Properties of Addition
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. Learn more here.
Adding 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.
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Imagine filling in the table 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.
| + | 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.
Identity Property of Addition
The sum of any number \(a\) and \(0\) is the number.
\(a + 0 = a\)
\(0 + a = a\)
Example
Find each sum:
- \(0 + 11\)
- \(42 + 0\)
Solution
| a. The first addend is zero. The sum of any number and zero is the number. | \(0 + 11 = 11\) |
| b. 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
Add:
- \(8 + 7\)
- \(7 + 8\)
Solution
| a. | |
| Add. | \(8 + 7\) |
| \(15\) |
| b. | |
| 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 (b), but in the reverse order. As a result, both sums are the same.
Example
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{array}{r} |
| Then add the digits in each place value.
Add the ones: \(8 + 1 = 9\) Add the tens: \(2 + 6 = 8\) |
\(\begin{array}{r} 28 \\ +\;61 \\ \hline 89 \\ \end{array}\) |
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.
This lesson is part of:
Introducing Numbers