Transcript Chapter 1
Chapter 1 Whole Numbers
Copyright © 2008 Pearson Addison-Wesley. All rights reserved.
Chapter 1 – Slide 1
Section 1.1
Introduction to Whole Numbers
Chapter 1 – Slide 2
Reading and Writing Whole Numbers
We
read
whole numbers in words, but we use the
digits
0, 1, 2, 3, 4, 5, 6, 7, 8, and 9 to
write
them.
We read the whole number
fifty-one,
but write it 51, which is called
standard form
.
Each of the digits in a whole number in standard form has a
place value
.
Chapter 1 – Slide 3
1-3
Reading and Writing Whole Numbers
The place value chart is shown below.
When we write large numbers we insert
commas
to separate the digits into groups of three, called
periods
.
Chapter 1 – Slide 4
1-4
Example
Identify the place value of the 8.
a. 508 b. 8,430,999 c. 6,800,000,002
1-5
Chapter 1 – Slide 5
Reading and Writing Whole Numbers
To Read a Whole Number
Working from left to right, • read the number in each period and then • name the period in place of the comma.
Chapter 1 – Slide 6
Example
How do you read the number 521,000,072?
Chapter 1 – Slide 7
1-7
Reading and Writing Whole Numbers
To Read a Whole Number
Working from left to right, • write the number named in each period and • replace the period in place of the comma.
Chapter 1 – Slide 8
Example
1.
Write the number six billion, twelve in standard form.
BILLIONS
O H
MILLIONS
T O H
THOUSANDS
T O H
ONES
T O 2.
The treasurer of a company write a check in the amount of three hundred thousand, two hundred eight. Using digits, how would she write this number?
Chapter 1 – Slide 9
1-9
Writing Whole Numbers in Expanded Form
Expanded form
of a number can be written using the number and its place value of its digits. The place value chart is shown below.
5,293 = 5 thousands + 2 hundreds + 9 tens + 3 ones Expanded form = 5000 + 200 + 90 + 3
BILLIO NS
O H
MILLIONS
T O H
THOUSANDS
T O 5
ONES
H 2 T O 9 3
Chapter 1 – Slide 10
Example
Write 803 in expanded form.
Write 8,407,800 in expanded form:
1-11
Chapter 1 – Slide 11
Rounding Whole Numbers
1-12
Chapter 1 – Slide 12
Rounding Whole Numbers
1-13
Chapter 1 – Slide 13
Example
Round 89,541 to: a. the nearest thousand b. the nearest hundred.
The Robinson’s are having new windows installed. The price is $12,870. How much is this to the nearest thousand dollars?
Chapter 1 – Slide 14
1-14
Example
Write in words the amount of money taken in by
The Lord of the Rings: The Two Towers
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Chapter 1 – Slide 15
Example
Round to the nearest ten million dollars the world total for
The Lord of the Rings: The Two Towers
.
Chapter 1 – Slide 16
Section 1.2
Adding and Subtracting Whole Numbers
Chapter 1 – Slide 17
Identities and Properties
The Identity Property of Addition
The sum of a number and zero is the original number.
3 + 0 = 3 or 0 + 5 = 5
The Commutative Property of Addition
Changing the order in which two numbers are added does not affect their sum.
3 + 2 = 2 + 3 5 = 5
Chapter 1 – Slide 18
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Identities and Properties
The Associative Property of Addition
When adding three numbers, regrouping addends gives the same sum. Note that the parentheses tell us which numbers to add first.
(4 + 7) + 2 = 4 + (7 + 2) 11 + 2 = 4 + 9 13 = 13
Chapter 1 – Slide 19
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Adding Whole Numbers
We add whole numbers by arranging the numbers vertically, keeping the digits with the same place value in the same column. Then we add the digits in each column. When the sum of the digits in a column is greater than 9, we must
regroup
and carry, because only a single digit can occupy a single space.
Chapter 1 – Slide 20
1-20
Example
1.
2.
3.
Add 56 and 39.
Add: 8,935 + 478 + 2,825 What is the perimeter of the region marked off for the construction of a brick patio?
18 feet 27 feet
Chapter 1 – Slide 21
1-21
Subtracting Whole Numbers
We write the whole numbers underneath one another, lined up on the right, so each column contains digits with the same place value. Keep the following properties of subtraction in mind.
• When we subtract a number from itself, the result is 0: 6 – 6 = 0 • When we subtract 0 from a number, the result is the original number: 32 – 0 = 32
Chapter 1 – Slide 22
1-22
Subtracting Whole Numbers
1-23
Chapter 1 – Slide 23
Example
1.
2.
3.
Subtract: 219 – 58 Find the difference between 400 and 174.
