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1.3: Add Whole Numbers (Part 1)

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    4971
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    Learning Objectives
    • Use addition notation
    • Model addition of whole numbers
    • Add whole numbers without models
    • Translate word phrases to math notation
    • Add whole numbers in applications
    Be Prepared!

    Before you get started, take this readiness quiz.

    1. What is the number modeled by the base-\(10\) blocks? If you missed this problem, review Example 1.1.2.

    An image consisting of three items. The first item is two squares of 100 blocks each, 10 blocks wide and 10 blocks tall. The second item is one horizontal rod containing 10 blocks. The third item is 5 individual blocks.

    Figure \(\PageIndex{1}\)

    1. Write the number three hundred forty-two thousand six using digits? If you missed this problem, review Example 1.1.6.

    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:

    \[3 + 4 \nonumber \]

    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.

    Definition: Addition Notation

    To describe addition, we can use symbols and words.

    Operation Notation Expression Read as Result
    Addition + 3+4 three plus four the sum of 3 and 4
    Example \(\PageIndex{1}\): Translation

    Translate from math notation to words:

    1. \(7 + 1\)
    2. \(12 + 14\)

    Solution

    1. 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.
    2. 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.
    Exercise \(\PageIndex{1}\)

    Translate from math notation to words:

    1. \(8 + 4\)
    2. \(18 + 11\)
    Answer a

    eight plus four; the sum of eight and four

    Answer b

    eighteen plus eleven; the sum of eighteen and eleven

    Exercise \(\PageIndex{2}\)

    Translate from math notation to words:

    1. \(21 + 16\)
    2. \(100 + 200\)
    Answer a

    twenty-one plus sixteen; the sum of twenty-one and sixteen

    Answer b

    one hundred plus two hundred; the sum of one hundred and two hundred

    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.

    CNX_BMath_Figure_01_02_019_img-02.png

    Then we model the second number with 4 blocks.

    CNX_BMath_Figure_01_02_019_img-03.png

    Count the total number of blocks.

    CNX_BMath_Figure_01_02_019_img-04.png

    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 \(3 + 4 = 7\).

    Example \(\PageIndex{2}\): Model

    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. CNX_BMath_Figure_01_02_016_img-02.png
    Model the second number with 6 blocks. CNX_BMath_Figure_01_02_016_img-03.png
    Count the total number of blocks CNX_BMath_Figure_01_02_016_img-04.png
      There are 8 blocks in all, so 2 + 6 = 8.
    Exercise \(\PageIndex{3}\): Model

    Model: \(3 + 6\).

    Answer

    Exercise 1.2.3.png

    Exercise \(\PageIndex{4}\)

    Model: \(5 + 1\).

    Answer

    Exercise 1.2.4.png

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

    Example \(\PageIndex{3}\): model

    Model the addition \(5 + 8\).

    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. CNX_BMath_Figure_01_02_017_img-02.png
    Model the second number with 8 blocks. CNX_BMath_Figure_01_02_017_img-03.png
    Count the result. There are more than 10 blocks so we exchange 10 ones blocks for 1 tens rod. CNX_BMath_Figure_01_02_017_img-04.png
    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.

    Exercise \(\PageIndex{5}\)

    Model the addition: \(5 + 7\)

    Answer

    Exercise 1.2.5.png

    Exercise \(\PageIndex{6}\)

    Model the addition: \(6 + 8\).

    Answer

    Exercise 1.2.6.png

    Example \(\PageIndex{4}\): model

    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    
    Exercise \(\PageIndex{7}\)

    Model each addition: \(15 + 27\).

    Answer

    Exercise 1.2.7.png

    Exercise \(\PageIndex{8}\)

    Model each addition: \(16 + 29\).

    Answer

    Exercise 1.2.8.png

    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 \(\PageIndex{1}\) 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.

    Table \(\PageIndex{1}\)
    + 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.

    Definition: Identity Property of Addition

    The sum of any number \(a\) and \(0\) is the number.

    \[a + 0 = a\]

    \[0 + a = a\]

    Example \(\PageIndex{5}\): add

    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
    1. The second addend is zero. The sum of any number and zero is the number.
    42 + 0 = 42
    Exercise \(\PageIndex{9}\)

    Find each sum:

    1. \(0 + 19\)
    2. \(39 + 0\)
    Answer a

    \(0+19=19\)

    Answer b

    \(39+0=39\)

    Exercise \(\PageIndex{10}\)

    Find each sum:

    1. \(0 + 24\)
    2. \(57 + 0\)
    Answer a

    \(0+24=24\)

    Answer b

    \(57+0=57\)

    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.

    Definition: Commutative Property of Addition

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

    \[a + b = b + a\]

    Example \(\PageIndex{6}\): add

    Add:

    1. \(8 + 7\)
    2. \(7 + 8\)

    Solution

    1. \(\begin{align*} 8+7 & \\ 15 & \end{align*}\)
    2. \(\begin{align*} 7 + 8 & \\ 15 & \end{align*}\)
    Exercise \(\PageIndex{11}\)

    Add: \(9 + 7\) and \(7 + 9\).

