# 1.4: Addition of Whole Numbers

Learning Objectives

• be able to add whole numbers
• be able to use the calculator to add one whole number to another

Suppose we have two collections of objects that we combine together to form a third collection. For example,

We are combining a collection of four objects with a collection of three objects to obtain a collection of seven objects.

The process of combining two or more objects (real or intuitive) to form a third, the total, is called addition.

In addition, the numbers being added are called addends or terms, and the total is called the sum. The plus symbol (+) is used to indicate addition, and the equal symbol (=) is used to represent the word "equal." For example, 4 + 3 = 7 means "four added to three equals seven."

## Addition Visualized on the Number Line

Addition is easily visualized on the number line. Let's visualize the addition of 4 and 3 using the number line.

To find 4 + 3

1. Start at 0.
2. Move to the right 4 units. We are now located at 4.
3. From 4, move to the right 3 units. We are now located at 7.

Thus, 4 + 3 = 7

We'll study the process of addition by considering the sum of 25 and 43.

We write this as 68.

We can suggest the following procedure for adding whole numbers using this example.

The Process of Adding Whole Numbers

The process:

1. Write the numbers vertically, placing corresponding positions in the same column.
(egin{array} {r} {25} {underline{+43}} end{array})
2. Add the digits in each column. Start at the right (in the ones position) and move to the left, placing the sum at the bottom.
(egin{array} {r} {25} {underline{+43}} {68} end{array})

Caution

Confusion and incorrect sums can occur when the numbers are not aligned in columns properly. Avoid writing such additions as

(egin{array} {l} {25} {underline{+43}} end{array})

(egin{array} {r} {25} {underline{+43 }} end{array})

Sample Set A

Solution

(egin{array} {r} {276} {underline{+103}} {379} end{array}) (egin{array} {r} {6 + 3 = 9.} {7 + 0 = 7.} {2 + 1 = 3.} end{array})

Sample Set A

Solution

(egin{array} {r} {1459} {underline{+130}} {1589} end{array}) (egin{array} {r} {9 + 0 = 9.} {5 + 3 = 8.} {4 + 1 = 5.} {1 + 0 = 1.} end{array})

In each of these examples, each individual sum does not exceed 9. We will examine individual sums that exceed 9 in the next section.

Practice Set A

Perform each addition. Show the expanded form in problems 1 and 2.

88

Practice Set A

5,527

Practice Set A

267,166

It often happens in addition that the sum of the digits in a column will exceed 9. This happens when we add 18 and 34. We show this in expanded form as follows.

Notice that when we add the 8 ones to the 4 ones we get 12 ones. We then convert the 12 ones to 1 ten and 2 ones. In vertical addition, we show this conversion by carrying the ten to the tens column. We write a 1 at the top of the tens column to indicate the carry. This same example is shown in a shorter form as follows:

8 + 4 = 12 Write 2, carry 1 ten to the top of the next column to the left.

Sample Set B

Perform the following additions. Use the process of carrying when needed.

Solution

(egin{array} {lcl} {5 + 8 = 13} & & { ext{Write 3, carry 1 ten.}} {1 + 7 + 5 = 13} & & { ext{Write 3, carry 1 hundred.}} {1 + 8 + 3 = 12} & & { ext{Write 2, carry 1 thousand.}} {1 + 1 = 2} & & {} end{array})

The sum is 2233.

Sample Set B

Solution

(egin{array} {lcl} {8 + 6 = 14} & & { ext{Write 4, carry 1 ten.}} {1 + 0 + 4 = 5} & & { ext{Write the 5 (nothing to carry).}} {2 + 9 = 11} & & { ext{Write 1, carry one thousand.}} {1 + 9 + 4 = 14} & & { ext{Write 4, carry one ten thousand}.} {1 + 8 = 9} & & {} end{array})

The sum is 94,154.

Sample Set B

Solution

(egin{array} {lcl} {8 + 5 = 13} & & { ext{Write 3, carry 1 ten.}} {1 + 3 + 9 = 13} & & { ext{Write 3, carry 1 hundred.}} {1 + 0 = 1} & & {} end{array})

As you proceed with the addition, it is a good idea to keep in mind what is actually happening.

The sum is 133.

Sample Set B

Find the sum 2648, 1359, and 861.

Solution

(egin{array} {lcl} {8 + 9 + 1 = 18} & & { ext{Write 8, carry 1 ten.}} {1 + 4 + 5 + 6 = 16} & & { ext{Write 6, carry 1 hundred.}} {1 + 6 + 3 + 8 = 18} & & { ext{Write 8, carry 1 thousand.}} {1 + 2 + 1 = 4} & & {} end{array})

The sum is 4,868.

Numbers other than 1 can be carried as illustrated in next example.

Sample Set B

Find the sum of the following numbers.

Solution

(egin{array} {lcl} {6 + 5 + 1 + 7 = 19} & & { ext{Write 9, carry the 1.}} {1 + 1 + 0 + 5 + 1 = 8} & & { ext{Write 8.}} {0 + 9 + 9 + 8 = 26} & & { ext{Write 6, carry the 2.}} {2 + 8 + 9 + 8 + 6 = 33} & & { ext{Write 3, carry the 3}.} {3 + 7 + 3 + 5 = 18} & & { ext{Write 8, carry the 1.}} {1 + 8 = 9} & & { ext{Write 9.}} end{array})

The sum is 983,689.

