{"id":4600,"date":"2020-04-21T00:19:06","date_gmt":"2020-04-21T00:19:06","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/mathforliberalartscorequisite\/chapter\/multiplying-whole-numbers\/"},"modified":"2025-09-02T22:19:39","modified_gmt":"2025-09-02T22:19:39","slug":"multiplying-whole-numbers","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/mathforliberalartscorequisite\/chapter\/multiplying-whole-numbers\/","title":{"raw":"Multiplying Whole Numbers: Notation and Modeling","rendered":"Multiplying Whole Numbers: Notation and Modeling"},"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 multiplication<\/li>\r\n \t<li>Model multiplication of whole numbers<\/li>\r\n<\/ul>\r\n<\/div>\r\n&nbsp;\r\n<h2>Use Multiplication Notation<\/h2>\r\nSuppose you were asked to count all these pennies shown below.\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24215519\/CNX_BMath_Figure_01_04_001.png\" alt=\"An image of 3 horizontal rows of pennies, each row containing 8 pennies.\" \/>\r\nWould you count the pennies individually? Or would you count the number of pennies in each row and add that number [latex]3[\/latex] times.\r\n<p style=\"text-align: center;\">[latex]8+8+8[\/latex]<\/p>\r\n<p style=\"text-align: left;\">Multiplication is a way to represent repeated addition. So instead of adding [latex]8[\/latex] three times, we could write a multiplication expression.<\/p>\r\n<p style=\"text-align: center;\">[latex]3\\times 8[\/latex]<\/p>\r\n<p style=\"text-align: left;\">We call each number being multiplied a factor and the result the product. We read [latex]3\\times 8[\/latex] as <em>three times eight<\/em>, and the result as <em>the product of three and eight<\/em>.<\/p>\r\n<p style=\"text-align: left;\">There are several symbols that represent multiplication. These include the symbol [latex]\\times [\/latex] as well as the dot, [latex]\\cdot [\/latex] , and parentheses [latex]\\left(\\right)[\/latex]<\/p>\r\n\r\n<div class=\"textbox shaded\">\r\n<h3>Operation Symbols for Multiplication<\/h3>\r\nTo describe multiplication, we can use symbols and words.\r\n<table id=\"fs-id1187222\" class=\"unnumbered\" 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>Operation<\/th>\r\n<th>Notation<\/th>\r\n<th>Expression<\/th>\r\n<th>Read as<\/th>\r\n<th>Result<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr valign=\"top\">\r\n<td>[latex]\\text{Multiplication}[\/latex]<\/td>\r\n<td>[latex]\\times [\/latex]\r\n\r\n[latex]\\cdot [\/latex]\r\n\r\n[latex]\\left(\\right)[\/latex]<\/td>\r\n<td>[latex]3\\times 8[\/latex]\r\n\r\n[latex]3\\cdot 8[\/latex]\r\n\r\n[latex]3\\left(8\\right)[\/latex]<\/td>\r\n<td>[latex]\\text{three times eight}[\/latex]<\/td>\r\n<td>[latex]\\text{the product of 3 and 8}[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nTranslate from math notation to words:\r\n<ol id=\"eip-id1168287342411\" class=\"circled\">\r\n \t<li>[latex]7\\times 6[\/latex]<\/li>\r\n \t<li>[latex]12\\cdot 14[\/latex]<\/li>\r\n \t<li>[latex]6\\left(13\\right)[\/latex]<\/li>\r\n<\/ol>\r\nSolution\r\n<ol>\r\n \t<li>We read this as <em>seven times six<\/em> and the result is <em>the product of seven and six<\/em>.<\/li>\r\n \t<li>We read this as <em>twelve times fourteen<\/em> and the result is <em>the product of twelve and fourteen<\/em>.<\/li>\r\n \t<li>We read this as <em>six times thirteen<\/em> and the result is <em>the product of six and thirteen<\/em>.<\/li>\r\n<\/ol>\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\n[ohm_question hide_question_numbers=1]144396[\/ohm_question]\r\n\r\n<\/div>\r\n&nbsp;\r\n\r\nIn the video below we present more ways that multiplication is expressed with words, and how to translate the words to math notation.\r\n\r\nhttps:\/\/youtu.be\/KavmzEwvh1g\r\n\r\n&nbsp;\r\n<h2>Model Multiplication of Whole Numbers<\/h2>\r\nThere are many ways to model multiplication. Unlike in the previous sections where we used [latex]\\text{base - 10}[\/latex] blocks, here we will use counters to help us understand the meaning of multiplication. A counter is any object that can be used for counting. We will use round blue counters.\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nModel: [latex]3\\times 8[\/latex].