pre-activity adding and subtracting whole numbers …10 this result matches the number in the...

43

PRE-ACTIVITY

PREPARATION

Max began his day off with $23 cash in his wallet. At the bank, he deposited his $248 paycheck and transferred $550 from his college savings account into his checking account. He drove to campus and paid the business offi ce his $470 tuition balance and the bookstore $283 for textbooks and supplies. How much money remained to buy both gas and groceries on his way home? To calculate the result of these transactions, he used both addition and subtraction.

Computing totals and fi nding differences are important basic skills to master because professionals are presumed competent in both processes; and the practicality of being able to add and subtract when there is no calculator at hand cannot be ignored.

Furthermore, a review of the basic properties of addition is an important starting point to learning the “whys” of the math that you do and may take for granted in your everyday calculations.

• Master the addition of whole numbers.

• Master the subtraction of whole numbers.

Adding and Subtracting Whole Numbers

NEW TERMS TO LEARN

addends operation

borrow/borrowing regrouping

carry/carrying subtrahend

difference sum

estimate total

estimation validate

minuend validation

PREVIOUSLY USED

methodology

place value

rounding

LLEARNINGEARNING OOBJECTIVESBJECTIVES

TTERMINOLOGYERMINOLOGY

Section 1.3

44 Chapter 1 — Whole Numbers

BBUILDING UILDING MMATHEMATICAL ATHEMATICAL LLANGUAGEANGUAGE

Addition, subtraction, multiplication, and division are the four basic mathematical operations. Each operation follows a step-by-step process (a methodology), based upon set mathematical principles, to calculate a result from the numbers you are given.

Addition

The numbers you add are called addends. The answer is called the total or the sum.

Consider adding the whole number 7 to the whole number 5 for a sum of 12.Symbolically, the addition might be written 5 + 7 or 5

Both may be read in any of the following ways: “fi ve plus seven,” “seven added to fi ve,” “the sum of fi ve and seven,” “seven more than fi ve,” “the total of fi ve and seven,” “add seven to fi ve,” and “increase fi ve by seven”

The following characteristics of addition, which will always be true, are known as the Mathematical Properties of Addition. A simple example is given for each.

+7

Commutative Property of Addition

Two numbers can be added in either order without affecting their sum.

Example: 4 + 5 = 5 + 4 = 9

Identity Property of Addition

The sum of any number and zero (0) is that number.

Example: 5 + 0 = 5

Associative Property of Addition

When adding three numbers, the numbers can be grouped in different ways without affecting their sum.

Example: (2 + 3) + 4 = 2 + (3 + 4)

5 + 4 = 2 + 7 = 9

} }

Mathematical Properties of Addition

12

addendaddendsum

45Section 1.3 — Adding and Subtracting Whole Numbers

Subtraction

The operation of subtraction involves only two numbers. The number you subtract is the subtrahend. The number you subtract it from is the minuend. The resulting answer is called the difference.

Symbolically, subtraction may be written horizontally, in which case the minuend is the fi rst number and the subtrahend is second. In 8 – 3, the minuend is 8; the subtrahend is 3.

When written vertically, the minuend is the top number and the subtrahend is the bottom number as in

Both may be read in any of the following ways:“eight minus three,” “three subtracted from eight,” “the difference of eight and three,” “three less than eight,” “three from eight,” “subtract three from eight,” or “decrease eight by three”

Since subtraction is addition’s opposite operation, your knowledge of the basic subtraction facts derives from your knowledge of the basic addition facts. For example, you know that 12 – 5 = 7 because you know that 7 + 5 = 12.

The table below presents the basic addition facts you must know confi dently for profi ciency, speed, and accuracy in addition. The box where two addends intersect gives their sum.

+ 0 1 2 3 4 5 6 7 8 9

0 0 1 2 3 4 5 6 7 8 9

1 1 2 3 4 5 6 7 8 9 10

2 2 3 4 5 6 7 8 9 10 11

3 3 4 5 6 7 8 9 10 11 12

4 4 5 6 7 8 9 10 11 12 13

5 5 6 7 8 9 10 11 12 13 14

6 6 7 8 9 10 11 12 13 14 15

7 7 8 9 10 11 12 13 14 15 16

8 8 9 10 11 12 13 14 15 16 17

9 9 10 11 12 13 14 15 16 17 18

addend

addend

8

3

5

−minuendsubtrahend

difference


Before moving on to the methodologies for adding and subtracting whole numbers, you should familiarize yourself with a process that this book will refer to as validation and with the practice of estimation.