Sample Set B

The number of students enrolled at Riemann College in the years 1984, 1985, 1986, and 1987 was 10,406, 9,289, 10,108, and 11,412, respectively. What was the total number of students en­rolled at Riemann College in the years 1985, 1986, and 1987?

Solution

We can determine the total number of students enrolled by adding 9,289, 10,108, and 11,412, the number of students enrolled in the years 1985, 1986, and 1987.

The total number of students enrolled at Riemann College in the years 1985, 1986, and 1987 was 30,809.

Practice Set B

Perform each addition. For the next three problems, show the expanded form.

87

(egin{array} {l} { ext{= 7 tens + 1 ten + 7 ones}} { ext{= 8 tens + 7 ones}} { ext{= 87}} end{array})

Practice Set B

561

(egin{array} {r} { ext{= 4 hundreds + 15 tens + 1 ten + 1 one}} { ext{= 4 hundreds + 16 tens + 1 one}} { ext{= 4 hundreds + 1 hundred + 6 tens + 1 one}} { ext{= 5 hundreds + 6 tens + 1 one}} { ext{= 561}} end{array})

Practice Set B

115

(egin{array} {l} { ext{= 10 tens + 1 ten + 5 ones}} { ext{= 11 tens + 5 ones}} { ext{= 11 hundred + 1 ten + 5 ones}} { ext{= 115}} end{array})

Practice Set B

153,525

For the next three problems, find the sums.

Practice Set B

(egin{array} {r} {57} {26} {underline{ 84}} end{array})

167

Practice Set B

(egin{array} {r} {847} {825} {underline{ 796}} end{array})

2,468

Practice Set B

(egin{array} {r} {16,945} {8,472} {387,721} {21,059} {underline{ 629}} end{array})

434,826

## Calculators

Calculators provide a very simple and quick way to find sums of whole numbers. For the two problems in Sample Set C, assume the use of a calculator that does not require the use of an ENTER key (such as many Hewlett-Packard calculators).

Sample Set C

Use a calculator to find each sum.

 34 + 21 Display Reads Type 34 34 Press + 34 Type 21 21 Press = 55

Solution

The sum is 55.

Sample Set C

 106 + 85 + 322 + 406 Display Reads Type 106 106 The calculator keeps a running subtotal Press + 106 Type 85 85 Press = 191 (leftarrow) 106 + 85 Type 322 322 Press + 513 (leftarrow) 191 + 322 Type 406 406 Press = 919 (leftarrow) 513 + 406

The sum is 919.

Practice Set C

Use a calculator to find the following sums.

62 + 81 + 12

155

Practice Set C

9,261 + 8,543 + 884 + 1,062

19,750

Practice Set C

10,221 + 9,016 + 11,445

30,682

## Exercises

For the following problems, perform the additions. If you can, check each sum with a calculator.

Exercise (PageIndex{1})

14 + 5

19

Exercise (PageIndex{2})

12 + 7

Exercise (PageIndex{3})

46 + 2

48

Exercise (PageIndex{4})

83 + 16

Exercise (PageIndex{5})

77 + 21

98

Exercise (PageIndex{6})

(egin{array} {r} {321} {underline{+ 84}} end{array})

Exercise (PageIndex{7})

(egin{array} {r} {916} {underline{+ 62}} end{array})

978

Exercise (PageIndex{8})

(egin{array} {r} {104} {underline{+561}} end{array})

Exercise (PageIndex{9})

(egin{array} {r} {265} {underline{+103}} end{array})

368

Exercise (PageIndex{10})

552 + 237

Exercise (PageIndex{11})

8,521 + 4,256

12,777

Exercise (PageIndex{12})

(egin{array} {r} {16,408} {underline{+ 3,101}} end{array})

Exercise (PageIndex{13})

(egin{array} {r} {16,515} {underline{+42,223}} end{array})

58,738

Exercise (PageIndex{14})

616,702 + 101,161

Exercise (PageIndex{15})

43,156,219 + 2,013,520

45,169,739

Exercise (PageIndex{16})

17 + 6

Exercise (PageIndex{17})

25 + 8

33

Exercise (PageIndex{18})

(egin{array} {r} {84} {underline{+ 7}} end{array})

Exercise (PageIndex{19})

(egin{array} {r} {75} {underline{+ 6}} end{array})

81

Exercise (PageIndex{20})

36 + 48

Exercise (PageIndex{21})

74 + 17

91

Exercise (PageIndex{22})

486 + 58

Exercise (PageIndex{23})

743 + 66

809

Exercise (PageIndex{24})

381 + 88

Exercise (PageIndex{25})

(egin{array} {r} {687} {underline{+175}} end{array})

862

Exercise (PageIndex{26})

(egin{array} {r} {931} {underline{+853}} end{array})

Exercise (PageIndex{27})

1,428 + 893

2,321

Exercise (PageIndex{28})

12,898 + 11,925

Exercise (PageIndex{29})

(egin{array} {r} {631,464} {underline{+509,740}} end{array})

1,141,204

Exercise (PageIndex{30})

