{"id":9229,"date":"2017-05-02T19:44:39","date_gmt":"2017-05-02T19:44:39","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/prealgebra\/?post_type=chapter&#038;p=9229"},"modified":"2024-04-29T18:42:22","modified_gmt":"2024-04-29T18:42:22","slug":"notation-and-modeling-division-of-whole-numbers","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/wm-developmentalemporium\/chapter\/notation-and-modeling-division-of-whole-numbers\/","title":{"raw":"Notation and Modeling Division of Whole Numbers","rendered":"Notation and Modeling Division of Whole Numbers"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Outcomes<\/h3>\r\n<ul>\r\n \t<li>Use symbols and words to represent division of numbers<\/li>\r\n \t<li>Model division of whole numbers<\/li>\r\n<\/ul>\r\n<\/div>\r\n&nbsp;\r\n<h2>Use Division Notation<\/h2>\r\nSo far we have explored addition, subtraction, and multiplication. Now let\u2019s consider division. Suppose you have [latex]12[\/latex] cookies and want to package them in bags with [latex]4[\/latex] cookies in each bag. How many bags would we need?\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24215605\/CNX_BMath_Figure_01_05_001.png\" alt=\"An image of three rows of four cookies to show twelve cookies.\" \/>\r\nYou might put [latex]4[\/latex] cookies in the first bag, [latex]4[\/latex] in the second bag, and so on, until you run out of cookies. Doing it this way, you would fill [latex]3[\/latex] bags.\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24215607\/CNX_BMath_Figure_01_05_042.png\" alt=\"An image of 3 bags of cookies, each bag containing 4 cookies.\" \/>\r\nIn other words, starting with the [latex]12[\/latex] cookies, you would take away, or subtract, [latex]4[\/latex] cookies at a time. Division is a way to represent repeated subtraction, just as multiplication represents repeated addition.\r\n\r\nInstead of subtracting [latex]4[\/latex] repeatedly, we can write\r\n<p style=\"text-align: center;\">[latex]12\\div 4[\/latex]<\/p>\r\nWe read this as <em>twelve divided by four<\/em> and the result is the quotient of [latex]12[\/latex] and [latex]4[\/latex]. The quotient is [latex]3[\/latex], because we can subtract [latex]4[\/latex] from [latex]12[\/latex] exactly [latex]3[\/latex] times. We call the number being divided, the dividend, and the number dividing it, the divisor. In this case, the dividend is [latex]12[\/latex] and the divisor is [latex]4[\/latex].\r\n\r\nIn the past, you may have used the notation [latex]4\\overline{)12}[\/latex] , but this division also can be written as [latex]12\\div 4, 12\\text{\/}4, \\frac{12}{4}[\/latex]. In each case, the [latex]12[\/latex] is the <strong>dividend<\/strong> and the [latex]4[\/latex] is the <strong>divisor<\/strong>.\r\n\r\n&nbsp;\r\n<div class=\"textbox shaded\">\r\n<h3>Operation Symbols for Division<\/h3>\r\nTo represent and describe division, we can use symbols and words.\r\n<table id=\"eip-id3140794\" class=\"unnumbered\" style=\"width: 85%;\" summary=\"A table with 5 columns and 1 row. The first column is labeled \">\r\n<thead>\r\n<tr valign=\"top\">\r\n<th style=\"width: 16%;\">Operation<\/th>\r\n<th style=\"width: 22.8571%;\">Notation<\/th>\r\n<th style=\"width: 25.7143%;\">Expression<\/th>\r\n<th style=\"width: 17.2857%;\">Read as<\/th>\r\n<th style=\"width: 18%;\">Result<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr valign=\"top\">\r\n<td style=\"width: 16%;\">[latex]\\text{Division}[\/latex]<\/td>\r\n<td style=\"width: 22.8571%;\">[latex]\\div [\/latex]\r\n\r\n[latex]\\frac{a}{b}[\/latex]\r\n\r\n[latex]b\\overline{)a}[\/latex]\r\n\r\n[latex]a\/b[\/latex]<\/td>\r\n<td style=\"width: 25.7143%;\">[latex]12\\div 4[\/latex]\r\n\r\n[latex]\\frac{12}{4}[\/latex]\r\n\r\n[latex]4\\overline{)12}[\/latex]\r\n\r\n[latex]12\/4[\/latex]<\/td>\r\n<td style=\"width: 17.2857%;\">[latex]\\text{Twelve divided by four}[\/latex]<\/td>\r\n<td style=\"width: 18%;\">[latex]\\text{the quotient of 12 and 4}[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n&nbsp;\r\n\r\nDivision 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.