{"id":618,"date":"2023-03-09T14:53:37","date_gmt":"2023-03-09T14:53:37","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/?post_type=chapter&#038;p=618"},"modified":"2025-08-23T01:00:27","modified_gmt":"2025-08-23T01:00:27","slug":"whole-numbers-fresh-take","status":"web-only","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/chapter\/whole-numbers-fresh-take\/","title":{"raw":"Whole Numbers: Fresh Take","rendered":"Whole Numbers: Fresh Take"},"content":{"raw":"<section class=\"textbox learningGoals\">\r\n<ul>\r\n\t<li>Identify whole numbers and counting numbers<\/li>\r\n\t<li>Write whole numbers in words<\/li>\r\n\t<li>Round whole numbers<\/li>\r\n\t<li>Add, subtract, multiply, and divide whole numbers<\/li>\r\n<\/ul>\r\n<\/section>\r\n<h2>Whole Numbers and Place Value<\/h2>\r\n<div class=\"textbox shaded\">\r\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\r\n<p><strong>Counting numbers<\/strong> start with [latex]1[\/latex] and continue.<\/p>\r\n<p><strong>Whole numbers<\/strong> are the counting numbers and zero.<\/p>\r\n<p>Finding the <strong>place value<\/strong> of a whole number primarily revolves around recognizing the value each digit holds based on its position in the number. Starting from the right, the first digit represents ones, followed by tens, hundreds, thousands, and so on as we move leftwards.<\/p>\r\n<\/div>\r\n<section class=\"textbox seeExample\">Which of the following are\r\n\r\n<ol style=\"list-style-type: decimal;\">\r\n\t<li>counting numbers?<\/li>\r\n\t<li>whole numbers?<\/li>\r\n<\/ol>\r\n<p style=\"padding-left: 60px;\">[latex]0,\\frac{1}{4},3,5.2,15,105[\/latex]<\/p>\r\n<p>[reveal-answer q=\"664749\"]Show Solution[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"664749\"]<\/p>\r\n<ol>\r\n\t<li>The counting numbers start at [latex]1[\/latex], so [latex]0[\/latex] is not a counting number. The numbers [latex]3,15,\\text{and }105[\/latex] are all counting numbers.<\/li>\r\n\t<li>Whole numbers are counting numbers and [latex]0[\/latex]. The numbers [latex]0,3,15,\\text{and }105[\/latex] are whole numbers.<\/li>\r\n<\/ol>\r\n<p>The numbers [latex]\\frac{1}{4}[\/latex] and [latex]5.2[\/latex] are neither counting numbers nor whole numbers. We will discuss these numbers later.<br \/>\r\n[\/hidden-answer]<\/p>\r\n<\/section>\r\n<p>Money gives us another, more familiar model of place value. Suppose a wallet contains three [latex]$100[\/latex] bills, seven [latex]$10[\/latex] bills, and four\u00a0[latex]$1[\/latex] bills. The amounts are summarized in the image below. How much money is in the wallet?<\/p>\r\n<center>\r\n[caption id=\"\" align=\"aligncenter\" width=\"597\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24215158\/CNX_BMath_Figure_01_01_002.png\" alt=\"Three groups of dollar bills. The first group is labeled 'Three $100 bills. 3 times $100 = $300,' the second is labeled 'Seven $10 bills. 7 times $10 = $70,' and the third is labeled 'Four $1 bills. 4 times $1 = $4.'\" width=\"597\" height=\"219\" \/> Figure 1. How much money is in this wallet?[\/caption]\r\n<\/center>\r\n<p>&nbsp;<\/p>\r\n<p>Find the total value of each kind of bill, and then add to find the total. The wallet contains\u00a0[latex]$374[\/latex].<\/p>\r\n<center>\r\n[caption id=\"\" align=\"aligncenter\" width=\"126\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24215200\/CNX_BMath_Figure_01_01_003_img.png\" alt=\"$300 + $70 + $4 adding up to $374.\" width=\"126\" height=\"90\" \/> Figure 2. Add the money in the wallet to get $374[\/caption]\r\n<\/center>\r\n<p>&nbsp;<\/p>\r\n<section class=\"textbox watchIt\"><iframe src=\"\/\/plugin.3playmedia.com\/show?mf=10374280&amp;p3sdk_version=1.10.1&amp;p=20361&amp;pt=375&amp;video_id=f3YjmQFSxl0&amp;video_target=tpm-plugin-stmep37d-f3YjmQFSxl0\" width=\"800px\" height=\"450px\" frameborder=\"0\" marginwidth=\"0px\" marginheight=\"0px\"><\/iframe>\r\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Quantitative+Reasoning+-+2023+Build\/Transcriptions\/Finding+the+Value+of+the+Underlined+Digit+%7C+Whole+Number+Place+Value+%7C+Math+with+Mr+J.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cFinding the Value of the Underlined Digit | Whole Number Place Value | Math with Mr. J\u201d here (opens in new window).<\/a><\/p>\r\n<\/section>\r\n<p>The video below shows more examples of how to determine the place value of a digit in a number.<\/p>\r\n<section class=\"textbox watchIt\">\r\n<p>[embed]https:\/\/www.youtube.com\/embed\/0-4rLGvVEOI[\/embed]<\/p>\r\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Quantitative+Reasoning+-+2023+Build\/Transcriptions\/Examples_+Determining+Place+Value.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cExamples: Determining Place Value\u201d here (opens in new window).<\/a><\/p>\r\n<\/section>\r\n<h3>Use Place Value to Write Whole Numbers<\/h3>\r\n<div class=\"textbox shaded\">\r\n<p><strong>The Main Idea<\/strong><\/p>\r\n<p><span style=\"font-size: 14pt;\"><strong>Name a whole number in words<\/strong><\/span><\/p>\r\n<ol style=\"list-style-type: decimal;\">\r\n\t<li>Starting at the digit on the left, name the number in each period, followed by the period name. Do not include the period name for the ones.<\/li>\r\n\t<li>Use commas in the number to separate the periods.<\/li>\r\n<\/ol>\r\n<p><strong><span style=\"font-size: 14pt;\">Write a whole number from words<\/span><\/strong><\/p>\r\n<ol style=\"list-style-type: decimal;\">\r\n\t<li>Identify the words that indicate periods. (Remember the ones period is never named).<\/li>\r\n\t<li>Draw three blanks to indicate the number of places needed in each period. Separate the periods by commas.<\/li>\r\n\t<li>Name the number in each period and place the digits in the correct place value position.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<section class=\"textbox seeExample\">A state budget was about [latex]\\text{\\$77}[\/latex] billion. Write the budget in standard form.<br \/>\r\n[reveal-answer q=\"595048\"]Show Solution[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"595048\"]\r\n\r\n<p>Identify the periods. In this case, only two digits are given and they are in the billions period. To write the entire number, write zeros for all of the other periods.<\/p>\r\n<center><img class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24215228\/CNX_BMath_Figure_01_01_018_img.png\" alt=\"An image with four blocks of text pointing to numerical values. The first block of text reads 'billions,' then '77 billion,' then an arrow to the number 77 with a blank spot to the left of it. The second block of text reads 'millions' with an arrow pointing to 000. The third block of text reads 'thousands,' with an arrow pointing to 000. The fourth block of text reads 'ones' with an arrow pointing to 000.\" width=\"688\" height=\"166\" \/><\/center>\r\n<p>&nbsp;<\/p>\r\n<p>So the budget was about [latex]\\text{\\$77,000,000,000.}[\/latex][\/hidden-answer]<\/p>\r\n<\/section>\r\n<p>&nbsp;<\/p>\r\n<section class=\"textbox seeExample\">Write each number in standard form:\r\n\r\n<ol>\r\n\t<li id=\"fs-id1800228\">The closest distance from Earth to Mars is about [latex]34[\/latex] million miles.<\/li>\r\n\t<li>The total weight of an aircraft carrier is [latex]204[\/latex] million pounds.<\/li>\r\n<\/ol>\r\n<p>[reveal-answer q=\"198073\"]Show Solution[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"198073\"]<\/p>\r\n<ol>\r\n\t<li>[latex]34,000,000[\/latex] miles<\/li>\r\n\t<li>[latex]204,000,000[\/latex] pounds<\/li>\r\n<\/ol>\r\n<p>[\/hidden-answer]<\/p>\r\n<\/section>\r\n<p>The video below shows more examples of how to use place value to write the name of a whole number.<\/p>\r\n<section class=\"textbox watchIt\">\r\n<p>[embed]https:\/\/www.youtube.com\/embed\/H9HpkvY_iaE[\/embed]<\/p>\r\n<p>You can view the <a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Quantitative+Reasoning+-+2023+Build\/Transcriptions\/Example_+Write+a+Whole+Number+in+Digits+from+Words.