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<h2 data-type="title">Adding Whole Numbers Without Models</h2>
<p id="fs-id2629808">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.</p>
<div class="wp-caption alignnone" id="attachment_101877" style="width: 800px"><a href="https://data.ulearngo.com/assets/f7385f2a-afba-41c4-9ce8-19ae9c0b09aa"><img alt="addition-square" class="size-full wp-image-101877" height="800" src="https://data.ulearngo.com/assets/f7385f2a-afba-41c4-9ce8-19ae9c0b09aa" width="800"/></a><p class="wp-caption-text">Image credit: ebay.co.uk</p></div>
<p id="fs-id2137623">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.</p>
<table class="column-header" id="fs-id2300206" summary="No Summary">
<thead>
<tr valign="middle">
<th data-align="center" scope="col">+</th>
<th data-align="center" scope="col">0</th>
<th data-align="center" scope="col">1</th>
<th data-align="center" scope="col">2</th>
<th data-align="center" scope="col">3</th>
<th data-align="center" scope="col">4</th>
<th data-align="center" scope="col">5</th>
<th data-align="center" scope="col">6</th>
<th data-align="center" scope="col">7</th>
<th data-align="center" scope="col">8</th>
<th data-align="center" scope="col">9</th>
</tr>
</thead>
<tbody>
<tr valign="middle">
<td data-align="center">0</td>
<td data-align="center">0</td>
<td data-align="center">1</td>
<td data-align="center">2</td>
<td data-align="center">3</td>
<td data-align="center">4</td>
<td data-align="center">5</td>
<td data-align="center">6</td>
<td data-align="center">7</td>
<td data-align="center">8</td>
<td data-align="center">9</td>
</tr>
<tr valign="middle">
<td data-align="center">1</td>
<td data-align="center">1</td>
<td data-align="center">2</td>
<td data-align="center">3</td>
<td data-align="center">4</td>
<td data-align="center">5</td>
<td data-align="center">6</td>
<td data-align="center">7</td>
<td data-align="center">8</td>
<td data-align="center">9</td>
<td data-align="center">10</td>
</tr>
<tr valign="middle">
<td data-align="center">2</td>
<td data-align="center">2</td>
<td data-align="center">3</td>
<td data-align="center">4</td>
<td data-align="center">5</td>
<td data-align="center">6</td>
<td data-align="center">7</td>
<td data-align="center">8</td>
<td data-align="center">9</td>
<td data-align="center">10</td>
<td data-align="center">11</td>
</tr>
<tr valign="middle">
<td data-align="center">3</td>
<td data-align="center">3</td>
<td data-align="center">4</td>
<td data-align="center">5</td>
<td data-align="center">6</td>
<td data-align="center">7</td>
<td data-align="center">8</td>
<td data-align="center">9</td>
<td data-align="center">10</td>
<td data-align="center">11</td>
<td data-align="center">12</td>
</tr>
<tr valign="middle">
<td data-align="center">4</td>
<td data-align="center">4</td>
<td data-align="center">5</td>
<td data-align="center">6</td>
<td data-align="center">7</td>
<td data-align="center">8</td>
<td data-align="center">9</td>
<td data-align="center">10</td>
<td data-align="center">11</td>
<td data-align="center">12</td>
<td data-align="center">13</td>
</tr>
<tr valign="middle">
<td data-align="center">5</td>
<td data-align="center">5</td>
<td data-align="center">6</td>
<td data-align="center">7</td>
<td data-align="center">8</td>
<td data-align="center">9</td>
<td data-align="center">10</td>
<td data-align="center">11</td>
<td data-align="center">12</td>
<td data-align="center">13</td>
<td data-align="center">14</td>
</tr>
<tr valign="middle">
<td data-align="center">6</td>
<td data-align="center">6</td>
<td data-align="center">7</td>
<td data-align="center">8</td>
<td data-align="center">9</td>
<td data-align="center">10</td>
<td data-align="center">11</td>
<td data-align="center">12</td>
<td data-align="center">13</td>
<td data-align="center">14</td>
<td data-align="center">15</td>
</tr>
<tr valign="middle">
<td data-align="center">7</td>
<td data-align="center">7</td>
<td data-align="center">8</td>
<td data-align="center">9</td>
<td data-align="center">10</td>
<td data-align="center">11</td>
<td data-align="center">12</td>
<td data-align="center">13</td>
<td data-align="center">14</td>
<td data-align="center">15</td>
<td data-align="center">16</td>
</tr>
<tr valign="middle">
<td data-align="center">8</td>
<td data-align="center">8</td>
<td data-align="center">9</td>
<td data-align="center">10</td>
<td data-align="center">11</td>
<td data-align="center">12</td>
<td data-align="center">13</td>
<td data-align="center">14</td>
<td data-align="center">15</td>
<td data-align="center">16</td>
<td data-align="center">17</td>
</tr>
<tr valign="middle">
