Download Adding Whole Numbers and Perimeter - Clement

Chapter One
Whole Numbers and
Introduction to Algebra
Written by Sue C. Little
North Harris College
Offered with thanks by Mr. Roberts
Clement Middle School, RUSD
Place Value and
Names for Numbers
Ten-millions
Millions
Hundred-thousands
Ten-thousands
Thousands
Hundreds
Tens
Ones
Hundred-millions
Billions
Ten-billions
Hundred-billions
The position of each digit in a
number determines its place
value.
3
5
6
8
9
4
0
2
Ten-millions
Millions
Hundred-thousands
Ten-thousands
Thousands
Hundreds
Tens
Ones
Hundred-millions
Billions
Ten-billions
Hundred-billions
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.
Billions
Millions Thousands
Ones
3
5
6
8
9
4
0
2
Ten-millions
Millions
Hundred-thousands
Ten-thousands
Thousands
Hundreds
Tens
Ones
Hundred-millions
Billions
Ten-billions
Hundred-billions
To write a whole number in words, write the
number in each period followed by the name
of the period.
3
5
6
8
9
4
0
2
thirty-five million, six hundred eighty-nine
thousand, four hundred two
Helpful Hint
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.
Standard Form
4,786
=
Expanded Form
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.
Adding Whole Numbers
and Perimeter
3
+
4
=
7
addend addend sum
Addition Property of 0
The sum of 0 and any number is that
number.
8+0=8
Commutative Property of
Addition
Changing the order of two addends
does not change their sum.
4+2=2+4
Associative Property of
Addition
Changing the grouping of addends does
not change their sum.
3 + (4 + 2) = (3 + 4) + 2
Descriptions of problems solved through
addition may include any of these key
words or phrases:
Key Words
Examples
Symbols
added to
3 added to 9
3+9
plus
5 plus 22
5 + 22
more than
7 more than 8
7+8
total
total of 6 and 5
6+5
increased by
16 increased by 7
16 + 7
sum
sum of 50 and 11
50 + 11
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
Finding the Perimeter of a Polygon
The perimeter of a polygon is the
distance around the polygon.
Subtracting Whole
Numbers
Subtraction is finding the
difference of two numbers.
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
Descriptions of problems solved by
subtraction may include any of these key
words or phrases:
Key Words
Examples
Symbols
subtract
subtract 3 from 9
9-3
difference
difference of 8 and 2
8-2
less
12 less 8
12 - 8
take away
14 take away 9
14 - 9
decreased by
subtracted from
16 decreased by 7
5 subtracted from 9
16 - 7
9-5
Rounding and
Estimating
Rounding a whole number
means approximating it.
20
23
30
23 rounded to the nearest ten is 20.
40
48
50
48 rounded to the nearest ten is 50.
10
15
20
15 rounded to the nearest ten is 20.
Rounding Whole Numbers to a
Given Place Value
 Step
1. Locate the digit to the right
of the given place value.
 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,
replace it and each digit to its right
by 0.
Making estimates is often the quickest
way to solve real-life problems when
their solutions do not need to be exact.
Multiplying Whole
Numbers and Area
Multiplication is repeated addition
with a different notation.
4 + 4 + 4 + 4 + 4 = 5 x 4 = 20
5 fours
factor product
Multiplication Property of 0
The product of 0 and any number is 0.
90=0
06=0
Multiplication Property of 1
The product of 1 and any number
is that same number.
91=9
16=6
Commutative Property of
Multiplication
Changing the order of two factors
does not change their product.
63=36
Associative Property of
Multiplication
Changing the grouping of factors
does not change their product.
5  ( 2  3) = (5  2)  3
Distributive Property
Multiplication distributes over addition.
5(3 + 4) = 5  3 + 5  4
There are several words or phrases that
indicate the operation of multiplication.
Some of these are as follows:
Key Words
Examples
Symbols
multiply
multiply 4 by 3
43
product
times
product of 2 and 5 2  5
7 times 6
76
1
1 square inch
Area
1
5 inches
3 inches
Area of a rectangle = length  width
= (5 inches)(3 inches)
= 15 square inches
Dividing Whole
Numbers
Division is the process of separating
a quantity into equal parts.
quotient
20
5
4
dividend
6
3 18
14  2  7
divisor
Division Properties of 1
The quotient of any number
and that same number is 1.
1
6
1 5 5 771
6

The quotient of any number and 1
is that same number.
5
6
6 15
1
7 1 7
Division Properties of 0
The quotient of 0 and any
number (except 0) is 0.
0
0
 0 5 0 0 7  0
6

The quotient of any number and 0
is not a number. We say that
6
0 5 7  0 are undefined (u).
0
Here are some key words and phrases that
indicate the operation of division.
Key Words
Examples
Symbols
divide
divide 15 by 3
15  3
quotient
quotient of 12 and 6
12
6
divided by
8 divided by 4
48
divided or
shared equally
$20 divided equally
among 5 people
20  5
How do you find an average?
A student’s 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.
Exponents and Order
of Operations
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
base
3
5
exponent
base
3
5
exponent
Read as “three to the fifth power” or
“the fifth power of three.”
This is called exponential notation.
The exponent, 5, indicates how many
times the base, 3, is a factor.
3•3•3•3•3
3 is a factor 5 times
1
4
4=
is read as “four to the first power.”
44= 4
is read as “four to the second power
or four squared.”
3
444= 4
is read as “four to the third power
or four cubed.”
4  4  4  4 = 44
is read as “four to the fourth power.”
2
To evaluate exponential notation,
we write the expression as a
product and then find the value of
the product.
3 = 3 • 3 • 3 • 3 • 3 = 243
5
Order of Operations
1. Do 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.
Introduction to
Variables and
Algebraic Expressions
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
Keywords and phrases suggesting
addition, subtraction, multiplication,
or division.
Addition
Subtraction
Multiplication
Division
sum
difference
product
quotient
plus
minus
times
divided by
added to
subtracted from
multiply
into
more than
less than
twice
per
increased by
decreased by
of
for every
total
less
double
slower than
faster than