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cExample: Write a Whole Number in Digits from Words\u201d here (opens in new window).<\/a><\/p>\r\n<\/section>\r\n<h2>Rounding Whole Numbers<\/h2>\r\n<div class=\"textbox shaded\">\r\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\r\n<p><strong>Rounding<\/strong> is a method used to shorten or simplify numbers by adjusting them to the closest specified value.<\/p>\r\n<p>To round a whole number to a specific place value:<\/p>\r\n<ol id=\"eip-id1168287215567\" class=\"stepwise\">\r\n\t<li>Locate the given place value. All digits to the left of that place value do not change.<\/li>\r\n\t<li>Underline the digit to the right of the given place value.<\/li>\r\n\t<li>Determine if this digit is greater than or equal to [latex]5[\/latex].<br \/>\r\n<ul id=\"fs-id2486248\">\r\n\t<li>Yes\u2014add [latex]1[\/latex] to the digit in the given place value.<\/li>\r\n\t<li>No\u2014do not change the digit in the given place value.<\/li>\r\n<\/ul>\r\n<\/li>\r\n\t<li>Replace all digits to the right of the given place value with zeros.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<section class=\"textbox proTip\">If the digit to the right of your given place value is [latex]5[\/latex] or above, give your given place value a shove. If the digit to the right of your given place value is [latex]4[\/latex] or less, let your given place value rest.<\/section>\r\n<p>Watch the video below for more examples of how to round whole numbers to a given place value.<\/p>\r\n<section class=\"textbox watchIt\">\r\n<p>[embed]https:\/\/www.youtube.com\/embed\/tpFCYyTi1hc[\/embed]<\/p>\r\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Quantitative+Reasoning+-+2023+Build\/Transcriptions\/Ex_+Rounding+to+the+Various+Place+Values+-+Number+Line+_+Formal+Rules.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cEx: Rounding to the Various Place Values - Number Line \/ Formal Rules\u201d here (opens in new window).<\/a><\/p>\r\n<\/section>\r\n<h2>Addition of Whole Numbers<\/h2>\r\n<div class=\"textbox shaded\">\r\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\r\n<p><strong>Addition<\/strong> of whole numbers involves combining the values of two or more numbers, using the plus (+) sign as the notation.<\/p>\r\n<table>\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>Addition<\/td>\r\n<td>[latex]+[\/latex]<\/td>\r\n<td>[latex]3+4[\/latex]<\/td>\r\n<td>three plus four<\/td>\r\n<td>the sum of [latex]3[\/latex] and [latex]4[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<p>&nbsp;<\/p>\r\n<p>The<strong> identity property of addition<\/strong> states that when a number is added to zero, the original number remains unchanged. This underscores the fact that zero is the 'identity' for addition.<\/p>\r\n<p>The <strong>commutative property of addition<\/strong>, on the other hand, affirms that the order in which numbers are added does not affect the result, that is, for any two whole numbers [latex]a[\/latex] and [latex]b[\/latex], [latex]a + b[\/latex] equals [latex]b + a[\/latex].<\/p>\r\n<\/div>\r\n<p>Addition is really just counting. This is easy to see by modeling. We will model addition with base-[latex]10[\/latex] blocks. Remember, a block represents [latex]1[\/latex] and a rod represents [latex]10[\/latex]. Let\u2019s start by modeling the addition expression we just considered, [latex]3+4[\/latex].<br \/>\r\nEach addend is less than [latex]10[\/latex], so we can use ones blocks.<\/p>\r\n<table>\r\n<tbody>\r\n<tr>\r\n<td>We start by modeling the first number with [latex]3[\/latex] blocks.<\/td>\r\n<td><img class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24215314\/CNX_BMath_Figure_01_02_019_img-02.png\" alt=\"Three individual blocks\" width=\"186\" height=\"51\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Then we model the second number with [latex]4[\/latex] blocks.<\/td>\r\n<td><img class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24215315\/CNX_BMath_Figure_01_02_019_img-03.png\" alt=\"Three individual blocks next to four individual blocks\" width=\"186\" height=\"51\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Count the total number of blocks.<\/td>\r\n<td><img class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24215316\/CNX_BMath_Figure_01_02_019_img-04.png\" alt=\"7 individual blocks\" width=\"186\" height=\"51\" \/><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<p><span style=\"font-size: 10pt;\"><strong>Figure 3. Model these blocks to add 3 and 4<\/strong><\/span><\/p>\r\n<p>There are [latex]7[\/latex] blocks in all. We use an equal sign [latex]\\text{(=)}[\/latex] to show the sum. A math sentence that shows that two expressions are equal is called an equation. We have shown that. [latex]3+4=7[\/latex].<\/p>\r\n<section class=\"textbox seeExample\">Model the addition [latex]2+6[\/latex].<br \/>\r\n[reveal-answer q=\"771692\"]Show Solution[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"771692\"]<br \/>\r\n[latex]2+6[\/latex] means the sum of [latex]2[\/latex] and [latex]6[\/latex]<br \/>\r\nEach addend is less than [latex]10[\/latex], so we can use ones blocks.\r\n\r\n<table>\r\n<tbody>\r\n<tr>\r\n<td>Model the first number with [latex]2[\/latex] blocks.<\/td>\r\n<td><img class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24215317\/CNX_BMath_Figure_01_02_016_img-02.png\" alt=\"Two individual blocks\" width=\"204\" height=\"41\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Model the second number with [latex]6[\/latex] blocks.<\/td>\r\n<td><img class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24215318\/CNX_BMath_Figure_01_02_016_img-03.png\" alt=\"Two individual blocks next to 6 individual blocks\" width=\"204\" height=\"45\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Count the total number of blocks<\/td>\r\n<td><img class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24215319\/CNX_BMath_Figure_01_02_016_img-04.png\" alt=\"8 individual blocks\" width=\"204\" height=\"46\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>There are [latex]8[\/latex] blocks in all, so [latex]2+6=8[\/latex].<\/td>\r\n<td>&nbsp;<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<p>[\/hidden-answer]<\/p>\r\n<\/section>\r\n<p>When the result is [latex]10[\/latex] or more ones blocks, we will exchange the [latex]10[\/latex] blocks for one rod.<\/p>\r\n<p>Next we will model adding two digit numbers.<\/p>\r\n<section class=\"textbox seeExample\">Model each addition:\r\n\r\n<ol style=\"list-style-type: decimal;\">\r\n\t<li>[latex]15+27[\/latex]<\/li>\r\n\t<li>[latex]16+29[\/latex]<\/li>\r\n<\/ol>\r\n<p>[reveal-answer q=\"586067\"]Show Solution[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"586067\"]<\/p>\r\n<ol style=\"list-style-type: decimal;\">\r\n\t<li><img class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24215334\/CNX_BMath_Figure_01_02_014_img.png\" alt=\"4 horizontal rods of 10 blocks each and two individual blocks. The whole thing is labeled 15+27=42\" width=\"168\" height=\"109\" \/>\r\n<p>&nbsp;<\/p>\r\n<\/li>\r\n\t<li><img class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24215336\/CNX_BMath_Figure_01_02_015_img.png\" alt=\"4 horizontal rods of 10 blocks each and five individual blocks. The whole thing is labeled 16+29=45.\" width=\"218\" height=\"108\" \/><\/li>\r\n<\/ol>\r\n<p>[\/hidden-answer]<\/p>\r\n<\/section>\r\n<p>Watch the video below for another example of how to use base [latex]10[\/latex] blocks to model the sum of two whole numbers.<\/p>\r\n<section class=\"textbox watchIt\">\r\n<p>[embed]https:\/\/www.youtube.com\/embed\/_smXQAPbygQ[\/embed]<\/p>\r\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Quantitative+Reasoning+-+2023+Build\/Transcriptions\/Model+Addition+of+Two+Digit+Whole+Numbers+Using+Base+Ten+Blocks.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cModel Addition of Two Digit Whole Numbers Using Base Ten Blocks\u201d here (opens in new window).