The Power of Validation

Validation is a name for the process of “checking your work” to confi rm that your fi nal answer is correct. In this course, you will learn a variety of methodologies and techniques to solve basic math problems without the use of a calculator. As you will discover, the practice of validation is particularly useful for this approach.

Once you have mastered a process and can apply it appropriately, any mistakes you happen to make in your calculations will most often be due to basic math fact errors, careless or sloppy notation in your process, or occasional copying errors when you set up the original problem. Perhaps you are a student whose area for improvement is to minimize those types of errors in your work; perhaps you are a student who, once you have mastered a process, considers accuracy as your strength. In either case, validation has proven itself one of the most powerful tools you can use to improve your performance in mathematics. Therefore, the methodologies and techniques in this book will include a fi nal step or present general guidelines on how to validate your answer.

You might think that merely doing a problem over again is suffi cient. However, the most effective validation technique is one that uses a different process to verify the accuracy of the answer as well as link the result to the original problem in some manner. For example,

Solve: 10 + 28

Process: 10

+ 28

38 Answer

Validation: 38

– 28

10 This result matches the number in the original problem.

When computing the answer to a basic addition, subtraction, multiplication, or division problem, using the opposite operation is the preferred validation technique. As in the above example, use subtraction to validate addition. To validate subtraction, use addition. Check your multiplication answers with division; and use division to validate multiplication.

The remainder of this book will introduce additional validation techniques for the methodologies that build upon these four basic operations.


How Estimation Can Help

To build confi dence in your answers, you can also take advantage of another mathematical practice called estimation.

Before you even begin, you can often approximate or predict the answer to a given math problem by simplifying its numbers to do an easy mental calculation. This technique of estimation gives you a good idea of what the answer should look like and assures that your answer is reasonable. In general, estimation involves rounding the numbers to simplify them.

Think of how many times you may have already used estimation in practical situations:

• estimating what a car repair or a new set of tires will cost

• estimating how much time to schedule for a task (such as study time for math class preparation)

• estimating how many cans of paint to buy for the exterior of your house

• estimating gas consumption and its cost for a long road trip

• estimating how much each person should pay to equally share the tab for a restaurant meal

• estimating the cost of the items in your grocery cart

There are instances when you might choose to overestimate “to be on the safe side.” For example, in the grocery store you might round each price up to the nearest fi fty cents or dollar to save yourself the embarrassment at the checkout lane of overspending the $30 in your wallet when the actual fi gure is computed.

Without the expressed intention to overestimate, there are many ways to estimate answers to given math problems. Some give closer approximations than others.

One method is to round each number to its largest place value to get an estimated answer.

For example, estimate the answer to 473 + 214 + 185.

After rounding each addend to its hundreds place, the estimated answer is simply 500 + 200 + 200 or 900.

The actual answer is 872, and 872 is close enough to the estimated answer 900 to be reasonable.

To determine that 872 is precisely correct, you would validate your answer by subtraction.

continued on the next page


It is important to re-emphasize that estimation gives only an assurance of reasonableness. Validation is the process that assures accuracy. Had your answer to the previous problem been 1672, it would not have been close enough to 900 to be reasonable, an indication that you most likely made an error in your calculation in the hundreds column. On the other hand, if your answer was 882, which is reasonably close to 900, you would not have detected your error until you validated the answer and found it to be in the tens column.

Consider another addition problem: 54,723 + 4,196 + 803

Following this same method of rounding each number to its largest place value, the estimated answer would be 50,000 + 4,000 + 800 or 54,800. The actual sum 59,722 might be described as “in the ballpark” or reasonable. At least the ten-thousands place is correct.

Again, estimation is just that—an approximation that gives some sense of what the actual answer ought to be. Rounding each addend to its nearest thousand might have made a better estimate:

54,000 + 4,000 + 1,000, or 59,000. An important feature of estimation, remember, is to simplify the problem to make it relatively easy to compute mentally.

Unless specifi cally instructed as to the method to follow, it will be your decision on when and how to estimate an answer effectively and effi ciently.

Estimate the answer to Example 1 in the following Methodology for Adding Whole Numbers: 8148 + 709 + 3896

You have options:

Round each number to its largest place value:

8000 + 700 + 4000 = 12,700

or Round each number to the nearest thousand:

8000 + 1000 + 4000 = 13,000

or Round each number to its nearest hundred:

8100 + 700 + 3900 = 12,700 (although this option is not quite as simple to compute mentally)

Estimate the answer to Example 2 in the following methodology. (Once solved, come back and compare your answer to your estimated answer.)


