1.1 Introduction to Whole Numbers

1.1 INTRODUCTION TO WHOLE NUMBERS

Objective A - Whole Numbers

Defined as: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12…

Numbers can be graphed (or “pictured”) on a number line:

To graph a number, you fill in a dot at the location for that number.

Ex. Graph the number 5 on the number line.

When you compare two numbers on a number line, the smaller number is always the number on the ______. Therefore, the larger number is always to the ______.

Inequalities

There are two inequality symbols: and

The symbol means “is greater than.”

The symbol means “is less than.”

Ex. 16 > 5 Ex. 3 < 11

Objective B - Writing Whole Numbers in Words and in Standard Form

Digits: There are 10 digits. They are: 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9

The position of a digit in a number is its place value.

Place values are shown below: See place-value chart in text!

_____

Hundred-Billions

Ten-Billions

Billions

/ /

Hundred-Millions

Ten-Millions

Millions

/ /

Hundred- Thousands

Ten- Thousands

Thousands

/ /

Hundreds

Tens

Ones

When reading a number, there is no “and” read until you reach a decimal point (Chapter 3).

Ex. Write 1,617,250 in words.

Ex. Write “five billion two hundred twenty-seven million three hundred forty thousand six hundred one” in standard form.

Objective C - Writing Whole Numbers in Expanded Form

· To write a number in expanded form means you take each digit in the number and write its value based on its position (place value), and then add these values together.

· In expanded form: 2,951 = 2,000 + 900 + 50 + 1

· In expanded form: 704,536 = 700,000 + 4,000 + 500 + 30 + 6

Ex. Write 2,611,097,105 in expanded form:

Objective D - Rounding a Number to a Given Place Value

· Pay attention to which place value you being asked to round.

1. Locate the digit in that place value.

2. Look at the digit to its right.

a. If the digit to the right is 0, 1, 2, 3, or 4 → round down. (Meaning, leave the located digit the same.)

b. If the digit to the right is 5, 6, 7, 8, or 9 → round up. (Meaning, increase the located digit by one.)

3. Fill in the rest of the place value positions with zeros to the right of the located

digit so it will be in the correct position.

Ex. Round to the nearest tens:

a. 29 b. 1,697

Ex. Round to the nearest thousands:

a. 2,907 b. 67,482

1.2 ADDITION OF WHOLE NUMBERS

Objective A – Adding Whole Numbers

· Review the basic addition facts for this section. If you have a hard time remembering them, flash cards made on 3 X 5 index cards are a great method to practice them.

o Sum: the “answer” to an addition problem.

o Addend: the numbers that are being added.

· There are specific properties of Addition:

1. Addition Property of Zero

2. Commutative Property of Addition See Vocabulary Sheet

3. Associative Property of Addition

· When you add whole numbers, line up the decimal points and add the digits that are in the same place value column.

· When you add a column of digits and the sum is a 2-digit number (greater than 9), you will need to “carry” the digit on the left over to the next column and add it to the digits in that place value column.

Ex. Add 63,721 + 854 + 1,062

1.3 SUBTRACTION OF WHOLE NUMBERS

Objective A - Subtracting Whole Numbers without Borrowing

· Subtract without Borrowing

o When you subtract whole numbers, line up the decimal points and subtract the digits that are in the same place value column.

Minuend – Subtrahend = Difference Also: Minuend

- Subtrahend

Difference

Ex. Subtract 965 – 431

· There is a way to check your answers to a subtraction problem:

o Add the Subtrahend + Difference and see if that equals the Minuend

o That is, since 7 – 3 = 4, you can check it by adding 3 + 4 = 7

Objective B - Subtracting Whole Numbers with Borrowing

· Subtract With Borrowing

o If the digit in the subtrahend is greater than the digit in the minuend, then you must borrow.

o When you subtract a column of digits, and the lower digit is larger than the upper digit, you will need to “borrow” from the digit on the left. Add the borrowed amount to the upper digit, and then subtract the digits in that place value column normally.

Ex. Subtract 347 – 172 Ex. Subtract 2,361 – 679

Words that mean subtraction:

15 less 5 15 – 5

6 less than 11 11 – 6

9 decreased by 1 9 – 1

difference between 7 and 5 7 – 5

subtract 4 from 12 12 – 4

· Subtract With a Zero in the Minuend

o Requires Repeated Borrowing

Ex. Subtract 4902 – 2957

1.4 MULTIPLICATION OF WHOLE NUMBERS

Objective A – Multiplying Whole Numbers

· Multiplying by a Single Digit

o Factors: the numbers that are being multiplied

o Product: the result (or solution) to a multiplication problem

· There are specific properties of Multiplication:

1. Multiplication Property of Zero

2. Multiplication Property of One

See Vocabulary Sheet

3. Commutative Property of Multiplication

4. Associative Property of Multiplication

· Multiplication is merely shorthand for the times when you need to add the same number many times.

· 3 + 3 + 3 + 3 means that you are adding 3 together 4 times, so we write.

· You will need to review the basic multiplication facts.

a. See text to refresh you memory of these.

b. Flash cards will help if you are struggling.