<\/a><\/p>\r\n<\/section>\r\n<h2>Subtraction of Whole Numbers<\/h2>\r\n<div class=\"textbox shaded\">\r\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\r\n<p><strong>Subtraction<\/strong> of whole numbers is the process of taking away one number from another to find the difference, often represented with a minus (-) sign.<\/p>\r\n<table>\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>Subtraction<\/td>\r\n<td>[latex]-[\/latex]<\/td>\r\n<td>[latex]7 - 3[\/latex]<\/td>\r\n<td>seven minus three<\/td>\r\n<td>the difference of [latex]7[\/latex] and [latex]3[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n<p>A model can help us visualize the process of subtraction much as it did with addition. Again, we will use [latex]\\text{base - 10}[\/latex] blocks. Remember a block represents\u00a0[latex]1[\/latex] and a rod represents\u00a0[latex]10[\/latex]. Let\u2019s start by modeling the subtraction expression we just considered, [latex]7 - 3[\/latex].<\/p>\r\n<table>\r\n<tbody>\r\n<tr>\r\n<td>We start by modeling the first number, [latex]7[\/latex].<\/td>\r\n<td><img class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24215416\/CNX_BMath_Figure_01_03_018_img-02.png\" alt=\"7 individual blocks\" width=\"199\" height=\"49\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Now take away the second number, [latex]3[\/latex]. We'll circle [latex]3[\/latex] blocks to show that we are taking them away.<\/td>\r\n<td><img class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24215417\/CNX_BMath_Figure_01_03_018_img-03.png\" alt=\"3 individual blocks in a circle and 4 individual blocks outside of it\" width=\"199\" height=\"69\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Count the number of blocks remaining.<\/td>\r\n<td><img class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24215418\/CNX_BMath_Figure_01_03_018_img-04.png\" alt=\"4 individual blocks\" width=\"199\" height=\"27\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>There are [latex]4[\/latex] ones blocks left.<\/td>\r\n<td>We have shown that [latex]7 - 3=4[\/latex] .<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<section><strong><span style=\"font-size: 10pt;\">Figure 4. Model these blocks to figure out 7 minus 3<\/span><\/strong><\/section>\r\n<section class=\"textbox seeExample\">Model the subtraction: [latex]8 - 2[\/latex]<br \/>\r\n[reveal-answer q=\"250813\"]Show Solution[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"250813\"][latex]8 - 2[\/latex] means the difference of [latex]8[\/latex] and [latex]2[\/latex]. Model the first, [latex]8[\/latex]. Take away the second number, [latex]2[\/latex]. Count the number of blocks remaining. There are [latex]6[\/latex] ones blocks left. We have shown that [latex]8 - 2=6[\/latex] .[\/hidden-answer]<\/section>\r\n<p>As we did with addition, we can describe the models as ones blocks and tens rods, or we can simply say ones and tens. Next we will model subtracting two digit numbers.<\/p>\r\n<section class=\"textbox seeExample\">Model each subtraction:\r\n\r\n<ol style=\"list-style-type: decimal;\">\r\n\t<li>[latex]42 - 27[\/latex]<\/li>\r\n\t<li>[latex]45 - 29[\/latex]<\/li>\r\n<\/ol>\r\n<p>[reveal-answer q=\"586167\"]Show Solution[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"586167\"]<\/p>\r\n<ol style=\"list-style-type: decimal;\">\r\n\t<li><img class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24215439\/CNX_BMath_Figure_01_03_016_img.png\" alt=\"One rod of 10 blocks and 5 individual blocks outside of a circle and two rods of 10 blocks and 7 individual blocks in a circle. The whole thing is labeled 42-27=15\" width=\"243\" height=\"166\" \/>\r\n<p>&nbsp;<\/p>\r\n<\/li>\r\n\t<li><img class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24215440\/CNX_BMath_Figure_01_03_017_img.png\" alt=\"One rod of 10 blocks and 6 individual blocks outside of a circle and two rods of 10 blocks and 9 individual blocks in a circle. The whole thing is labeled 45-29=16.\" width=\"245\" height=\"185\" \/><\/li>\r\n<\/ol>\r\n<p>[\/hidden-answer]<\/p>\r\n<\/section>\r\n<p>Watch the video below to see more examples of using base 10 blocks to model subtraction.<\/p>\r\n<section class=\"textbox watchIt\">\r\n<p>[embed]https:\/\/www.youtube.com\/embed\/FQI0IgA-7fI[\/embed]<\/p>\r\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Quantitative+Reasoning+-+2023+Build\/Transcriptions\/Model+Subtraction+of+Two+Digit+Whole+Numbers+Using+Base+Ten+Blocks.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cModel Subtraction of Two Digit Whole Numbers Using Base Ten Blocks\u201d here (opens in new window).<\/a><\/p>\r\n<\/section>\r\n<p>Watch the video below to see another example of subtracting whole numbers by lining up place values.<\/p>\r\n<section class=\"textbox watchIt\">\r\n<p>[embed]https:\/\/www.youtube.com\/embed\/hneqy1EGACs[\/embed]<\/p>\r\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Quantitative+Reasoning+-+2023+Build\/Transcriptions\/Example_+Subtracting+Whole+Numbers.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cExample: Subtracting Whole Numbers\u201d here (opens in new window).<\/a><\/p>\r\n<\/section>\r\n<h2>Multiplying Whole Numbers<\/h2>\r\n<div class=\"textbox shaded\">\r\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\r\n<p><strong>Multiplication<\/strong> of whole numbers is a process where one number, known as a factor, is repeatedly added to itself based on the value of the second factor, resulting in a total called the product.<\/p>\r\n<table>\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>\r\n<p>[latex]\\times [\/latex]<\/p>\r\n<p>[latex]\\cdot [\/latex]<\/p>\r\n<p>[latex]\\left(\\right)[\/latex]<\/p>\r\n<\/td>\r\n<td>\r\n<p>[latex]3\\times 8[\/latex]<\/p>\r\n<p>[latex]3\\cdot 8[\/latex]<\/p>\r\n<p>[latex]3\\left(8\\right)[\/latex]<\/p>\r\n<\/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<p>&nbsp;<\/p>\r\n<p>The <strong>multiplication property of zero<\/strong> states the product of any number and [latex]0[\/latex] is [latex]0[\/latex].<\/p>\r\n<p>The <strong>identity property of multiplication<\/strong> states the product of any number and [latex]1[\/latex] is the number.<\/p>\r\n<p>The <strong>commutative property of multiplication<\/strong> states that changing the order of the factors does not change their product.<\/p>\r\n<\/div>\r\n<p>Just like with subtraction and addition, modeling multiplication can help with understanding what is going on. There are many ways to model multiplication. Unlike in the previous sections where we used base-[latex]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. Let\u2019s start by modeling the multiplication expression we just considered, [latex]3\\times 8[\/latex].<\/p>\r\n<p>To model the product [latex]3\\times 8[\/latex], we\u2019ll start with a row of [latex]8[\/latex] counters.<\/p>\r\n<center>\r\n[caption id=\"\" align=\"alignnone\" width=\"212\"]<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.\" width=\"212\" height=\"25\" \/> Figure 5. 8 counters[\/caption]\r\n<\/center>\r\n<p>&nbsp;<\/p>\r\n<p>The other factor is [latex]3[\/latex], so we\u2019ll make [latex]3[\/latex] rows of [latex]8[\/latex] counters.<\/p>\r\n<center>\r\n[caption id=\"\" align=\"alignnone\" width=\"212\"]<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.\" width=\"212\" height=\"80\" \/> Figure 6. 3 rows of 8 counters[\/caption]\r\n<\/center>\r\n<p>&nbsp;<\/p>\r\n<p>Now we can count the result. There are [latex]24[\/latex] counters in all.<\/p>\r\n<center>[latex]3\\times 8=24[\/latex]<\/center>\r\n<p>&nbsp;<\/p>\r\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.