\r\n[reveal-answer q=\"661924\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"661924\"]\r\n\r\nSolution\r\nTo model the product [latex]3\\times 8[\/latex], we\u2019ll start with a row 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\/24215521\/CNX_BMath_Figure_01_04_002_img.png\" alt=\"An image of a horizontal row of 8 counters.\" \/>\r\nThe other factor is [latex]3[\/latex], so we\u2019ll make [latex]3[\/latex] rows 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\/24215523\/CNX_BMath_Figure_01_04_003_img.png\" alt=\"An image of 3 horizontal rows of counters, each row containing 8 counters.\" \/>\r\nNow we can count the result. There are [latex]24[\/latex] counters in all.\r\n[latex]3\\times 8=24[\/latex]\r\n\r\nIf you look at the counters sideways, you\u2019ll see that we could have also made [latex]8[\/latex] rows of [latex]3[\/latex] counters. The product would have been the same. We\u2019ll get back to this idea later.\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 each multiplication: [latex]4\\times 6[\/latex]\r\n[reveal-answer q=\"294903\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"294903\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24215524\/CNX_BMath_Figure_01_04_004_img.png\" alt=\"No Alt Text\" \/>[\/hidden-answer]\r\n\r\n&nbsp;\r\n\r\nModel each multiplication: [latex]5\\times 7[\/latex]\r\n[reveal-answer q=\"382631\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"382631\"]\r\n\r\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24215525\/CNX_BMath_Figure_01_04_005_img.png\" alt=\"No Alt Text\" \/>[\/hidden-answer]\r\n\r\n<\/div>\r\n&nbsp;","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li>Use symbols and words to represent multiplication<\/li>\n<li>Model multiplication of whole numbers<\/li>\n<\/ul>\n<\/div>\n<p>&nbsp;<\/p>\n<h2>Use Multiplication Notation<\/h2>\n<p>Suppose you were asked to count all these pennies shown below.<\/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\/24215519\/CNX_BMath_Figure_01_04_001.png\" alt=\"An image of 3 horizontal rows of pennies, each row containing 8 pennies.\" \/><br \/>\nWould you count the pennies individually? Or would you count the number of pennies in each row and add that number [latex]3[\/latex] times.<\/p>\n<p style=\"text-align: center;\">[latex]8+8+8[\/latex]<\/p>\n<p style=\"text-align: left;\">Multiplication is a way to represent repeated addition. So instead of adding [latex]8[\/latex] three times, we could write a multiplication expression.<\/p>\n<p style=\"text-align: center;\">[latex]3\\times 8[\/latex]<\/p>\n<p style=\"text-align: left;\">We call each number being multiplied a factor and the result the product. We read [latex]3\\times 8[\/latex] as <em>three times eight<\/em>, and the result as <em>the product of three and eight<\/em>.<\/p>\n<p style=\"text-align: left;\">There are several symbols that represent multiplication. These include the symbol [latex]\\times[\/latex] as well as the dot, [latex]\\cdot[\/latex] , and parentheses [latex]\\left(\\right)[\/latex]<\/p>\n<div class=\"textbox shaded\">\n<h3>Operation Symbols for Multiplication<\/h3>\n<p>To describe multiplication, we can use symbols and words.<\/p>\n<table id=\"fs-id1187222\" class=\"unnumbered\" summary=\"A table with 5 columns and 1 row. The first column is labeled\">\n<thead>\n<tr valign=\"top\">\n<th>Operation<\/th>\n<th>Notation<\/th>\n<th>Expression<\/th>\n<th>Read as<\/th>\n<th>Result<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr valign=\"top\">\n<td>[latex]\\text{Multiplication}[\/latex]<\/td>\n<td>[latex]\\times[\/latex]<\/p>\n<p>[latex]\\cdot[\/latex]<\/p>\n<p>[latex]\\left(\\right)[\/latex]<\/td>\n<td>[latex]3\\times 8[\/latex]<\/p>\n<p>[latex]3\\cdot 8[\/latex]<\/p>\n<p>[latex]3\\left(8\\right)[\/latex]<\/td>\n<td>[latex]\\text{three times eight}[\/latex]<\/td>\n<td>[latex]\\text{the product of 3 and 8}[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Translate from math notation to words:<\/p>\n<ol id=\"eip-id1168287342411\" class=\"circled\">\n<li>[latex]7\\times 6[\/latex]<\/li>\n<li>[latex]12\\cdot 14[\/latex]<\/li>\n<li>[latex]6\\left(13\\right)[\/latex]<\/li>\n<\/ol>\n<p>Solution<\/p>\n<ol>\n<li>We read this as <em>seven times six<\/em> and the result is <em>the product of seven and six<\/em>.<\/li>\n<li>We read this as <em>twelve times fourteen<\/em> and the result is <em>the product of twelve and fourteen<\/em>.