Ex. Multiply 67 5 Ex. Multiply 294 7

Objective B Multiplying Larger Whole Numbers

· Patterns:

o Handy way to multiply fast!

o If one of the factors has one or more zeros at the end, you can multiply the non-zero parts and then attach the same number of zeros to the product as the total number of zeros in the factors.

Ex. Multiply 50 5 Ex. Multiply 11 2,000

Multiplying Numbers That Do Not End With Zeros:

Ex. Multiply 29 56 Ex. Multiply 328 45

Words that mean multiplication:

5 times 7 5 7

product of 8 and 6 8 6

7 multiplied by 9 7 9

1.5 DIVISION OF WHOLE NUMBERS

Objective A – Dividing Whole Numbers

1. Dividing by a Single Digit with No Remainder in the Quotient.

2. Division is related to multiplication.

· Since 2 3 = 6, 6 3 = 2

· You are finding how many groupings of 3 there are in 6

Dividend Divisor = Quotient Also:

· To check a division problem: Quotient Divisor = Dividend

· For example: 8 4 = 2

· To check this: 4 2 = 8

3. Important Quotients

o Any whole number, except zero, divided by itself is one

o Any whole number divided by 1 equals that same number

o Zero divided by any whole number (other than zero) equals zero

o Any number divided by zero – CANNOT BE DONE!!!

· Division by zero is not allowed

· There is NO number whose product with zero equals 8?

Ex. Divide 2120 4 Ex. Divide 480 6

Objective B – Dividing by a Single Digit with a Remainder in the Quotient

· Often it is NOT possible to divide the Dividend into a whole number without a Remainder in the Quotient.

Ex. Divide 32 5

Ex. Divide 1,632 7

Objective C – Dividing by Larger Whole Numbers

· When the divisor has more than one digit.

o Estimate each step by using the first digit of the divisor.

o If that product is too large, lower the guess by one and try again.

o Use the following method:

§ / 3 • 33 = 99 → to large, try again
§ / 2 • 33 = 66 < 89 → works!
Repeat process with 8 • 33 = 264 → to large, try again
7 • 33 = 231 < 234 → works!

Ex. Divide 70 Ex. Divide 33

1.6 EXPONENTS AND THE ORDER OF OPERATIONS AGREEMENT

Objective A – Simplifying Expressions

1. Exponents are shorthand notation for repeated multiplication of the same factor.

2. → called factored form and can be written as

o Read as, “3 to the 4th power” or “3 to the 4th”

3. For the above example, 3 is the base and 4 is the exponent.

4. There are “nicknames” for two exponents:

o An exponent of 2: 52 can be read as “5 to the second” or “5 squared.

o An exponent of 3: 53 can be read as “5 to the third” or “5 cubed.”

Evaluate the following:

Ex. 83 Ex. 72 Ex. 23 • 42 =

Ex. Write as a power of 10: Ex. Simplify 62 • 24 =

10 • 10 • 10 • 10 • 10 =

Objective B – Order of Operations to Simplify an Expression

· The order of operations is employed for simplifying expressions.

· More than one operation can occur in a numerical expression.

o Therefore, the answer may different depending on the order in which the operations are performed.

· Example: OR

ORDER OF OPERATIONS

· There is a correct order to follow if there are several operations involved in a problem.

· It is easy to remember using the acronym: Please Excuse My Dear Aunt Sally

· P refers to Parentheses – do all operations in parentheses first (includes ( ), { } or [ ].)

· E refers to Exponents – simplify exponents next.

· M refers to multiplication and division → do them in order from left to right

· D (multiplication is NOT automatically done before division)

· A refers to addition and subtraction → do them in order from left to right

· S (addition is NOT automatically done before subtraction)

Ex. Ex.

1.7 PRIME NUMBERS AND FACTORING

Objective A - Factoring

· Factoring Numbers

o Whole-number factors of a number divide a number evenly.

o NO remainder is left-over after division.

Ex. List whole number factors of 28:

· Here are some rules to help you find the factors of a number

o If the last digit of a number is 0, 2, 4, 6, or 8, then 2 is a factor of the number.

o If the sum of the digits of a number is divisible by 3, then 3 is a factor of this number.

o If the last digit of the number is 0 or 5, then 5 is a factor of the number.

Ex. Find all the factors of 12. Ex. Find all the factors of 50.

Objective B – Prime Factorization of a Number

· Finding the Prime Factor of a Number

· Prime number = a number is prime if its only whole number factors are 1 and itself

o Examples include 3 and 7

· Composite number = a number that is not prime is composite.

o From above, 12, 28, and 50 are composite numbers.

o Let’s make a list of the prime numbers less than 50

· The prime factorization of a number means that you write that number as the product of its prime factors.

· The textbook uses a “T-diagram” to find a number’s prime factorization.

Ex. 12

· Another method of finding a number’s prime factors is called the “cake method.”

Ex.

· Start by dividing the number by any prime number that is a factor of the number

o “2” in the example above

· You then divide the quotient by another prime number until the result is 1

· All of the resulting divisors are the prime factors

Ex. Find the prime factorization of 40. Ex. Find the prime factorization of 65

Ex. Find the prime factorization of 89. Ex. Find the prime factorization of 225.