{"id":4624,"date":"2020-04-21T00:19:10","date_gmt":"2020-04-21T00:19:10","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/mathforliberalartscorequisite\/chapter\/using-models-to-represent-fractions-and-mixed-numbers\/"},"modified":"2023-03-23T00:02:02","modified_gmt":"2023-03-23T00:02:02","slug":"using-models-to-represent-fractions-and-mixed-numbers","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/slcc-mathforliberalartscorequisite\/chapter\/using-models-to-represent-fractions-and-mixed-numbers\/","title":{"raw":"Using Models to Represent Fractions and Mixed Numbers","rendered":"Using Models to Represent Fractions and Mixed Numbers"},"content":{"raw":"\n\n<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n \t<li>Write fractions that represent portions of objects<\/li>\n \t<li>Identify the numerator and the denominator of a fraction<\/li>\n \t<li>Use fraction circles to make wholes given<\/li>\n \t<li>Use models to visualize improper fractions and mixed numbers.<\/li>\n<\/ul>\n<\/div>\n<h3>Representing Parts of a Whole as Fractions<\/h3>\nAndy and Bobby love pizza. On Monday night, they share a pizza equally. How much of the pizza does each one get? Are you thinking that each boy gets half of the pizza? That\u2019s right. There is one whole pizza, evenly divided into two parts, so each boy gets one of the two equal parts.\n\nIn math, we write [latex]\\Large{\\frac{1}{2}}[\/latex] to mean one out of two parts.\n\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220650\/CNX_BMath_Figure_04_01_001_img.png\" alt=\"An image of a round pizza sliced vertically down the center, creating two equal pieces. Each piece is labeled as one half.\">\nOn Tuesday, Andy and Bobby share a pizza with their parents, Fred and Christy, with each person getting an equal amount of the whole pizza. How much of the pizza does each person get? There is one whole pizza, divided evenly into four equal parts. Each person has one of the four equal parts, so each has [latex]\\Large{\\frac{1}{4}}[\/latex] of the pizza.\n\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220652\/CNX_BMath_Figure_04_01_002_img.png\" alt=\"An image of a round pizza sliced vertically and horizontally, creating four equal pieces. Each piece is labeled as one fourth.\">\nOn Wednesday, the family invites some friends over for a pizza dinner. There are a total of [latex]12[\/latex] people. If they share the pizza equally, each person would get [latex]\\Large{\\frac{1}{12}}[\/latex] of the pizza.\n\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220653\/CNX_BMath_Figure_04_01_003_img.png\" alt=\"An image of a round pizza sliced into twelve equal wedges. Each piece is labeled as one twelfth.\">\n<div class=\"textbox shaded\">\n<h3>Fractions<\/h3>\nA fraction is written [latex]\\Large{\\frac{a}{b}}[\/latex], where [latex]a[\/latex] and [latex]b[\/latex] are integers and [latex]b\\ne 0[\/latex]. In a fraction, [latex]a[\/latex] is called the numerator and [latex]b[\/latex] is called the denominator.\n\n<\/div>\nA fraction is a way to represent parts of a whole. The denominator [latex]b[\/latex] represents the number of equal parts the whole has been divided into, and the numerator [latex]a[\/latex] represents how many parts are included. The denominator, [latex]b[\/latex], cannot equal zero because division by zero is undefined.\n\nIn the image below, the circle has been divided into three parts of equal size. Each part represents [latex]\\Large{\\frac{1}{3}}[\/latex] of the circle. This type of model is called a fraction circle. Other shapes, such as rectangles, can also be used to model fractions.\n\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220654\/CNX_BMath_Figure_04_01_004.png\" alt=\"A circle is divided into three equal wedges. Each piece is labeled as one third.\">\nDoing the Manipulative Mathematics activity Model Fractions will help you develop a better understanding of fractions, their numerators and denominators.\n\nWhat does the fraction [latex]\\Large{\\frac{2}{3}}[\/latex] represent? The fraction [latex]\\Large{\\frac{2}{3}}[\/latex] means two of three equal parts.\n\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220656\/CNX_BMath_Figure_04_01_005_img.png\" alt=\"A circle is divided into three equal wedges. Two of the wedges are shaded.\">\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\nName the fraction of the shape that is shaded in each of the figures.\n\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220658\/CNX_BMath_Figure_04_01_006_img.png\" alt=\"In part \">\n\nSolution:\nWe need to ask two questions. First, how many equal parts are there? This will be the denominator. Second, of these equal parts, how many are shaded? This will be the numerator.\n\n[latex]\\begin{array}{cccc}\\text{How many equal parts are there?}\\hfill &amp; &amp; &amp; \\text{There are eight equal parts}\\text{.}\\hfill \\\\ \\text{How many are shaded?}\\hfill &amp; &amp; &amp; \\text{Five parts are shaded}\\text{.}\\hfill \\end{array}[\/latex]\nFive out of eight parts are shaded. Therefore, the fraction of the circle that is shaded is [latex]\\Large{\\frac{5}{8}}[\/latex].\n\n[latex]\\begin{array}{cccc}\\text{How many equal parts are there?}\\hfill &amp; &amp; &amp; \\text{There are nine equal parts}\\text{.}\\hfill \\\\ \\text{How many are shaded?}\\hfill &amp; &amp; &amp; \\text{Two parts are shaded}\\text{.}\\hfill \\end{array}[\/latex]\nTwo out of nine parts are shaded. Therefore, the fraction of the square that is shaded is [latex]\\Large{\\frac{2}{9}}[\/latex].\n\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try it<\/h3>\n[embed]https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=145974&amp;theme=oea&amp;iframe_resize_id=ohm1[\/embed]\n\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\nShade [latex]\\frac{3}{4}[\/latex] of the circle.\n\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220702\/CNX_BMath_Figure_04_01_008_img.png\" alt=\"An image of a circle.