<\/p>\r\n<section class=\"textbox seeExample\">Model each multiplication:\r\n\r\n<ol style=\"list-style-type: decimal;\">\r\n\t<li>[latex]4\\times 6[\/latex]<\/li>\r\n\t<li>[latex]5\\times 7[\/latex]<\/li>\r\n<\/ol>\r\n<p>[reveal-answer q=\"586061\"]Show Solution[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"586061\"]<\/p>\r\n<ol>\r\n\t<li><img class=\"alignnone\" 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=\"4 rows of 6 counters\" width=\"123\" height=\"88\" \/>\r\n<p>&nbsp;<\/p>\r\n<\/li>\r\n<\/ol>\r\n<p>\u00a0 \u00a0 \u00a02. <img class=\"alignnone\" 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=\"5 rows of 7 counters\" width=\"144\" height=\"110\" \/><\/p>\r\n<p>[\/hidden-answer]<\/p>\r\n<\/section>\r\n<p>In the video below, we summarize the concepts covered in multiplying whole numbers including the multiplication property of zero, the identity property of multiplication, and the commutative property of multiplication.<\/p>\r\n<section class=\"textbox watchIt\">\r\n<p>[embed]https:\/\/www.youtube.com\/embed\/kW7JBfplJGE[\/embed]<\/p>\r\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Quantitative+Reasoning+-+2023+Build\/Transcriptions\/Multiplying+Whole+Numbers.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cMultiplying Whole Numbers\u201d here (opens in new window).<\/a><\/p>\r\n<\/section>\r\n<h2>Dividing Whole Numbers<\/h2>\r\n<div class=\"textbox shaded\">\r\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\r\n<p><strong>Division<\/strong> of whole numbers involves partitioning a given quantity, known as the dividend, into equal parts, with the number of these parts represented by the divisor, resulting in the quotient.<\/p>\r\n<table>\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{Division}[\/latex]<\/td>\r\n<td>\r\n<p>[latex]a \\div b[\/latex]<\/p>\r\n<p>[latex]\\frac{a}{b}[\/latex]<\/p>\r\n<p>[latex]b\\overline{)a}[\/latex]<\/p>\r\n<p>[latex]a\/b[\/latex]<\/p>\r\n<\/td>\r\n<td>\r\n<p>[latex]12\\div 4[\/latex]<\/p>\r\n<p>[latex]\\frac{12}{4}[\/latex]<\/p>\r\n<p>[latex]4\\overline{)12}[\/latex]<\/p>\r\n<p>[latex]12\/4[\/latex]<\/p>\r\n<\/td>\r\n<td>[latex]\\text{Twelve divided by four}[\/latex]<\/td>\r\n<td>[latex]\\text{the quotient of 12 and 4}[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<p>&nbsp;<\/p>\r\n<p>The <strong>division properties of one<\/strong> states dividing any number, except [latex]0[\/latex], by itself produces a quotient of [latex]1[\/latex] and any number divided by [latex]1[\/latex] produces a quotient of the number.<\/p>\r\n<p>The <strong>division properties of zero<\/strong> states any number divided by zero is undefined, while zero divided by any number (except zero) is always zero.<\/p>\r\n<\/div>\r\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. Let\u2019s start by modeling the division expression we just considered, [latex]24\\div 8[\/latex].<\/p>\r\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 \/>\r\nModel the dividend. Start with [latex]24[\/latex] counters.<\/p>\r\n<center>\r\n[caption id=\"\" align=\"alignnone\" width=\"301\"]<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.\" width=\"301\" height=\"114\" \/> Figure 7. 24 counters, randomly placed[\/caption]\r\n<\/center>\r\n<p>&nbsp;<\/p>\r\n<p>The divisor tell us the number of counters we want in each group. Form groups of [latex]8[\/latex] counters.<\/p>\r\n<center>\r\n[caption id=\"\" align=\"alignnone\" width=\"312\"]<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.\" width=\"312\" height=\"132\" \/> Figure 8. Group those counters into 3 bubbles[\/caption]\r\n<\/center>\r\n<p>&nbsp;<\/p>\r\n<p>Count the number of groups. There are [latex]3[\/latex] groups.<\/p>\r\n<center>[latex]24\\div 8=3[\/latex]<\/center>\r\n<p>&nbsp;<\/p>\r\n<section class=\"textbox seeExample\">Model each division:\r\n\r\n<ol style=\"list-style-type: decimal;\">\r\n\t<li>[latex]24\\div 6[\/latex]<\/li>\r\n\t<li>[latex]42\\div 7[\/latex]<\/li>\r\n<\/ol>\r\n<p>[reveal-answer q=\"586069\"]Show Solution[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"586069\"]<\/p>\r\n<ol style=\"list-style-type: decimal;\">\r\n\t<li><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=\"24 counters grouped into 4 sets of 6\" width=\"186\" height=\"182\" \/>\u00a0\u00a0[latex]24\\div 6=4[\/latex]\r\n\r\n<p>&nbsp;<\/p>\r\n<\/li>\r\n\t<li><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=\"42 counters grouped in 6 groups of 7\" width=\"242\" height=\"178\" \/>\u00a0 [latex]42\\div 7=6[\/latex]<\/li>\r\n<\/ol>\r\n<p>[\/hidden-answer]<\/p>\r\n<\/section>\r\n<p>In the video below we show another way to model division using area.<\/p>\r\n<section class=\"textbox watchIt\">\r\n<p>[embed]https:\/\/www.youtube.com\/embed\/jKHAsIcEolM[\/embed]<\/p>\r\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Quantitative+Reasoning+-+2023+Build\/Transcriptions\/Division+of+Whole+Numbers+using+Area+(No+Remainder).txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cDivision of Whole Numbers using Area (No Remainder)\u201d here (opens in new window).<\/a><\/p>\r\n<\/section>\r\n<p>In the video below we show another example of using long division.<\/p>\r\n<section class=\"textbox watchIt\">\r\n<p>[embed]https:\/\/www.youtube.com\/embed\/KvVhaB5mqr8[\/embed]<\/p>\r\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Quantitative+Reasoning+-+2023+Build\/Transcriptions\/Ex_+Long+Division+-+Two+Digit+Divided+by+One+Digit+(No+Remainder).txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cEx: Long Division - Two Digit Divided by One Digit (No Remainder)\u201d here (opens in new window).<\/a><\/p>\r\n<\/section>\r\n<p>Watch this video for another example of how to use long division to divide a four digit whole number by a two digit whole number.<\/p>\r\n<section class=\"textbox watchIt\">\r\n<p>[embed]https:\/\/www.youtube.com\/embed\/V7Korf09iWI[\/embed]<\/p>\r\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Quantitative+Reasoning+-+2023+Build\/Transcriptions\/Example_+Dividing+Whole+Numbers+without+a+Remainder.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cExample: Dividing Whole Numbers without a Remainder\u201d here (opens in new window).<\/a><\/p>\r\n<\/section>\r\n<p>Watch the video below for another example of how to use long division to divide whole numbers when there is a remainder.<\/p>\r\n<section class=\"textbox watchIt\">\r\n<p>[embed]https:\/\/www.youtube.com\/embed\/UPUcShGCBOs[\/embed]<\/p>\r\n<p>You can view the <a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Quantitative+Reasoning+-+2023+Build\/Transcriptions\/Example_+Dividing+Whole+Numbers+with+a+Remainder.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cExample: Dividing Whole Numbers with a Remainder\u201d here (opens in new window).<\/a><\/p>\r\n<\/section>\r\n<p>For a review of adding, subtracting, multiplying, and division of whole numbers watch the following video.<\/p>\r\n<section class=\"textbox watchIt\"><iframe src=\"\/\/plugin.3playmedia.com\/show?mf=10387435&amp;p3sdk_version=1.10.1&amp;p=20361&amp;pt=375&amp;video_id=HAhg0uXn9GA&amp;video_target=tpm-plugin-a42jw4sm-HAhg0uXn9GA\" width=\"800px\" height=\"450px\" frameborder=\"0\" marginwidth=\"0px\" marginheight=\"0px\"><\/iframe>\r\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Quantitative+Reasoning+-+2023+Build\/Transcriptions\/Whole+Number+Operations+%7C+Adding%2C+Subtracting%2C+Multiplying%2C+and+Dividing.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cWhole Number Operations | Adding, Subtracting, Multiplying, and Dividing\u201d here (opens in new window).<\/a><\/p>\r\n<\/section>","rendered":"<section class=\"textbox learningGoals\">\n<ul>\n<li>Identify whole numbers and counting numbers<\/li>\n<li>Write whole numbers in words<\/li>\n<li>Round whole numbers<\/li>\n<li>Add, subtract, multiply, and divide whole numbers<\/li>\n<\/ul>\n<\/section>\n<h2>Whole Numbers and Place Value<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\n<p><strong>Counting numbers<\/strong> start with [latex]1[\/latex] and continue.<\/p>\n<p><strong>Whole numbers<\/strong> are the counting numbers and zero.<\/p>\n<p>Finding the <strong>place value<\/strong> of a whole number primarily revolves around recognizing the value each digit holds based on its position in the number. Starting from the right, the first digit represents ones, followed by tens, hundreds, thousands, and so on as we move leftwards.