, people used a number
system based on bars and dots. Several numbers are shown below. Look
for a pattern and write the number 18 in the Mayan system. 3 5 8 10 13 15 27. What’s the Error? A student was asked to write the next three numbers
in the pattern 96, 48, 24, 12, . . . . The student’s response was 6, 2, 1. Describe
and correct the student’s error.
28. Write About It A school chess club meets every Tuesday during the
month of March. March 1 falls on a Sunday. Explain how to use a number
pattern to find all the dates when the club meets.
29. Challenge Find the 83rd number in the pattern 5, 10, 15, 20, 25, . . . . 30. Multiple Choice Which is the missing number in the pattern
2, 6, , 54, 162, . . . ?
A 10 B 18 C 30 D 48 31. Gridded Response Find the next number in the pattern 9, 11, 15, 21, 29, 39, . . . .
Round each number to the nearest ten. (Previous course)
32. 61 33. 88 34. 105 35. 2,019 36. 11,403 Round each number to the nearest hundred. (Previous course)
37. 91 38. 543 39. 952 40. 4,050 41. 23,093 1-1 Numbers and Patterns 9 m707_c01_010_013 12/29/05 11:38 AM 1-2
Learn to represent
numbers by using
exponents. Vocabulary
power
exponent Page 10 Exponents
A DNA molecule makes a copy of
itself by splitting in half. Each half
becomes a molecule that is identical
to the original. The molecules
continue to split so that the two
become four, the four become
eight, and so on.
Each time DNA copies itself, the
number of molecules doubles.
After four copies, the number of
molecules is 2 2 2 2 16. base This multiplication can also be
written as a power , using a base
and an exponent. The exponent
tells how many times to use the
base as a factor.
Read 24 as “the
fourth power of 2”
or “2 to the fourth
power.” The structure of DNA can be
compared to a twisted ladder. Exponent Base EXAMPLE 1 Evaluating Powers
Find each value.
52
52 5 5
25 Use 5 as a factor 2 times. 26
26 2 2 2 2 2 2
64 Use 2 as a factor 6 times. 251
251 25 Any number to the first power is equal
to that number. Any number to the zero power, except zero, is equal to 1.
60 1 100 1 190 1 Zero to the zero power is undefined, meaning that it does not exist. 10 Chapter 1 Algebraic Reasoning m707_c01_010_013 1/7/06 4:09 PM Page 11 To express a whole number as a power, write the number as the
product of equal factors. Then write the product using the base and
an exponent. For example, 10,000 10 10 10 10 104. EXAMPLE 2 Expressing Whole Numbers as Powers
Write each number using an exponent and the given base. EXAMPLE Earth Science 3 49, base 7
49 7 7
72 7 is used as a factor 2 times. 81, base 3
81 3 3 3 3
34 3 is used as a factor 4 times. Earth Science Application
The Richter scale measures an
earthquake’s strength, or magnitude.
Each category in the table is 10 times
stronger than the next lower category.
For example, a large earthquake is
10 times stronger than a moderate
earthquake. How many times stronger
is a great earthquake than a moderate
one? Earthquake Strength
Category Magnitude Moderate 5 Large 6 Major 7 Great 8 An earthquake with a magnitude of 6 is 10 times stronger than one
with a magnitude of 5.
An earthquake
measuring 7.2 on the
Richter scale struck
Duzce, Turkey, on
November 12, 1999. An earthquake with a magnitude of 7 is 10 times stronger than one
with a magnitude of 6.
An earthquake with a magnitude of 8 is 10 times stronger than one
with a magnitude of 7.
10 10 10 103 1,000
A great earthquake is 1,000 times stronger than a moderate one. Think and Discuss
1. Describe a relationship between 35 and 36.
2. Tell which power of 8 is equal to 26. Explain.
3. Explain why any number to the first power is equal to
that number. 1-2 Exponents 11 m707_c01_010_013 12/29/05 1-2 11:38 AM Page 12 Exercises KEYWORD: MS7 1-2
KEYWORD: MS7 Parent GUIDED PRACTICE
See Example 1 Find each value.
1. 25 See Example 2 2. 33 4. 91 5. 106 Write each number using an exponent and the given base.
6. 25, base 5 See Example 3 3. 62 7. 16, base 4 8. 27, base 3 9. 100, base 10 10. Earth Science On the Richter scale, a great earthquake is 10 times
stronger than a major one, and a major one is 10 times stronger than a large
one. How many times stronger is a great earthquake than a large one? INDEPENDENT PRACTICE
See Example 1 See Example 2 See Example 3 Find each value.
11. 112 12. 35 13. 83 14. 43 15. 34 16. 25 17. 51 18. 23 19. 53 20. 301 Write each number using an exponent and the given base.
21. 81, base 9 22. 4, base 4 23. 64, base 4 24. 1, base 7 25. 32, base 2 26. 128, base 2 27. 1,600, base 40 28. 2,500, base 50 29. 100,000, base 10 30. In a game, a contestant had a starting score of one point. He tripled his
score every turn for four turns. Write his score after four turns as a power.
Then find his score. PRACTICE AND PROBLEM SOLVING
Extra Practice
See page 724. Give two ways to represent each number using powers.
31. 81 32. 16 33. 64 34. 729 35. 625 Compare. Write , , or .
