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Model the addition: 5 + 7 .


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Model the addition: 6 + 8 .


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Next we will model adding two digit numbers.

Model the addition: 17 + 26 .

Solution

17 + 26 means the sum of 17 and 26.

Model the 17. 1 ten and 7 ones CNX_BMath_Figure_01_02_018_img-02.png
Model the 26. 2 tens and 6 ones CNX_BMath_Figure_01_02_018_img-03.png
Combine. 3 tens and 13 ones CNX_BMath_Figure_01_02_018_img-04.png
Exchange 10 ones for 1 ten. 4 tens and 3 ones
40 + 3 = 43
CNX_BMath_Figure_01_02_018_img-05.png
We have shown that 17 + 26 = 43
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Model each addition: 15 + 27 .


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Model each addition: 16 + 29 .


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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 [link] 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

Find each sum:

  1. 0 + 11
  2. 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
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Find each sum:

  1. 0 + 19
  2. 39 + 0

  1. 0 + 19 = 19
  2. 39 + 0 = 39
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Find each sum:

  1. 0 + 24
  2. 57 + 0

  1. 0 + 24 = 24
  2. 57 + 0 = 57
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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

Add:

  1. 8 + 7
  2. 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 just by recognizing that the addends were the same as in part , but in the reverse order. As a result, both sums are the same.

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Add: 9 + 7 and 7 + 9 .

9 + 7 = 16 ; 7 + 9 = 16

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Add: 8 + 6 and 6 + 8 .

8 + 6 = 14 ; 6 + 8 = 14

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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. 28 + 61 ____
Then add the digits in each place value.
Add the ones: 8 + 1 = 9
Add the tens: 2 + 6 = 8
28 + 61 ____ 89
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Add: 32 + 54 .

32 + 54 = 86

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Add: 25 + 74 .

25 + 74 = 99

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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. [link] 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.

Practice Key Terms 1

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Source:  OpenStax, Prealgebra. OpenStax CNX. Jul 15, 2016 Download for free at http://legacy.cnx.org/content/col11756/1.9
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