\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nTranslate from math notation to words.\r\n1. [latex]64\\div 8[\/latex]\r\n\r\n2. [latex]\\frac{42}{7}[\/latex]\r\n\r\n3. [latex]4\\overline{)28}[\/latex]\r\n\r\nSolution\r\n<ul id=\"eip-id1168289650120\">\r\n \t<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>\r\n \t<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>\r\n \t<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>\r\n<\/ul>\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\n[embed]https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=144450&amp;theme=oea&amp;iframe_resize_id=mom1[\/embed]\r\n\r\n<\/div>\r\nIn the video below, we present more examples of how to express division with words, and how to translate those words into math notation.\r\n\r\nhttps:\/\/youtu.be\/WxJxY4aJ9Vk\r\n\r\n&nbsp;\r\n<h2>Model Division of Whole Numbers<\/h2>\r\nAs we did with multiplication, we will model division using counters. The operation of division helps us organize items into equal groups as we start with the number of items in the dividend and subtract the number in the divisor repeatedly.\r\n\r\nDoing the Manipulative Mathematics activity, Model Division of Whole Numbers, will help you develop a better understanding of dividing whole numbers.\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nModel the division: [latex]24\\div 8[\/latex]\r\n\r\n[reveal-answer q=\"12486\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"12486\"]\r\n\r\nSolution\r\n\r\nTo find the quotient [latex]24\\div 8[\/latex], we want to know how many groups of [latex]8[\/latex] are in [latex]24[\/latex].\r\nModel the dividend. Start with [latex]24[\/latex] counters.\r\n\r\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24215608\/CNX_BMath_Figure_01_05_003_img.png\" alt=\"An image of 24 counters placed randomly.\" \/>\r\nThe divisor tell us the number of counters we want in each group. Form groups of [latex]8[\/latex] counters.\r\n\r\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24215610\/CNX_BMath_Figure_01_05_004_img.png\" alt=\"An image of 24 counters, all contained in 3 bubbles, each bubble containing 8 counters.\" \/>\r\nCount the number of groups. There are [latex]3[\/latex] groups.\r\n[latex]24\\div 8=3[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\nModel: [latex]24\\div 6[\/latex]\r\n\r\n[reveal-answer q=\"294014\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"294014\"]\r\n\r\n<img class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24215612\/CNX_BMath_Figure_01_05_006_img.png\" alt=\"An image showing 24 circles, grouped into 4 groups of 6 by irregular shapes drawn around the circles.  \" width=\"186\" height=\"182\" \/>\u00a0\u00a0[latex]24\\div 6=4[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\nModel: [latex]42\\div 7[\/latex]\r\n\r\n[reveal-answer q=\"440138\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"440138\"]\r\n\r\n<img class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24215614\/CNX_BMath_Figure_01_05_007_img.png\" alt=\"An image showing 42 circles, grouped into 6 groups of 7 by irregular shapes drawn around the circles.  \" width=\"242\" height=\"178\" \/>\u00a0 [latex]42\\div 7=6[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nIn the video below, we show another way to model division using area.\r\n\r\nhttps:\/\/youtu.be\/jKHAsIcEolM","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li>Use symbols and words to represent division of numbers<\/li>\n<li>Model division of whole numbers<\/li>\n<\/ul>\n<\/div>\n<p>&nbsp;<\/p>\n<h2>Use Division Notation<\/h2>\n<p>So far we have explored addition, subtraction, and multiplication. Now let\u2019s consider division. Suppose you have [latex]12[\/latex] cookies and want to package them in bags with [latex]4[\/latex] cookies in each bag. How many bags would we need?<\/p>\n<p><img decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24215605\/CNX_BMath_Figure_01_05_001.png\" alt=\"An image of three rows of four cookies to show twelve cookies.