(egin{array} {r} {805,996} {underline{+ 98,516}} end{array})

Exercise (PageIndex{31})

(egin{array} {r} {38,428,106} {underline{+522,936,005}} end{array})

561,364,111

Exercise (PageIndex{32})

5,288,423,100 + 16,934,785,995

Exercise (PageIndex{33})

98,876,678,521,402 + 843,425,685,685,658

942,302,364,207,060

Exercise (PageIndex{34})

41 + 61 + 85 + 62

Exercise (PageIndex{35})

21 + 85 + 104 + 9 + 15

234

Exercise (PageIndex{36})

(egin{array} {r} {116} {27} {110} {110} {underline{+ 8}} end{array})

Exercise (PageIndex{37})

(egin{array} {r} {75,206} {4,152} {underline{+16,007}} end{array})

95,365

Exercise (PageIndex{38})

(egin{array} {r} {8,226} {143} {92,015} {8} {487,553} {underline{+ 5,218}} end{array})

Exercise (PageIndex{39})

(egin{array} {r} {50,006} {1,005} {100,300} {20,008} {1,000,009} {underline{+ 800,800}} end{array})

1,972,128

Exercise (PageIndex{40})

(egin{array} {r} {616} {42,018} {1,687} {225} {8,623,418} {12,506,508} {19} {2,121} {underline{ 195,643}} end{array})

For the following problems, perform the additions and round to the nearest hundred.

Exercise (PageIndex{41})

(egin{array} {r} {1,468} {underline{2,183}} end{array})

3,700

Exercise (PageIndex{42})

(egin{array} {r} {928,725} {underline{ 15,685}} end{array})

Exercise (PageIndex{43})

(egin{array} {r} {82,006} {underline{ 3,019,528}} end{array})

3,101,500

Exercise (PageIndex{44})

(egin{array} {r} {18,621} {underline{ 5,059}} end{array})

Exercise (PageIndex{45})

(egin{array} {r} {92} {underline{ 48}} end{array})

100

Exercise (PageIndex{46})

(egin{array} {r} {16} {underline{ 37}} end{array})

Exercise (PageIndex{47})

(egin{array} {r} {21} {underline{ 16}} end{array})

0

Exercise (PageIndex{48})

(egin{array} {r} {11,171} {22,749} {underline{ 12,248}} end{array})

Exercise (PageIndex{49})

(egin{array} {r} {240} {280} {210} {underline{ 310}} end{array})

1000

Exercise (PageIndex{50})

(egin{array} {r} {9,573} {101,279} {underline{ 122,581}} end{array})

For the next five problems, replace the letter mm with the whole number that will make the addition true.

Exercise (PageIndex{51})

(egin{array} {r} {62} {underline{+ m}} {67} end{array})

5

Exercise (PageIndex{52})

(egin{array} {r} {106} {underline{+ m}} {113} end{array})

Exercise (PageIndex{53})

(egin{array} {r} {432} {underline{+ m}} {451} end{array})

19

Exercise (PageIndex{54})

(egin{array} {r} {803} {underline{+ m}} {830} end{array})

Exercise (PageIndex{55})

(egin{array} {r} {1,893} {underline{+ m}} {1,981} end{array})

88

Exercise (PageIndex{56})

The number of nursing and related care facilities in the United States in 1971 was 22,004. In 1978, the number was 18,722. What was the total num­ber of facilities for both 1971 and 1978?

Exercise (PageIndex{57})

The number of persons on food stamps in 1975, 1979, and 1980 was 19,179,000, 19,309,000, and 22,023,000, respectively. What was the total number of people on food stamps for the years 1975, 1979, and 1980?

60,511,000

Exercise (PageIndex{58})

The enrollment in public and nonpublic schools in the years 1965, 1970, 1975, and 1984 was 54,394,000, 59,899,000, 61,063,000, and 55,122,000, respectively. What was the total en­rollment for those years?

Exercise (PageIndex{59})

The area of New England is 3,618,770 square miles. The area of the Mountain states is 863,563 square miles. The area of the South Atlantic is 278,926 square miles. The area of the Pacific states is 921,392 square miles. What is the total area of these regions?

5,682,651 square miles

Exercise (PageIndex{60})

In 1960, the IRS received 1,188,000 corporate income tax returns. In 1965, 1,490,000 returns were received. In 1970, 1,747,000 returns were received. In 1972 —1977, 1,890,000; 1,981,000; 2,043,000; 2,100,000; 2,159,000; and 2,329,000 re­turns were received, respectively. What was the total number of corporate tax returns received by the IRS during the years 1960, 1965, 1970, 1972 —1977?

Exercise (PageIndex{61})

Find the total number of scientists employed in 1974.

1,190,000

Exercise (PageIndex{62})

Find the total number of sales for space vehicle systems for the years 1965-1980.

Exercise (PageIndex{63})

Find the total baseball attendance for the years 1960-1980.

271,564,000

Exercise (PageIndex{64})

Find the number of prosecutions of federal officials for 1970-1980.

For the following problems, try to add the numbers mentally.