<\/li>\n<li>We read this as <em>six times thirteen<\/em> and the result is <em>the product of six and thirteen<\/em>.<\/li>\n<\/ol>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm144396\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=144396&theme=oea&iframe_resize_id=ohm144396\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>&nbsp;<\/p>\n<p>In the video below we present more ways that multiplication is expressed with words, and how to translate the words to math notation.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-1\" title=\"The Language of Multiplication\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/KavmzEwvh1g?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p>&nbsp;<\/p>\n<h2>Model Multiplication of Whole Numbers<\/h2>\n<p>There are many ways to model multiplication. Unlike in the previous sections where we used [latex]\\text{base - 10}[\/latex] blocks, here we will use counters to help us understand the meaning of multiplication. A counter is any object that can be used for counting. We will use round blue counters.<\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Model: [latex]3\\times 8[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q661924\">Show Solution<\/span><\/p>\n<div id=\"q661924\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution<br \/>\nTo model the product [latex]3\\times 8[\/latex], we\u2019ll start with a row 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\/24215521\/CNX_BMath_Figure_01_04_002_img.png\" alt=\"An image of a horizontal row of 8 counters.\" \/><br \/>\nThe other factor is [latex]3[\/latex], so we\u2019ll make [latex]3[\/latex] rows 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\/24215523\/CNX_BMath_Figure_01_04_003_img.png\" alt=\"An image of 3 horizontal rows of counters, each row containing 8 counters.\" \/><br \/>\nNow we can count the result. There are [latex]24[\/latex] counters in all.<br \/>\n[latex]3\\times 8=24[\/latex]<\/p>\n<p>If you look at the counters sideways, you\u2019ll see that we could have also made [latex]8[\/latex] rows of [latex]3[\/latex] counters. The product would have been the same. We\u2019ll get back to this idea later.<\/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 each multiplication: [latex]4\\times 6[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q294903\">Show Solution<\/span><\/p>\n<div id=\"q294903\" class=\"hidden-answer\" style=\"display: none\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24215524\/CNX_BMath_Figure_01_04_004_img.png\" alt=\"No Alt Text\" \/><\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<p>Model each multiplication: [latex]5\\times 7[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q382631\">Show Solution<\/span><\/p>\n<div id=\"q382631\" class=\"hidden-answer\" style=\"display: none\">\n<p><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24215525\/CNX_BMath_Figure_01_04_005_img.png\" alt=\"No Alt Text\" \/><\/div>\n<\/div>\n<\/div>\n<p>&nbsp;<\/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-4600\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Original<\/div><ul class=\"citation-list\"><li>Question ID 144396. <strong>Authored by<\/strong>: Lumen Learning. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><\/ul><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><li>The Language of Multiplication. <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com). <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/KavmzEwvh1g\">https:\/\/youtu.be\/KavmzEwvh1g<\/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: 144396. <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 <\/section>","protected":false},"author":17533,"menu_order":12,"template":"","meta":{"_candela_citation":"[{\"type\":\"original\",\"description\":\"Question ID 144396\",\"author\":\"Lumen Learning\",\"organization\":\"\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"cc\",\"description\":\"The Language of Multiplication\",\"author\":\"James Sousa (Mathispower4u.com)\",\"organization\":\"\",\"url\":\"https:\/\/youtu.be\/KavmzEwvh1g\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"cc-attribution\",\"description\":\"Prealgebra\",\"author\":\"\",\"organization\":\"OpenStax\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"Download for free at 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