36. 42 15 40. 10,000 105 37. 23 32 38. 64 43 39. 83 74 41. 65 3,000 42. 93 36 43. 54 73 44. To find the volume of a cube, find the third power of the length of an edge
of the cube. What is the volume of a cube that is 6 inches long on an edge?
45. Patterns Domingo decided to save $0.03 the first day and to triple the
amount he saves each day. How much will he save on the seventh day?
46. Life Science A newborn panda cub weighs an average of 4 ounces. How
many ounces might a one-year-old panda weigh if its weight increases by
the power of 5 in one year? 12 Chapter 1 Algebraic Reasoning m707_c01_010_013 12/29/05 11:38 AM Page 13 47. Social Studies If the populations of the
cities in the table double every 10 years,
what will their populations be in 2034?
48. Critical Thinking Explain why 63 36. City Population (2004) Yuma, AZ 86,070 Phoenix, AZ 1,421,298 49. Hobbies Malia is making a quilt with a pattern of rings. In the
center ring, she uses four stars. In each of the next three rings, she
uses three times as many stars as in the one before. How many stars does
she use in the fourth ring? Write the answer using a power and find its
value.
Order each set of numbers from least to greatest.
50. 29, 23, 62, 16, 35 51. 43, 33, 62, 53, 101 52. 72, 24, 80, 102, 18 53. 2, 18, 34, 161, 0 54. 52, 21, 112, 131, 19 55. 25, 33, 9, 52, 81 56. Life Science The cells of some kinds of
bacteria divide every 30 minutes. If you begin
with a single cell, how many cells will there be
after 1 hour? 2 hours? 3 hours?
57. What’s the Error? A student wrote 64
as 8 2. How did the student apply exponents
incorrectly?
58. Write About It Is 25 greater than or less
than 33? Explain your answer.
59. Challenge What is the length of the edge
of a cube if its volume is 1,000 cubic meters? Bacteria divide by pinching in two. This
process is called binary fission. 60. Multiple Choice What is the value of 46?
A 24 B 1,024 C 4,096 D 16,384 J 82 61. Multiple Choice Which of the following is NOT equal to 64?
F 64 G 43 H 26 62. Gridded Response Simplify 23 32.
Simplify. (Previous course)
63. 15 27 5 3 11 16 7 4 64. 2 6 5 7 100 1 75 65. 2 9 8 12 6 8 5 6 7 66. 9 30 4 1 4 1 7 5 Identify a possible pattern. Use the pattern to write the next three numbers. (Lesson 1-1)
67. 100, 91, 82, 73, 64, . . . 68. 17, 19, 22, 26, 31, . . . 69. 2, 6, 18, 54, 162, . . . 1-2 Exponents 13 m707_c01_014_017 12/29/05 11:38 AM 1-3
Learn to identify, convert, and compare
metric units. Page 14 Metric Measurements
The Micro Flying Robot II is the world’s
lightest helicopter. Produced in Japan
in 2004, the robot is 85 millimeters tall
and has a mass of 8.6 grams.
You can use the following benchmarks
to help you understand millimeters,
grams, and other metric units.
Metric Unit
Length Mass Capacity EXAMPLE 1 Benchmark Millimeter (mm) Thickness of a dime Centimeter (cm) Width of your little finger Meter (m) Width of a doorway Kilometer (km) Length of 10 football fields Milligram (mg) Mass of a grain of sand Gram (g) Mass of a small paperclip Kilogram (kg) Mass of a textbook Milliliter (mL) Amount of liquid in an eyedropper Liter (L) Amount of water in a large water bottle Kiloliter (kL) Capacity of 2 large refrigerators Choosing the Appropriate Metric Unit
Choose the most appropriate metric unit for each measurement.
Justify your answer.
The length of a car
Meters—the length of a car is similar to the width of several
doorways.
The mass of a skateboard
Kilograms—the mass of a skateboard is similar to the mass
of several textbooks.
The recommended dose of a cough syrup
Milliliters—one dose of cough syrup is similar to the amount
of liquid in several eyedroppers. 14 Chapter 1 Algebraic Reasoning m707_c01_014_017 1/7/06 4:10 PM Page 15 Prefixes:
Milli- means
“thousandth”
Centi- means
“hundredth”
Kilo- means
“thousand” The prefixes of metric units correlate to place values in the base-10 number
system. The table shows how metric units are based on powers of 10.
1,000 10 1 Thousands Hundreds Tens Ones Kilo- 100 Hecto- 0.1
Tenths Deca- Base unit Deci- 0.01 0.001 Hundredths Thousandths
Centi- Milli- You can convert units within the metric system by multiplying or
dividing by powers of 10. To convert to a smaller unit, you must
multiply. To convert to a larger unit, you must divide. EXAMPLE 2 Converting Metric Units
Convert each measure.
510 cm to meters
510 cm (510 100) m 100 cm 1 m, so divide by 100.
5.1 m
Move the decimal point 2 places left: 510.
2.3 L to milliliters
2.3 L (2.3 1,000) mL 1 L 1,000 mL, so multiply by 1,000.
2,300 mL
Move the decimal point 3 places right: 2.300 EXAMPLE 3 Using Unit Conversion to Make Comparisons
Mai and Brian are ...

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