Exercise (PageIndex{65})

(egin{array} {r} {5} {5} {3} {underline{ 7}} end{array})

20

Exercise (PageIndex{66})

(egin{array} {r} {8} {2} {6} {underline{ 4}} end{array})

Exercise (PageIndex{67})

(egin{array} {r} {9} {1} {8} {5} {underline{ 2}} end{array})

25

Exercise (PageIndex{68})

(egin{array} {r} {5} {2} {5} {8} {3} {underline{ 7}} end{array})

Exercise (PageIndex{69})

(egin{array} {r} {6} {4} {3} {1} {6} {7} {9} {underline{ 4}} end{array})

40

Exercise (PageIndex{70})

(egin{array} {r} {20} {underline{ 30}} end{array})

Exercise (PageIndex{71})

(egin{array} {r} {15} {underline{ 35}} end{array})

50

Exercise (PageIndex{72})

(egin{array} {r} {16} {underline{ 14}} end{array})

Exercise (PageIndex{73})

(egin{array} {r} {23} {underline{ 27}} end{array})

50

Exercise (PageIndex{74})

(egin{array} {r} {82} {underline{ 18}} end{array})

Exercise (PageIndex{75})

(egin{array} {r} {36} {underline{ 14}} end{array})

50

Exercise (PageIndex{76})

Each period of numbers has its own name. From right to left, what is the name of the fourth period?

Exercise (PageIndex{77})

In the number 610,467, how many thousands are there?

0

Exercise (PageIndex{78})

Write 8,840 as you would read it.

Exercise (PageIndex{79})

Round 6,842 to the nearest hundred.

6,800

Exercise (PageIndex{80})

Round 431,046 to the nearest million.

Find here an unlimited supply of printable worksheets for addition of whole numbers and integers, including both horizontal and vertical problems, missing number problems, customized number range, and more. The worksheets are available both in PDF and html formats, are highly customizable, and include an answer key.

By controlling the range in the generator below, you can use negative numbers (integers), numbers less than 10, very large numbers, and so on. You can limit the numbers to be multiples of ten, hundred, thousand, or a multiple of any other number by using the "Step" option in the generator.

The option Switch addends randomly switches the order of the numbers to be added. For example, if you set addend 1 to be multiples of ten and addend 2 to be multiples of hundred, you will get such problems as 20 + 300 and 400 + 70.

Or, if you set the addend 1 to be negative and addend 2 positive, the option Switch addends randomly makes the negative number sometimes to be first, sometimes second.

To create integer addition problems where the negative numbers occur in any order, set one addend to be negative, another positive, the third one from negative to positive range (such as -10 to 10), and use the option Switch addends randomly.

To create problems about place value, let the range for the first addend to be 0-9, for the second addend from 0 to 90 with step 10, for the third addend from 0 to 900 with step 100, and so on. Then tick the bode for Switch addends randomly and Missing addend options.

Find all of our addition worksheets, from adding by counting objects to addition of multiple large numbers in columns.

K5 Learning offers free worksheets, flashcards and inexpensive workbooks for kids in kindergarten to grade 5. We help your children build good study habits and excel in school.

K5 Learning offers free worksheets, flashcards and inexpensive workbooks for kids in kindergarten to grade 5. We help your children build good study habits and excel in school.

## Examples

Multiply the denominator 2 and whole number 1.

Now, take the denominator 2 as a common denominator for the sum (1 + 2)

Multiply the denominator 2 and whole number 10.

Now, take the denominator 2 as a common denominator for the sum (3 + 20)

Multiply the denominator 3 and whole number 5.

Now, take the denominator 2 as a common denominator for the sum (15 + 2)

Multiply the denominator 8 and whole number 9.

Now, take the denominator 8 as a common denominator for the sum (7 + 72)

Multiply the denominator 8 and whole number 7.

Now, take the denominator 8 as a common denominator for the sum (56 + 5)

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## 1.4: Addition of Whole Numbers

The Improving Mathematics Education in Schools (TIMES) Project

Number and Algebra : Module 7Years : 4-7

• An understanding of the Hindu-Arabic notation and place value as applied
to whole numbers (see module, Using place value to write numbers) .
• Counting on by ones and skip-counting.
• An appreciation that addition can be modelled by combining sets of objects,
and also by movement to the right on a number line.
• An understanding of, and fluency with, addition of two single-digit numbers.
• Experience with the use of doubles, near-doubles and tens complements in
• An appreciation of the commutativity and associativity of addition.

For example, 3 + 7 = 7 + 3 (commutative law for addition) and (2 + 3) + 7 = 2 + (3 + 7) (associative law for addition).

For example, 10 = 1 + 9 = 2 + 8 = 3 + 7 etc.

Numeracy and literacy are essential skills in modern society. Of the four arithmetic operations on numbers, addition is the most natural and, historically, was the first operation developed. The ability to add numbers in your head is used in everyday life, when you play or watch sport and when you buy a couple of items at the shops.

While there are many labour-saving devices that will do calculations, a student will not develop a number-sense or a fluency with operations if they move to algorithms and calculators too quickly.

Formal or written algorithms are useful when larger numbers make mental calculations difficult. While there are many ways to solve problems using arithmetic, the commonly taught algorithms have remained in constant use because they are accurate and efficient.

Once an understanding of numbers has been developed, we can use calculators and computers with some confidence that any data-entry errors that are inconsistent with our number sense will be identified. A relatively common example of someone working without a sense of number is the person at the check-out who tries to charge you a large sum for an inexpensive item simply because the cash register tells them to, without pausing to think that perhaps the code for the item was incorrect.