\">\n[reveal-answer q=\"323057\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"323057\"]\n\nSolution\nThe denominator is [latex]4[\/latex], so we divide the circle into four equal parts \u24d0.\nThe numerator is [latex]3[\/latex], so we shade three of the four parts \u24d1.\n\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220704\/CNX_BMath_Figure_04_01_009.png\" alt=\"In \">\n[latex]\\Large{\\frac{3}{4}}[\/latex] of the circle is shaded.\n\n[\/hidden-answer]\n\n<\/div>\n&nbsp;\n<div class=\"textbox key-takeaways\">\n<h3>Try it<\/h3>\nShade [latex]\\Large{\\frac{6}{8}}[\/latex] of the circle.\n\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220705\/CNX_BMath_Figure_04_01_010_img.png\" alt=\"A circle is divided into eight equal pieces.\">\n[reveal-answer q=\"621333\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"621333\"]\n\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220706\/CNX_BMath_Figure_04_01_011_img.png\" alt=\"A circle is shown divided into 8 pieces, of which 6 are shaded.\">\n\n[\/hidden-answer]\n\n&nbsp;\n\nShade [latex]\\Large{\\frac{2}{5}}[\/latex] of the rectangle.\n\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220707\/CNX_BMath_Figure_04_01_012_img.png\" alt=\"A rectangle is divided vertically into five equal pieces.\">\n[reveal-answer q=\"452323\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"452323\"]\n\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220708\/CNX_BMath_Figure_04_01_013_img.png\" alt=\"A rectangle is divided into 5 sections, of which 2 are shaded.\">\n\n[\/hidden-answer]\n\n<\/div>\nWatch the following video to see more examples of how to write fractions given a model.\n\nhttps:\/\/youtu.be\/c_yIA4OQ4qA\n\nIn earlier examples, we used circles and rectangles to model fractions. Fractions can also be modeled as manipulatives called fraction tiles, as shown in the image below. Here, the whole is modeled as one long, undivided rectangular tile. Beneath it are tiles of equal length divided into different numbers of equally sized parts.\n\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220710\/CNX_BMath_Figure_04_01_014.png\" alt=\"One long, undivided rectangular tile is shown, labeled \">\nWe\u2019ll be using fraction tiles to discover some basic facts about fractions. Refer to the fraction tiles above&nbsp;to answer the following questions:\n<table id=\"fs-id7569845\" class=\"unnumbered\" style=\"width: 85%\" summary=\".\">\n<tbody>\n<tr>\n<td>How many [latex]\\Large{\\frac{1}{2}}[\/latex] tiles does it take to make one whole tile?<\/td>\n<td>It takes two halves to make a whole, so two out of two is [latex]{\\Large\\frac{2}{2}}=1[\/latex].<\/td>\n<\/tr>\n<tr>\n<td>How many [latex]\\Large{\\frac{1}{3}}[\/latex] tiles does it take to make one whole tile?<\/td>\n<td>It takes three thirds, so three out of three is [latex]{\\Large{\\frac{3}{3}}}=1[\/latex].<\/td>\n<\/tr>\n<tr>\n<td>How many [latex]\\Large{\\frac{1}{4}}[\/latex] tiles does it take to make one whole tile?<\/td>\n<td>It takes four fourths, so four out of four is [latex]{\\Large\\frac{4}{4}}=1[\/latex].<\/td>\n<\/tr>\n<tr>\n<td>How many [latex]\\Large{\\frac{1}{6}}[\/latex] tiles does it take to make one whole tile?<\/td>\n<td>It takes six sixths, so six out of six is [latex]{\\Large\\frac{6}{6}}=1[\/latex].<\/td>\n<\/tr>\n<tr>\n<td>What if the whole were divided into [latex]24[\/latex] equal parts? (We have not shown fraction tiles to represent this, but try to visualize it in your mind.) How many [latex]\\Large{\\frac{1}{24}}[\/latex] tiles does it take to make one whole tile?<\/td>\n<td>It takes [latex]24[\/latex] twenty-fourths, so [latex]{\\Large\\frac{24}{24}}=1[\/latex].<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\nThis leads us to the <em>Property of One<\/em>.\n<div class=\"textbox shaded\">\n<h3>Property of One<\/h3>\nAny number, except zero, divided by itself is one.\n\n[latex]{\\Large\\frac{a}{a}}=1\\left(a\\ne 0\\right)[\/latex]\n\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\nUse fraction circles to make wholes using the following pieces:\n<ol>\n \t<li>[latex]4[\/latex] fourths<\/li>\n \t<li>[latex]5[\/latex] fifths<\/li>\n \t<li>[latex]6[\/latex] sixths<\/li>\n<\/ol>\n[reveal-answer q=\"447874\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"447874\"]\n\nSolution\n\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220712\/CNX_BMath_Figure_04_01_015_img.png\" alt=\"Three circles are shown. The circle on the left is divided into four equal pieces. The circle in the middle is divided into five equal pieces. The circle on the right is divided into six equal pieces. Each circle says \">\n\n[\/hidden-answer]\n\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try it<\/h3>\nUse fraction circles to make wholes with the following pieces: [latex]3[\/latex] thirds.\n[reveal-answer q=\"412196\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"412196\"]\n\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220714\/CNX_BMath_Figure_04_01_016_img.png\" alt=\"A circle is shown. It is divided into 3 equal pieces. All 3 pieces are shaded.\">\n\n[\/hidden-answer]\n\n&nbsp;\n\nUse fraction circles to make wholes with the following pieces: [latex]8[\/latex] eighths.\n[reveal-answer q=\"205673\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"205673\"]\n\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220715\/CNX_BMath_Figure_04_01_017_img.png\" alt=\"A circle is divided into 8 sections, of which all are shaded.\">\n\n[\/hidden-answer]\n\n<\/div>\nWhat if we have more fraction pieces than we need for [latex]1[\/latex] whole? We\u2019ll look at this in the next example.\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\nUse fraction circles to make wholes using the following pieces:\n<ol>\n \t<li>[latex]3[\/latex] halves<\/li>\n \t<li>[latex]8[\/latex] fifths<\/li>\n \t<li>[latex]7[\/latex] thirds<\/li>\n<\/ol>\n[reveal-answer q=\"938641\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"938641\"]\n\nSolution\n1. [latex]3[\/latex] halves make [latex]1[\/latex] whole with [latex]1[\/latex] half left over.