Whole Numbers and Introduction to Algebra

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Whole Numbers and Introduction to Algebra
Chapter One

• 1.2 Place Value and Names for Numbers
• 1.3 Adding Subtracting Whole Numbers, and
  Perimeter
• 1.4 Rounding Estimating
• 1.5 Multiplying Whole Numbers and Area
• 1.6 Dividing Whole Numbers
• 1.7 Exponents and Order of Operations
• 1.8 Introduction to Variables, Algebraic
  Expressions Equations

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Place Value and Names for Numbers
Section 1.2
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The position of each digit in a number determines
its place value.
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A whole number such as 35,689,402 is written in
standard form. The columns separate the digits
into groups of threes. Each group of three digits
is a period.
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To write a whole number in words, write the
number in each period followed by the name of the
period.
thirty-five million, six hundred eighty-nine
thousand, four hundred two
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The name of the ones period is not used when
reading and writing whole numbers. Also, the
word and is not used when reading and writing
whole numbers. It is used when reading and
writing mixed numbers and some decimal values as
shown later.
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Standard Form
Expanded Form
4,786 4000 700 80 6
The place value of a digit can be used to write a
number in expanded form. The expanded form of a
number shows each digit of the number with its
place value.
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Comparing Whole Numbers
We can picture whole numbers as equally spaced
points on a line called the number line.
0
5
4
1
2
3
A whole number is graphed by placing a dot on the
number line. The graph of 4 is shown.
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Comparing Numbers . . .
For any two numbers graphed on a number line, the
number to the right is the greater number, and
the number to the left is the smaller number.
2 is to the left of 5, so 2 is less than 5
5 is to the right of 2, so 5 is greater than 2
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Comparing Numbers . . .
2 is less than 5 can be written in symbols as 2 lt
5 5 is greater than 2 is written as 5 gt 2
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One way to remember the meaning of the inequality
symbols lt and gt is to think of them as
arrowheads pointing toward the smaller number.
For example, 2 lt 5 and 5 gt 2 are both true
statements.
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Reading Tables
Most Medals Olympic Winter (1924 2002) Games
Source The Sydney Morning Herald Flags courtesy
of www.theodora.com/flags used with permission
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Adding and Subtracting Whole Numbers and
Perimeter
Section 1.3

• 3 4 7
• addend addend sum

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Addition Property of 0
The sum of 0 and any number is that number. 8
0 8 and 0 8 8
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Commutative Property of Addition
Changing the order of two addends does not change
their sum. 4 2 6 and 2 4 6
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Associative Property of Addition
Changing the grouping of addends does not change
their sum. 3 (4 2) 3 6 9 and (3 4)
2 7 2 9
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Subtraction Properties of 0
The difference of any number and that same number
is 0. 9 - 9 0 The difference of any number
and 0 is the same number. 7 - 0 7
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A polygon is a flat figure formed by line
segments connected at their ends.
Geometric figures such as triangles, squares, and
rectangles are called polygons.
triangle
square
rectangle
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Finding the Perimeter of a Polygon
The perimeter of a polygon is the distance
around the polygon.
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Descriptions of problems solved through addition
may include any of these key words or phrases
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Descriptions of problems solved by subtraction
may include any of these key words or phrases
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Be careful when solving applications that suggest
subtraction. Although order does not matter when
adding, order does matter when subtracting. For
example, 10 3 and 3 10 do not simplify to the
same number.
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Since subtraction and addition are reverse
operations, dont forget that a subtraction
problem can be checked by adding.
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Reading a Bar Graph
Source The internet Movie Database and Cuadra
Associates Movie Star Database, 2003.
The graph shows the ratings of Best Picture
nominees since 1984.
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Rounding and Estimating
Section 1.4

• Rounding a whole number means approximating it.

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23 rounded to the nearest ten is 20.
48 rounded to the nearest ten is 50.
15 rounded to the nearest ten is 20.
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Rounding Whole Numbers to a Given Place Value

• Step 1. Locate the digit to the right of the
  given place value to be rounded.
• Step 2. If this digit is 5 or greater, add 1 to
  the digit in the given place value and replace
  each digit to its right by 0.
• Step 3. If this digit is less than 5, keep the
  digit in the given place value and replace each
  digit to its right by 0.