Developing a solid understanding of addition is essential for understanding later ideas and topics including other arithmetic operations and algebra.

Addition algorithms should not be introduced until students have started to develop a familiarity with basic addition. This can be developed by giving students hands-on experiences, including the use of manipulatives and number lines, and practice with mental strategies for addition based on the basic properties of numbers.

Some mental strategies are more useful than others depending on the numbers used. Several levels of mathematical sophistication are evident amongst the selection of strategies explained here.

A review of the addition of single-digit numbers is essential to ensure students have achieved fluency. Lack of fluency is a serious impediment to their mathematical development because the addition of two single-digit numbers appears as an embedded process in so many arithmetic and algebraic calculations.

Adding a single-digit number to a two-digit number without carrying

The first step is to understand that this case simplifies to the case of adding two single-digit numbers. Using hands-on materials is necessary in the early stages. Students then need to mentally apply decomposition and associativity to produce arguments such as the following.

When children are using the number line, we can identify children that are still counting on by ones

from those that are skip-counting by fives.

Adding a single-digit number to a two-digit number with carrying

Once the previous case is mastered, students should be progressed to the extra complication of the need to carry a ten. In the first instance, students would use tens complements as illustrated below.

On the number line, this corresponds to jumping to the first number, then jumping to the nearest ten above it, then jumping the rest of the way. The mental strategy essentially involves calculating the size of this last jump.

Alternative strategies should also be investigated. Different strategies should be recognized as equally valid and their relative merits discussed. In particular, students should be introduced to the process used in the standard algorithm in an informal way.

We observe that this argument reduces to two applications of adding two single-digit numbers, with one of the additions taking place in the tens column.

Adding two two-digit numbers with no carries involved

Mental strategies for adding two-digit numbers usually involve decomposing one of them and reducing the problem to one, or a combination of the cases already discussed. We illustrate this with the example 24 + 15.

This approach corresponds to

24 + 15 = 24 + 5 + 10 = 29 + 10 = 39

This is the approach that is formalised in the standard algorithm. On the number line, this corresponds to skip-counting as illustrated below.

This involves the calculation

24 + 15 = 24 + 10 + 5 = 34 + 5 = 39

This is a valid approach. There is a formal algorithm known as the Hindu scratch method . This will be considered later in this module. Indeed, developmentally it often comes before the previous technique.

On the number line this corresponds to implementing the second and third jumps above in the opposite order.

Adding two two-digit numbers with carrying involved.

The next level of complication involves introducing carrying. We illustrate various techniques using 28 + 15.

28 + 15 = 28 + 5 + 10 = 33 + 10 = 43

This technique requires revisiting the tens after the ones have been dealt with. Algorithmically, it is implemented as the Hindu scratch method described later in this module.

In this technique we decompose one number to create a tens complement for the other. This can usually be done in more than one way. For example

28 + 15 = 28 + 2 + 13 = 30 + 13 = 43

28 + 15 = 23 + 5 + 15 = 23 + 20 = 43

Write the following numbers on the whiteboard.

60132142189755

Write the numbers 0 to 20 on stickers and place them randomly around a beach ball. Pass the beach ball around the class. The person who catches the ball adds the number that lands nearest their right thumb to one of the numbers selected by the teacher from the list above and states which of the strategies for addition they have used.

A variation might be for the teacher to choose the strategy to be used.

Addition satisfies various properties that make calculations easier. The most commonly known law is the commutative law that says, for example, that

It is a mistake to take commutativity for granted or think of it as a trivial observation. Note that subtraction is not commutative (3 − 4 &ne 4 − 3). In particular, we observe a geometric difference between 3 + 4 and 4 + 3 on the following number lines even though the arithmetic outcome is the same.

3 + 4 corresponds to

4 + 3 corresponds to

An algorithm works most efficiently if it uses a small number of steps that apply in all situations. So algorithms do not resort to techniques, such as the use of near-doubles, that are efficient for a few cases but useless in the majority of cases. The benefit of an algorithm is that it can become an automated process that, once understood, provides an accurate and efficient means to find an answer. No algorithm will help you to add two single-digit numbers. It is essential that students are fluent with the addition of two single-digit numbers before embarking on any formal algorithm for addition.

As soon as you start using the standard algorithm to add more than two numbers, you need to be able to add a single-digit number to a two-digit number in the implementation of the algorithm.

As a procedure the standard algorithm works in the following steps:

• Align the digits in the numbers into columns according to place value.
• Draw a line under the last number you are adding and put a + somewhere to note which operation you are performing.
• Starting from the rightmost column and working from right to left, perform the following subprocedure for each column.
• Add the digits in the column, including any carry digits.
• Write the units digit of your answer in the same column, but under the line.
• Make a note of any carry digits in the next column to the left.

Depending on where you mark your carry digits, the standard algorithm comes in versions exemplified by

The digits are aligned in columns to ensure that like terms are added. In the standard algorithm, the location of the carry digits are habitual, as is the noting and location of the + sign.

Rather than give students slabs of ‘add-ups’ to do, the following method of finding palindromes requires the use of an addition algorithm.