\n\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220716\/CNX_BMath_Figure_04_01_018_img.png\" alt=\"Two circles are shown, both divided into two equal pieces. The circle on the left has both pieces shaded and is labeled as \">\n2. [latex]8[\/latex] fifths make [latex]1[\/latex] whole with [latex]2[\/latex] fifths left over.\n\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220718\/CNX_BMath_Figure_04_01_019_img.png\" alt=\"Two circles are shown, both divided into five equal pieces. The circle on the left has all five pieces shaded and is labeled as \">\n3. [latex]7[\/latex] thirds make [latex]2[\/latex] wholes with [latex]2[\/latex] thirds left over.\n\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220720\/CNX_BMath_Figure_04_01_020_img.png\" alt=\"Three circles are shown, all divided into three equal pieces. The two circles on the left have all three pieces shaded and are labeled with ones. The circle on the right has one piece shaded and is labeled as one third.\">\n\n[\/hidden-answer]\n\n<\/div>\n&nbsp;\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\nUse fraction circles to make wholes with the following pieces: [latex]5[\/latex] thirds.\n[reveal-answer q=\"364407\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"364407\"]\n\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220722\/CNX_BMath_Figure_04_01_021_img.png\" alt=\"Two circles are shown. Each is divided into three sections. All of the first circle is shaded. 2 out of 3 sections of the second circle are shaded.\">\n\n[\/hidden-answer]\n\n&nbsp;\n\nUse fraction circles to make wholes with the following pieces: [latex]5[\/latex] halves.\n[reveal-answer q=\"779741\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"779741\"]\n\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220723\/CNX_BMath_Figure_04_01_022_img.png\" alt=\"Three circles are shown. Each is divided into two sections. The first two circles are completely shaded. Half of the third circle is shaded.\">\n\n[\/hidden-answer]\n\n&nbsp;\n\n<\/div>\n<h2>Model Improper Fractions and Mixed Numbers<\/h2>\nIn an earlier example, you had eight equal fifth pieces. You used five of them to make one whole, and you had three fifths left over. Let us use fraction notation to show what happened. You had eight pieces, each of them one fifth, [latex]{\\Large\\frac{1}{5}}[\/latex], so altogether you had eight fifths, which we can write as [latex]{\\Large\\frac{8}{5}}[\/latex]. The fraction [latex]{\\Large\\frac{8}{5}}[\/latex] is one whole, [latex]1[\/latex], plus three fifths, [latex]{\\Large\\frac{3}{5}}[\/latex], or [latex]1{\\Large\\frac{3}{5}}[\/latex], which is read as <em>one and three-fifths<\/em>.\n\nThe number [latex]1{\\Large\\frac{3}{5}}[\/latex] is called a mixed number. A mixed number consists of a whole number and a fraction.\n<div class=\"textbox shaded\">\n<h3>Mixed Numbers<\/h3>\nA mixed number consists of a whole number [latex]a[\/latex] and a fraction [latex]{\\Large\\frac{b}{c}}[\/latex] where [latex]c\\ne 0[\/latex]. It is written as follows.\n<p style=\"text-align: center\">[latex]a{\\Large\\frac{b}{c}}\\text{, }c\\ne 0[\/latex]<\/p>\n\n<\/div>\nFractions such as [latex]{\\Large\\frac{5}{4}},{\\Large\\frac{3}{2}},{\\Large\\frac{5}{5}}[\/latex], and [latex]{\\Large\\frac{7}{3}}[\/latex] are called improper fractions. In an improper fraction, the numerator is greater than or equal to the denominator, so its value is greater than or equal to one. When a fraction has a numerator that is smaller than the denominator, it is called a proper fraction, and its value is less than one. Fractions such as [latex]{\\Large\\frac{1}{2}},{\\Large\\frac{3}{7}}[\/latex], and [latex]{\\Large\\frac{11}{18}}[\/latex] are proper fractions.\n<div class=\"textbox shaded\">\n<h3>Proper and Improper Fractions<\/h3>\nThe fraction [latex]{\\Large\\frac{a}{b}}[\/latex] is a proper fraction if [latex]a&lt;b[\/latex] and an improper fraction if [latex]a\\ge b[\/latex].\n\n<\/div>\n&nbsp;\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\nName the improper fraction modeled. Then write the improper fraction as a mixed number.\n\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220725\/CNX_BMath_Figure_04_01_023_img.png\" alt=\"Two circles are shown, both divided into three equal pieces. The circle on the left has all three pieces shaded. The circle on the right has one piece shaded.\">\n\n[reveal-answer q=\"670905\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"670905\"]\n\nSolution:\nEach circle is divided into three pieces, so each piece is [latex]{\\Large\\frac{1}{3}}[\/latex] of the circle. There are four pieces shaded, so there are four thirds or [latex]{\\Large\\frac{4}{3}}[\/latex]. The figure shows that we also have one whole circle and one third, which is [latex]1{\\Large\\frac{1}{3}}[\/latex]. So, [latex]{\\Large\\frac{4}{3}}=1{\\Large\\frac{1}{3}}[\/latex].[\/hidden-answer]\n\n<\/div>\n&nbsp;\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n[ohm_question]145976[\/ohm_question]\n\n[ohm_question]145977[\/ohm_question]\n\n<\/div>\n&nbsp;\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\nDraw a figure to model [latex]{\\Large\\frac{11}{8}}[\/latex].\n[reveal-answer q=\"992194\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"992194\"]\n\nSolution:\nThe denominator of the improper fraction is [latex]8[\/latex]. Draw a circle divided into eight pieces and shade all of them. This takes care of eight eighths, but we have [latex]11[\/latex] eighths. We must shade three of the eight parts of another circle.\n\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220730\/CNX_BMath_Figure_04_01_026_img.png\" alt=\"Two circles are shown, both divided into eight equal pieces. The circle on the left has all eight pieces shaded and is labeled as eight eighths. The circle on the right has three pieces shaded and is labeled as three eighths. The diagram indicates that eight eighths plus three eighths is one plus three eighths.\">\nSo, [latex]{\\Large\\frac{11}{8}}=1{\\Large\\frac{3}{8}}[\/latex].