\" \/><br \/>\nYou might put [latex]4[\/latex] cookies in the first bag, [latex]4[\/latex] in the second bag, and so on, until you run out of cookies. Doing it this way, you would fill [latex]3[\/latex] bags.<\/p>\n<p><img decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24215607\/CNX_BMath_Figure_01_05_042.png\" alt=\"An image of 3 bags of cookies, each bag containing 4 cookies.\" \/><br \/>\nIn other words, starting with the [latex]12[\/latex] cookies, you would take away, or subtract, [latex]4[\/latex] cookies at a time. Division is a way to represent repeated subtraction, just as multiplication represents repeated addition.<\/p>\n<p>Instead of subtracting [latex]4[\/latex] repeatedly, we can write<\/p>\n<p style=\"text-align: center;\">[latex]12\\div 4[\/latex]<\/p>\n<p>We read this as <em>twelve divided by four<\/em> and the result is the quotient of [latex]12[\/latex] and [latex]4[\/latex]. The quotient is [latex]3[\/latex], because we can subtract [latex]4[\/latex] from [latex]12[\/latex] exactly [latex]3[\/latex] times. We call the number being divided, the dividend, and the number dividing it, the divisor. In this case, the dividend is [latex]12[\/latex] and the divisor is [latex]4[\/latex].<\/p>\n<p>In the past, you may have used the notation [latex]4\\overline{)12}[\/latex] , but this division also can be written as [latex]12\\div 4, 12\\text{\/}4, \\frac{12}{4}[\/latex]. In each case, the [latex]12[\/latex] is the <strong>dividend<\/strong> and the [latex]4[\/latex] is the <strong>divisor<\/strong>.<\/p>\n<p>&nbsp;<\/p>\n<div class=\"textbox shaded\">\n<h3>Operation Symbols for Division<\/h3>\n<p>To represent and describe division, we can use symbols and words.<\/p>\n<table id=\"eip-id3140794\" class=\"unnumbered\" style=\"width: 85%;\" summary=\"A table with 5 columns and 1 row. The first column is labeled\">\n<thead>\n<tr valign=\"top\">\n<th style=\"width: 16%;\">Operation<\/th>\n<th style=\"width: 22.8571%;\">Notation<\/th>\n<th style=\"width: 25.7143%;\">Expression<\/th>\n<th style=\"width: 17.2857%;\">Read as<\/th>\n<th style=\"width: 18%;\">Result<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr valign=\"top\">\n<td style=\"width: 16%;\">[latex]\\text{Division}[\/latex]<\/td>\n<td style=\"width: 22.8571%;\">[latex]\\div[\/latex]<\/p>\n<p>[latex]\\frac{a}{b}[\/latex]<\/p>\n<p>[latex]b\\overline{)a}[\/latex]<\/p>\n<p>[latex]a\/b[\/latex]<\/td>\n<td style=\"width: 25.7143%;\">[latex]12\\div 4[\/latex]<\/p>\n<p>[latex]\\frac{12}{4}[\/latex]<\/p>\n<p>[latex]4\\overline{)12}[\/latex]<\/p>\n<p>[latex]12\/4[\/latex]<\/td>\n<td style=\"width: 17.2857%;\">[latex]\\text{Twelve divided by four}[\/latex]<\/td>\n<td style=\"width: 18%;\">[latex]\\text{the quotient of 12 and 4}[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<p>&nbsp;<\/p>\n<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 &#8220;<em>of&#8221;<\/em> and &#8220;<em>and&#8221;<\/em> to identify the numbers.<\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Translate from math notation to words.<br \/>\n1. [latex]64\\div 8[\/latex]<\/p>\n<p>2. [latex]\\frac{42}{7}[\/latex]<\/p>\n<p>3. [latex]4\\overline{)28}[\/latex]<\/p>\n<p>Solution<\/p>\n<ul id=\"eip-id1168289650120\">\n<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>\n<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>\n<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>\n<\/ul>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm144450\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=144450&#38;theme=oea&#38;iframe_resize_id=ohm144450&#38;show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>In the video below, we present more examples of how to express division with words, and how to translate those words into math notation.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-1\" title=\"The Language of Division\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/WxJxY4aJ9Vk?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p>&nbsp;<\/p>\n<h2>Model Division of Whole Numbers<\/h2>\n<p>As we did with multiplication, we will model division using counters. The operation of division helps us organize items into equal groups as we start with the number of items in the dividend and subtract the number in the divisor repeatedly.