    Answer

    \(9+7=16; 7+9=16\)

    Exercise \(\PageIndex{12}\)

    Add: \(8 + 6\) and \(6 + 8\).

    Answer

    \(8+6=14; 6+8=14\)

    Example \(\PageIndex{7}\): add

    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.png
    Then add the digits in each place value. Add the ones: 8 + 1 = 9. 28+61.png
    Add the tens: 2 + 6 = 8. 89
    Exercise \(\PageIndex{13}\)

    Add: \(32 + 54\).

    Answer

    \(32+54=86\)

    Exercise \(\PageIndex{14}\)

    Add: \(25 + 74\).

    Answer

    \(25+74=99\)

    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. Figure \(\PageIndex{2}\) 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”.

    Figure \(\PageIndex{2}\)

    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

    Step 1. Write the numbers so each place value lines up vertically.

    Step 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.

    Step 3. Continue adding each place value from right to left, adding each place value and carrying if needed.

    Example \(\PageIndex{8}\): add

    Add: \(43 + 69\).

    Solution

    Write the numbers so the digits line up vertically. 43+69.png
    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. 43+69(2).png
    Now add the tens: 1 + 4 + 6 = 11. Write the 11 in the sum. 43+69(3).png
    Exercise \(\PageIndex{15}\)

    Add: \(35 + 98\).

    Answer

    \(35+98=133\)

    Exercise \(\PageIndex{16}\)

    Add: \(72 + 89\).

    Answer

    \(72+89=161\)

    Example \(\PageIndex{9}\): add

    Add: \(324 + 586\).

    Solution

    Write the numbers so the digits line up vertically. CNX_BMath_Figure_01_02_020-01.png
    Add the digits in each place value. Add the ones: 4 + 6 = 10. Write the 0 in the ones place in the sum and carry the 1 ten to the tens place. CNX_BMath_Figure_01_02_020-02.png
    Add the tens: 1 + 2 + 8 = 11. Write the 1 in the tens place in the sum and carry the 1 hundred to the hundreds. CNX_BMath_Figure_01_02_020-03.png
    Add the hundreds: 1 + 3 + 5 = 9. Write the 9 in the hundreds place. CNX_BMath_Figure_01_02_020-04.png
    Exercise \(\PageIndex{17}\)

    Add: \(456 + 376\).

    Answer

    \(456+376=832\)

    Exercise \(\PageIndex{18}\)

    Add: \(269 + 578\).

    Answer

    \(269+578=847\)

    Example \(\PageIndex{10}\): add

    Add: \(1,683 + 479\).

    Solution

    Write the numbers so the digits line up vertically. 1683+479(1).png
    Add the digits in each place value  
    Add the ones: 3 + 9 = 12. Write the 2 in the ones place of the sum and carry the 1 ten to the tens place. 1683+479(2).png
    Add the tens: 1 + 7 + 8 = 16. Write the 6 in the tens place and carry the 1 hundred to the hundreds place. 1683+479(3).png
    Add the hundreds: 1 + 6 + 4 = 11. Write the 1 in the hundreds place and carry the 1 thousand to the thousands place 1683+479(4).png
    Add the thousands 1 + 1 = 2. Write the 2 in the thousands place of the sum. 1683+479(5).png

    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.

    Exercise \(\PageIndex{19}\)

    Add: \(4,597 + 685\).

    Answer

    \(4,597+685=5,282\)

    Exercise \(\PageIndex{20}\)

    Add: \(5,837 + 695\).

    Answer

    \(5,837+695=6,532\)

    Example \(\PageIndex{11}\): add

    Add: \(21,357 + 861 + 8,596\).

    Solution

    Write the numbers so the place values line up vertically. Ex 1.21(1).png
    Add the digits in each place value.  
    Add the ones: 7 + 1 + 6 = 14. Write the 4 in the ones place of the sum and carry the 1 to the tens place. Ex 1.21(2).png
    Add the tens: 1 + 5 + 6 + 9 = 21. Write the 1 in the tens place and carry the 2 to the hundreds place. Ex 1.21(3).png
    Add the hundreds: 2 + 3 + 8 + 5 = 18. Write the 8 in the hundreds place and carry the 1 to the thousands place. Ex 1.21(4).png
    Add the thousands 1 + 1 + 8 = 10. Write the 0 in the thousands place and carry the 1 to the ten thousands place. Ex 1.21(5).png
    Add the ten-thousands 1 + 2 = 3. Write the 3 in the ten thousands place in the sum. Ex 1.21(6).png

    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.

    Exercise \(\PageIndex{21}\)

    Add: \(46,195 + 397 + 6,281\).

    Answer

    \(46,195 + 397 + 6,281=52,873\)

    Exercise \(\PageIndex{22}\)

    Add: \(53,762 + 196 + 7,458\).

    Answer

    \(53,762 + 196 + 7,458=61,416\)

    Contributors and Attributions

    • Lynn Marecek (Santa Ana College) and MaryAnne Anthony-Smith (formerly of Santa Ana College). This content produced by OpenStax and is licensed under a Creative Commons Attribution License 4.0 license.

    This page titled 1.3: Add Whole Numbers (Part 1) is shared under a CC BY 4.0 license and was authored, remixed, and/or curated by OpenStax.

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