<\/p>\n<\/div>\n<section class=\"textbox seeExample\">Which of the following are<\/p>\n<ol style=\"list-style-type: decimal;\">\n<li>counting numbers?<\/li>\n<li>whole numbers?<\/li>\n<\/ol>\n<p style=\"padding-left: 60px;\">[latex]0,\\frac{1}{4},3,5.2,15,105[\/latex]<\/p>\n<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q664749\">Show Solution<\/button><\/p>\n<div id=\"q664749\" class=\"hidden-answer\" style=\"display: none\">\n<ol>\n<li>The counting numbers start at [latex]1[\/latex], so [latex]0[\/latex] is not a counting number. The numbers [latex]3,15,\\text{and }105[\/latex] are all counting numbers.<\/li>\n<li>Whole numbers are counting numbers and [latex]0[\/latex]. The numbers [latex]0,3,15,\\text{and }105[\/latex] are whole numbers.<\/li>\n<\/ol>\n<p>The numbers [latex]\\frac{1}{4}[\/latex] and [latex]5.2[\/latex] are neither counting numbers nor whole numbers. We will discuss these numbers later.\n<\/div>\n<\/div>\n<\/section>\n<p>Money gives us another, more familiar model of place value. Suppose a wallet contains three [latex]$100[\/latex] bills, seven [latex]$10[\/latex] bills, and four\u00a0[latex]$1[\/latex] bills. The amounts are summarized in the image below. How much money is in the wallet?<\/p>\n<div style=\"text-align: center;\">\n<figure style=\"width: 597px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24215158\/CNX_BMath_Figure_01_01_002.png\" alt=\"Three groups of dollar bills. The first group is labeled 'Three $100 bills. 3 times $100 = $300,' the second is labeled 'Seven $10 bills. 7 times $10 = $70,' and the third is labeled 'Four $1 bills. 4 times $1 = $4.'\" width=\"597\" height=\"219\" \/><figcaption class=\"wp-caption-text\">Figure 1. How much money is in this wallet?<\/figcaption><\/figure>\n<\/div>\n<p>&nbsp;<\/p>\n<p>Find the total value of each kind of bill, and then add to find the total. The wallet contains\u00a0[latex]$374[\/latex].<\/p>\n<div style=\"text-align: center;\">\n<figure style=\"width: 126px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24215200\/CNX_BMath_Figure_01_01_003_img.png\" alt=\"$300 + $70 + $4 adding up to $374.\" width=\"126\" height=\"90\" \/><figcaption class=\"wp-caption-text\">Figure 2. Add the money in the wallet to get $374<\/figcaption><\/figure>\n<\/div>\n<p>&nbsp;<\/p>\n<section class=\"textbox watchIt\"><iframe loading=\"lazy\" src=\"\/\/plugin.3playmedia.com\/show?mf=10374280&amp;p3sdk_version=1.10.1&amp;p=20361&amp;pt=375&amp;video_id=f3YjmQFSxl0&amp;video_target=tpm-plugin-stmep37d-f3YjmQFSxl0\" width=\"800px\" height=\"450px\" frameborder=\"0\" marginwidth=\"0px\" marginheight=\"0px\"><\/iframe><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Quantitative+Reasoning+-+2023+Build\/Transcriptions\/Finding+the+Value+of+the+Underlined+Digit+%7C+Whole+Number+Place+Value+%7C+Math+with+Mr+J.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cFinding the Value of the Underlined Digit | Whole Number Place Value | Math with Mr. J\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<p>The video below shows more examples of how to determine the place value of a digit in a number.<\/p>\n<section class=\"textbox watchIt\">\n<p><iframe loading=\"lazy\" id=\"oembed-1\" title=\"Examples:  Determining Place Value\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/0-4rLGvVEOI?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Quantitative+Reasoning+-+2023+Build\/Transcriptions\/Examples_+Determining+Place+Value.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cExamples: Determining Place Value\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<h3>Use Place Value to Write Whole Numbers<\/h3>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea<\/strong><\/p>\n<p><span style=\"font-size: 14pt;\"><strong>Name a whole number in words<\/strong><\/span><\/p>\n<ol style=\"list-style-type: decimal;\">\n<li>Starting at the digit on the left, name the number in each period, followed by the period name. Do not include the period name for the ones.<\/li>\n<li>Use commas in the number to separate the periods.<\/li>\n<\/ol>\n<p><strong><span style=\"font-size: 14pt;\">Write a whole number from words<\/span><\/strong><\/p>\n<ol style=\"list-style-type: decimal;\">\n<li>Identify the words that indicate periods. (Remember the ones period is never named).<\/li>\n<li>Draw three blanks to indicate the number of places needed in each period. Separate the periods by commas.<\/li>\n<li>Name the number in each period and place the digits in the correct place value position.<\/li>\n<\/ol>\n<\/div>\n<section class=\"textbox seeExample\">A state budget was about [latex]\\text{\\$77}[\/latex] billion. Write the budget in standard form.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q595048\">Show Solution<\/button><\/p>\n<div id=\"q595048\" class=\"hidden-answer\" style=\"display: none\">\n<p>Identify the periods. In this case, only two digits are given and they are in the billions period. To write the entire number, write zeros for all of the other periods.<\/p>\n<div style=\"text-align: center;\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24215228\/CNX_BMath_Figure_01_01_018_img.png\" alt=\"An image with four blocks of text pointing to numerical values. The first block of text reads 'billions,' then '77 billion,' then an arrow to the number 77 with a blank spot to the left of it. The second block of text reads 'millions' with an arrow pointing to 000. The third block of text reads 'thousands,' with an arrow pointing to 000. The fourth block of text reads 'ones' with an arrow pointing to 000.\" width=\"688\" height=\"166\" \/><\/div>\n<p>&nbsp;<\/p>\n<p>So the budget was about [latex]\\text{\\$77,000,000,000.}[\/latex]<\/p><\/div>\n<\/div>\n<\/section>\n<p>&nbsp;<\/p>\n<section class=\"textbox seeExample\">Write each number in standard form:<\/p>\n<ol>\n<li id=\"fs-id1800228\">The closest distance from Earth to Mars is about [latex]34[\/latex] million miles.<\/li>\n<li>The total weight of an aircraft carrier is [latex]204[\/latex] million pounds.<\/li>\n<\/ol>\n<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q198073\">Show Solution<\/button><\/p>\n<div id=\"q198073\" class=\"hidden-answer\" style=\"display: none\">\n<ol>\n<li>[latex]34,000,000[\/latex] miles<\/li>\n<li>[latex]204,000,000[\/latex] pounds<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/section>\n<p>The video below shows more examples of how to use place value to write the name of a whole number.<\/p>\n<section class=\"textbox watchIt\">\n<p><iframe loading=\"lazy\" id=\"oembed-2\" title=\"Example:  Write a Whole Number in Digits from Words\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/H9HpkvY_iaE?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p>You can view the <a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Quantitative+Reasoning+-+2023+Build\/Transcriptions\/Example_+Write+a+Whole+Number+in+Digits+from+Words.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cExample: Write a Whole Number in Digits from Words\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<h2>Rounding Whole Numbers<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\n<p><strong>Rounding<\/strong> is a method used to shorten or simplify numbers by adjusting them to the closest specified value.<\/p>\n<p>To round a whole number to a specific place value:<\/p>\n<ol id=\"eip-id1168287215567\" class=\"stepwise\">\n<li>Locate the given place value. All digits to the left of that place value do not change.<\/li>\n<li>Underline the digit to the right of the given place value.<\/li>\n<li>Determine if this digit is greater than or equal to [latex]5[\/latex].\n<ul id=\"fs-id2486248\">\n<li>Yes\u2014add [latex]1[\/latex] to the digit in the given place value.<\/li>\n<li>No\u2014do not change the digit in the given place value.<\/li>\n<\/ul>\n<\/li>\n<li>Replace all digits to the right of the given place value with zeros.