MMETHODOLOGIESETHODOLOGIES

Adding Whole Numbers

Example 1: Add: 8148 + 709 + 3896 Example 2: Add: 9817 + 5403 + 296►►►►

Steps in the Methodology Example 1 Example 2

Step 1

Set up the problem.

Right align the numbers in columns according to place values.

Note: The order in which you list the whole numbers is your choice.

Step 2

Add each column.

Add each column, starting with the ones, then the tens, then the hundreds, and so on.

If the sum of the place value column is greater than 9, carry the tens digit of the sum to the next higher place value column.

Step 3

Present the answer.

Present your answer. 12,753

Step 4

Validate your answer.

Validate your answer by using the opposite operation—subtraction.

Start with your answer and, in succession, subtract the original addends. When all but one addend have been subtracted, the resulting number should match the remaining addend.

Use the numbers from the original problem statement, as this will help to detect transcription errors.

Validate by using subtraction twice. (See Methodology for Subtracting Whole Numbers.)

8,148 matches the remaining addend in the original problem.

? ? ?Why can you do this?

8148709

3896+

? ? ?Why do you do this?

8148709

3896

12753

1 1 2

+

1 2 7 5 33 8 9 68 8 5 7

0 1 6 4 11 1 1

−

88 5 770 9

8148

4 1

−

Try It!


? ? ?Why can you do Step 1?

The Commutative Property of Addition allows you to add a list of numbers in any order and the sum will always be the same.

The meaning behind the carrying process comes from an understanding of place values.

8148709

+38963

2In Example 1, the sum of the digits in the ones column is 23, meaning 23 ones or 2 tens and 3 ones. Write the 3 below the ones column. The 2 that you carry over to the tens column represents the 2 tens in the sum of the ones column.

When you add the tens column, you add 2 tens + 4 tens + 0 tens + 9 tens. The result is 15 tens (150), or 1 hundred and 5 tens. Write the 5 below the tens column and carry the 1 to the hundreds column.

28148709

+389653

1

The sum of the hundreds column is 17 hundreds (1700), or 1 thousand and 7 hundreds. Write the 7 under the hundreds column, and carry the 1 to the thousands column.2

8148709

+3896753

11

Finally, 1 thousand + 8 thousands + 3 thousands = 12 thousands, 2 in the thousands column and 1 in the ten-thousands column.

28148709

+389612753

11

You can estimate a difference in the same way you estimate a sum, by rounding the numbers appropriately. To estimate the answer to Example 1 in the following Methodology for Subtracting Whole Numbers, (5639 – 745), you have options:

Round each number to its largest place value: 6000 – 700, or 5300

or Since the second number is in the hundreds, round each to its nearest hundreds place: 5600 – 700, or 4900or Round each number to the nearest thousand:

6000 – 1000, or 5000

Estimate the answer to Example 2 in the space below. (Once solved, come back and compare your answer to your estimated answer.)

? ? ?

How Estimation Can Help in a Subtraction Problem

Why do you do Step 2?


? ? ? ?

? ? ? ?

Subtracting Whole Numbers

Example 1: Subtract: 5639 – 745 Example 2: Subtract: 7153 – 4237►►►►

How and why do you do Step 2?

Steps in the Methodology Example 1 Example 2

Step 1

Set up the problem.

Right align the numbers in columns according to place values, with the minuend as the top number.

Step 2

Subtract each column.

For each column, subtract the bottom digit (in the subtrahend) from the corresponding top digit (in the minuend).

Subtract column by column, working from the ones column to the left-most column.

If the digit in the top number is less than the digit in the bottom number, borrow from the next higher place value and subtract.

Step 3

Present the answer.

Present your answer. 4,894

Step 4

Validate your answer.

Validate your answer by using the opposite operation—addition.

Add the subtrahend (bottom or second number) to your answer. The result should match the minuend (top or fi rst number).

5639 matches the original fi rst number.

5639745−

How and why do you do this?

5 6 39745

4 8 94

4 5 11

−

4894745

5639

1 1

+

How does the borrowing process work and why can you borrow? As it was for addition, an understanding of place values is key. In fact, because of the following process, borrowing is sometimes referred to as regrouping place values.

Try It!

Special Case:

Borrowing when 0 is a digit in the top number (see pages 52 & 53, Models A & B)


5639–745

4

In Example 1, the ones column is straightforward. 9 – 5 = 4 in the ones column.

