<p>By the end of this lesson and the next few, you should be able to:</p>
<ul>
<li>Use division notation</li>
<li>Model division of whole numbers</li>
<li>Divide whole numbers</li>
<li>Translate word phrases to math notation</li>
<li>Divide whole numbers in applications</li>
</ul>
<div class="wp-caption alignnone" id="attachment_102022" style="width: 800px"><a href="https://data.ulearngo.com/assets/97962bc3-2b38-4825-858e-48662faddc75"><img alt="division" class="size-full wp-image-102022" height="541" src="https://data.ulearngo.com/assets/97962bc3-2b38-4825-858e-48662faddc75" width="800"/></a><p class="wp-caption-text">Image credit: Public Domain</p></div>
<h2>Using Division Notation</h2>
<p>So far we have explored addition, subtraction, and multiplication. Now let’s consider division. Suppose you have the \(12\) cookies in the figure below and want to package them in bags with \(4\) cookies in each bag. How many bags would we need?</p>
<figure><img alt="An image of three rows of four cookies to show twelve cookies." src="https://data.ulearngo.com/assets/2b737042-3787-4f48-a890-66a2b4eabf28"/></figure>
<p>You might put \(4\) cookies in first bag, \(4\) in the second bag, and so on until you run out of cookies. Doing it this way, you would fill \(3\) bags.</p>
<p><img alt="An image of 3 bags of cookies, each bag containing 4 cookies." src="https://data.ulearngo.com/assets/f2b420d7-bb2d-4d5b-8df4-dae371f91068"/></p>
<p>In other words, starting with the \(12\) cookies, you would take away, or subtract, \(4\) cookies at a time. Division is a way to represent repeated subtraction just as multiplication represents repeated addition.</p>
<p>Instead of subtracting \(4\) repeatedly, we can write</p>
<div data-label="">\(12÷4\)</div>
<p>We read this as <em>twelve divided by four</em> and the result is the <strong>quotient</strong> of \(12\) and \(4.\) The quotient is \(3\) because we can subtract \(4\) from \(12\) exactly \(3\) times. We call the number being divided the <strong>dividend</strong> and the number dividing it the <strong>divisor</strong>. In this case, the dividend is \(12\) and the divisor is \(4.\)</p>
<p>In the past you may have used the notation \(4\overline{)12}\), but this division also can be written as .\(12÷4,\phantom{\rule{0.2em}{0ex}}12\text{/}4,\frac{12}{4}.\) In each case the \(12\) is the dividend and the \(4\) is the divisor.</p>
<div class="ns-tutorial">
<div class="ns-school-defn2">
<h3>Operation Symbols for Division:</h3>
<p>To represent and describe division, we can use symbols and words.</p>
<table data-label="" summary="A table with 5 columns and 1 row. The first column is labeled “operation” and has the value “division”. The second column is labeled “notation” and has three values - the x multiplication symbol, the dot multiplication symbol, and the parenthesis multiplication symbol. The third column is labeled “expression” and has three values - 3 times 8 with the different multiplication symbols from the second column. The fourth column is labeled “read as” and has the value “three times eight”. The fifth column has the label “result” and has the value “the product of 3 and 8”.">
<thead>
<tr>
<th>Operation</th>
<th>Notation</th>
<th>Expression</th>
<th>Read as</th>
<th>Result</th>
</tr>
</thead>
<tbody>
<tr>
<td data-valign="top">\(\text{Division}\)</td>
<td data-valign="top">\(÷\)

<div> </div>

\(\frac{a}{b}\)

<div> </div>

\(b\overline{)a}\)

<div> </div>

\(a/b\)</td>
<td data-valign="top">\(12÷4\)

<div> </div>

\(\frac{12}{4}\)

<div> </div>

\(4\overline{)12}\)

<div> </div>

\(12/4\)</td>
<td data-valign="top">\(\text{Twelve divided by four}\)</td>
<td data-valign="top">\(\text{the quotient of 12 and 4}\)</td>
</tr>
</tbody>
</table>
<p>Division is performed on two numbers at a time. When translating from math notation to English words, or English words to math notation, look for the words <em>of</em> and <em>and</em> to identify the numbers.</p>
</div>
</div>
<div class="ns-school-emp">
<h3>Example</h3>
<div class="ns-school-defn">
<div data-type="example">
<section>
<div>
<section>
<div>
<p>Translate from math notation to words.</p>
<p>(a) \(64÷8\) (b) \(\frac{42}{7}\) (c) \(4\overline{)28}\)</p>
</div>
<div>
<section>
<h4>Solution</h4>
<ol data-labeled-item="true" style="list-style-type: lower-alpha;">
<li>We read this as <em>sixty-four divided by eight</em> and the result is <em>the quotient of sixty-four and eight</em>.</li>
<li>We read this as <em>forty-two divided by seven</em> and the result is <em>the quotient of forty-two and seven</em>.</li>
<li>We read this as <em>twenty-eight divided by four</em> and the result is <em>the quotient of twenty-eight and four</em>.</li>
</ol>
</section>
</div>
</section>
</div>
</section>
</div>
</div>
</div>
<div class="ns-tutorial">
<div class="ns-school-defn2">
<h3>Extra:</h3>
<div data-type="note">
<section></section>
<section>
<div>
<p>Translate from math notation to words:</p>
<p>(a) \(84÷7\) (b) \(\frac{18}{6}\) (c) \(8\overline{)24}\)</p>
</div>
<div>
<h4>Solution</h4>
<section>
<ol data-labeled-item="true" style="list-style-type: lower-alpha;">
<li>eighty-four divided by seven; the quotient of eighty-four and seven</li>
<li>eighteen divided by six; the quotient of eighteen and six.</li>
<li>twenty-four divided by eight; the quotient of twenty-four and eight</li>
</ol>
</section>
</div>
</section>
</div>
</div>
</div>
<div class="ns-tutorial">
<div class="ns-school-defn2">
<h3>Extra:</h3>
<div data-type="note">
<section></section>
<section>
<div>
<p>Translate from math notation to words:</p>
<p>(a) \(72÷9\) (b) \(\frac{21}{3}\) (c) \(6\overline{)54}\)</p>
</div>
<div>
<h4>Solution</h4>
<section>
<ol data-labeled-item="true" style="list-style-type: lower-alpha;">
<li>seventy-two divided by nine; the quotient of seventy-two and nine</li>
<li>twenty-one divided by three; the quotient of twenty-one and three</li>
<li>fifty-four divided by six; the quotient of fifty-four and six</li>
</ol>
</section>
</div>
</section>
</div>
</div>
</div>
<section></section>
<h2>Optional Video: Dividing Whole Numbers</h2>
<p>This video below by <a href="http://www.mathispower4u.com/">Mathispower4u</a> explains how to perform division using whole numbers.</p>
<p><iframe allowfullscreen="allowfullscreen" data-mce-fragment="1" frameborder="0" height="360" src="https://www.youtube.com/embed/nwTPFqWfosM?rel=0&amp;modestbranding=1&amp;cc_lang_pref=en&amp;cc_load_policy=1" width="640"></iframe></p>

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Division Notation

Mathematics studies measurement, relationships, and properties of quantities and sets, using numbers and symbols. Arithmetic, algebra, geometry, and calculus are popular topics in mathematics.

Mathematics

Explore the world of numbers and learn about their history, properties, and applications, including place value, addition, subtraction, multiplication, and division.


Division Notation

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