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Prealgebra, Chapter 1 - Whole Numbers

1.1 Whole Numbers and Place Value

Number Systems

The most basic reason that we need numbers:

Examples

Throughout history, numbers have been used

Egyptian hieroglyphs

Roman numerals

Numbers have been in use as long as civilization has been around.

Our number system:

This number system is in widespread use today, largely because it is superior to other numbersystems that have developed. In particular, mathematical operations are far easier to do whenusing our number system than they are when using something such as hieroglyphs or Romannumerals.

Place Value

The key to understanding our number system lies in understanding the fact that it is a place valuesystem.

Each digit in a number sits in a certain place, and each place has a certain value.

Example: 374

Understanding the concept of place value is the key to understanding our number system and howit works.


Expanded Form

If we understand place value, we can take a number, such as 297, and expand it.

297 =

This is called writing a number in expanded form. The more usual “297" is referred to asstandard form.

Examples: Write in expanded form.

815

4,372

The Place Values

Consider this very large number:

8,994,893,694,829

Each place has a name. Starting at the far right, we have

9 ones, 2 tens, 8 hundreds, 4 thousands,

9

6

3

9

8

4

9

9

8

Numbers this large do in fact show up in the real world. The number above represents the USGovernment debt, in dollars, at the time this was written.


Identifying Place Value

You should be able to look at a number and identify the place value of each digit.

Example: 27,803

2

7

8

0

3

Example: For the number 41,827,936

What is the place value of the 8?




Writing Numbers with Words

Commas are used to make large numbers easier to read. The commas group the digits intogroups of three, starting at the right. Each group has a name.

4,378,810,924

Instead of expanded form, where we expand each digit according to its place, we can expand eachgroup.

Thinking of the number in terms of groups is the key to writing it in words.

Example: 5,302,811

Note: We do not use the word “and”We do not write the name of the ones group at the end.


Why is writing numbers in words important? There are plenty of cases where numbers are writtenby hand. Sometimes, handwriting is messy and difficult to read.

308,215

The 0, when written by hand, can sometimes look like a 6. This, of course, changes the meaningof the number. Other handwriting mistakes can also be make. But it is usually important to get thenumber correct.

Example: Writing a check.

Example: 17, 821

Example: Three trillion, two hundred seventy two billion, eight hundred twelve million, sixhundred forty two thousand, nine hundred fifteen.

‘ Practice Problems‘ HW 1A


1.1 Practice

Write these numbers in expanded form.

1. 729

2. 5,634

Give the place value for the indicated digits.

3. The 8 in 819

4. The 5 in 1,075

Which digit tells the number of tens?

5. 45,432

6. 4,890

Which digit tells the number of ten thousands?

7. 96,584,273

8. 5,374,190

Write these numbers in words.

9. 76,540

10. 6,008,529

Give the numerical form of these numbers.

11. Seventeen thousand, two hundred seventy-two

12. Five billion, one hundred six million, seven thousand, nine hundred


Write the whole number in these sentences in standard form.

13. The earth is approximately ninety-three million miles from the sun.

14. Some new cars cost one hundred twenty-five thousand dollars.

Country Rank Size(1000's of square miles)

Russia 1 6591

Canada 2 3854

United States 3 3718

China 4 3704

Brazil 5 3286

Write answers in standard form using the data in the table.

15. What is the size of China?

16. What is the size of the United States?


1.2 Adding Whole Numbers

When we are counting, we generally start from 1.

1, 2, 3, 4, ...

This group of numbers is sometimes called the counting numbers or the natural numbers, simplybecause this is the way that we naturally count things. We start at 1 and we count up.

But the number zero is important, too. If we want to do more with the numbers than just counting,that it, if we want to add, subtract, multiply, and divide, the zero is necessary.

All of the counting numbers and zero

0, 1, 2, 3, ...

are referred to as whole numbers.

In this section, we will discuss the rules and techniques for adding whole numbers.

Adding things that are of like kind

The following point is very important. We can add numbers:

We can also add numbers that represent things

4 cats + 2 cats =

3 birds + 8 birds =

And this works for anything, as long as they are the same kind

20 dollars + 8 dollars =

7 miles + 9 miles =

But if they are not the same kind, we cannot add them

5 dollars + 3 cats =

2 birds + 20 miles =


Adding numbers

Definitions

3 + 8

the “plus” sign, or addition symbol

read, “three plus eight”

3 and 8 are referred to as addends.

3 + 8 = 11

11 is called the sum.

Adding numbers is often visualized with objects:

qqq + qq = qqqqq

Adding things in this way is fairly intuitive, and fits with our everyday experience.

Addition can also be represented on a number line. This is a little more abstract, but is just asimportant.