In the tens column, you cannot subtract 4 tens from 3 tens. However, you can borrow 1 hundred from the 6 hundreds in the next higher place value column. That leaves 5 hundreds in that column, so cross out the 6 and replace it with a 5. The 1 hundred that you borrowed must be renamed. It is, in fact, equal to 10 tens. Adding the 10 tens to the 3 tens already in the tens column results in 13 tens. Use a 1 in the tens column to keep track of the borrowing process. Notice that you have now regrouped the top number 5639 (5000 + 600 + 30 + 9) as 5000 + 500 + 130 + 9.

13 tens minus 4 tens equals 9 tens. Write the 9 under the tens column.

5 6 3 9–7 4 5

9 4

15

Moving to the left, you cannot subtract 7 hundreds from 5 hundreds, so borrow from the thousands column. The 1 thousand borrowed equals 10 hundreds. Make the proper notations for the regrouping. The 5 thousands are now 4 thousands and the 5 hundreds are now 15 hundreds.

Subtract 7 hundreds from 15 hundreds. Write the 8 under the hundreds column.

5 6 3 9–7 4 58 9 4

154 1

5 6 3 9–7 4 5

4 8 9 4

154 1

Moving to the thousands, 4 thousands minus 0 thousands equals 4 thousands.

MMODELSODELS

►►A Subtract 1,586 from 3,045.

30451586−

Step 1

3,000 – 2,000 = 1,000(or 3,000 – 1,600 = 1,400)Estimate:

Special Case: Borrowing when 0 is a Digit in the Top Number (minuend )


When you must borrow from the next higher place value and the digit in that place value is zero (0), borrow from the next higher place value(s), one place at a time, until you can get the top number into a form that will allow the necessary subtraction.

►►B

In the tens column, 3 is less than 8. You must borrow. However, there is a zero in the hundreds column. In order to borrow from the hundreds column, fi rst borrow from the thousands column.

1 thousand equals 10 hundreds. Cross out the 3, write a 2 in the thousands place, and write the 0 hundreds as 10 hundreds.

Next borrow 1 hundred from the 10 hundreds. Cross out the 10 and write a 9 in the hundreds place and write the 3 tens as 13 tens.

Continue the subtraction process. For the tens place, 13 – 8 = 5

for the hundreds place, 9 – 5 = 4

for the thousands place, 2 – 1 = 1

3 0 4 5–1 5 8 6

9

1312

313 0 4 5–1 5 8 61 4 5 9

12 19

Step 3 Answer: 1,459 It is reasonably close to the estimate.

Step 4 Validate:

matches the original number

14591586

3045

1 1 1

+

Subtract: 26,004 – 4,568 For this problem, perhaps the easiest mental estimate would be to round each number to its thousands place and subtract.

Estimate: 26,000 – 5,000 = 21,000.

For the ones column, borrow 1 from the 4 in the tens column, making 15 in the ones column, and 3 in the tens column.

15 – 6 = 9 as the digit in the ones column of the answer.

Step 2 3 0 4 5–1 5 8 6

9

13

Steps 1 & 2

Step 3 Answer: 21,436 It is reasonably close to the estimate.

Step 4 Validate:

matches the original number

2 6 0 0 4–4 5 6 8

2 1 4 3 6

5 19911

214364568

26 004

1 1 1

+

,


AADDRESSING DDRESSING CCOMMON OMMON EERRORSRRORS

15431

123−

Issue Incorrect Process Resolution Correct

Process Validation

Forgetting to add the carried digits when totaling a place value column

Use effective notation. Write down all the carried digits in the proper columns.

Incorrectly borrowing when zeros are in the minuend (top number) of a subtraction problem

Use complete notation to document borrowing as it proceeds one place value at a time.

Copying a problem incorrectly

Add: 34+98+31 Say the numbers when you write them.

Not using the original terms when validating

Incorrect validation of previous example:

The answer appears to be correct.

Always validate using the terms from the original problem, as this will help detect transcription errors.

Validation using the original terms:

No match. Check for transcription error.

See the previous issue for the correct addition.

See the previous issue for the correct validation.