<\/p>\n<p>Doing the Manipulative Mathematics activity, Model Division of Whole Numbers, will help you develop a better understanding of dividing whole numbers.<\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Model the division: [latex]24\\div 8[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q12486\">Show Solution<\/span><\/p>\n<div id=\"q12486\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution<\/p>\n<p>To find the quotient [latex]24\\div 8[\/latex], we want to know how many groups of [latex]8[\/latex] are in [latex]24[\/latex].<br \/>\nModel the dividend. Start with [latex]24[\/latex] counters.<\/p>\n<p><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24215608\/CNX_BMath_Figure_01_05_003_img.png\" alt=\"An image of 24 counters placed randomly.\" \/><br \/>\nThe divisor tell us the number of counters we want in each group. Form groups of [latex]8[\/latex] counters.<\/p>\n<p><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24215610\/CNX_BMath_Figure_01_05_004_img.png\" alt=\"An image of 24 counters, all contained in 3 bubbles, each bubble containing 8 counters.\" \/><br \/>\nCount the number of groups. There are [latex]3[\/latex] groups.<br \/>\n[latex]24\\div 8=3[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p>Model: [latex]24\\div 6[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q294014\">Show Solution<\/span><\/p>\n<div id=\"q294014\" class=\"hidden-answer\" style=\"display: none\">\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24215612\/CNX_BMath_Figure_01_05_006_img.png\" alt=\"An image showing 24 circles, grouped into 4 groups of 6 by irregular shapes drawn around the circles.\" width=\"186\" height=\"182\" \/>\u00a0\u00a0[latex]24\\div 6=4[\/latex]<\/p>\n<\/div>\n<\/div>\n<p>Model: [latex]42\\div 7[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q440138\">Show Solution<\/span><\/p>\n<div id=\"q440138\" class=\"hidden-answer\" style=\"display: none\">\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24215614\/CNX_BMath_Figure_01_05_007_img.png\" alt=\"An image showing 42 circles, grouped into 6 groups of 7 by irregular shapes drawn around the circles.\" width=\"242\" height=\"178\" \/>\u00a0 [latex]42\\div 7=6[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>In the video below, we show another way to model division using area.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-2\" title=\"Division of Whole Numbers using Area (No Remainder)\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/jKHAsIcEolM?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n\n\t\t\t <section class=\"citations-section\" role=\"contentinfo\">\n\t\t\t <h3>Candela Citations<\/h3>\n\t\t\t\t\t <div>\n\t\t\t\t\t\t <div id=\"citation-list-9229\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><li>The Language of Division. <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com). <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/WxJxY4aJ9Vk\">https:\/\/youtu.be\/WxJxY4aJ9Vk<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Division of Whole Numbers using Area (No Remainder). <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com). <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/jKHAsIcEolM\">https:\/\/youtu.be\/jKHAsIcEolM<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Question ID: 144450. <strong>Authored by<\/strong>: Alyson Day. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em>. <strong>License Terms<\/strong>: IMathAS Community License CC-BY + GPL<\/li><\/ul><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Specific attribution<\/div><ul class=\"citation-list\"><li>Prealgebra. <strong>Provided by<\/strong>: OpenStax. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em>. <strong>License Terms<\/strong>: Download for free at http:\/\/cnx.org\/contents\/caa57dab-41c7-455e-bd6f-f443cda5519c@9.757<\/li><\/ul><\/div>\n\t\t\t\t\t\t <\/div>\n\t\t\t\t\t <\/div>\n\t\t\t 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