A palindrome is a word, sentence or number that reads the same backwards as it does forwards. For example, Hannah, 2 437 342 and “Ma, I am a llama, I am!”

We can create palindromes by following a procedure that starts with almost any number.

Start with any positive integer, reverse it and add the two numbers. Repeat the procedure until the sum of the two numbers is a palindrome.

For example, 64 generates a palindrome in 2 steps:

Try starting with the numbers 12, 32, 39, 76, 79, 256 and 73 187.

It could take 6 or more steps to get to a palindrome, but while they are searching for a palindrome, your students are practising their addition!

Some numbers take a great many steps, for example 89 takes 24 steps to reach the palindrome 8 813 200 023 188. There are 12 numbers less than 1000 that lead to this palindrome. Other numbers such as 196 seem to never lead to a palindrome but this has not been proved.

A common early error is to misalign the columns. For example, miscalculating 278 + 54 by writing

Entering a two-digit number into a single column.

Another common error is to enter a two-digit number into a single column, thereby destroying the place-value alignment in the solution. For example,

Forgetting to add the carry digits in the calculation.

When implementing the algorithm to add two numbers, the most complicated process we face when adding a column of digits is the sum of two single-digit numbers. When we use the algorithm to add more than two numbers, we may have to use mental arithmetic to add a single-digit number to a two-digit number when adding the digits in a column. Consider the following example.

When adding the digits in the ones column we calculate 3 + 9 = 12 and then 12 + 6 = 18. Similarly, when adding the digits in the tens column we also need to use mental arithmetic to add a single-digit number to a two-digit number.

In some cases, the carry digits are greater than 1.

When we add a long list of numbers, the sum of a column may be a three-digit number. In this case we will need to add a single-digit number to a three-digit number, and the carry will be a two-digit number.

The Hindu scratch method starts from the left and adjusts previous terms as it progresses. It links naturally to mental arithmetic. It is hard to illustrate the method in a static way, but the final version of a calculation would look something like the following.

Start from the left.

If there is no carrying then it is impossible to distinguish the use of the Hindu scratch method from the standard algorithm by looking at the finished product. You can only tell the Hindu scratch method has been used if there is carrying involved. In particular, children often intuitively use the Hindu scratch method with nobody noticing until carry digits are needed. Students using the Hindu scratch method often have a good understanding of addition they are unlikely to have been taught the method and are likely to have developed it on their own.

Students found to be using the Hindu scratch method should not be told they are incorrect, but should be encouraged to use the standard algorithm as it is more efficient.

Further Mental Strategies &minus the associative law
and the any-order property

As well as being commutative, addition is associative, meaning that for all numbers a , b and c

Because of the associative law we have

The combined effect of commutativity and associativity can be described in the following way.

A list of whole numbers can be added two at a time in any order to give the same result.

We often use the any-order property in mental arithmetic, even when implementing the algorithm. For example, when calculating 71 + 68 + 49 + 32 most of us would naturally pair the tens complements to make the calculation easier:

Find these sums by pairing the tens complements and rearranging to make the calculation easier.

a 24 + 7 + 32 + 6 + 93 + 8 =

b 98 + 49 + 17 + 11 + 32 + 43 =

c 333 + 54 + 145 + 7 + 55 + 6 =

Addition is the foundation of arithmetic. One way to model multiplication of whole numbers is as repeated addition. Subtraction is the inverse process of addition and division is the inverse process of multiplication. Thus in a very real sense, mastery of addition underpins success in all of arithmetic.

A strong number-sense is an invaluable advantage in the understanding of algebra. In particular, the process of decomposing and recombining numbers aids the understanding of general algebraic manipulations. A strong grounding in arithmetic sets a student up for success in algebra.

Addition, in the sense of measuring the size of combined sets, was probably done as soon as people counted. Addition itself does not change 4+2 is six regardless of whether you write it as 6, VI or . Just as the history of number is really all about the development of numerals, the history of addition is mainly the history of the processes people have used to perform calculations.

The development of the Hindu-Arabic place-value notation enabled the implementation of efficient algorithms for arithmetic and was probably the main reason for the popularity and fast adoption of the notation.

The word algorithm is derived from the name of Muhammad al- Khwārizmī an Islamic astronomer and mathematician. In 825 AD he wrote a treatise entitled Book on Addition and Subtraction after the Method of the Indians . It was translated into Latin in the 12th century as Algoritmi de numero Indorum . The term Algoritmi probably referred to
al-Khwārizmī rather than a general procedure of calculation, but the name has stuck.

A History of Mathematics: An Introduction, 3rd Edition, Victor J. Katz, Addison-Wesley, (2008)

Mathematical Circus, Martin Gardner, Penguin, (1970)

The Improving Mathematics Education in Schools (TIMES) Project 2009-2011 was funded by the Australian Government Department of Education, Employment and Workplace Relations.

The views expressed here are those of the author and do not necessarily represent the views of the Australian Government Department of Education, Employment and Workplace Relations.

