The topics covered in this section are:

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.

To describe addition, we can use symbols and words.

#### 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.

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

#### 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.

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 $8$.

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 $26$.

## 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.

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.

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

Look at the pairs of sums.

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.

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

#### Example 6

• $8 + 7$
• $7 + 8$
Solution

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.

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.

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

#### Example 9

Add: $324 + 586$.

Solution

#### Example 10

Add: $1,683 + 479$.

Solution

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

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.

 Operation Words Example Expression Addition plussumincreased bymore thantotal ofadded 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.

#### Example 13

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

Solution

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

## 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.

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