1.5 Dividing Whole Numbers
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1.5 Dividing Whole Numbers
Remember there are three different ways to write division problems
412 ÷ 3 = 4 3 / 12 12/3 = 4
All of these represent the same problem:12 divided by 3 is 4.
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There are some terms that are special to division that we should be familiar with:
Quotient-the answer when we divideDividend-the number being divided Divisor-the number being divided into something
dividend / divisor = quotientdividend ÷ divisor = quotient quotientdivisor / dividend
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Division with zero
Zero divided by any number• Zero divided by any number
is always and forever ZERO
• 0/99 = 0• 0 ÷ 99 = 0
Any number divided by zero• Any number divided by zero
is always undefined. We cannot divide by zero. It is an illegal operation mathematically.
• 99/0• 99 ÷ 0
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Division and the number 1
Any number divided by itself• Any number divided by
itself is equal to one.
• 9/9 = 1• 99/99 = 1
Any number divided by 1• Any number divided by one
is equal to the number itself.
• 9/1 =9• 99/1 =99
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The connection
Multiplication and Division are closely related. We can go back and forth between the two operations.
20/4 = 5 so 4 × 5 = 20
72 ÷ 9 = 8 so 9 × 8 = 72
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We can also use multiplication to check division even when there is a remainder.
Take: divisor x quotient + remainder = dividend
458 ÷ 5 = 91 r 3divisor x quotient + remainder = dividend 5 x 91 + 3 = 455 + 3 = 458 It works!
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Divisibility rules
A number is divisible by:-2 if the ones digit is even-3 if the sum of the digits is divisible by 3-5 if it ends in 5 or 0-9 if the sum of the digits is divisible by 9-10 if it ends in 0
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1.6 Exponents
323 is your power or exponent;2 is your base
Read 2 to the 3rd power
The exponent tells how many times the base appears as a factor.
2 x 2 x 2 = 8
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1.6 Exponents
Any number to the zero power = 1
20 = 11000 = 1
One to any power = 118 = 1
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1.6 Exponents
Any number that has no exponent written has an understood exponent of one
2=21
100=1001
Zero to any power = 118 = 1
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1.6 Order of Operations
Order of Operations exists because when there is more than operation involved, if we do not have an agreed upon order to do things, we will not all come up with the same answer. The order of operations ensures that a problem has only one correct answer.
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1.6 Order of Operations
Parenthesis (or grouping symbols)ExponentsMultiplication or Division from Left to RightAddition or Subtraction from Left to Right
PEMDAS
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1.6 Order of Operations
In the parenthesis step, you may encounter nested parenthesis. Below you will see the same problem written two ways: once with nested parenthesis and the other with a variety of grouping symbols (including brackets, braces, and parenthesis).
(( 5 x ( 2 + 3 )) + 7 ) – 2OR
{[ 5 x ( 2 + 3 )] + 7 } - 2
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1.6 Order of Operations
Just a reminder that there are many ways to show multiplication. You will still see the “x” for times or multiply, but you will see other ways as well.3(2)(3)2(3)(2)3 2
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1.7 Rounding Whole Numbers
• To round we must remember place value
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1.7 Steps for roundingFind the place you are rounding to and
underline itLook at the digit to the right of the underlined
place-if it is 5 or higher, the underlined
number will go up;-if it is 4 or lower, the underline number
will stay the same. Change all the digits the right of the underlined
digit to zeros.
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1.7 Rounding examples
Round 478 to the nearest tenFind the tens place: 478 Look at the digit behind the 7The 8 will push the 7 up to an 8Fill in zeros480
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1.7 Rounding examples
Round 46352 to the nearest thousandFind the thousands place: 46352Look at the digit behind the 6The 3 will not push the 6 up – leave itFill in zeros46000
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1.7 Rounding examples
Round 4963 to the nearest hundredFind the hundreds place: 4963Look at the digit behind the 9The 6 will push the 9 up to a 10 which rolls over
and pushes the 4 up to a 5. watch out! Fill in zeros5000
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1.7Rounding for estimating purposes
Rounding to a given place-value
Round to hundreds place and add for an estimated answer.
949 900
759 800+ 525 + 500
2200
Front-end roundingFront-end round as
appropriate for an estimated answer.
3825 4000 72 70 565 600+2389 + 2000
6670
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1.8 Application Problems
In most word problems, there are usually one or more words that indicate a particular operation. Being able to pick out these words is a key skill in being able to solve word problems successfully.
What are some of these words?
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1.8 Application ProblemsWords for AdditionPlus, more, add, total, sum, increase, gainWords for SubtractionLess, difference, fewer, decrease, loss, minusWords for MultiplicationProduct, times, of, twice, double, tripleWords for DivisionQuotient, divide, perWords for EqualsIs, yields, results in, are
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1.8 Application Problems
Read the problem through once quicklyRead a second time, paying a bit more attention
to detailMake some notesTry to come up with a planDo the mathLabel your answer