The number line has certain characteristics:

Evenly spaced markingsLarger numbers to the rightThe zero point is called the origin.

Addition can be pictured as movement along the number line.

Example: 5 + 2

Example: 7 + 1 + 4


Basic Addition Facts

A table of the basic addition facts looks something like this:

0 1 2 3 4

0 0 1 2 3 4

1 1 2 3 4 5

2 2 3 4 5 6

3 3 4 5 6 7

4 4 5 6 7 8

And the values for the numbers at least up through 10 rows and 10 columns should be memorized.You should be able to recall use these facts quickly and easily.

Properties of Addition

The Commutative Property

If you think about the basic addition facts, you will see that

3 + 4 = 7

and also that

4 + 3 = 7

The same pattern holds for all of the numbers. If we are adding two numbers, it does not matterwhich one comes first. We say that addition is commutative. This simply means that the order ofthe numbers does not affect the answer. This fact is referred to as the commutative property ofaddition.

Examples:

8 + 2 =

3,479 + 12,892 =

The Associative Property

Sometimes it is useful to group numbers together. This is generally the case when we are addingmore than two numbers.

Think about adding these three numbers:

8 + 4 + 3


The associative property of addition says that the way in which the numbers are grouped does notaffect the sum. In other words, we could associate the 4 with the 8:

(8 + 4) + 3

or we could associate the 4 with the 3

8 + (4 + 3)

and we see that the sum is the same in each case.

Associating, or grouping, numbers, can come in handy when adding long lists of numbers. It isoften possible to find pairs or groups of numbers that add up to 10 or multiples of 10, which canallow the sum to be computed quickly and easily in your head.

Examples:

7 + 12 + 3 + 8

15 + 3 + 1 + 11 + 3 + 5 + 6

38 + 1 + 4 + 2 + 5 + 16 + 7 + 4

2 + 7 + 6 + 8 + 18 + 3 + 4 + 22 + 5 + 2 + 6

The Additive Identity Property

Adding zero to a number is the same as not adding anything at all.

5 + 0 =

0 + 213 =

The number 0 is called the identity for addition.


Adding Numbers with Multiple Digits

When add numbers that have multiple digits, we line up the columns so that all the digits of onecolumn have the same place value.

We add the numbers in each column. We can do this because the numbers in the ones columnrepresent “ones”, and the numbers in the tens column represent “tens”. In other words, all thenumbers in a given column represent the same thing.

This is similar to adding items that are of the same type. We can add apples and apples, or inchesand inches. The things we are adding have to be of the same kind. By the same reasoning, wecan add ones and ones, or tens and tens.

Examples:

Carrying

In many cases, the digits add up to a number that has more than one digit, and we have to carrya digit into the next column.

We always start at the right, in the ones place, and we work our way to the left, toward the higherplace values. In this example, when we add the 7 and the 8, we get 15, but we recognize that 15is really 1 ten plus 5 ones. The “1" has to go in the tens column because it does in fact representtens. So we put it there, usually be writing a “1" at the top of the tens column.

Sometimes we have to carry in multiple steps.


Examples:

Recognizing words that indicate addition

Often we encounter spoken or written statements of problems that have to be solvedmathematically. Someone might say, “That will cost $300 for the new sink and $125 for the mirrorand $48 for the faucet.” Situations like this are common. This is an English statement, but at itscore there is a mathematical problem to be solved.

We would write

We knew to add because of the use of the word and. There are other words and phrases that arecommonly used to indicate addition.

sum, plus, add to, more, more than, increased by, total, in addition to, additional

Example: In July, there were 12 students signed up for the class. During the month ofAugust, 6 more students signed up. What was the total number of students in theclass.

Application: The Perimeter of a Figure

The perimeter of a closed figure is defined as the distance around the figure. Finding the perimeterof a figure is usually a matter of adding up the lengths of all the sides.

Example:


Example:

Think about the perimeter of a rectangle:

Every rectangle has four sides, two that we call the width and two that we call the length. Insteadof referring to a specific rectangle, such as the one shown above, we can refer to a generalrectangle.

This could be any rectangle, not just the one shown. Whatever its actual length and width are, weknow that the perimeter will be

Perimeter =

And we usually write this with a more convenient notation:

This is an example of a ________________. The letters P, W, and L are _________________.

If we know the value of W and L, we can plug them into the formula and do the addition to find P.


Example: A rectangle has a width of 6 yd and a length of 20 yd. What is its perimeter?

Example: A field is 160 feet by 220 feet. You want to build a fence around the field. What isthe total length of fence that you need to enclose the field?