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Making estimates is often the quickest way to
solve real-life problems when their solutions do
not need to be exact.
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Estimation is useful to check for incorrect
answers when using a calculator. For example,
pressing a key too hard may result in a double
digit, while pressing a key too softly may result
in the number not appearing in the display.
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Multiplying Whole Numbers and Area
Section 1.5
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Multiplication is repeated addition with a
different notation.
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Multiplication Property of 0
The product of 0 and any number is 0. 9 ? 0 0
0 ? 6 0
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Multiplication Property of 1
The product of 1 and any number is that same
number. 9 ? 1 9 1 ? 6 6
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Commutative Property of Multiplication
Changing the order of two factors does not change
their product. 6 ? 3 18 and 3 ? 6 18
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Associative Property of Multiplication
Changing the grouping of factors does not change
their product. 5 ? ( 2 ? 3) 5 ? 6
30 and (5 ? 2) ? 3 10 ? 3 30
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Distributive Property
Multiplication distributes over addition. 5(3
4) 5 ? 3 5 ? 4
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Area
Area of a rectangle length ? width
(5 inches)(3 inches)
15 square inches
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Remember that perimeter (distance around a plane
figure) is measured in units. Area (space
enclosed by a plane figure) is measured in square
units.
Area (5 inches)(4 inches) 20 square inches
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There are several words or phrases that indicate
the operation of multiplication. Some of these
are as follows
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Dividing Whole Numbers
Section 1.6
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The process of separating a quantity into equal
parts is called division.
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Division Properties of 1
The quotient of any number and that same number
is 1.
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Division Properties of 1 . . .
The quotient of any number and 1 is that same
number.
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Division Properties of 0
The quotient of 0 and any number (except 0) is 0.
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Division Properties of 0 . . .
The quotient of any number and 0 is not a number.
We say that
are undefined.
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Since division and multiplication are reverse
operations, dont forget that a division problem
can be checked by multiplying.
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Here are some key words and phrases that indicate
the operation of division.
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How do you find an average?
A students prealgebra grades at the end of the
semester are 90, 85, 95, 70, 80, 100, 98, 82,
90, 90. How do you find his average?
Find the sum of the scores and then divide the
sum by the number of scores.
Sum 880
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Exponents and Order of Operations
Section 1.7
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An exponent is a shorthand notation for repeated
multiplication.
3 3 3 3 3
3 is a factor 5 times Using
an exponent, this product can be written as
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This is called exponential notation. The
exponent, 5, indicates how many times the base,
3, is a factor.
Read as three to the fifth power or the fifth
power of three.
3 3 3 3 3
3 is a factor 5 times
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Reading Exponential Notation
4
is read as four to the first power.
4 ? 4
is read as four to the second power or four
squared.
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Reading Exponential Notation . . .
4 ? 4 ? 4
is read as four to the third power or four
cubed.
4 ? 4 ? 4 ? 4
is read as four to the fourth power.
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Usually, an exponent of 1 is not written, so when
no exponent appears, we assume that the exponent
is 1. For example,
2 21 and 7 71.
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To evaluate an exponential expression, we write
the expression as a product and then find the
value of the product.
35 3 3 3 3 3 243
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An exponent applies only to its base. For example,
4 23 means 4 2 2 2.
Dont forget that 24 is not 2 4. 24
means repeated multiplication of the same factor.
24 2 2 2 2 16, whereas 2 4 8
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Order of Operations
1. Perform all operations within grouping symbols
such as parentheses or brackets. 2. Evaluate any
expressions with exponents. 3. Multiply or divide
in order from left to right. 4. Add or subtract
in order from left to right.
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Section 1.8
Introduction to Variables, Algebraic Expressions,
and Equations
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A combination of operations on letters
(variables) and numbers is called an algebraic
expression.
Algebraic Expressions 5 x 6 ? y 3
? y 4 x
4x means 4 ? x and xy means x ? y
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• Replacing a variable in an expression by a number
  and then finding the value of the expression is
  called evaluating the expression for the variable.

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Evaluating Algebraic Expressions
Evaluate x y for x 5 and y 2.
Replace x with 5 and y with 2 in x y.
x y ( ) ( )
5
2
7
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Equation

• Statements like 5 2 7 are called equations.
• An equation is of the form
  expression expression
• An equation can be labeled as

Equal sign
x 5 9
left side
right side
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Solving/Solution

• When an equation contains a variable, deciding
  which values of the variable make an equation a
  true statement is called solving an equation for
  the variable.
• A solution of an equation is a value for the
  variable that makes an equation a true statement.

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Solving/Solution ...

• Determine whether a number is a solution

Is -2 a solution of the equation 2y 1 -3?
Replace y with -2 in the equation.
2y 1 -3
?
2(-2) 1 -3
?
- 4 1 -3
-3 -3
True
Since -3 -3 is a true statement, -2 is a
solution of the equation.
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Solving/Solution ...

• Determine whether a number is a solution

Is 6 a solution of the equation 5x - 1 30?
Replace x with 6 in the equation.
5x - 1 30
?
5(6) - 1 30
?
30 - 1 30
29 30
False
Since 29 30 is a false statement, 6 is not a
solution of the equation.
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Solving/Solution...

• To solve an equation, we will use properties of
  equality to write simpler equations, all
  equivalent to the original equation, until the
  final equation has the form
• x number or number x
• Equivalent equations have the same solution.
• The word number above represents the solution
  of the original equation.

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Keywords and phrases suggesting addition,
subtraction, multiplication, division or equals.
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Translating Word Phrases into Expressions

• the product of 5 and a number
• 5x
• twice a number
• 2x
• a number decreased by 3
• n - 3
• a number increased by 2
• z 2
• four times a number
• 4w

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Additional Word Phrases into Algebraic
Expressions ...

• x 7
• three times the sum of a number and 7
• 3(x 7)
• the quotient of 5 and a number

the sum of a number and 7
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Remember that order is important when
subtracting. Study the order of numbers and
variables below.
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