\n\n[\/hidden-answer]\n\n<\/div>\n&nbsp;\n<div class=\"textbox key-takeaways\">\n<h3>Try it<\/h3>\nDraw a figure to model [latex]{\\Large\\frac{7}{6}}[\/latex]\n[reveal-answer q=\"924546\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"924546\"]\n\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220732\/CNX_BMath_Figure_04_01_027_img.png\" alt=\"Two circles are shown. Each is divided into six sections. All of the first circle is shaded and one section of the second circle is shaded.\">\n\n[\/hidden-answer]\n\n&nbsp;\n\nDraw a figure to model [latex]{\\Large\\frac{6}{5}}[\/latex]\n[reveal-answer q=\"203648\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"203648\"]\n\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220733\/CNX_BMath_Figure_04_01_028_img.png\" alt=\"Two circles are shown. Each is divided into five sections. All of the first circle is shaded and one section of the second circle is shaded.\">\n\n[\/hidden-answer]\n\n<\/div>\n&nbsp;\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\nUse a model to rewrite the improper fraction [latex]{\\Large\\frac{11}{6}}[\/latex] as a mixed number.\n[reveal-answer q=\"121702\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"121702\"]\n\nSolution:\nWe start with [latex]11[\/latex] sixths [latex]\\left({\\Large\\frac{11}{6}}\\right)[\/latex]. We know that six sixths makes one whole.\n<p style=\"padding-left: 30px\">[latex]{\\Large\\frac{6}{6}}=1[\/latex]<\/p>\nThat leaves us with five more sixths, which is [latex]{\\Large\\frac{5}{6}}[\/latex] (11 sixths minus 6 sixths is 5 sixths).\n<p style=\"padding-left: 30px\">So, [latex]{\\Large\\frac{11}{6}}=1{\\Large\\frac{5}{6}}[\/latex]<\/p>\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220735\/CNX_BMath_Figure_04_01_029_img.png\" alt=\"Two circles are shown, both divided into six equal pieces. The circle on the left has all six pieces shaded and is labeled as six sixths. The circle on the right has five pieces shaded and is labeled as five sixths. Below the circles, it says one plus five sixths, then six sixths plus five sixths equals eleven sixths, and one plus five sixths equals one and five sixths. It then says that eleven sixths equals one and five sixths.\">\n\n[\/hidden-answer]\n\n<\/div>\n&nbsp;\n<div class=\"textbox key-takeaways\">\n<h3>Try it<\/h3>\n[ohm_question]145982[\/ohm_question]\n\n<\/div>\nIn the next video we show another way to draw a model that represents a fraction. &nbsp;You will see example of both proper and improper fractions shown.\n\nhttps:\/\/youtu.be\/akyByv80Uzc\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\nUse a model to rewrite the mixed number [latex]1{\\Large\\frac{4}{5}}[\/latex] as an improper fraction.\n[reveal-answer q=\"852331\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"852331\"]\n\nSolution:\nThe mixed number [latex]1{\\Large\\frac{4}{5}}[\/latex] means one whole plus four fifths. The denominator is [latex]5[\/latex], so the whole is [latex]{\\Large\\frac{5}{5}}[\/latex]. Together five fifths and four fifths equals nine fifths.\nSo, [latex]1{\\Large\\frac{4}{5}}={\\Large\\frac{9}{5}}[\/latex]\n\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220738\/CNX_BMath_Figure_04_01_030_img.png\" alt=\"Two circles are shown, both divided into five equal pieces. The circle on the left has all five pieces shaded and is labeled as 5 fifths. The circle on the right has four pieces shaded and is labeled as 4 fifths. It then says that 5 fifths plus 4 fifths equals 9 fifths and that 9 fifths is equal to one plus 4 fifths.\">\n\n[\/hidden-answer]\n\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try it<\/h3>\n[ohm_question]145981[\/ohm_question]\n\n<\/div>\n\n","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li>Write fractions that represent portions of objects<\/li>\n<li>Identify the numerator and the denominator of a fraction<\/li>\n<li>Use fraction circles to make wholes given<\/li>\n<li>Use models to visualize improper fractions and mixed numbers.<\/li>\n<\/ul>\n<\/div>\n<h3>Representing Parts of a Whole as Fractions<\/h3>\n<p>Andy and Bobby love pizza. On Monday night, they share a pizza equally. How much of the pizza does each one get? Are you thinking that each boy gets half of the pizza? That\u2019s right. There is one whole pizza, evenly divided into two parts, so each boy gets one of the two equal parts.<\/p>\n<p>In math, we write [latex]\\Large{\\frac{1}{2}}[\/latex] to mean one out of two parts.<\/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\/24220650\/CNX_BMath_Figure_04_01_001_img.png\" alt=\"An image of a round pizza sliced vertically down the center, creating two equal pieces. Each piece is labeled as one half.\" \/><br \/>\nOn Tuesday, Andy and Bobby share a pizza with their parents, Fred and Christy, with each person getting an equal amount of the whole pizza. How much of the pizza does each person get? There is one whole pizza, divided evenly into four equal parts. Each person has one of the four equal parts, so each has [latex]\\Large{\\frac{1}{4}}[\/latex] of the pizza.<\/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\/24220652\/CNX_BMath_Figure_04_01_002_img.png\" alt=\"An image of a round pizza sliced vertically and horizontally, creating four equal pieces. Each piece is labeled as one fourth.\" \/><br \/>\nOn Wednesday, the family invites some friends over for a pizza dinner. There are a total of [latex]12[\/latex] people. If they share the pizza equally, each person would get [latex]\\Large{\\frac{1}{12}}[\/latex] of the pizza.<\/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\/24220653\/CNX_BMath_Figure_04_01_003_img.png\" alt=\"An image of a round pizza sliced into twelve equal wedges. Each piece is labeled as one twelfth.\" \/><\/p>\n<div class=\"textbox shaded\">\n<h3>Fractions<\/h3>\n<p>A fraction is written [latex]\\Large{\\frac{a}{b}}[\/latex], where [latex]a[\/latex] and [latex]b[\/latex] are integers and [latex]b\\ne 0[\/latex]. In a fraction, [latex]a[\/latex] is called the numerator and [latex]b[\/latex] is called the denominator.