<\/li>\n<\/ol>\n<\/div>\n<section class=\"textbox proTip\">If the digit to the right of your given place value is [latex]5[\/latex] or above, give your given place value a shove. If the digit to the right of your given place value is [latex]4[\/latex] or less, let your given place value rest.<\/section>\n<p>Watch the video below for more examples of how to round whole numbers to a given place value.<\/p>\n<section class=\"textbox watchIt\">\n<p><iframe loading=\"lazy\" id=\"oembed-3\" title=\"Ex: Rounding to the Various Place Values - Number Line \/ Formal Rules\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/tpFCYyTi1hc?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Quantitative+Reasoning+-+2023+Build\/Transcriptions\/Ex_+Rounding+to+the+Various+Place+Values+-+Number+Line+_+Formal+Rules.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cEx: Rounding to the Various Place Values &#8211; Number Line \/ Formal Rules\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<h2>Addition of Whole Numbers<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\n<p><strong>Addition<\/strong> of whole numbers involves combining the values of two or more numbers, using the plus (+) sign as the notation.<\/p>\n<table>\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>Addition<\/td>\n<td>[latex]+[\/latex]<\/td>\n<td>[latex]3+4[\/latex]<\/td>\n<td>three plus four<\/td>\n<td>the sum of [latex]3[\/latex] and [latex]4[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>&nbsp;<\/p>\n<p>The<strong> identity property of addition<\/strong> states that when a number is added to zero, the original number remains unchanged. This underscores the fact that zero is the &#8216;identity&#8217; for addition.<\/p>\n<p>The <strong>commutative property of addition<\/strong>, on the other hand, affirms that the order in which numbers are added does not affect the result, that is, for any two whole numbers [latex]a[\/latex] and [latex]b[\/latex], [latex]a + b[\/latex] equals [latex]b + a[\/latex].<\/p>\n<\/div>\n<p>Addition is really just counting. This is easy to see by modeling. We will model addition with base-[latex]10[\/latex] blocks. Remember, a block represents [latex]1[\/latex] and a rod represents [latex]10[\/latex]. Let\u2019s start by modeling the addition expression we just considered, [latex]3+4[\/latex].<br \/>\nEach addend is less than [latex]10[\/latex], so we can use ones blocks.<\/p>\n<table>\n<tbody>\n<tr>\n<td>We start by modeling the first number with [latex]3[\/latex] blocks.<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24215314\/CNX_BMath_Figure_01_02_019_img-02.png\" alt=\"Three individual blocks\" width=\"186\" height=\"51\" \/><\/td>\n<\/tr>\n<tr>\n<td>Then we model the second number with [latex]4[\/latex] blocks.<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24215315\/CNX_BMath_Figure_01_02_019_img-03.png\" alt=\"Three individual blocks next to four individual blocks\" width=\"186\" height=\"51\" \/><\/td>\n<\/tr>\n<tr>\n<td>Count the total number of blocks.<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24215316\/CNX_BMath_Figure_01_02_019_img-04.png\" alt=\"7 individual blocks\" width=\"186\" height=\"51\" \/><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p><span style=\"font-size: 10pt;\"><strong>Figure 3. Model these blocks to add 3 and 4<\/strong><\/span><\/p>\n<p>There are [latex]7[\/latex] blocks in all. We use an equal sign [latex]\\text{(=)}[\/latex] to show the sum. A math sentence that shows that two expressions are equal is called an equation. We have shown that. [latex]3+4=7[\/latex].<\/p>\n<section class=\"textbox seeExample\">Model the addition [latex]2+6[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q771692\">Show Solution<\/button><\/p>\n<div id=\"q771692\" class=\"hidden-answer\" style=\"display: none\">\n[latex]2+6[\/latex] means the sum of [latex]2[\/latex] and [latex]6[\/latex]<br \/>\nEach addend is less than [latex]10[\/latex], so we can use ones blocks.<\/p>\n<table>\n<tbody>\n<tr>\n<td>Model the first number with [latex]2[\/latex] blocks.<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24215317\/CNX_BMath_Figure_01_02_016_img-02.png\" alt=\"Two individual blocks\" width=\"204\" height=\"41\" \/><\/td>\n<\/tr>\n<tr>\n<td>Model the second number with [latex]6[\/latex] blocks.<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24215318\/CNX_BMath_Figure_01_02_016_img-03.png\" alt=\"Two individual blocks next to 6 individual blocks\" width=\"204\" height=\"45\" \/><\/td>\n<\/tr>\n<tr>\n<td>Count the total number of blocks<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24215319\/CNX_BMath_Figure_01_02_016_img-04.png\" alt=\"8 individual blocks\" width=\"204\" height=\"46\" \/><\/td>\n<\/tr>\n<tr>\n<td>There are [latex]8[\/latex] blocks in all, so [latex]2+6=8[\/latex].<\/td>\n<td>&nbsp;<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/section>\n<p>When the result is [latex]10[\/latex] or more ones blocks, we will exchange the [latex]10[\/latex] blocks for one rod.<\/p>\n<p>Next we will model adding two digit numbers.<\/p>\n<section class=\"textbox seeExample\">Model each addition:<\/p>\n<ol style=\"list-style-type: decimal;\">\n<li>[latex]15+27[\/latex]<\/li>\n<li>[latex]16+29[\/latex]<\/li>\n<\/ol>\n<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q586067\">Show Solution<\/button><\/p>\n<div id=\"q586067\" class=\"hidden-answer\" style=\"display: none\">\n<ol style=\"list-style-type: decimal;\">\n<li><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24215334\/CNX_BMath_Figure_01_02_014_img.png\" alt=\"4 horizontal rods of 10 blocks each and two individual blocks. The whole thing is labeled 15+27=42\" width=\"168\" height=\"109\" \/>\n<p>&nbsp;<\/p>\n<\/li>\n<li><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24215336\/CNX_BMath_Figure_01_02_015_img.png\" alt=\"4 horizontal rods of 10 blocks each and five individual blocks. The whole thing is labeled 16+29=45.\" width=\"218\" height=\"108\" \/><\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/section>\n<p>Watch the video below for another example of how to use base [latex]10[\/latex] blocks to model the sum of two whole numbers.<\/p>\n<section class=\"textbox watchIt\">\n<p><iframe loading=\"lazy\" id=\"oembed-4\" title=\"Model Addition of Two Digit Whole Numbers Using Base Ten Blocks\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/_smXQAPbygQ?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Quantitative+Reasoning+-+2023+Build\/Transcriptions\/Model+Addition+of+Two+Digit+Whole+Numbers+Using+Base+Ten+Blocks.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cModel Addition of Two Digit Whole Numbers Using Base Ten Blocks\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<h2>Subtraction of Whole Numbers<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\n<p><strong>Subtraction<\/strong> of whole numbers is the process of taking away one number from another to find the difference, often represented with a minus (-) sign.<\/p>\n<table>\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>Subtraction<\/td>\n<td>[latex]-[\/latex]<\/td>\n<td>[latex]7 - 3[\/latex]<\/td>\n<td>seven minus three<\/td>\n<td>the difference of [latex]7[\/latex] and [latex]3[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<p>A model can help us visualize the process of subtraction much as it did with addition. Again, we will use [latex]\\text{base - 10}[\/latex] blocks. Remember a block represents\u00a0[latex]1[\/latex] and a rod represents\u00a0[latex]10[\/latex]. Let\u2019s start by modeling the subtraction expression we just considered, [latex]7 - 3[\/latex].<\/p>\n<table>\n<tbody>\n<tr>\n<td>We start by modeling the first number, [latex]7[\/latex].<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24215416\/CNX_BMath_Figure_01_03_018_img-02.png\" alt=\"7 individual blocks\" width=\"199\" height=\"49\" \/><\/td>\n<\/tr>\n<tr>\n<td>Now take away the second number, [latex]3[\/latex]. We&#8217;ll circle [latex]3[\/latex] blocks to show that we are taking them away.