Using a Calculator

You need to be able to add numbers both by hand and by using a calculator. Different calculatorswork in different ways, but virtually all of them use the “+” key for addition. On some calculators,you may use the equals key, “=”, to compute the sum. Others may have an “Enter” key. Be surethat you know how to use your particular calculator.

‘ Practice Problems‘ HW 1B


1.2 Practice

1. Identify the addends and the sum in the equation: 3 + 7 = 10

Addends:

Sum:

Identify which property of addition is shown in the equations below.

2. (5 + 3) + 2 = 5 + (3 + 2)

3. 6 + 2 + 3 = 2 + 6 + 3

Add these numbers.

4. 5. 6. 7.

8. Which number is 216 more 9. What is the total of 123 and 7,421?than 3,771?

10. What is the sum of 56, 234, 1120, and 221?

11. Find the perimeter of the figure shown.


12. Find the perimeter of the figure shown.

13. A picture frame has the dimensions 7 in. by 5 in. What is its perimeter?

14. A clothing store sold 1,010 dresses in April, 965 dresses in May, and 982 dresses in June.How many dresses did the store sell?

15. Bob’s Burgers sold the following food items from December to February. Compute thetotals and fill in the information missing in the chart.

Food January February March Totals

Hot dogs 1,721 1,888 1,714

Hamburgers 1,432 1,516 1,499

French Fries 2,004 1,989 1,990

Sodas 2,138 2,210 2,511


1.3 Subtracting Whole Numbers

Subtraction is the opposite of addition. If addition means adding more, subtraction means takingsome away so that you have less.

The minus sign: - used to indicate subtraction

Terminology

Consider this example: 9 - 2

The number 9 is called the _______________________


9 - 2 = 7


The term difference fits with our everyday understanding of the term. If we want to knowthe difference between two numbers, we subtract the smaller from the larger.

Subtracting Numbers with Multiple Digits

When working with numbers with multiple digits, we arrange them in a manner similar that usedfor addition, with the place values lined up in the same column. We subtract the ones from theones, the tens from the tens, etc.

Examples:

Because subtraction is the opposite of addition, we can use addition to check our work and makesure our answer is correct.

We get an answer of 532. Therefore, 532 and 441 should add up to 973.

Check:


Recognizing words that indicate subtraction

Many situations in the real world require subtraction. When trying to figure something out or solvea problem, we need to be able to recognize when we need to subtract rather than add.

In a written or spoken problem, certain words or phrases often serve as clues that subtraction willbe involved.

subtract, minus, difference, less than, decreased by

Example: 4250 gallons of water were in the storage tank. Irrigating the garden decreased thewater supply by 120 gallons. How much water was left in the storage tank?

Sometimes the question “How much more...?” can indicate the need to subtract.

Example: Doug wants to purchase a new mountain bike for $475. He has saved up $320.How much more money does he need to save before he can purchase the bike?

Borrowing

When adding, we sometimes need to carry. When subtracting, we sometimes need to borrow. You have probably done subtraction problems that involve borrowing. Rather than simply followinga mechanical process called borrowing, you should try to understand why this process works.

Look at this problem:

We cannot subtract 5 from 3, because it is bigger than 3. We solve this problem by borrowing.

The eight in the above problem represents 8 tens. We can take one of these tens and add it to the3, making the 3 a 13. We still have 8 tens. One of them has simply been borrowed by the onescolumn.


Again, we can check our work:

Check:

Sometime we need to borrow multiple times.

Examples:

Applications

Plenty of real world situations involve subtraction.

Example: Three students are selling books to raise money for a charity. They need to sell atotal of 2500 books. John has sold 733 books. Mary has sold 815 books. Sue hassold 392. How many more books need to be sold for them to reach their goal?

Example: Joe has $852 in the bank. He withdraws $230 to purchase a new bike and $176 topurchase a new computer monitor. How much does he have left in the bank?

‘ Practice Problems for section 1.3


1.3 Practice

1. In the statement 6 – 5 = 1

6 is called the

5 is called the

1 is called the

Write the related addition statement.

Do the indicated subtraction.

2. 3. 4.

5. 6. 7.

8. What number is 34 less than 89? 9. What is the difference between 718and 291?

10. Jake is driving his Jeep over mountainous terrain near the Ocomat Gorge. Starting at anelevation of 1,435 ft, his elevation then decreases by 306 ft, increases by 152 ft, thendecreases of 75 ft. Find Jake’s final elevation.


11. Thomas has saved $297. He wants to buy a TV that costs $459. How much more moneydoes he need to save?

12. The difference between two numbers is 142. The greater number is 411. What is thesmaller number?

13. A certain high school basketball team scored 72, 66, 69, and 70 points in its first fourgames. That same team scored 81, 82, 75, and 94 points in its last four games. How manymore points did the team score in its last four games than in its first four games?