<\/p>\n<\/div>\n<p>A fraction is a way to represent parts of a whole. The denominator [latex]b[\/latex] represents the number of equal parts the whole has been divided into, and the numerator [latex]a[\/latex] represents how many parts are included. The denominator, [latex]b[\/latex], cannot equal zero because division by zero is undefined.<\/p>\n<p>In the image below, the circle has been divided into three parts of equal size. Each part represents [latex]\\Large{\\frac{1}{3}}[\/latex] of the circle. This type of model is called a fraction circle. Other shapes, such as rectangles, can also be used to model fractions.<\/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\/24220654\/CNX_BMath_Figure_04_01_004.png\" alt=\"A circle is divided into three equal wedges. Each piece is labeled as one third.\" \/><br \/>\nDoing the Manipulative Mathematics activity Model Fractions will help you develop a better understanding of fractions, their numerators and denominators.<\/p>\n<p>What does the fraction [latex]\\Large{\\frac{2}{3}}[\/latex] represent? The fraction [latex]\\Large{\\frac{2}{3}}[\/latex] means two of three equal parts.<\/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\/24220656\/CNX_BMath_Figure_04_01_005_img.png\" alt=\"A circle is divided into three equal wedges. Two of the wedges are shaded.\" \/><\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Name the fraction of the shape that is shaded in each of the figures.<\/p>\n<p><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220658\/CNX_BMath_Figure_04_01_006_img.png\" alt=\"In part\" \/><\/p>\n<p>Solution:<br \/>\nWe need to ask two questions. First, how many equal parts are there? This will be the denominator. Second, of these equal parts, how many are shaded? This will be the numerator.<\/p>\n<p>[latex]\\begin{array}{cccc}\\text{How many equal parts are there?}\\hfill & & & \\text{There are eight equal parts}\\text{.}\\hfill \\\\ \\text{How many are shaded?}\\hfill & & & \\text{Five parts are shaded}\\text{.}\\hfill \\end{array}[\/latex]<br \/>\nFive out of eight parts are shaded. Therefore, the fraction of the circle that is shaded is [latex]\\Large{\\frac{5}{8}}[\/latex].<\/p>\n<p>[latex]\\begin{array}{cccc}\\text{How many equal parts are there?}\\hfill & & & \\text{There are nine equal parts}\\text{.}\\hfill \\\\ \\text{How many are shaded?}\\hfill & & & \\text{Two parts are shaded}\\text{.}\\hfill \\end{array}[\/latex]<br \/>\nTwo out of nine parts are shaded. Therefore, the fraction of the square that is shaded is [latex]\\Large{\\frac{2}{9}}[\/latex].<\/p>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try it<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm145974\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=145974&#38;theme=oea&#38;iframe_resize_id=ohm145974&#38;show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Shade [latex]\\frac{3}{4}[\/latex] of the circle.<\/p>\n<p><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220702\/CNX_BMath_Figure_04_01_008_img.png\" alt=\"An image of a circle.\" \/><\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q323057\">Show Solution<\/span><\/p>\n<div id=\"q323057\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution<br \/>\nThe denominator is [latex]4[\/latex], so we divide the circle into four equal parts \u24d0.<br \/>\nThe numerator is [latex]3[\/latex], so we shade three of the four parts \u24d1.<\/p>\n<p><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220704\/CNX_BMath_Figure_04_01_009.png\" alt=\"In\" \/><br \/>\n[latex]\\Large{\\frac{3}{4}}[\/latex] of the circle is shaded.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>Try it<\/h3>\n<p>Shade [latex]\\Large{\\frac{6}{8}}[\/latex] of the circle.<\/p>\n<p><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220705\/CNX_BMath_Figure_04_01_010_img.png\" alt=\"A circle is divided into eight equal pieces.\" \/><\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q621333\">Show Solution<\/span><\/p>\n<div id=\"q621333\" 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\/24220706\/CNX_BMath_Figure_04_01_011_img.png\" alt=\"A circle is shown divided into 8 pieces, of which 6 are shaded.\" \/><\/p>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<p>Shade [latex]\\Large{\\frac{2}{5}}[\/latex] of the rectangle.<\/p>\n<p><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220707\/CNX_BMath_Figure_04_01_012_img.png\" alt=\"A rectangle is divided vertically into five equal pieces.\" \/><\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q452323\">Show Solution<\/span><\/p>\n<div id=\"q452323\" 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\/24220708\/CNX_BMath_Figure_04_01_013_img.png\" alt=\"A rectangle is divided into 5 sections, of which 2 are shaded.\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>Watch the following video to see more examples of how to write fractions given a model.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-1\" title=\"Ex:  Determine the Fraction Modeled\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/c_yIA4OQ4qA?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p>In earlier examples, we used circles and rectangles to model fractions. Fractions can also be modeled as manipulatives called fraction tiles, as shown in the image below. Here, the whole is modeled as one long, undivided rectangular tile. Beneath it are tiles of equal length divided into different numbers of equally sized parts.<\/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\/24220710\/CNX_BMath_Figure_04_01_014.png\" alt=\"One long, undivided rectangular tile is shown, labeled\" \/><br \/>\nWe\u2019ll be using fraction tiles to discover some basic facts about fractions. Refer to the fraction tiles above&nbsp;to answer the following questions:<\/p>\n<table id=\"fs-id7569845\" class=\"unnumbered\" style=\"width: 85%\" summary=\".\">\n<tbody>\n<tr>\n<td>How many [latex]\\Large{\\frac{1}{2}}[\/latex] tiles does it take to make one whole tile?