### Learning Outcomes

• Identify and use the identity property of addition
• Identify and use the commutative property of addition
• Add multiple-digit numbers using columns for place value

## 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 the table below by adding each row number along the left side to each column number across the top. You can use this table for reference, but it will make your work faster and easier if you have the sums memorized.

+ $0$ $1$ $2$ $3$ $4$ $5$ $6$ $7$ $8$ $9$
$0$ $0$ $1$ $2$ $3$ $4$ $5$ $6$ $7$ $8$ $9$
$1$ $1$ $2$ $3$ $4$ $5$ $6$ $7$ $8$ $9$ $10$
$2$ $2$ $3$ $4$ $5$ $6$ $7$ $8$ $9$ $10$ $11$
$3$ $3$ $4$ $5$ $6$ $7$ $8$ $9$ $10$ $11$ $12$
$4$ $4$ $5$ $6$ $7$ $8$ $9$ $10$ $11$ $12$ $13$
$5$ $5$ $6$ $7$ $8$ $9$ $10$ $11$ $12$ $13$ $14$
$6$ $6$ $7$ $8$ $9$ $10$ $11$ $12$ $13$ $14$ $15$
$7$ $7$ $8$ $9$ $10$ $11$ $12$ $13$ $14$ $15$ $16$
$8$ $8$ $9$ $10$ $11$ $12$ $13$ $14$ $15$ $16$ $17$
$9$ $9$ $10$ $11$ $12$ $13$ $14$ $15$ $16$ $17$ $18$

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.

$\begin{array}{}\\ a+0=a\\ 0+a=a\end{array}$

### example

Find each sum:

1. $0+11$
2. $42+0$

Solution

 1.  The first addend is zero. The sum of any number and zero is the number. $0+11=11$ 2. The second addend is zero. The sum of any number and zero is the number. $42+0=42$

### try it

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.

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

$a+b=b+a$

### example

1. $8+7$
2. $7+8$

Did you notice that changing the order of the addends did not change their sum? We could have immediately known the sum from part 2 just by recognizing that the addends were the same as in part 1, but in the reverse order. As a result, both sums are the same.

### example

Add: $28+61$

### try it

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 $\text{base - 10}$ model to find out.

The graphic 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.

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

Add: $43+69$.

### example

Add: $324+586$.

### example

Add: $1,683+479$.

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

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

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.

### try it

Watch the video below for another example of how to add three whole numbers by lining up place values.