14. Fredrick scored 97 points on a history test. Sara scored 19 fewer points on the same test.How many points did Sara score on the test?

15. John bought a desk chair for $89. Greg bought an identical desk chair to the one Johnbought for $125. How much more did Greg pay?

16. The land area of Georgia is 59,425 sq. mi. The land area for Florida is 65,755 sq. mi. Howmuch larger is Florida than Georgia?


17. Evaluate these expressions:

(a) 12 – (8 – 1) (b) (12 – 8) – 1

Do they give the same answer? What can you conclude about subtraction and theassociative property?

18. This problem involves a bank account and money withdrawn (subtracted) from the previousamount. Fill in the blanks.

Original balance $ 7291st withdrawal $ 2342nd withdrawal $ 1673rd withdrawal $ 94

a) Find the balance after b) Find the balance after c) Find the balance after the first withdrawal. the second withdrawal. the third withdrawal.

Work the following problems on your calculator.

19. 22.

21. 156,705 – 78,892 =

22. Subtract 743 from the sum of 365 and 458.


1.4 Rounding, Estimating, and Ordering Numbers

Rounding

In most cases, mathematics is exact. 2 + 2 = 4. The answer is exactly 4. Not just some numberclose to 4, but exactly 4.

In many cases, however, we don’t need exact numbers, only estimates. Sometimes, in fact,estimates make more sense than exact numbers.

Think about the distance between two cities:Atlanta and Jacksonville. If you want to drive fromAtlanta to Jacksonville, you will drive approximately310 miles. The exact distance depends on wherein Atlanta you start, and where in Jacksonville youfinish. To find the exact distance, we would needtwo specific points, such as the distance from theboundary line of one city to the boundary line of theother, or the distance from the courthouse of onecity to the courthouse of another. Or maybe youneed to know the distance from your house inAtlanta to your Grandmother’s house inJacksonville. Without specifying two specificpoints, trying to find the exact distance isimpossible. It makes more sense to speak of theapproximate distance between the cities, which isabout 310 miles.

The distance 310 miles is the distance between the cities, rounded to the nearest 10 miles. It mayalso make sense to approximate the distance to the nearest hundred miles, in which case we wouldsay that the distance from Atlanta to Jacksonville is 300 miles. When you say something like this,people generally understand that you are giving an approximate number.

Approximating a number by using a simpler number that is close to the original number is calledrounding.

We can visualize the process of rounding by using a number line.

The number line shown here is marked off in increments of 10. The number 84 is also shown. Itis apparent that 84 is closer to 80 than it is to 90. We say that 84 rounded to the nearest 10 is 80.

Shown here is a number line calibrated in increments of 100. The number 527 is seen to be closerto 500 than it is to 600, so we say that 527 rounded to the nearest hundred is 500.


The number line shown here is marked in increments of 1000. The number 52,775 is closer to53,000 than it is to 52,000. We say that 52,775 rounded to the nearest thousand is 53,000.

Rules for Rounding

The rules for rounding are simple, and should make sense if you understand what rounding is.1. Find the place you want to round to.2. If the next digit is a 5 or higher, round up. Otherwise round down.

That’s it. Some practice, of course, will help.

Examples: Round to the nearest 10.

51 52 53 54 55 56

Round to the nearest 10

621 625 628


628 684 98 34


27,811 27,500 27,499

Rounding up a 9

Examples: Round to the nearest 100

2921 2973 43,984 29,978


497 2398 192 198


Estimating

In some cases, exact numbers are not needed and an estimate is sufficient.

Suppose, for example, that you need to know the number of people seated in a baseball stadium.You do not need to know the exact number, just an estimate that is accurate to the nearesthundred. You have the following information from the ticket sales:

In section A: 128 peopleIn section B: 188 peopleIn section C: 417 peopleIn section D: 530 peopleIn section E: 312 peopleIn section F: 694 people

It would take some time to add all the numbers up. There would be a lot of carrying. Mistakesmight be made. Even typing all the numbers into the calculator could take a little while, and theremay be typographical errors.

Instead, you can get a quick estimate of the total to the nearest 100 by rounding each number tothe nearest 100 and then adding.

A:

B:

C:

D:

E:

F:

And you can probably compute the sum in your head without too much difficulty.

A warning:

Note that this method works well, and gives reasonably accurate results when some of thenumbers get rounded up and some get rounded down. Rounding the numbers make the finalanswer inaccurate, but if some of the numbers are rounded up and some are rounded down, thenthese inaccuracies tend to cancel each other out. This is usually the case. If we have a randomsample of numbers, and we are trying to round, say, to the nearest hundred, some of the numberswill probably be a little above an even hundred, and will round down while other numbers will bea little below an even hundred and will round up. The rounding errors tend to cancel each otherout. Realize that it is possible, however, for most or all of the errors to be in the same direction.That is, most of the numbers could, by chance, end up rounding up (or down), and the errors couldaccumulate to a large total. This method, therefore, while reasonably good, is not perfect.