## Whole Numbers Worksheets Building a Strong Base for Further Education

The study of math and especially algebra is based on whole numbers and so, students in their early years should be introduced well to whole numbers. All the further education of not only math, but almost all subjects is more or less based on the knowledge of these numbers. Whole numbers are those countless numbers which start from zero and have no decimals or fractions. You can introduce whole numbers to children with rounding whole numbers worksheets. By working on these worksheets, children are well acquainted to the whole numbers. They also should be introduced to adding and subtracting whole numbers, and also, multiplying and dividing whole numbers, which too you can do with the whole numbers worksheets. The initial years of school will bring much for children to work on, for which the whole number worksheets will build a strong base.

## 1.4: Addition of Whole Numbers

Explanation :-
Addition is Commutative for Whole Numbers, this means that even if we change the order of numbers in addition expression, the result remains same. This property is also known as Commutativity for Addition of Whole numbers

Commutative Property for Addition of Whole Numbers can be further understood with the help of following examples :-

Example 1 = Explain Commutative Property for addition of whole numbers 5 & 7 in addition expression ?
Answer = Given Whole Numbers = 5, 7 and their two orders are as follows :-
Order 1 = 5 + 7 = 12
Order 2 = 7 + 5 = 12
As, in both the orders the result is same i.e 12
So, we can say that Addition is Commutative for Whole Numbers.

Example 2 = Explain Commutative Property for addition of whole numbers 23 & 43 in addition expression ?
Answer = Given Whole Numbers = 23, 43 and their two orders are as follows :-
Order 1 = 23 + 43 = 66
Order 2 = 43 + 23 = 66
As, in both the orders the result is same i.e 66
So, we can say that Addition is Commutative for Whole Numbers.

Example 3 = Explain Commutative Property for addition of whole numbers 20 & 4.
Answer = Given Whole numbers = 20, 4 and their two orders are as follows :-
Order 1 = 20 + 4 = 24
Order 2 = 4 + 20 = 24
As, in both the orders the result is same i.e 24,
So, we can say that Addition is Commutative for Integers.

## 1.4: Addition of Whole Numbers

a) estimate sums, differences, products, and quotients of whole numbers

b) add, subtract, and multiply whole numbers

c) divide whole numbers, finding quotients with and without remainders and

d) solve single-step and multistep addition, subtraction, and multiplication problems with whole numbers.

Computation and Estimation

Probability, Statistics, Patterns, Functions, and Algebra

Words and Definitions

(3.4) estimate and solve single step and multi step addition 4 digit numbers or less with or without regrouping.

Difference – The answer to a subtraction problem

Number Sentence – An equation 3+4=7

Rounding - Reducing the digits in a number while trying to keep it's value similar

Estimation – Finding a value that is close enough to the correct answer

Other words/phrases to consider:

A little more than, Between, Closer to, Compatible numbers

Smartboard: (See file below)

Adding and Subtracting Senteo and SMART Response

Understanding the Standard

Essential Understandings

Essential Knowledge and Skills

· Addition is the combining of quantities it uses the following terms:

· Subtraction is the inverse of addition it yields the difference between two numbers and uses the following terms:

· Before adding or subtracting with paper and pencil, addition and subtraction problems in horizontal form should be rewritten in vertical form by lining up the places vertically.

· Using base-10 materials to model and stimulate discussion about a variety of problem situations helps students understand regrouping and enables them to move from the concrete to the abstract. Regrouping is used in addition and subtraction algorithms. In addition, when the sum in a place is 10 or more, place value is used to regroup the sums so that there is only one digit in each place. In subtraction, when the number (minuend) in a place is not enough from which to subtract, regrouping is required.

· Develop and use strategies to estimate whole number sums and differences and to judge the reasonableness of such results.

· Understand that addition and subtraction are inverse operations.

· Understand that division is the operation of making equal groups or equal shares. When the original amount and the number of shares are known, divide to find the size of each share. When the original amount and the size of each share are known, divide to find the number of shares.

· Understand that multiplication and division are inverse operations.

· Understand various representations of division and the terms used in division are dividend, divisor, and quotient.

dividend ¸ divisor = quotient

· Understand how to solve single-step and multistep problems using whole number operations.

· When is it more appropriate to estimate differences than to compute them?

· What are some strategies to use to estimate differences, and how do we decide which to use?

· What situations call for the computation of differences?

· How can place value understandings be used to devise strategies to compute differences, products?

· How can we use number sense to model the reasonableness of an estimation or computation?

· How can we use the inverse relationships between addition and subtraction to solve problems?

· Determine the sum or difference of two whole numbers, each 999,999 or less, in vertical form with or without regrouping.

· Determine the sum or difference of two whole numbers, each 999,999 or less, in horizontal form with or without regrouping.

· Find the sum or difference of two whole numbers, each 999,999 or less, using paper and pencil.

· Find the sum or difference of two whole numbers, each 999,999 or less, using a calculator.

## Properties Of Addition - Definition with Examples

There are four properties of addition of whole numbers.

Whole numbers

Natural numbers (Counting numbers), including 0, form the set of whole numbers.

Closure Property:

The sum of the addition of two or more whole numbers is always a whole number.

Whole Number + Whole Number = Whole Number

For example, 2 + 4 = 6

Commutative Property

When we add two or more whole numbers, their sum is the same regardless of the order of the addends.

Example 1: 2 + 4 = 4 + 2 = 6

Associative Property

When three or more numbers are added, the sum is the same regardless of the grouping of the addends.