<td data-align="center">9</td>
<td data-align="center">9</td>
<td data-align="center">10</td>
<td data-align="center">11</td>
<td data-align="center">12</td>
<td data-align="center">13</td>
<td data-align="center">14</td>
<td data-align="center">15</td>
<td data-align="center">16</td>
<td data-align="center">17</td>
<td data-align="center">18</td>
</tr>
</tbody>
</table>
<p id="fs-id1516824">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.</p>
<div class="ns-tutorial">
<div class="ns-school-defn2">
<h3>Identity Property of Addition</h3>
<p id="fs-id1960203">The sum of any number \(a\) and \(0\) is the number.</p>
<h4>\(a + 0 = a\)</h4>
<h4>\(0 + a = a\)</h4>
<div class="equation unnumbered" data-label="" data-type="equation" id="fs-id1858891"> </div>
</div>
</div>
<div class="ns-school-emp">
<h3>Example</h3>
<div class="ns-school-defn">
<p id="fs-id2287930">Find each sum:</p>
<ol style="list-style-type: lower-alpha;">
<li>\(0 + 11\)</li>
<li>\(42 + 0\)</li>
</ol>
<h4>Solution</h4>
<table class="unnumbered unstyled" id="eip-629" summary="Solution for A: The first addend is zero. The sum of any number and zero is the number. 0+11=11 Solution for B: The second addend is zero. The sum of any number and zero is the number. 42 + 0 = 42">
<tbody>
<tr>
<td>a. The first addend is zero. The sum of any number and zero is the number.</td>
<td>\(0 + 11 = 11\)</td>
</tr>
<tr>
<td>b. The second addend is zero. The sum of any number and zero is the number.</td>
<td>\(42 + 0 = 42\)</td>
</tr>
</tbody>
</table>
<div class="circled" data-list-type="enumerated" data-number-style="arabic" data-type="list" id="eip-id1168289479427"> </div>
</div>
</div>
<p id="fs-id2777667">Look at the pairs of sums.</p>
<table class="unnumbered unstyled" data-label="" id="eip-id1168287208889" summary=" ">
<tbody>
<tr>
<td>\(2 + 3 = 5\)</td>
<td>\(3 + 2 = 5\)</td>
</tr>
<tr>
<td>\(4 + 7 = 11\)</td>
<td>\(7 + 4 = 11\)</td>
</tr>
<tr>
<td>\(8 + 9 = 17\)</td>
<td>\(9 + 8 = 17\)</td>
</tr>
</tbody>
</table>
<p id="fs-id2785833">Notice that when the order of the <strong><span class="no-emphasis" data-type="term">addends</span></strong> 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.</p>
<div class="ns-tutorial">
<div class="ns-school-defn2">
<h3>Commutative Property of Addition</h3>
<p id="fs-id1884628">Changing the order of the addends \(a\) and \(b\) does not change their sum.</p>
<p>\(a + b = b + a\)</p>
</div>
</div>
<div class="ns-school-emp">
<h3>Example</h3>
<div class="ns-school-defn">
<p>Add:</p>
<ol style="list-style-type: lower-alpha;">
<li>\(8 + 7\)</li>
<li>\(7 + 8\)</li>
</ol>
<h4>Solution</h4>
<table class="unnumbered unstyled" data-label="" id="eip-id1168288224694" summary=" ">
<tbody>
<tr>
<td>a.</td>
<td> </td>
</tr>
<tr>
<td>Add.</td>
<td>\(8 + 7\)</td>
</tr>
<tr>
<td> </td>
<td>\(15\)</td>
</tr>
</tbody>
</table>
<table class="unnumbered unstyled" data-label="" id="eip-id11688224694" summary=" ">
<tbody>
<tr>
<td>b.</td>
<td> </td>
</tr>
<tr>
<td>Add.</td>
<td>\(7 + 8\)</td>
</tr>
<tr>
<td> </td>
<td>\(15\)</td>
</tr>
</tbody>
</table>
<p id="fs-id3218755">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.</p>
</div>
</div>
<div class="ns-school-emp">
<h3>Example</h3>
<div class="ns-school-defn">
<p>Add: \(28 + 61.\)</p>
<h4>Solution</h4>
<p id="fs-id1806085">To add numbers with more than one digit, it is often easier to write the numbers vertically in columns.</p>
<table class="unnumbered unstyled" id="eip-id1168288293873" summary="a">
<tbody>
<tr>
<td>Write the numbers so the ones and tens digits line up vertically.</td>
<td>
<p>\(\begin{array}{r}<br/>
 28 \\<br/>
 \underline{+\;61} \\<br/>
 \end{array}\)</p>
</td>
</tr>
<tr>
<td>Then add the digits in each place value.

<p>Add the ones: \(8 + 1 = 9\)</p>
<p>Add the tens: \(2 + 6 = 8\)</p>
</td>
<td>\(\begin{array}{r}<br/>
 28 \\<br/>
 +\;61 \\<br/>
 \hline<br/>
 89 \\<br/>
 \end{array}\)</td>
</tr>
</tbody>
</table>
</div>
</div>
<p>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.</p>
<p><a href="https://data.ulearngo.com/assets/20611cfc-5232-4f08-ba7b-889a6eb64266"><img alt="17-plus-26" class="size-full wp-image-100463 alignnone" height="146" src="https://data.ulearngo.com/assets/20611cfc-5232-4f08-ba7b-889a6eb64266" width="619"/></a></p>
<p id="fs-id2297149">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.</p>
<p>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.</p>

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Properties of Addition

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Properties of Addition

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