A Bad Example:

Estimate the sum of the following numbers to the nearest hundred by rounding eachnumber to the nearest hundred and then adding.

472

251

160

278

354

155

770

When we add the rounded numbers, we get 500 + 300 + 200 + 300 + 400 + 200 + 800,which equals 2700.

The actual value is 2440. Rounded to the nearest hundred we should have ended up with2400. We are off by 300! This is because all of the numbers ended up rounding up, so ourresult was skewed to high side. This is usually not the case. Realize, however, that it ispossible for this method of rapid estimating to produce incorrect results.

Example: Suppose you go to the hardware store and select the following items

light bulbs: $15.95batteries: $3.95screwdriver: $2.95tape: $6.95bolts: $8.15boards: $4.25

You have $50 in your wallet. Round each price to the nearest dollar and make aquick estimate of the total. Do you have enough money to purchase all of theitems?


Ordering Numbers on the Number Line

The number line is typically drawn with smaller numbers to the left and larger numbers to the right.

If we plot two numbers on the number line, 3 and 7 for example. We can see that

7 is to the right of 3

3 is to the left of 7

We can write these facts using the greater than and less than symbols.

Example: Plot the numbers 6 and 8 on the number line. Then write two mathematicalstatements about the numbers, with one statement using the greater than symboland one using the less than symbol.

Examples: Complete the following statements by writing < or > so that each statement istrue.

4 _____ 8 6000 _____ 5000

75 _____ 57 0 _____ 4

‘ Practice Problems for section 1.4‘ HW 1C


1.4 Practice

Round each number to the indicated place.

1. 89 to the nearest ten 2. 412 to the nearest hundred

3. 1,283 to the nearest ten 4. 6,120 to the nearest hundred

5. 48,973 to the nearest hundred 6. 26,099 to the nearest thousand

7. 34,500 to the nearest thousand 8. 149,992 to the nearest ten thousand

9. 34,499,238 to the nearest million

Estimate the answers by rounding to the indicated place. Then, check the reasonableness of theestimate by working the problem.

The nearest ten:

10. 11.

The nearest hundred:

12. 13.


Estimate the answers by rounding to the indicated place. Then, check the reasonableness of theestimate by working the problem.

The nearest thousand:

14. 15.

16. Complete the following statements with greater than (>) or less than (<) signs.

200 ___ 70 19 ___ 56 900 ___ 899

17. Michael went to a bookstore and bought these items: Science textbook, $39.88; Math textbook,$45.99; History book, $19.98; Gift card, $10.00. Estimate the total cost by rounding the numbersto the nearest dollar.

18. Lisa went to the grocery store and bought these items: Milk, $3.59; Eggs, $2.79; Bread, $3.49,Cookies, $3.09. Estimate the total cost by rounding the numbers to the nearest dollar.

19. A number rounded to the nearest hundred is 500. What is the largest possible number it couldbe? What is the smallest possible number it could be?

20. A certain number rounded to the nearest ten is 800. What is the largest possible number itcould be? What is the smallest possible number it could be?


1.5 Multiplying Whole Numbers

Multiplication can be thought of as repeated addition.

The × symbol is used to indicate multiplication. However, as we get into algebra, we use variables,and the letter x is commonly used as a variable. To avoid confusion, the × symbol is not often usedto represent multiplication. Instead, a dot is used:

When we multiply, as in the example above, 5 and 3 are called the ___________________

and 15 is called the ______________________.

The Multiplication Table, or Basic Multiplication Facts.

0 1 2 3 4

0 0 0 0 0 0

1 0 1 2 3 4

2 0 2 4 6 8

3 0 3 6 9 12

4 0 4 8 12 16

You should know the multiplication table out through at least 10. These should be memorized, andyou should be able to recall and use these facts quickly.

Example: Use multiplication to find the number of parking spaces in the parking area shown here.


Multiplying by using the Distributive Property

Sometimes relatively large numbers can be multiplied quickly and easily without a calculator.Knowing the distributive property is the key.

Example:

Example:

Multiplication using the Short Method

You should already be familiar with the procedure for multiplying large numbers. Just as theprocedures for addition, it is based on placed value and involved carrying.

Example:

Example:

Multiplication with 0 and 1

Any number multiplied by 1 is simply that same number.

= __________

= ___________

This concept can be stated with a variable:


Any number multiplied by 0 is simply 0.