For example (4 + 2) + 3 = (4 + 3) + 2

Here, the addends are 2, 4 and 3. So, as per the associative property, the sum of the three numbers will remain the same, no matter how we group them.

On adding zero to any number, the sum remains the original number. Adding 0 to a number does not change the value of the number.

For example, 3 + 0 = 3

Addition of two whole numbers except for zero will always give a bigger number.

When you add numbers (except 0) on a number line, the result will always shift you to the right.

Here you will find some simple information and advice about Fraction of a Whole Number.

At the bottom of this page you will also find two printable resource sheets which explain about how to calculate fractions in a little more detail.

Before you start learning to calculate fractions of numbers, you should be able to work out fractions of shapes.

### How to find a fraction of a whole number

Here are the two easy steps for finding the fraction of a number:

Step 1 - Find the unit fraction by dividing the number by the denominator

Step 2 - Multiply by the numerator .

You should have now found your fraction of a number!

Finding a fraction of a whole number is the same as multiplying the fraction by the whole number.

[ <4 over 5> of 30 is the same as <4 over 5> imes 30 ]

### Examples of Fraction of a Whole Number

#### Example 1) [ Find of 24 ]

A unit fraction is a fraction where the numerator is equal to 1.

To find the unit fraction of a number, you need to divide the number by the denominator.

This gives us: [ <1 over 6> of 24 = 24 ÷ 6 = 4 ]

To find five-sixths, we need to multiply our answer by the numerator which is 5.

So [ <5 over 6> of 24 = ( <1 over 6> of 24) imes 5 = 4 imes 5 = 20 ]

#### Example 2) [ Find of 35]

To find the unit fraction, we need to divide the number by the denominator.

This gives us: [ <1 over 7> of 35 = 35 ÷ 7 = 5 ]

To find three-sevenths, we need to multiply our answer by the numerator which is 3.

So [ <3 over 7> of 35 = ( <1 over 7> of 35) imes 3 = 5 imes 3 = 15 ]

#### Example 3) [ Find of $230] To find the unit fraction, we need to divide the number by the denominator. This gives us: [ <1 over 10> of$230 = $230 ÷ 10 =$23 ]

To find three-tenths, we need to multiply our answer by the numerator which is 3.

So [ <3 over 10> of $230 = ( <1 over 10> of$230) imes 3 ] and [ ( <1 over 10> of $230) imes 3 =$23 imes 3 = $69 ] Final answer [ <3 over 10> of$230 = \$69 ]

### How to calculate fractions - the algebra.

For those of you who like to see things in Algebra. this is what it looks like

If we want to work out: [ of a number n ]

First we work out: [ <1 over b> of n = or n ÷ b ]

Next we need to multiply this by the numerator a.

### How do you find fractions of a number support sheet

This printable support sheet below gives a little more detail about finding fractions of numbers including a step-by-step visual guide to how and why it works.

### Fraction of a whole number worksheets

• Fractions of numbers Sheet 1
• Fractions of numbers Sheet 2
• Fractions of numbers Sheet 3
• Fractions of numbers Sheet 4
• Fractions of numbers Sheet 5
• Fractions of numbers Sheet 6

### Fraction of a whole number problems

These problems all involve finding the fraction of a whole number.

There are 3 versions of each sheet:

• Sheets 1a and 2a are the easiest. They mainly involve finding simple unit fractions of small numbers.
• Sheets 1b and 2b are a little harder. They involve finding (mainly unit) fractions of larger numbers.
• Sheets 1c and 2c are hardest. They involve finding non-unit fractions of larger numbers.

### More Recommended Math Worksheets

Take a look at some more of our worksheets similar to these.

### Practice your Fraction of a Number skills online

Want to practice your skills online?

We have a fraction of a whole number practice area where you can practice finding different fractions of numbers.

### More Fractions of a Whole Number Worksheets

We also have a page of 3rd grade worksheets about finding unit fractions of whole numbers.

Unit fractions are fractions with a numerator of 1. The sheets are easier than those on this page.

There is also a randomly worksheet generator for you to make you own fractions of a number worksheets to meet your needs.

• develop an understanding of fractions as parts of a whole
• know how to calculate unit fractions of a range of numbers.

### Learning Fractions Math Help Page

Here you will find the Math Salamanders free online Math help pages about Fractions.

There is a wide range of help pages including help with:

• fraction definitions
• equivalent fractions
• converting improper fractions
• how to add and subtract fractions
• how to convert fractions to decimals and percentages
• how to simplify fractions.

### Fractions of a Whole Number Online Quiz

Our quizzes have been created using Google Forms.

At the end of the quiz, you will get the chance to see your results by clicking 'See Score'.

This will take you to a new webpage where your results will be shown. You can print a copy of your results from this page, either as a pdf or as a paper copy.

For incorrect responses, we have added some helpful learning points to explain which answer was correct and why.

The quizzes are anonymous, and we do not collect any personal data from them. We do collect the results from the quizzes which we use to help us to develop our resources.

This quick quiz tests your knowledge and skill at finding proper fractions of a range of numbers with our online quiz.

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### Math-Salamanders.com

The Math Salamanders hope you enjoy using these free printable Math worksheets and all our other Math games and resources.