<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24215417\/CNX_BMath_Figure_01_03_018_img-03.png\" alt=\"3 individual blocks in a circle and 4 individual blocks outside of it\" width=\"199\" height=\"69\" \/><\/td>\n<\/tr>\n<tr>\n<td>Count the number of blocks remaining.<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24215418\/CNX_BMath_Figure_01_03_018_img-04.png\" alt=\"4 individual blocks\" width=\"199\" height=\"27\" \/><\/td>\n<\/tr>\n<tr>\n<td>There are [latex]4[\/latex] ones blocks left.<\/td>\n<td>We have shown that [latex]7 - 3=4[\/latex] .<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<section><strong><span style=\"font-size: 10pt;\">Figure 4. Model these blocks to figure out 7 minus 3<\/span><\/strong><\/section>\n<section class=\"textbox seeExample\">Model the subtraction: [latex]8 - 2[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q250813\">Show Solution<\/button><\/p>\n<div id=\"q250813\" class=\"hidden-answer\" style=\"display: none\">[latex]8 - 2[\/latex] means the difference of [latex]8[\/latex] and [latex]2[\/latex]. Model the first, [latex]8[\/latex]. Take away the second number, [latex]2[\/latex]. Count the number of blocks remaining. There are [latex]6[\/latex] ones blocks left. We have shown that [latex]8 - 2=6[\/latex] .<\/div>\n<\/div>\n<\/section>\n<p>As we did with addition, we can describe the models as ones blocks and tens rods, or we can simply say ones and tens. Next we will model subtracting two digit numbers.<\/p>\n<section class=\"textbox seeExample\">Model each subtraction:<\/p>\n<ol style=\"list-style-type: decimal;\">\n<li>[latex]42 - 27[\/latex]<\/li>\n<li>[latex]45 - 29[\/latex]<\/li>\n<\/ol>\n<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q586167\">Show Solution<\/button><\/p>\n<div id=\"q586167\" class=\"hidden-answer\" style=\"display: none\">\n<ol style=\"list-style-type: decimal;\">\n<li><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24215439\/CNX_BMath_Figure_01_03_016_img.png\" alt=\"One rod of 10 blocks and 5 individual blocks outside of a circle and two rods of 10 blocks and 7 individual blocks in a circle. The whole thing is labeled 42-27=15\" width=\"243\" height=\"166\" \/>\n<p>&nbsp;<\/p>\n<\/li>\n<li><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24215440\/CNX_BMath_Figure_01_03_017_img.png\" alt=\"One rod of 10 blocks and 6 individual blocks outside of a circle and two rods of 10 blocks and 9 individual blocks in a circle. The whole thing is labeled 45-29=16.\" width=\"245\" height=\"185\" \/><\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/section>\n<p>Watch the video below to see more examples of using base 10 blocks to model subtraction.<\/p>\n<section class=\"textbox watchIt\">\n<p><iframe loading=\"lazy\" id=\"oembed-5\" title=\"Model Subtraction of Two Digit Whole Numbers Using Base Ten Blocks\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/FQI0IgA-7fI?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Quantitative+Reasoning+-+2023+Build\/Transcriptions\/Model+Subtraction+of+Two+Digit+Whole+Numbers+Using+Base+Ten+Blocks.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cModel Subtraction of Two Digit Whole Numbers Using Base Ten Blocks\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<p>Watch the video below to see another example of subtracting whole numbers by lining up place values.<\/p>\n<section class=\"textbox watchIt\">\n<p><iframe loading=\"lazy\" id=\"oembed-6\" title=\"Example:  Subtracting Whole Numbers\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/hneqy1EGACs?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Quantitative+Reasoning+-+2023+Build\/Transcriptions\/Example_+Subtracting+Whole+Numbers.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cExample: Subtracting Whole Numbers\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<h2>Multiplying Whole Numbers<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\n<p><strong>Multiplication<\/strong> of whole numbers is a process where one number, known as a factor, is repeatedly added to itself based on the value of the second factor, resulting in a total called the product.<\/p>\n<table>\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>\n[latex]\\times[\/latex]<br \/>\n[latex]\\cdot[\/latex]<br \/>\n[latex]\\left(\\right)[\/latex]\n<\/td>\n<td>\n[latex]3\\times 8[\/latex]<br \/>\n[latex]3\\cdot 8[\/latex]<br \/>\n[latex]3\\left(8\\right)[\/latex]\n<\/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<p>&nbsp;<\/p>\n<p>The <strong>multiplication property of zero<\/strong> states the product of any number and [latex]0[\/latex] is [latex]0[\/latex].<\/p>\n<p>The <strong>identity property of multiplication<\/strong> states the product of any number and [latex]1[\/latex] is the number.<\/p>\n<p>The <strong>commutative property of multiplication<\/strong> states that changing the order of the factors does not change their product.<\/p>\n<\/div>\n<p>Just like with subtraction and addition, modeling multiplication can help with understanding what is going on. There are many ways to model multiplication. Unlike in the previous sections where we used base-[latex]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. Let\u2019s start by modeling the multiplication expression we just considered, [latex]3\\times 8[\/latex].<\/p>\n<p>To model the product [latex]3\\times 8[\/latex], we\u2019ll start with a row of [latex]8[\/latex] counters.<\/p>\n<div style=\"text-align: center;\">\n<figure style=\"width: 212px\" class=\"wp-caption alignnone\"><img loading=\"lazy\" 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.\" width=\"212\" height=\"25\" \/><figcaption class=\"wp-caption-text\">Figure 5. 8 counters<\/figcaption><\/figure>\n<\/div>\n<p>&nbsp;<\/p>\n<p>The other factor is [latex]3[\/latex], so we\u2019ll make [latex]3[\/latex] rows of [latex]8[\/latex] counters.<\/p>\n<div style=\"text-align: center;\">\n<figure style=\"width: 212px\" class=\"wp-caption alignnone\"><img loading=\"lazy\" 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.\" width=\"212\" height=\"80\" \/><figcaption class=\"wp-caption-text\">Figure 6. 3 rows of 8 counters<\/figcaption><\/figure>\n<\/div>\n<p>&nbsp;<\/p>\n<p>Now we can count the result. There are [latex]24[\/latex] counters in all.<\/p>\n<div style=\"text-align: center;\">[latex]3\\times 8=24[\/latex]<\/div>\n<p>&nbsp;<\/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.<\/p>\n<section class=\"textbox seeExample\">Model each multiplication:<\/p>\n<ol style=\"list-style-type: decimal;\">\n<li>[latex]4\\times 6[\/latex]<\/li>\n<li>[latex]5\\times 7[\/latex]<\/li>\n<\/ol>\n<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q586061\">Show Solution<\/button><\/p>\n<div id=\"q586061\" class=\"hidden-answer\" style=\"display: none\">\n<ol>\n<li><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" 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=\"4 rows of 6 counters\" width=\"123\" height=\"88\" \/>\n<p>&nbsp;<\/p>\n<\/li>\n<\/ol>\n<p>\u00a0 \u00a0 \u00a02. <img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" 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=\"5 rows of 7 counters\" width=\"144\" height=\"110\" \/><\/p>\n<\/div>\n<\/div>\n<\/section>\n<p>In the video below, we summarize the concepts covered in multiplying whole numbers including the multiplication property of zero, the identity property of multiplication, and the commutative property of multiplication.<\/p>\n<section class=\"textbox watchIt\">\n<p><iframe loading=\"lazy\" id=\"oembed-7\" title=\"Multiplying Whole Numbers\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/kW7JBfplJGE?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Quantitative+Reasoning+-+2023+Build\/Transcriptions\/Multiplying+Whole+Numbers.