= _____

= _____

= _____

= _____

= _____

This concept can also be stated with a variable:

Multiplying with Units

If we multiply a number that has units, the untis remain in the answer.

Example: = ____________

= ____________

Multiplying numbers with two or more digits

We have a procedure for multiplying large numbers.

Examples:


Sometimes a zero in a number will allow us to speed up the process:

Example:

A repeated digit can also allow us to speed up the process:

Example

Examples:

Application

Multiplication, of course, shows up in many places in the real world.

Example: Suppose we own a car rental company. We have 47 cars in the lot. We need toput 14 gallons of gas in each car. How much gas will we need?


Multiplying with the number 10

Because we have a base 10 number system (ten digits), multiplication with the number 10 isparticularly easy. Multiplying a number by 10, by 100, by 1000, or by any other power of 10 issimply a matter of adding zeros onto the end of the number.

= ____________

= ____________

= ____________

Our procedure for multiplying larger numbers can also be simplified when working with any numberthat ends in 0.

Examples:

Try these: 378 x 4000 420 x 9000 270 x 5200


Multiplication and Estimating

Sometimes we don’t need exact numbers. Instead, an estimate may be sufficient.

Example: The campground has 215 campsites. Each campsite needs about 4¼ gallons ofwater per day. What is the total daily water usage for the campground?

Note that in this case it makes sense to round and use estimated values. The exactnumbers aren’t really meaningful. The campground might not be 100% occupied, and evenif it was, every campsite would not necessarily be using exactly 4¼ gallons of water eachday. “Exact” numbers would necessarily be any better than rounded numbers in this case.

Note also that we can get large errors at times if both factors round up or both round down asignificant amount.

A Bad Example: There will be 150 people at the picnic. Each person will eat about 2½ hotdogs. How many hot dogs do we need to buy to get ready for the picnic?

The answer we get from estimating, in this case, is pretty far from the actual result of 375from multiplying the numbers without estimating. This is because both factors rounded up,and did so significantly.

A Bad Example: There are 143 kids at the summer camp. We will buy each one a t-shirt for$13.50. Round the numbers to get a quick estimate of the total cost.

Again, our answer is pretty far from the actual calculated value of $1930.50. The point isthat we have to be careful when rounding and estimating. We should be aware of thepossibility of large error when both numbers round in the same direction, and when therounding is significant.


Multiplication using the Associative and Commutative Properties

When multiplying, the ordering and the grouping does not affect the answer.

We can sometimes use these concepts to multiply numbers more quickly and easily.

Examples:

Try these:

Area

Multiplication is useful when we need to calculate area. Area is used to measure the amount ofsurface.

Example:

The area of a table top tells us how much useful surface there is on the top of the table.

Area is typically measured in square units.


Square inches, , would be used to measure relatively small objects, such as a piece of paper.

Fabric is typically measured in square yards, . In the metric system, a square meter, , is

just a little bigger than a square yard. The area of a large city or lake might be measure in square

kilometers, , or square miles, . The city of Atlanta is a little over 130 square miles.

Think about the top of a table that is 2 feet wide and 4 feet long.

In the diagram, we can count the square feet. There are 8.

We can get the same result by multiplying: . Note the units on the factors and

the answer.

Example: A driveway is 12 feet wide and 70 feet long. What is the area?

In general, the area of a rectangular region will always be found by the equation

This is a formula, which can be written concisely using variables:

If the rectangle is a square, we can write:


Areas of odd-shaped figures

Example: Find the area of the figure shown

Three ways to solve this:

Example: Find the area of the figure shown

Application

You are buying tiles to tile your kitchen floor, shown in the diagram. The tiles cost $7.60per square foot. How much will the tiles for the project cost.

‘ Practice Problems for section 1.5‘ HW 1D


1.5 Practice

1. If , we call 3 and 4 the ______________ of 12, and we call 12

the _______________ of 3 and 4.

2. Find the number of one square foot tiles in a floor that is 13 feet long and 8 feet wide.

Multiply.

2. 3. 4. 5.

6. Name each property that is illustrated.

7. A painter can paint 740 square feet per hour. How many square feet can he paint in sixhours?

8. Katerina decides to buy a motorcycle. She pays $4,500 as a down payment, plus shemakes monthly payments of $141 dollars per month for three years. What is the total costof the motorcycle?


9. You buy the New York Yankees baseball team and need to hire a new staff. The tableshows how much each person will need to paid.

Job Hourly Cost

Bookkeeper $40

Secretary $42

Vice-President $51

Accountant $49

Custodian $30

For a game day, you will need 17 custodians and one person for each of the otherpositions. You will need everyone there for nine hours. What is the total cost of your stafffor a game day?