The junior class donated 365 cans of food to the food drive. The senior class donated 286 cans. How many more cans did the junior class donate?
Chapter 1 – Slide 24
1-24
Example
Attendance
Charles Pickney 24 Congaree Swamp 96 Cowpens National Kings Mountain 213 257 Fort Sumter 0 50 100 150 200 250
Number of Visitors (thousands)
http://www.scprt.com/files/Research/National_and_State_Parks.htm
300 319 350 Which park had the greatest number of visitors?
Chapter 1 – Slide 25
1-25
Example
Attendance
Charles Pickney 24 Congaree Swamp 96 Cowpens National Kings Mountain 213 257 Fort Sumter 0 50 100 150 200 250
Number of Visitors (thousands)
http://www.scprt.com/files/Research/National_and_State_Parks.htm
300 319 350 How many visitors were there at Fort Sumter and Kings Mountain?
Chapter 1 – Slide 26
1-26
Estimating Sums and Differences
An estimation can be used to check an answer and see if your answer is “close” to the exact answer.
Chapter 1 – Slide 27
1-27
Example
1.
Compute the sum 8,935 + 478 + 2,825 . Check by estimation. 2.
Subtract 2,387 from 7,329. Check by estimating.
Chapter 1 – Slide 28
1-28
1-29
Section 1.3
Multiplying Whole Numbers
Chapter 1 – Slide 29
The Meaning and Properties of Multiplication
Multiplication is repeated addition. For example, suppose you buy 5 packages of crayons for your child and each package has 6 crayons.
+ + + +
1-30
6 + 6 + 6 + 6 + 6 30 crayons 6 5 = 30 The parts of a product, that is the 6 and 5, are called
factors
.
Chapter 1 – Slide 30
Identities and Properties
The Identity Property of Multiplication
The product of any number and 1 is that number. 3 1 = 3 or 12 1 = 12
The Multiplication Property of 0
The product of any number and 0 is 0. 3 0 = 0 or 12 0 = 0
Chapter 1 – Slide 31
1-31
Identities and Properties
The Commutative Property of Multiplication
Changing the order in which two numbers are multiplied does not affect their product.
3 2 = 2 6 = 6 3
The Associative Property of Multiplication
When multiplying three numbers, regrouping the factors gives the same product. (4 7) 28 2 = 4 2 = 4 (7 14 2) 56 = 56
Chapter 1 – Slide 32
1-32
Multiplying Whole Numbers
To multiply whole numbers with reasonable speed, you must commit to memory the products of all single-digit whole numbers.
Chapter 1 – Slide 33
1-33
Example
1.
2.
3.
4.
Multiply: 76 · 6 Multiply: 400 60 5 ft Calculate the area of the home office. Multiply: (17)(4)(3) 9 ft 8 ft 14 ft
Chapter 1 – Slide 34
1-34
Estimating Sums and Differences
An estimation can be used to check an answer and see if your answer is “close” to the exact answer.
1.
2.
Examples
Multiply 412 by 198. Check the answer by estimating.
A class planning their class trip saved $3000 for theatre tickets. Each ticket costs $62, and a total of 28 tickets are needed. By estimating, decide if the class has set aside enough money for the tickets
Chapter 1 – Slide 35
1-35
1-36
Section 1.4
Dividing Whole Numbers
Chapter 1 – Slide 36
The Meaning and Properties of Division
In a division problem, the number that is being used to divide another number is called the
divisor
. The number being divided is the
dividend
. The result is the
quotient
. We can also think of division as the
opposite
(
inverse
) of multiplication.
Chapter 1 – Slide 37
1-37
Example
Divide and check: 3024 ÷ 6.
49, 021 Compute Then check your answer.
7
Chapter 1 – Slide 38
1-38
Remainders
When a division problem results in a remainder as well as a quotient, we use this relationship for checking. (Quotient × Divisor) + Remainder = Dividend We will often write the results of a division problem as
Chapter 1 – Slide 39
1-39
Example
1.
Find the quotient of 23,399 and 4. Then check.
2.
1,867 Compute and check.
23 3.
4.
Find the quotient and remainder of 12,861 and 63. Then check.
Divide and check: 9,000 ÷ 30.
Chapter 1 – Slide 40
1-40
Checking by Estimating
As for other operations, estimating is an important skill for division. Checking a quotient by estimating is faster than checking it by multiplication, although less exact. And in some division problems, we only need an approximate answer.
Example
An office building has an area of 329,479 square feet. If there are 9 floors in the building, estimate the square footage of each floor.
Chapter 1 – Slide 41
1-41
Section 1.5
Exponents, Order of Operations, and Averages
Chapter 1 – Slide 42
1-42
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Exponents
Writing an expression in
exponential form
provides a shorthand method for representing repeated multiplication of the same factor.