<\/td>\n<td>It takes two halves to make a whole, so two out of two is [latex]{\\Large\\frac{2}{2}}=1[\/latex].<\/td>\n<\/tr>\n<tr>\n<td>How many [latex]\\Large{\\frac{1}{3}}[\/latex] tiles does it take to make one whole tile?<\/td>\n<td>It takes three thirds, so three out of three is [latex]{\\Large{\\frac{3}{3}}}=1[\/latex].<\/td>\n<\/tr>\n<tr>\n<td>How many [latex]\\Large{\\frac{1}{4}}[\/latex] tiles does it take to make one whole tile?<\/td>\n<td>It takes four fourths, so four out of four is [latex]{\\Large\\frac{4}{4}}=1[\/latex].<\/td>\n<\/tr>\n<tr>\n<td>How many [latex]\\Large{\\frac{1}{6}}[\/latex] tiles does it take to make one whole tile?<\/td>\n<td>It takes six sixths, so six out of six is [latex]{\\Large\\frac{6}{6}}=1[\/latex].<\/td>\n<\/tr>\n<tr>\n<td>What if the whole were divided into [latex]24[\/latex] equal parts? (We have not shown fraction tiles to represent this, but try to visualize it in your mind.) How many [latex]\\Large{\\frac{1}{24}}[\/latex] tiles does it take to make one whole tile?<\/td>\n<td>It takes [latex]24[\/latex] twenty-fourths, so [latex]{\\Large\\frac{24}{24}}=1[\/latex].<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>This leads us to the <em>Property of One<\/em>.<\/p>\n<div class=\"textbox shaded\">\n<h3>Property of One<\/h3>\n<p>Any number, except zero, divided by itself is one.<\/p>\n<p>[latex]{\\Large\\frac{a}{a}}=1\\left(a\\ne 0\\right)[\/latex]<\/p>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Use fraction circles to make wholes using the following pieces:<\/p>\n<ol>\n<li>[latex]4[\/latex] fourths<\/li>\n<li>[latex]5[\/latex] fifths<\/li>\n<li>[latex]6[\/latex] sixths<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q447874\">Show Solution<\/span><\/p>\n<div id=\"q447874\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution<\/p>\n<p><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220712\/CNX_BMath_Figure_04_01_015_img.png\" alt=\"Three circles are shown. The circle on the left is divided into four equal pieces. The circle in the middle is divided into five equal pieces. The circle on the right is divided into six equal pieces. Each circle says\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try it<\/h3>\n<p>Use fraction circles to make wholes with the following pieces: [latex]3[\/latex] thirds.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q412196\">Show Solution<\/span><\/p>\n<div id=\"q412196\" 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\/24220714\/CNX_BMath_Figure_04_01_016_img.png\" alt=\"A circle is shown. It is divided into 3 equal pieces. All 3 pieces are shaded.\" \/><\/p>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<p>Use fraction circles to make wholes with the following pieces: [latex]8[\/latex] eighths.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q205673\">Show Solution<\/span><\/p>\n<div id=\"q205673\" 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\/24220715\/CNX_BMath_Figure_04_01_017_img.png\" alt=\"A circle is divided into 8 sections, of which all are shaded.\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>What if we have more fraction pieces than we need for [latex]1[\/latex] whole? We\u2019ll look at this in the next example.<\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Use fraction circles to make wholes using the following pieces:<\/p>\n<ol>\n<li>[latex]3[\/latex] halves<\/li>\n<li>[latex]8[\/latex] fifths<\/li>\n<li>[latex]7[\/latex] thirds<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q938641\">Show Solution<\/span><\/p>\n<div id=\"q938641\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution<br \/>\n1. [latex]3[\/latex] halves make [latex]1[\/latex] whole with [latex]1[\/latex] half left over.<\/p>\n<p><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220716\/CNX_BMath_Figure_04_01_018_img.png\" alt=\"Two circles are shown, both divided into two equal pieces. The circle on the left has both pieces shaded and is labeled as\" \/><br \/>\n2. [latex]8[\/latex] fifths make [latex]1[\/latex] whole with [latex]2[\/latex] fifths left over.<\/p>\n<p><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220718\/CNX_BMath_Figure_04_01_019_img.png\" alt=\"Two circles are shown, both divided into five equal pieces. The circle on the left has all five pieces shaded and is labeled as\" \/><br \/>\n3. [latex]7[\/latex] thirds make [latex]2[\/latex] wholes with [latex]2[\/latex] thirds left over.<\/p>\n<p><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220720\/CNX_BMath_Figure_04_01_020_img.png\" alt=\"Three circles are shown, all divided into three equal pieces. The two circles on the left have all three pieces shaded and are labeled with ones. The circle on the right has one piece shaded and is labeled as one third.\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p>Use fraction circles to make wholes with the following pieces: [latex]5[\/latex] thirds.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q364407\">Show Solution<\/span><\/p>\n<div id=\"q364407\" 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\/24220722\/CNX_BMath_Figure_04_01_021_img.png\" alt=\"Two circles are shown. Each is divided into three sections. All of the first circle is shaded. 2 out of 3 sections of the second circle are shaded.\" \/><\/p>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<p>Use fraction circles to make wholes with the following pieces: [latex]5[\/latex] halves.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q779741\">Show Solution<\/span><\/p>\n<div id=\"q779741\" 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\/24220723\/CNX_BMath_Figure_04_01_022_img.png\" alt=\"Three circles are shown. Each is divided into two sections. The first two circles are completely shaded. Half of the third circle is shaded.