Estimate by rounding each factor to the nearest hundred and then multiplying.

10. 11.

Multiply and write your answer with the proper units.

12. 13.

14. True or false?

15. True or false?


16. Find the area of the figure shown.

17. Find the area of the figure shown

18. Find the area of the figure shown.


1.6 Division

Division is most easily understood with an example.

Example: 24 marbles are divided evenly among 4 children.

We can see that each child will get 6 marbles.

Notation:

Division can be thought of as repeated subtraction.

We subtract 4 a total of 6 times, so 24 ÷ 4 = 6.

In the above example, 24 is the ________________, 4 is the ________________,

and 6 is the ________________.

Units

When dividing a number that has units (a denominate number) by another number, the unitsremain.

Example: 600 ft ÷ 12 =

If both numbers have units, then the answer has units accordingly.

Example: A car drives 100 miles in 2 hours. We can find the average speed by dividing thedistance by the time.


Recognizing words that indicate division

In addition to the obvious mathematical terms, some other words are commonly used to indicatedivision:

divide, divided evenly, split evenly, how much . . . each, how many . . . each

Example: 28 tons of rice is divided among 4 trucks. How much rich goes into each truck?

Checking division with multiplication

The answer to a division problem can be checked by multiplying.

63 ÷ 7 =

Dividing and leaving a remainder

Numbers are not always exactly divisible by the divisor. For example, 22 ÷ 5.

We have a remainder, a part that is left over after all the repeated subtraction.

When we check our division by multiplying, we have to add the remainder back in.

Example: 43 ÷ 8


Division and Zero

0 divided by any number is simply 0.

0 ÷ 8 = ________

0 ÷ 1000 = ________

0 ÷ 2,834,296,401 = ________

Division by 0 is undefined.

5 ÷ 0 = ________________

Long Division

The process of long division should also already be familiar to you. Right now we will look at howthe process works as a shortcut for repeated subtraction.

Example:


Example:

Long division with a two digit divisor

Example:

Example:


Division, Estimation, and Rounding

People rarely do long division by hand any more because the process is cumbersome, timeconsuming, and error prone. These days, calculators are readily available and are generallypreferred.

Estimating answers to division problems, however, is much easier and faster and can be veryuseful. We can round numbers so that they are more easily divided and get an estimate of ananswer that will hopefully be reasonably close.

Example: A flight from New York to London travels 3471 miles. A typical commercial airlinertravels at about 530 miles per hour. How many hours should the trip take?

When we divide 3000 by 500, we get 6, which is reasonably close to the value of6.55 that would result from dividing the numbers.

Note once again that estimated answers are reasonable in this case. The “exact” answerswould not really be exact. Different planes travel at different speeds. There are differentroutes that have different distances. Wind affects the speed of the aircraft. To estimatethe answer instead of trying to compute it exactly is a very reasonable approach.

Note also that the usual warnings about errors arising from rounding also apply in this case.

Division on a Calculator

Try these division problems on your calculator:

357 ÷ 17 = ____________

357 ÷ 0 = ____________

0 ÷ 20 = ____________

50 ÷ 7 = _____________

Note that when there is a remainder, the calculator expresses it as a decimal.

‘ Practice Problems for section 1.6‘ HW 1E, HW 1F


1.6 Practice

1. In the equation 144 ÷ 16 = 9, the number 144 is the _______________, the number

16 is the _______________ , and the number 9 is the _______________ .

2. Jim is driving 9 different people to various places, one person at a time. Each of the 9people spends an equal amount of time in the car. If the total time spent on the drives is45 hours, how much time is each person in the car.

Divide. Write your answers with the correct units.

3. 21 days ÷ 3

4. 5280 feet ÷ 15

5. 24 miles ÷ 6 hours

6. 108 dollars ÷ 9

Divide using long division.

7. 48 ÷ 7 8. 59 ÷ 6

9 79 ÷ 4 10. 0 ÷ 12


Divide using long division.

11. 12.

13. 14.

15. John jogged around a square city park, finishing exactly where he started. If he jogged atotal of 3600 feet, how long was each side of the park?

16. In his race car, Dave must complete 7 laps in 11 minutes 19 seconds. How much timedoes he have for each lap?

17. Eighty major league baseball players are contributing to build a new wing for a hospital.The project will cost 1.4 million dollars. How much should each player contribute?


18. At the end of the year, Susan’s company has $10,989 left over. She decides to distributethe money equally to her 37 employees. How much will each employee receive?

19. The fuel tank on Jake’s boat holds 104 gallons of fuel. If he drives 2392 miles on a fulltank, what is the gas mileage of his boat in miles per gallon?

Estimate the answers to the following problems by rounding the numbers first and then dividing.