Definition
An
exponent
(or
power
) is a number that indicates how many times another number (called the
base
) is used as a factor.
3 • 3 • 3 • 3 • 3 = 3 5
Chapter 1 – Slide 43
1-43
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Example
1.
2.
a.
Compute: 1 7 3.
b.
13 2 Write 8 3 ∙ 4 2 in standard form and evaluate.
4.
Rewrite 4 ∙ 4 ∙ 9 ∙ 9 ∙ 9 ∙ 9 ∙ 9 in exponential form.
Approximately 10,000 seedlings were planted in a state forest. Express this number in terms of a power of 10.
Chapter 1 – Slide 44
1-44
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1-45
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Chapter 1 – Slide 45
Example
1.
2.
3.
4.
Evaluate: 34 – 9 ∙ 3.
Find the value of 7 + 3 ∙ (4 ∙ 6 2 ).
Find the value of 7 + 3 ∙ (4 ∙ 6 2 ).
Simplify: 7 2
1-46
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Chapter 1 – Slide 46
Averages
Definition
The
average
(or
mean
) of a set of numbers is the sum of those numbers divided by however many numbers are in the set.
Example
What is the average of 87, 95, and 88?
Chapter 1 – Slide 47
1-47
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Example
The following shows the high temperatures in Virginia during one week in November. a. What is the average temperature for the week?
b. Which day(s) has a temperature higher than the average temperature.
1-48 High Temp.
Sun. Mon. Tues. Wed. Thurs. Fri
42°F 49°F 53°F 39°F 30°F
Sat
41°F 54°F
Chapter 1 – Slide 48
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Calculator Examples
1.
Evaluate 27 3 using your calculator.
2.
Evaluate 5 + 9 ÷ 3 × 2 by hand and check using your calculator.
1-49
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Chapter 1 – Slide 49
Section 1.6
More on Solving Word Problems
1-50
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Chapter 1 – Slide 50
Solving Word Problems
To Solve Word Problems
• Read the problem carefully • Choose a strategy (such as drawing a picture, breaking up the question, substituting simpler numbers, or making a table).
• Decide which basic operation(s) are relevant and then translate the words into mathematical symbols.
• Perform the operations.
• Check the solution to see if the answer is reasonable. If it is not, start again by rereading the problem.
Chapter 1 – Slide 51
1-51
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Four Basic Operations
Operation × ÷ + − Meaning
Combining Taking away Adding repeatedly Splitting up
Chapter 1 – Slide 52
1-52
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Clue Words
1-53
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Chapter 1 – Slide 53
Drawing a Picture
Sketching even a rough representation of a problem, can provide insight into its solution.
Example:
At Greenfield High School, there are 292 freshmen, 213 sophomores, and 524 juniors. If there are 1,036 total students, how many seniors are there in the school?
Freshmen 292
Greenfield High School
Sophomore Junior Senior 213 254 Total 1036
Chapter 1 – Slide 54
1-54
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Breaking Up the Question
Another effective problem-solving strategy is to break up the given question into a chain of simpler questions.
1-55
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Chapter 1 – Slide 55
Example
On her way to work, Melinda must travel through 18 traffic lights. If she is stopped by 5, how many more traffic lights did she get a green light than a red light?
How many traffic lights were green? How many did she get stopped by?
How many more traffic lights were green than red?
Chapter 1 – Slide 56
1-56
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Substituting Simpler Numbers
A word problem involving large numbers often seems difficult just because of these numbers. A good problem-solving strategy is to consider first the identical problem but with simpler numbers.
1-57
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Chapter 1 – Slide 57
Example
Dinner tickets for a benefit are sold at $12 each. How many dinner tickets must be sold before the benefit profits if the break even amount for the cost of food is $2,700?
To determine the operation, substitute a simpler number such as $24 for the break even amount. Because it is a “fit in” question, we must divide $24 by $12. Going back to the original problem, we see that we must divide $2,700 by 12.
Chapter 1 – Slide 58
1-58
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Making a Table
When a word problem involves many numbers, organizing the numbers in a table often leads to a solution.
1-59
Example
A semi truck driver must travel 1,372 miles to its destination. If the driver travels 65 miles in an hour, how many miles are remaining after 8 hours?
Chapter 1 – Slide 59
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Making A Table - Continued
1-60 After Hour
4 5 1 2 3 6 7 8
Remaining Miles
1,372 – 65 = 1,307 1,307 – 65 = 1,242 1,242 – 65 = 1,117 1,112 – 65 = 1,047 1,047 – 65 = 982 Copyright © 2008 Pearson Addison-Wesley. All rights reserved.
Chapter 1 – Slide 60