\" \/><\/p>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<\/div>\n<h2>Model Improper Fractions and Mixed Numbers<\/h2>\n<p>In an earlier example, you had eight equal fifth pieces. You used five of them to make one whole, and you had three fifths left over. Let us use fraction notation to show what happened. You had eight pieces, each of them one fifth, [latex]{\\Large\\frac{1}{5}}[\/latex], so altogether you had eight fifths, which we can write as [latex]{\\Large\\frac{8}{5}}[\/latex]. The fraction [latex]{\\Large\\frac{8}{5}}[\/latex] is one whole, [latex]1[\/latex], plus three fifths, [latex]{\\Large\\frac{3}{5}}[\/latex], or [latex]1{\\Large\\frac{3}{5}}[\/latex], which is read as <em>one and three-fifths<\/em>.<\/p>\n<p>The number [latex]1{\\Large\\frac{3}{5}}[\/latex] is called a mixed number. A mixed number consists of a whole number and a fraction.<\/p>\n<div class=\"textbox shaded\">\n<h3>Mixed Numbers<\/h3>\n<p>A mixed number consists of a whole number [latex]a[\/latex] and a fraction [latex]{\\Large\\frac{b}{c}}[\/latex] where [latex]c\\ne 0[\/latex]. It is written as follows.<\/p>\n<p style=\"text-align: center\">[latex]a{\\Large\\frac{b}{c}}\\text{, }c\\ne 0[\/latex]<\/p>\n<\/div>\n<p>Fractions such as [latex]{\\Large\\frac{5}{4}},{\\Large\\frac{3}{2}},{\\Large\\frac{5}{5}}[\/latex], and [latex]{\\Large\\frac{7}{3}}[\/latex] are called improper fractions. In an improper fraction, the numerator is greater than or equal to the denominator, so its value is greater than or equal to one. When a fraction has a numerator that is smaller than the denominator, it is called a proper fraction, and its value is less than one. Fractions such as [latex]{\\Large\\frac{1}{2}},{\\Large\\frac{3}{7}}[\/latex], and [latex]{\\Large\\frac{11}{18}}[\/latex] are proper fractions.<\/p>\n<div class=\"textbox shaded\">\n<h3>Proper and Improper Fractions<\/h3>\n<p>The fraction [latex]{\\Large\\frac{a}{b}}[\/latex] is a proper fraction if [latex]a<b[\/latex] and an improper fraction if [latex]a\\ge b[\/latex].\n\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Name the improper fraction modeled. Then write the improper fraction as a mixed number.<\/p>\n<p><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220725\/CNX_BMath_Figure_04_01_023_img.png\" alt=\"Two circles are shown, both divided into three equal pieces. The circle on the left has all three pieces shaded. The circle on the right has one piece shaded.\" \/><\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q670905\">Show Solution<\/span><\/p>\n<div id=\"q670905\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution:<br \/>\nEach circle is divided into three pieces, so each piece is [latex]{\\Large\\frac{1}{3}}[\/latex] of the circle. There are four pieces shaded, so there are four thirds or [latex]{\\Large\\frac{4}{3}}[\/latex]. The figure shows that we also have one whole circle and one third, which is [latex]1{\\Large\\frac{1}{3}}[\/latex]. So, [latex]{\\Large\\frac{4}{3}}=1{\\Large\\frac{1}{3}}[\/latex].<\/p><\/div>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm145976\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=145976&theme=oea&iframe_resize_id=ohm145976&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<p><iframe loading=\"lazy\" id=\"ohm145977\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=145977&theme=oea&iframe_resize_id=ohm145977&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Draw a figure to model [latex]{\\Large\\frac{11}{8}}[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q992194\">Show Solution<\/span><\/p>\n<div id=\"q992194\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution:<br \/>\nThe denominator of the improper fraction is [latex]8[\/latex]. Draw a circle divided into eight pieces and shade all of them. This takes care of eight eighths, but we have [latex]11[\/latex] eighths. We must shade three of the eight parts of another circle.<\/p>\n<p><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220730\/CNX_BMath_Figure_04_01_026_img.png\" alt=\"Two circles are shown, both divided into eight equal pieces. The circle on the left has all eight pieces shaded and is labeled as eight eighths. The circle on the right has three pieces shaded and is labeled as three eighths. The diagram indicates that eight eighths plus three eighths is one plus three eighths.\" \/><br \/>\nSo, [latex]{\\Large\\frac{11}{8}}=1{\\Large\\frac{3}{8}}[\/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>Draw a figure to model [latex]{\\Large\\frac{7}{6}}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q924546\">Show Solution<\/span><\/p>\n<div id=\"q924546\" 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\/24220732\/CNX_BMath_Figure_04_01_027_img.png\" alt=\"Two circles are shown. Each is divided into six sections. All of the first circle is shaded and one section of the second circle is shaded.\" \/><\/p>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<p>Draw a figure to model [latex]{\\Large\\frac{6}{5}}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q203648\">Show Solution<\/span><\/p>\n<div id=\"q203648\" 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\/24220733\/CNX_BMath_Figure_04_01_028_img.png\" alt=\"Two circles are shown. Each is divided into five sections. All of the first circle is shaded and one section of the second circle is shaded.\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Use a model to rewrite the improper fraction [latex]{\\Large\\frac{11}{6}}[\/latex] as a mixed number.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q121702\">Show Solution<\/span><\/p>\n<div id=\"q121702\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution:<br \/>\nWe start with [latex]11[\/latex] sixths [latex]\\left({\\Large\\frac{11}{6}}\\right)[\/latex]. We know that six sixths makes one whole.<\/p>\n<p style=\"padding-left: 30px\">[latex]{\\Large\\frac{6}{6}}=1[\/latex]<\/p>\n<p>That leaves us with five more sixths, which is [latex]{\\Large\\frac{5}{6}}[\/latex] (11 sixths minus 6 sixths is 5 sixths).