20. 901 ÷ 18 21. 7192 ÷ 53


1.7 Exponents and Order of Operations

Exponents

Multiplication can be thought of repeated addition. But what if we have repeated multiplication?Exponents are a way to express repeated multiplication.

can be written as ______

In the above expression, the 5 is called the ________________

and the 4 is called the ____________________________________________

This is read as “five to the power of four" or “five to the fourth power.”

Examples:

Write the following in exponential notation:

= ________ (7)(7)(7)(7) = ________

Evaluate the following:

= ________ = ________ = ________

Remember that exponents are not the same thing as ordinary multiplication. is not the same

thing as

= ________ = ________

Powers of 0 and 1

Any number raised to the power of 1 is simply that same number.

= ________ = ________

This concept can be stated with a variable:

This should make sense. The exponent tells us the number of times that the base is used as afactor. If it is used just one time, then we simply have that number.


We also need to know what happens when the exponent is 0. The result is not necessarilyintuitive, but mathematicians have defined any base to the power of 0 is equal to 1.

= ________ = ________

or, the concept can be stated in general using a variable:

Again, this doesn’t necessarily make intuitive sense, but it is true by definition.

Powers of 10

Powers of 10 are important because we have a base 10 number system.

=

=

=

=

=

. . .

=

One particular power of 10 has a special name

Order of Operations

Consider the following expression:

12 - 6 ÷ 2

How do we evaluate this? Should we do the subtraction first, or the division?

( 12 - 6 ) ÷ 2 or 12 - ( 6 ÷ 2 )


The rules for the order of operations:

Do multiplication and division, left to right, and then addition and subtraction, left to right.

Examples:

=

=

=

÷ 3 =

Order of Operations with Exponents

There is an additional rule when exponents are involved:

Exponents are evaluated before multiplication and division.

We need this rule because of situations like this:

Without a known rule, we have an ambiguous expression. Is this

or

The Order of Operations:

1. Evaluate exponents first2. Then do multiplication and division, left to right3. Then do addition and subtraction, left to right

Examples


Parentheses

Operations within parentheses are done first. For example,

Within parentheses, we obey the order of operations.

Examples:

÷ 2

÷ 2 )

÷ 2

20 ÷

40 ÷

Order of Operations on a Calculator

Most calculators today are capable of dealing with large expressions that contain many parts. Mostalso understand the correct order of operations.

Enter on your calculator. Examine the result. If it is 17, then the calculator is performing

the order of operations correctly (multiplication before addition).

Most calculators also have parentheses.

Enter the following on your calculator. Examine the results.

‘ Practice Problems for section 1.7‘ HW 1G, HW 1H


1.7 Practice

Evaluate each of the following.

1. 2.

3. 4.

5. 6.

7. 8.

9. 10.

11. 12.

12. 13. 9 ÷ 3 + 8

14. 18 - 16 ÷ 2 15. (18 - 16) ÷ 2

16. 14 + 6 ÷ 3 17. 60 ÷ 4 × 5


18. 42 ÷ 7 - 15 ÷ 5 19.

20. 21.

22. 23. 60 ÷ (12 - 7 + 1)

24. Are the numbers 1, 2, and 3 a Pythagorean Triple? Why or why not?

25. Are the numbers 7, 24, and 25 a Pythagorean Triple? Why or why not?

26. Are the numbers 1, 1, and a Pythagorean Triple? Why or why not?

27. Is equal to ?


1.8 An Introduction to Equations

An expression is either a number or a collection of numbers together with arithmetic operators suchas +, -, ×, and ÷.

Examples: ÷ 2

An equation will have an equals sign in it.

Examples:

An equation is a mathematical statement. It states something. That is, it tells us something.

Just like a statement or sentence in English, a mathematical statement can be either true or false.

Examples:

Translating Sentences into Equations

Math is used to solve problems. Many problems in the real world can be solved mathematically.Usually, we will have a statement of the problem in English, and we need to translate the Englishstatement into a mathematical equation.

Right now we will practice translating very simply English statements into simple equations. Laterwe will translate more complicated, and more useful, statements.

Examples: Translate the following sentences into mathematical equations.

Five squared minus ten is fifteen.

Twenty four divided by eight is equal to twelve minus nine.

Six less than nineteen is ten plus three.


1.8 Practice

1. Label each of the following as an equation or an expression.

a)

b)

c)

d)

2. Determine whether each of the following statement is true or false.

a)

b)

c)

d)

e)

3. Write each of the following as a mathematical expression or equation..

a) Four times three squared

b) Six less than twenty-five.

c) Five squared plus three is twenty eight.

d) Seventeen is equivalent to the cube of three subtracted from the product of four andeleven.

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