8 6 7 2–1 3 9 37 2 7 9

1165

7 2 7 9–2 7 4 84 5 3 1

16

1 7 0 0 8– 5 4 2 91 1 6 8 9

6 1111 7 0 0 8– 5 4 2 91 1 5 7 9

6 19911

34

31

154

89+

1

1 3 2– 9 8

3 4

0 112

1 2 3– 9 8

2 5

0 111

1 2 3– 8 9

3 4

0 111

4

3 9 3

1

5 3 12 7 4 8

+1

8 6 7 2

1 1453127481393

7572

+

1 14 2 9

1

5 7 9+5

1 7 0 0 8

1 1

349831

163

1

+

16331

132−

15431

123−

15431

123−

+2 48

327

39

7

1 66 8

7 0 0

8

10017 4

8 9

67

318

hape c rr

e answs e

he pear

rrec


PPREPARATION REPARATION IINVENTORYNVENTORY

Before proceeding, you should have an understanding of each of the following:

the terminology and notation associated with adding and subtracting whole numbers

the mathematical property that gives you the fl exibility to choose the order in which to add numbers

the process and meaning of “carrying” in addition

the process and meaning of “borrowing” in subtraction

the validation of addition by successive subtractions

the validation of subtraction by addition

56

ACTIVITY

PPERFORMANCE ERFORMANCE CCRITERIARITERIA

• Adding any group of whole numbers – neatness of presentation – validation of the answer

• Subtracting any two whole numbers – neatness of presentation – validation of the answer

CCRITICAL RITICAL TTHINKING HINKING QQUESTIONSUESTIONS

1. What does it mean to “carry” from one column to another?

2. What does it mean to “borrow” from a higher place value column?

3. What techniques should you use to make sure that “carrying” and “borrowing” are done without making errors?

Adding and Subtracting Whole Numbers

Section 1.3


4. What does it mean to validate your answer?

5. What is the most effective way to validate a math computation (addition and subtraction)?

6. For validation of addition, what mathematical property allows you to subtract the addends in any order?

7. What is the procedure to follow when it is necessary to “borrow” and the digit in the next higher place value is zero?

8. How can the Associative Property of Addition be used to increase speed when adding a column of numbers?

continued on the next page


TTIPS FOR IPS FOR SSUCCESSUCCESS

• Use graph paper or lined paper turned sideways to help align place value columns accurately.

• To focus on the intended operation, always include the operation sign in your set-up of the problem.

• Show all of your work neatly and legibly with proper carrying and borrowing notation.

• Know confi dently all combinations of single digit numbers and their related subtraction facts; work to increase your profi ciency, speed, and accuracy.

• Always validate!

DDEMONSTRATE EMONSTRATE YYOUR OUR UUNDERSTANDINGNDERSTANDING

2. Perform the indicated operation in each of the following. Validate your answers.

Problem Worked Solution Validation

a) 71 + 34 + 306 + 43

1. Estimate the answers by rounding the numbers to their largest place values, before you add or subtract them.

a) 73 + 29 + 67 + 12 + 98

b) 890,035 – 456,180

c) 8349 + 3901 + 1982 + 4110

d) 50,123 – 2,650

9. How can estimation help strengthen your performance when doing math computations?



b) 386 + 407 + 34 + 267

c) 989 + 19 + 1346 + 4

d) Find the sum of 14,326 and 3,724.

e) Subtract: 3476 – 1998



f) Subtract 359 from 20,008.

g) Subtract: 494,830 – 398,751

h) 3,012,010 – 12,036


In the second column, identify the error(s) you fi nd in each of the following worked solutions. If the answer appears to be correct, validate it in the second column and label it “Correct.” If the worked solution is incorrect, solve the problem correctly in the third column and validate your answer in the last column.

Worked SolutionWhat is Wrong Here?

Identify Errors or Validate Correct Process Validation

1) Add: 1,392 + 64,351 + 5,470 + 1,382

Carrying is done incorrectly.

There should be a one (1) that is carried to the thousands column and a one (1) in the ten thousands column; not two ones in the ten thousands column.

2) Add:

2003 + 49 + 182 + 1927

3) Add:

623 + 42 + 537 + 97

IDENTIFY AND CORRECT THE ERRORSIDENTIFY AND CORRECT THE ERRORS

139264351

54701382

72,595

1

1

2

1

72,595–1,38271213–547065743–64351

1392

11

6 1

6 10

Answer: 72,595


Add or subtract as indicated. Validate your answers.

1. 8205 + 356 + 649

2. 5768 + 3470

3. 23 + 92 + 78 + 65 + 11 + 84

4. Find the sum of 1,948 and 7,659.

5. 2556 – 847

6. 41,006 – 2,898

7. 2100 – 703

8. Subtract 1,492 from 30,010.

ADDITIONAL EXERCISESADDITIONAL EXERCISES

Worked Solution Identify Errors or Validate Correct Process Validation

4) Subtract: 503,504 – 498,658

pre-activity adding and subtracting whole numbers …10 this result matches the number in the...

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