Overview
How does math play a role in everyday life? You can use math in many ways: to find an address, make change, or count calories. You can also use it to balance your checkbook, measure ingredients, or take medication. Clearly, math skills are important. In particular, this course teaches you how to add, subtract, multiply, and divide whole numbers and fractions. Moreover, throughout the course, you will apply these topics to real-life situations. Therefore, the goal of this course is to help you develop basic math skills you need for daily living and further studies.
This course includes twelve lessons. Lesson 1 introduces whole numbers. Lessons 2 through 5 explain how to add, subtract, multiply, and divide whole numbers. Lesson 6 describes certain relationships among whole numbers. Lessons 7 and 8 introduce fractions and mixed numbers. Lessons 9 and 10 explain how to multiply and divide fractions. Finally, Lessons 11 and 12 describe how to add and subtract fractions.
This course also includes some special features. “Math Tips” supplement key information in each lesson. “Math in Real Life” sections apply math skills to everyday activities.
For clarity, the lessons are written from the point of view of a student using either print or a braillewriter. Keep this in mind if you are using a slate and stylus to complete your work for this course. For example, the lessons tell you to write numbers from left to right. Slate and stylus users, however, will write numbers in reverse, from right to left.
The practice exercises in each lesson are for your personal development only. Do not send your responses to your Hadley instructor. Rather, check your comprehension by comparing your answers with those provided. You can always contact your instructor, however, to clarify concepts. You are also required to submit twelve assignments, one at the end of each lesson. These assignments enable your instructor to measure your understanding of the material presented in the course.
If you are ready to explore practical math, begin Lesson1: What Are Whole Numbers?
Overview v
Lesson 1: What Are Whole Numbers?
Which one comes first? Which one comes last? Many things form an order: calendar days, check numbers, and street addresses, to name a few. You can also compare items: Which soap brand has the lowest price? Which player scored the most points? Lesson 1 introduces some characteristics and basic uses of whole numbers. Understanding these numbers helps you develop the math skills you need for daily living and further studies.
Objectives
After completing this lesson, you will be able to
a. define whole numbers
b. indicate place value
c. change whole numbers from words to digits and vice versa
d. tell if a whole number is greater than, less than, or equal to another whole number
e. put whole numbers in order from least to greatest and vice versa
f. round a whole number to the nearest ten, hundred, or thousand
Key Terms
The following terms appear in this lesson. Familiarize yourself with their meanings so you can use them in your course work.
digits: symbols that represent numbers: 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9
equal sign (=): the symbol that means “equivalent to,” or “having the same value”
even numbers: whole numbers that have 0, 2, 4, 6,
or 8 as the last digit
greater-than sign (>): the symbol that means “greater than,” or “having larger value”
less-than sign (<): the symbol that means “less than,” or “having smaller value”
odd numbers: whole numbers that have 1, 3, 5, 7, or 9as the last digit
place value: the value of a digit, which varies depending on the digit's place in a number
rename: to indicate the place value of each digit in a number
round: to change a number to the nearest place value, as specified (i.e., to the nearest ten, hundred, thousand, or so on)
whole numbers: digits starting with zero and going on indefinitely, with each one being one more than the previous one; used to count or tell how many
About Whole Numbers
This section defines whole numbers. It also identifies the difference between even and odd numbers.
People use whole numbers to count, or to tell how many. To write whole numbers, you use digits, which are the numeric symbols 0, 1, 2, 3, 4, 5, 6, 7, 8, and9. Whole numbers start with 0 and continue into the millions, billions, and beyond. The following is a list of whole numbers from 0 to 50:
0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50
A single- or one-digit number has only one digit in it. Some examples are 1, 3, and 8. Double- or two-digit numbers have two digits in them. For example, 12, 25, and 78 are all two-digit numbers.
The whole numbers that end in 0, 2, 4, 6, or 8 are called even numbers. For example, the number 72 is even because it ends in 2. The whole numbers that end in 1, 3, 5, 7, or 9 are called odd numbers. For example, the number 83 is odd because it ends in 3. In order, whole numbers form a pattern: an odd number follows each even number and vice versa.
You also can tell whether larger numbers are odd or even. For example, 795 is odd number because it ends in 5. The number 824 is even because it ends in 4.
Math TipThe last digit on the right helps you determine if a number is odd or even.
Example 1
How would you write the whole numbers from 10 to20?
Remember, each whole number is one more than the previous number. So the whole numbers from 10 to 20 are 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, and20.
Example 2
Suppose you have the following list of numbers:
1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, and 15. Which of these numbers are odd numbers?
Odd numbers end in 1, 3, 5, 7, or 9. The odd numbers in this list are 1, 3, 5, 7, 9, 11, 13, and 15.
Example 3
Consider this list of whole numbers:
30, 31, 32, 33, 34, 35, 36, 37, 38, 39, and 40. Which are the even numbers?
Even numbers end in 0, 2, 4, 6, and 8. The even numbers in this list are 30, 32, 34, 36, 38, and 40.
Math in Real Life
You use whole numbers in your everyday life. For example, you probably write or tell people your address. Most street addresses have a number and a street name, such as 421 Elm Street.
In many cities, addresses with odd numbers are on one side of the street, and addresses with even numbers are on the other side. For example, the number 701 is odd, and the number 728 is even. So 701 Pine Street and 728 Pine Street usually would be on opposite sides of the street.
Consider the following real-life example:
Jeffrey is on his summer vacation, and this is his first time visiting relatives in another city. He’s planning how to travel to different homes. In this city, odd-numbered addresses are on the east side of the street, and even ones are on the west.
What side of the street is each of the following addresses on?