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cMultiplying Whole Numbers\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<h2>Dividing Whole Numbers<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\n<p><strong>Division<\/strong> of whole numbers involves partitioning a given quantity, known as the dividend, into equal parts, with the number of these parts represented by the divisor, resulting in the quotient.<\/p>\n<table>\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{Division}[\/latex]<\/td>\n<td>\n[latex]a \\div b[\/latex]<br \/>\n[latex]\\frac{a}{b}[\/latex]<br \/>\n[latex]b\\overline{)a}[\/latex]<br \/>\n[latex]a\/b[\/latex]\n<\/td>\n<td>\n[latex]12\\div 4[\/latex]<br \/>\n[latex]\\frac{12}{4}[\/latex]<br \/>\n[latex]4\\overline{)12}[\/latex]<br \/>\n[latex]12\/4[\/latex]\n<\/td>\n<td>[latex]\\text{Twelve divided by four}[\/latex]<\/td>\n<td>[latex]\\text{the quotient of 12 and 4}[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>&nbsp;<\/p>\n<p>The <strong>division properties of one<\/strong> states dividing any number, except [latex]0[\/latex], by itself produces a quotient of [latex]1[\/latex] and any number divided by [latex]1[\/latex] produces a quotient of the number.<\/p>\n<p>The <strong>division properties of zero<\/strong> states any number divided by zero is undefined, while zero divided by any number (except zero) is always zero.<\/p>\n<\/div>\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. Let\u2019s start by modeling the division expression we just considered, [latex]24\\div 8[\/latex].<\/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<div style=\"text-align: center;\">\n<figure style=\"width: 301px\" class=\"wp-caption alignnone\"><img loading=\"lazy\" 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.\" width=\"301\" height=\"114\" \/><figcaption class=\"wp-caption-text\">Figure 7. 24 counters, randomly placed<\/figcaption><\/figure>\n<\/div>\n<p>&nbsp;<\/p>\n<p>The divisor tell us the number of counters we want in each group. Form groups of [latex]8[\/latex] counters.<\/p>\n<div style=\"text-align: center;\">\n<figure style=\"width: 312px\" class=\"wp-caption alignnone\"><img loading=\"lazy\" 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.\" width=\"312\" height=\"132\" \/><figcaption class=\"wp-caption-text\">Figure 8. Group those counters into 3 bubbles<\/figcaption><\/figure>\n<\/div>\n<p>&nbsp;<\/p>\n<p>Count the number of groups. There are [latex]3[\/latex] groups.<\/p>\n<div style=\"text-align: center;\">[latex]24\\div 8=3[\/latex]<\/div>\n<p>&nbsp;<\/p>\n<section class=\"textbox seeExample\">Model each division:<\/p>\n<ol style=\"list-style-type: decimal;\">\n<li>[latex]24\\div 6[\/latex]<\/li>\n<li>[latex]42\\div 7[\/latex]<\/li>\n<\/ol>\n<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q586069\">Show Solution<\/button><\/p>\n<div id=\"q586069\" class=\"hidden-answer\" style=\"display: none\">\n<ol style=\"list-style-type: decimal;\">\n<li><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=\"24 counters grouped into 4 sets of 6\" width=\"186\" height=\"182\" \/>\u00a0\u00a0[latex]24\\div 6=4[\/latex]\n<p>&nbsp;<\/p>\n<\/li>\n<li><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=\"42 counters grouped in 6 groups of 7\" width=\"242\" height=\"178\" \/>\u00a0 [latex]42\\div 7=6[\/latex]<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/section>\n<p>In the video below we show another way to model division using area.<\/p>\n<section class=\"textbox watchIt\">\n<p><iframe loading=\"lazy\" id=\"oembed-8\" 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<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Quantitative+Reasoning+-+2023+Build\/Transcriptions\/Division+of+Whole+Numbers+using+Area+(No+Remainder).txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cDivision of Whole Numbers using Area (No Remainder)\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<p>In the video below we show another example of using long division.<\/p>\n<section class=\"textbox watchIt\">\n<p><iframe loading=\"lazy\" id=\"oembed-9\" title=\"Ex: Long Division - Two Digit Divided by One Digit (No Remainder)\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/KvVhaB5mqr8?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Quantitative+Reasoning+-+2023+Build\/Transcriptions\/Ex_+Long+Division+-+Two+Digit+Divided+by+One+Digit+(No+Remainder).txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cEx: Long Division &#8211; Two Digit Divided by One Digit (No Remainder)\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<p>Watch this video for another example of how to use long division to divide a four digit whole number by a two digit whole number.<\/p>\n<section class=\"textbox watchIt\">\n<p><iframe loading=\"lazy\" id=\"oembed-10\" title=\"Example:  Dividing Whole Numbers without a Remainder\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/V7Korf09iWI?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Quantitative+Reasoning+-+2023+Build\/Transcriptions\/Example_+Dividing+Whole+Numbers+without+a+Remainder.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cExample: Dividing Whole Numbers without a Remainder\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<p>Watch the video below for another example of how to use long division to divide whole numbers when there is a remainder.<\/p>\n<section class=\"textbox watchIt\">\n<p><iframe loading=\"lazy\" id=\"oembed-11\" title=\"Example:  Dividing Whole Numbers with a Remainder\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/UPUcShGCBOs?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p>You can view the <a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Quantitative+Reasoning+-+2023+Build\/Transcriptions\/Example_+Dividing+Whole+Numbers+with+a+Remainder.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cExample: Dividing Whole Numbers with a Remainder\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<p>For a review of adding, subtracting, multiplying, and division of whole numbers watch the following video.<\/p>\n<section class=\"textbox watchIt\"><iframe loading=\"lazy\" src=\"\/\/plugin.3playmedia.com\/show?mf=10387435&amp;p3sdk_version=1.10.1&amp;p=20361&amp;pt=375&amp;video_id=HAhg0uXn9GA&amp;video_target=tpm-plugin-a42jw4sm-HAhg0uXn9GA\" width=\"800px\" height=\"450px\" frameborder=\"0\" marginwidth=\"0px\" marginheight=\"0px\"><\/iframe><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Quantitative+Reasoning+-+2023+Build\/Transcriptions\/Whole+Number+Operations+%7C+Adding%2C+Subtracting%2C+Multiplying%2C+and+Dividing.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cWhole Number Operations | Adding, Subtracting, Multiplying, and Dividing\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n","protected":false},"author":15,"menu_order":13,"template":"","meta":{"_candela_citation":"[{\"type\":\"copyrighted_video\",\"description\":\"Revision and Adaptation\",\"author\":\"\",\"organization\":\"Lumen Learning\",\"url\":\"\",\"project\":\"\",\"license\":\"arr\",\"license_terms\":\"\"},{\"type\":\"copyrighted_video\",\"description\":\"Finding the Value of the Underlined Digit | Whole Number Place Value | Math with Mr. J\",\"author\":\"Math with Mr. J\",\"organization\":\"\",\"url\":\"https:\/\/youtu.be\/f3YjmQFSxl0\",\"project\":\"\",\"license\":\"arr\",\"license_terms\":\"\"},{\"type\":\"copyrighted_video\",\"description\":\"Examples: Determining Place Value\",\"author\":\"James Sousa (Mathispower4u.com)\",\"organization\":\"\",\"url\":\"https:\/\/youtu.be\/0-4rLGvVEOI\",\"project\":\"\",\"license\":\"arr\",\"license_terms\":\"\"},{\"type\":\"copyrighted_video\",\"description\":\"Example: Write a Whole Number in Digits from Words\",\"author\":\"James Sousa (Mathispower4u.com)\",\"organization\":\"\",\"url\":\"https:\/\/youtu.be\/H9HpkvY_iaE\",\"project\":\"\",\"license\":\"arr\",\"license_terms\":\"\"},{\"type\":\"copyrighted_video\",\"description\":\"Ex: Rounding to the Various Place Values - 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