Add Whole Numbers

1.2 Add Whole Numbers

The topics covered in this section are:

  1. Use addition notation
  2. Model addition of whole numbers
  3. Add whole numbers without models
  4. Translate word phrases to math notation
  5. Add whole numbers in applications

1.2.1 Use Addition Notation

A college student has a part-time job. Last week he worked $3$ hours on Monday and $4$ hours on Friday. To find the total number of hours he worked last week, he added $3$ and $4$.

The operation of addition combines numbers to get a sum. The notation we use to find the sum of $3$ and $4$ is:

$3 + 4$

We read this as three plus four and the result is the sum of three and four. The numbers $3$ and $4$ are called the addends. A math statement that includes numbers and operations is called an expression.

Addition Notation

To describe addition, we can use symbols and words.

OperationNotationExpressionRead asResult
Addition$+$$3+4$three plus fourthe sum of $3$ and $4$

Example 1

Translate from math notation to words:

  • 7+1
  • 12-14
Solution
  • The expression consists of a plus symbol connecting the addends $7$ and $1$. We read this as seven plus one. The result is the sum of seven and one.
  • The expression consists of a plus symbol connecting the addends $12$ and $14$. We read this as twelve plus fourteen. The result is the sum of twelve and fourteen.

1.2.2 Model Addition of Whole Numbers

Addition is really just counting. We will model addition with base-$10$ blocks. Remember, a block represents $1$ and a rod represents $10$. Let’s start by modeling the addition expression we just considered, $3+4$.

Each addend is less than $10$ so we can use ones blocks.

We start by modeling the first number with $3$ blocks.
Then we model the second number with $4$ blocks.
Count the total number of blocks.

There are $7$ blocks in all. We use an equal sign ($=$) to show the sum. A math sentence that shows that two expressions are equal is called an equation. We have shown that $3+4=7$.

Manipulative Mathematics

Doing the Manipulative Mathematics activity “Model Addition of Whole Numbers” will help you develop a better understanding of adding whole numbers.

Example 2

Model the addition $2+6$.

Solution

$2+6$ means the sum of $2$ and $6$.
Each addend is less than $10$, so we can use ones blocks.

Model the first number with $2$ blocks.
Model the second number with $6$ blocks.
Count the total number of blocks.
There are $8$ blocks in all, so $2+6=8$.

When the result is $10$ or more ones blocks, we will exchange the $10$ blocks for one rod.

Example 3

Model the addition of $5+8$.

Solution

$5+8$ means the sum of $5$ and $8$.

Each addend is less than $10$, se we can use ones blocks.
Model the first number with $5$ blocks.
Model the second number with $8$ blocks.
Count the result. There are more than $10$ blocks so we exchange $10$ ones blocks for $1$ tens rod.
Now we have $1$ ten and $3$ ones, which is $13$.$5+8=13$

Notice that we can describe the models as ones blocks and tens rods, or we can simply say ones and tens. From now on, we will use the shorter version but keep in mind that they mean the same thing.

Next we will model adding two digit numbers.

Example 4

Model the addition: $17+26$.

Solution

$17+26$ means the sum of $17$ and $26$.

Model the $17$.$1$ ten and $7$ ones.
Model the $26$.$2$ tens and $6$ ones.
Combine.$3$ tens and $13$ ones.
Exchange $10$ ones for $1$ ten.$4$ tens and $3$ ones.
$40+3=43$
We have shown that $17+26=43$

1.2.3 Add Whole Numbers Without Models

Now that we have used models to add numbers, we can move on to adding without models. Before we do that, make sure you know all the one digit addition facts. You will need to use these number facts when you add larger numbers.

Imagine filling in Table 1.1 by adding each row number along the left side to each column number across the top. Make sure that you get each sum shown. If you have trouble, model it. It is important that you memorize any number facts you do not already know so that you can quickly and reliably use the number facts when you add larger numbers.

+0123456789
00123456789
112345678910
2234567891011
33456789101112
445678910111213
5567891011121314
66789101112131415
778910111213141516
8891011121314151617
99101112131415161718
Table 1.1

Did you notice what happens when you add zero to a number? The sum of any number and zero is the number itself. We call this the Identity Property of Addition. Zero is called the additive identity.

Identity Property of Addition

The sum of any number $a$ and $0$ is the number.
$a+0=a$
$0+a=a$

Example 5

Find each sum:

  • $0 + 11$
  • $42 + 0$
Solution
The first addend is zero. The sum of any number and zero is the number.$0+11=11$
The second addend is zero. The sum of any number and zero is the number.$42+0=42$

Look at the pairs of sums.

$2+3=5$$3+2=5$
$4+7=11$$7+4=11$
$8+9=17$$9+8=17$

Notice that when the order of the addends is reversed, the sum does not change. This property is called the Commutative Property of Addition, which states that changing the order of the addends does not change their sum.