<\/p>\n<p style=\"padding-left: 30px\">So, [latex]{\\Large\\frac{11}{6}}=1{\\Large\\frac{5}{6}}[\/latex]<\/p>\n<p><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220735\/CNX_BMath_Figure_04_01_029_img.png\" alt=\"Two circles are shown, both divided into six equal pieces. The circle on the left has all six pieces shaded and is labeled as six sixths. The circle on the right has five pieces shaded and is labeled as five sixths. Below the circles, it says one plus five sixths, then six sixths plus five sixths equals eleven sixths, and one plus five sixths equals one and five sixths. It then says that eleven sixths equals one and five sixths.\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>Try it<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm145982\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=145982&theme=oea&iframe_resize_id=ohm145982&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>In the next video we show another way to draw a model that represents a fraction. &nbsp;You will see example of both proper and improper fractions shown.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-2\" title=\"Draw Models of Fractions and Explain the Meaning of the Fraction\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/akyByv80Uzc?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Use a model to rewrite the mixed number [latex]1{\\Large\\frac{4}{5}}[\/latex] as an improper fraction.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q852331\">Show Solution<\/span><\/p>\n<div id=\"q852331\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution:<br \/>\nThe mixed number [latex]1{\\Large\\frac{4}{5}}[\/latex] means one whole plus four fifths. The denominator is [latex]5[\/latex], so the whole is [latex]{\\Large\\frac{5}{5}}[\/latex]. Together five fifths and four fifths equals nine fifths.<br \/>\nSo, [latex]1{\\Large\\frac{4}{5}}={\\Large\\frac{9}{5}}[\/latex]<\/p>\n<p><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220738\/CNX_BMath_Figure_04_01_030_img.png\" alt=\"Two circles are shown, both divided into five equal pieces. The circle on the left has all five pieces shaded and is labeled as 5 fifths. The circle on the right has four pieces shaded and is labeled as 4 fifths. It then says that 5 fifths plus 4 fifths equals 9 fifths and that 9 fifths is equal to one plus 4 fifths.\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try it<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm145981\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=145981&theme=oea&iframe_resize_id=ohm145981&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\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-4624\">\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 145976, 145977, 145974, 145981, 145982, . <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>. <strong>License Terms<\/strong>: IMathAS Community License CC-BY + GPL<\/li><\/ul><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><li>Ex: Determine the Fraction Modeled. <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com). <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/c_yIA4OQ4qA\">https:\/\/youtu.be\/c_yIA4OQ4qA<\/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>Draw Models of Fractions and Explain the Meaning of the Fraction. <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com). <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/akyByv80Uzc\">https:\/\/youtu.be\/akyByv80Uzc<\/a>. <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, 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":4,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc-attribution\",\"description\":\"Prealgebra\",\"author\":\"\",\"organization\":\"OpenStax\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"Download for free at http:\/\/cnx.org\/contents\/caa57dab-41c7-455e-bd6f-f443cda5519c@9.757\"},{\"type\":\"original\",\"description\":\"Question ID 145976, 145977, 145974, 145981, 145982, \",\"author\":\"Lumen Learning\",\"organization\":\"\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"IMathAS Community License CC-BY + GPL\"},{\"type\":\"cc\",\"description\":\"Ex: Determine the Fraction Modeled\",\"author\":\"James Sousa (Mathispower4u.com)\",\"organization\":\"\",\"url\":\"https:\/\/youtu.be\/c_yIA4OQ4qA\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"cc\",\"description\":\"Draw Models of Fractions and Explain the Meaning of the Fraction\",\"author\":\"James Sousa (Mathispower4u.com)\",\"organization\":\"\",\"url\":\"https:\/\/youtu.be\/akyByv80Uzc\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"}]","CANDELA_OUTCOMES_GUID":"","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-4624","chapter","type-chapter","status-publish","hentry"],"part":4619,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/slcc-mathforliberalartscorequisite\/wp-json\/pressbooks\/v2\/chapters\/4624","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/slcc-mathforliberalartscorequisite\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/slcc-mathforliberalartscorequisite\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/slcc-mathforliberalartscorequisite\/wp-json\/wp\/v2\/users\/17533"}],"version-history":[{"count":1,"href":"https:\/\/courses.lumenlearning.com\/slcc-mathforliberalartscorequisite\/wp-json\/pressbooks\/v2\/chapters\/4624\/revisions"}],"predecessor-version":[{"id":5392,"href":"https:\/\/courses.lumenlearning.com\/slcc-mathforliberalartscorequisite\/wp-json\/pressbooks\/v2\/chapters\/4624\/revisions\/5392"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/slcc-mathforliberalartscorequisite\/wp-json\/pressbooks\/v2\/parts\/4619"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/slcc-mathforliberalartscorequisite\/wp-json\/pressbooks\/v2\/chapters\/4624\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/slcc-mathforliberalartscorequisite\/wp-json\/wp\/v2\/media?parent=4624"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/slcc-mathforliberalartscorequisite\/wp-json\/pressbooks\/v2\/chapter-type?post=4624"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/slcc-mathforliberalartscorequisite\/wp-json\/wp\/v2\/contributor?post=4624"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/slcc-mathforliberalartscorequisite\/wp-json\/wp\/v2\/license?post=4624"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}