· Uncle Ralph and Aunt Clothilde
510 Central Street
Their house is on the west side: 510 is an even number.
· Cousin Tiffany
201 Skokie Boulevard
Their house is on the east side: 201 is an odd number.
Practice Exercise 1–1
1. Write the whole numbers from 20 to 30.
2. Write the even numbers from 30 to 40.
3. Write the odd numbers from 40 to 50.
The following addresses are in a city where even-numbered addresses are on one side of the street, and odd-numbered addresses are on the other. For each pair, tell whether or not the addresses are on the same side of the street and explain your answer.
4. 910 Avenue A and 916 Avenue A
5. 566 Ohio Street and 567 Ohio Street
6. Li moves from 129 Oak Street to 144 Oak Street. Is she still on the same side of the street?
7. Tony lives at 801 Ocean Drive. Tina lives at 710 Ocean Drive. Do they live on the same side of the street?
Answers 1–1
1. The whole numbers from 20 to 30 are 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, and 30.
2. The even numbers from 30 to 40 are 30, 32, 34, 36, 38, and 40.
3. The odd numbers from 40 to 50 are 41, 43, 45, 47, and 49.
4. They are on the same side of the street. The numbers 910 and 916 are both even.
5. They are on opposite sides of the street. The number 566 is even and 567 is odd.
6. No, Li moved across the street. The number 129 is odd and 144 is even.
7. No, Tony and Tina live on different sides of Ocean Drive. The number 801 is odd, and the number 710 is even.
This section defined whole numbers. It also discussed the difference between even and odd numbers. Why not use this information when you try to locate addresses?
Place Value
Numbers consist of digits. Place value refers to the value of a digit, which varies depending on its place in a number. You may find place-value charts helpful. When solving math problems, use place-value columns.
Place-Value Charts
When determining the place value of digits in a number, work from right to left. The places in a number from right to left are ones, tens, hundreds, thousands, ten thousands, hundred thousands, millions, ten millions, hundred millions, and so on. Starting from the right, a comma comes after every third digit in a number.
Math TipTo find the place value of a number, start with the ones place and work toward the left.
Consider the following numbers. In each of them, the 6stands for a different value, as noted:
· 67
The 6 means 6 tens.
· 605
The 6 means 6 hundreds.
· 26
The 6 means 6 ones.
A simple chart can help illustrate the place value of each digit in these numbers. A place-value chart is made up of columns, each for a specific place value. Going from right to left, the chart has a column for ones, tens, hundreds, and so forth.
Hundreds / Tens / Ones6 / 7
6 / 0 / 5
2 / 6
If asked the value of a particular digit, you can use a place-value chart to find the answer. For example, what is the value of the digit 2 in the number 26? In this case, the digit 2 is in the tens column, or the second column from the right. Therefore, the digit 2 means 2tens.
You can also use this chart to rename an entire number. When you rename a number, you are saying the same number but in a different way by showing the place value of each digit:
· 67 = 6 tens + 7 ones
· 605 = 6 hundreds + 5 ones
(Note that you can skip, or not mention, a zero when renaming a number.)
· 26 = 2 tens + 6 ones
Math Tip
Think of the word rename. The prefix re- means “again.” For example, the words redo, rebuild, and refill mean “do again,” “build again,” and “fill again.” When you rename a number, you’re naming it again in a way that makes the number easier to use. Whether you call the number 67 or 6 tens plus 7 ones, you are talking about the same number!
Example
Consider this number:
4,328,765
You can use a place-value chart to show the value of its digits:
Millions / HundredThousands / Ten
Thousands / Thousands / Hundreds / Tens / Ones
4, / 3 / 2 / 8, / 7 / 6 / 5
If someone asks you the value of a particular digit of this number, you can use the chart to find the answer. For example, what is the value of the digit 8 in the number 4,328,765? The digit 8 is in the thousands column, or the fourth column from the right; therefore, it means 8 thousands.
Using this chart, you can also rename the entire number to show the place value of each digit, as follows:
4,328,765 = 4 millions + 3 hundred thousands + 2ten thousands + 8 thousands + 7 hundreds + 6tens + 5 ones
When solving place-value problems, you may want to make charts like the ones shown here. Some people prefer to memorize the order of the place values (ones, tens, hundreds, thousands, etc.). Use the method that you feel most comfortable with.
Math TipStarting from the right, a comma comes after every third digit in a number.
Place-Value Columns
When you solve math problems, consider the place value of the numbers you use. Often, you need to put numbers in columns according to place value. To do so, vertically line up the ones with the ones, the tens with the tens, and so forth. Consider how you line up these three numbers vertically:
67
605
26
All the ones digits are lined up one on top of the other, and the same is true for the other place-value digits. Remember place-value columns when you solve problems later in this course.
Example
Consider how you line up these numbers vertically:
10,245
5,627
831
All the ones digits are lined up one on top of the other, and the same is true for the tens, hundreds, thousands, and ten thousands. In this case, the 5 of 10,245 and the 7of 5,627 and the 1 of 831 all are in the ones place. So they all line up, one on top of the other, in the ones column.
Math in Real Life
You may encounter place values on the job, at school, at the store, and elsewhere. Consider the following examples:
· Tia needs a new washing machine. A machine costs 435 British pounds. How many hundreds are in the cost?
Rename the amount: 435 equals 4 hundreds plus 3tens and 5 ones. So 4 hundreds are in the cost.
· The Carlson Corporation has an annual budget of $3,270,500. How many millions are in the budget?
Rename the amount: 3,270,500 equals 3 millions + 2hundred thousands + 7 ten thousands + 5hundreds. Therefore, 3 millions are in the budget.