Commutative Property of Addition

Changing the order of the addends $a$ and $b$ does not change their sum.
$a+b=b+a$

Example 6

Add:

  • $8 + 7$
  • $7 + 8$
Solution
Add.$8+7$
$15$
Add.$7+8$
$15$

Did you notice that changing the order of the addends did not change their sum? We could have immediately known the sum from part b just by recognizing that the addends were the same as in part a, but in the reverse order. As a result, both sums are the same.

Example 7

Add: $28 + 61$.

Solution

To add numbers with more than one digit, it is often easier to write the numbers vertically in columns.

Write the numbers so the ones and tens digits line up vertically.$\begin{align*} 28& \\ + \ 61& \end{align*}$
Then add the digits in each place value. 
Add the ones: $8+1=9$
Add the tens: $2+6=8$
$\begin{align*} 28& \\ + \ 61& \\ \overline{89} \end{align*}$

In the previous example, the sum of the ones and the sum of the tens were both less than $10$. But what happens if the sum is $10$ or more? Let’s use our base-$10$ model to find out. The figure below shows the addition of $17$ and $26$ again.

An image containing two groups of items. The left group includes 1 horizontal rod with 10 blocks and 7 individual blocks 2 horizontal rods with 10 blocks each and 6 individual blocks. The label to the left of this group of items is “17 + 26 =”. The right group contains two items. Four horizontal rods containing 10 blocks each. Then, 3 individual blocks. The label for this group is “17 + 26 = 43”.

When we add the ones, $7 + 6$, we get $13$ ones. Because we have more than $10$ ones, we can exchange $10$ of the ones for $1$ ten. Now we have $4$ tens and $3$ ones. Without using the model, we show this as a small red $1$ above the digits in the tens place.

When the sum in a place value column is greater than $9$ we carry over to the next column to the left. Carrying is the same as regrouping by exchanging. For example, $10$ ones for $1$ ten or $10$ tens for $1$ hundred.

How to: Add Whole Numbers

  1. Write the numbers so each place value lines up vertically.
  2. Add the digits in each place value. Work from right to left starting with the ones place. If a sum in a place value is more than $9$ carry to the next place value.
  3. Continue adding each place value from right to left, adding each place value and carrying if needed.

Example 8

Add: $43 + 69$.

Solution
Write the numbers so the digits line up vertically.$\begin{align*} 43& \\ \underline{+69} \end{align*}$
Add the digits in each place.
Add the ones $3+9=12$
Write the $2$ in the ones place in the sum.
Add the $1$ ten to the tens place.
$\begin{align*} \overset{1}{4}3& \\ \underline{+69}& \\ 2& \end{align*}$
Now add the tens: $1+4+6=11$
Write $11$ in the sum.
$\begin{align*} \overset{1}{4}3& \\ {+69}& \\ \overline{112}& \end{align*}$

Example 9

Add: $324 + 586$.

Solution
Write the numbers so the digits line up vertically.$\begin{align*} 324& \\ \underline{+ \ 586}& \end{align*}$
Add the digits in each place.
Add the ones: $4+6=10$
Write $0$ in the ones place in the sum and carry $1$ ten to the tens place.
$\begin{align*} 3\overset{1}{2}4& \\ \underline{+ \ 586}& \\ 0& \end{align*}$
Add the tens: $1+2+8=11$
Write $1$ in the tens place in the sum and carry $1$ hundreds to the hundreds.
$\begin{align*} \overset{1}{3}\overset{1}{2}4& \\ \underline{+ \ 586}& \\ 10& \end{align*}$
Add the hundreds: $1+3+5=9$
Write $9$ in the hundreds place.
$\begin{align*} \overset{1}{3}\overset{1}{2}4& \\ \underline{+ \ 586}& \\ 910& \end{align*}$

Example 10

Add: $1,683 + 479$.

Solution
Write the numbers so the digits line up vertically.$\begin{align*} 1,683& \\ \underline{+ \ 479}& \end{align*}$
Add the digits in each place.
Add the ones: $3+9=12$
Write $2$ in the ones place in the sum and carry $1$ ten to the tens place.
$\begin{align*} 1,6\overset{1}{8}3& \\ \underline{+ \ 479}& \\ 2& \end{align*}$
Add the tens: $1+7+8=16$
Write $6$ in the tens place and carry $1$ hundreds to the hundreds.
$\begin{align*} 1,\overset{1}{6}\overset{1}{8}3& \\ \underline{+ \ 479}& \\ 62& \end{align*}$
Add the hundreds: $1+6+4=11$
Write $1$ in the hundreds place and carry $1$ thousand to the thousands place.
$\begin{align*} \overset{1}{1},\overset{1}{6}\overset{1}{8}3& \\ \underline{+ \ 479}& \\ 162& \end{align*}$
Add the thousands: $1+1=2$
Write $2$ in the thousands place of the sum.
$\begin{align*} \overset{1}{1},\overset{1}{6}\overset{1}{8}3& \\ \underline{+ \ 479}& \\ 2,162& \end{align*}$

When the addends have different numbers of digits, be careful to line up the corresponding place values starting with the ones and moving toward the left.

Example 11

Add: $21,357 + 861 + 8,596$.

Solution
Write the numbers so the digits line up vertically.$\begin{align*} 21,357& \\ 861&\\ \underline{+ \ 8,596}& \end{align*}$
Add the digits in each place.
Add the ones: $7+1+6=14$
Write $4$ in the ones place in the sum and carry $1$ ten to the tens place.
$\begin{align*} 21,3\overset{1}{5}7& \\ 861&\\ \underline{+ \ 8,569}& \\ 4& \end{align*}$
Add the tens: $1+5+6+9=21$
Write $1$ in the tens place and carry $2$ hundreds to the hundreds.
$\begin{align*} 21,\overset{2}{3}\overset{1}{5}7& \\ 861&\\ \underline{+ \ 8,596}& \\ 14& \end{align*}$
Add the hundreds: $2+3+8+5=18$
Write $8$ in the hundreds place and carry $1$ thousand to the thousands place.
$\begin{align*} 2\overset{1}{1},\overset{2}{3}\overset{1}{5}7& \\ 861&\\ \underline{+ \ 8,596}& \\ 814& \end{align*}$
Add the thousands: $1+1+8=10$
Write $0$ in the thousands place and carry $1$ to the ten thousands place.
$\begin{align*} \overset{1}{2}\overset{1}{1},\overset{2}{3}\overset{1}{5}7& \\ 861&\\ \underline{+ \ 8,596}& \\ 0814& \end{align*}$
Add the ten thousands: $1+2=3$
Write $3$ in the ten thousands in the sum.
$\begin{align*} \overset{1}{2}\overset{1}{1},\overset{2}{3}\overset{1}{5}7& \\ 861&\\ \underline{+ \ 8,596}& \\ 30,814& \end{align*}$

This example had three addends. We can add any number of addends using the same process as long as we are careful to line up the place values correctly.

1.2.4 Translate Word Phrases to Math Notation

Earlier in this section, we translated math notation into words. Now we’ll reverse the process. We’ll translate word phrases into math notation. Some of the word phrases that indicate addition are listed in Table 1.2 below.

OperationWordsExampleExpression
Additionplus
sum
increased by
more than
total of
added to
$1$ plus $2$
the sum of $3$ and $4$
$5$ increased by $6$
$8$ more than $7$
the total of $9$ and $5$
$6$ added to $4$
$1+2$
$3+4$
$5+6$
$7+8$
$9+5$
$4+6$

Example 12

Translate and simplify: the sum of $19$ and $23$.

Solution

The word sum tells us to add. The words of $19$ and $23$ tell us the addends.

The sum of $19$ and $23$
Translate.$19+23$
Add.$42$
The sum of $19$ and $23$ is $42$.

Example 13

Translate and simplify: $28$ increased by $31$.

Solution

The word increased tells us to add. The numbers given are the addends.

$28$ increased by $31$.
Translate.$28+31$
Add.$59$
So $28$ increased by $31$ is $59$.

1.2.5 Add Whole Numbers in Applications

Now that we have practiced adding whole numbers, let’s use what we’ve learned to solve real-world problems. We’ll start by outlining a plan. First, we need to read the problem to determine what we are looking for. Then we write a word phrase that gives the information to find it. Next we translate the word phrase into math notation and then simplify. Finally, we write a sentence to answer the question.

Example 14

Hao earned grades of $87, 93, 68, 95$, and $89$ on the five tests of the semester. What is the total number of points he earned on the five tests?

Solution

We are asked to find the total number of points on the tests.

Write a phrase.the sum of points on the tests
Translate to math notation.$87+93+68+95+89$
Then we simplify by adding.
Since there are several numbers, we will write them vertically.$\begin{align*} \overset{3}87&\\ 93&\\ 68&\\ 95&\\ \underline{+ \ 89}&\\ 432 \end{align*}$
Write a sentence to answer the question.Hao earned a total of $432$ points.

Notice that we added points, so the sum is $432$ points. It is important to include the appropriate units in all answers to applications problems.

Some application problems involve shapes. For example, a person might need to know the distance around a garden to put up a fence or around a picture to frame it. The perimeter is the distance around a geometric figure. The perimeter of a figure is the sum of the lengths of its sides.

Example 15

Find the perimeter of the patio shown.

Solution
We are asked to find the perimeter.
Write a phrase.the sum of the sides
Translate to math notation.$4+6+2+3+2+9$
Simplify by adding.$26$
Write a sentence to answer the question.
We added feet, so the sum is $26$ feet.The perimeter of the patio is $26$ feet.
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