{"id":3791,"date":"2020-02-08T20:22:25","date_gmt":"2020-02-08T20:22:25","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/mathforlibscoreq\/?post_type=chapter&#038;p=3791"},"modified":"2021-02-05T23:54:44","modified_gmt":"2021-02-05T23:54:44","slug":"using-models-to-define-slope","status":"web-only","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/slcc-mathforliberalartscorequisite\/chapter\/using-models-to-define-slope\/","title":{"raw":"Using Models to Define Slope","rendered":"Using Models to Define Slope"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Outcomes<\/h3>\r\n<ul>\r\n \t<li>Use a model to determine slope<\/li>\r\n \t<li>Create a model of slope<\/li>\r\n<\/ul>\r\n<\/div>\r\n<p>In this section, we will explore the concepts of slope.<\/p>\r\n<p>Using rubber bands on a geoboard gives a concrete way to model lines on a coordinate grid. By stretching a rubber band between two pegs on a geoboard, we can discover how to find the slope of a line. And when you ride a bicycle, you <u>feel<\/u> the slope as you pump uphill or coast downhill.<\/p>\r\nWe\u2019ll start by stretching a rubber band between two pegs to make a line as shown in the image below.\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/25224529\/CNX_BMath_Figure_11_04_001.png\" alt=\"The figure shows a grid of evenly spaced dots. There are 5 rows and 5 columns. There is a rubber band style loop connecting the point in column 1 row 4 and the point in column 4 row 2.\" \/>\r\nDoes it look like a line?\r\n\r\nNow we stretch one part of the rubber band straight up from the left peg and around a third peg to make the sides of a right triangle as shown in the image below. We carefully make a [latex]90^ \\circ [\/latex] angle around the third peg, so that one side is vertical and the other is horizontal.\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/25224530\/CNX_BMath_Figure_11_04_002.png\" alt=\"The figure shows a grid of evenly spaced dots. There are 5 rows and 5 columns. There is a rubber band style triangle connecting three of the three points at column 1 row 2, column 1 row 4,and column 4 row 2.\" \/>\r\nTo find the slope of the line, we measure the distance along the vertical and horizontal legs of the triangle. The vertical distance is called the <em>rise<\/em> and the horizontal distance is called the <em>run<\/em>, as shown below.\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/25224531\/CNX_BMath_Figure_11_04_003.png\" alt=\"This figure shows two arrows. The first arrow is vertical and is labeled \" \/>\r\nTo help remember the terms, it may help to think of the images shown below.\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/25224533\/CNX_BMath_Figure_11_04_004.png\" alt=\"...\" \/>\r\nOn our geoboard, the rise is [latex]2[\/latex] units because the rubber band goes up [latex]2[\/latex] spaces on the vertical leg. See the image below.\r\n\r\nWhat is the run? Be sure to count the spaces between the pegs rather than the pegs themselves! The rubber band goes across [latex]3[\/latex] spaces on the horizontal leg, so the run is [latex]3[\/latex] units.\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/25224536\/CNX_BMath_Figure_11_04_005.png\" alt=\"The figure shows a grid of evenly spaced dots. There are 5 rows and 5 columns. There is a rubber band style triangle connecting three of the three points at column 1 row 2, column 1 row 4, and column 4 row 2. The triangle has a rise of 2 units and a run of 3 units.\" \/>\r\nThe slope of a line is the ratio of the rise to the run. So the slope of our line is [latex]{\\Large\\frac{2}{3}}[\/latex]. In mathematics, the slope is always represented by the letter [latex]m[\/latex].\r\n<div class=\"textbox shaded\">\r\n<h3>Slope of a line<\/h3>\r\nThe slope of a line is [latex]m={\\Large\\frac{\\text{rise}}{\\text{run}}}[\/latex].\r\nThe rise measures the vertical change and the run measures the horizontal change.\r\n\r\n<\/div>\r\nWhat is the slope of the line on the geoboard in the image above?\r\n<p style=\"text-align: center\">[latex]m={\\Large\\frac{\\text{rise}}{\\text{run}}}[\/latex]\r\n[latex]m={\\Large\\frac{2}{3}}[\/latex]<\/p>\r\n<p style=\"text-align: center\">The line has slope [latex]{\\Large\\frac{2}{3}}[\/latex]<\/p>\r\nWhen we work with geoboards, it is a good idea to get in the habit of starting at a peg on the left and connecting to a peg to the right. Then we stretch the rubber band to form a right triangle.\r\n\r\nIf we start by going up the rise is positive, and if we stretch it down the rise is negative. We will count the run from left to right, just like you read this paragraph, so the run will be positive.\r\n\r\nSince the slope formula has rise over run, it may be easier to always count out the rise first and then the run.\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nWhat is the slope of the line on the geoboard shown?\r\n\r\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/25224537\/CNX_BMath_Figure_11_04_006.png\" alt=\"The figure shows a grid of evenly spaced dots. There are 5 rows and 5 columns. There is a rubber band style loop connecting the point in column 1 row 5 and the point in column 5 row 2.\" \/>\r\n\r\nSolution\r\nUse the definition of slope.\r\n[latex]m={\\Large\\frac{\\text{rise}}{\\text{run}}}[\/latex]\r\n\r\nStart at the left peg and make a right triangle by stretching the rubber band up and to the right to reach the second peg.\r\nCount the rise and the run as shown.\r\n\r\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/25224538\/CNX_BMath_Figure_11_04_007.png\" alt=\"The figure shows a grid of evenly spaced dots. There are 5 rows and 5 columns. There is a rubber band style triangle connecting three of the three points at column 1 row 2, column 1 row 5,and column 5 row 2.\" \/>\r\n[latex]\\begin{array}{cccc}\\text{The rise is }3\\text{ units}.\\hfill &amp; &amp; &amp; m={\\Large\\frac{3}{\\text{run}}}\\hfill \\\\ \\text{The run is}4\\text{ units}.\\hfill &amp; &amp; &amp; m={\\Large\\frac{3}{4}}\\hfill \\\\ &amp; &amp; &amp; \\text{The slope is }{\\Large\\frac{3}{4}}\\hfill \\end{array}[\/latex]\r\n\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nWhat is the slope of the line on the geoboard shown?\r\n\r\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/25224542\/CNX_BMath_Figure_11_04_010.png\" alt=\"The figure shows a grid of evenly spaced dots. There are 5 rows and 5 columns. There is a rubber band style loop connecting the point in column 1 row 3 and the point in column 4 row 4.\" \/>\r\n[reveal-answer q=\"698295\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"698295\"]\r\n\r\nSolution\r\nUse the definition of slope.\r\n[latex]m={\\Large\\frac{\\text{rise}}{\\text{run}}}[\/latex]\r\n\r\nStart at the left peg and make a right triangle by stretching the rubber band to the peg on the right. This time we need to stretch the rubber band down to make the vertical leg, so the rise is negative.\r\n\r\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/25224543\/CNX_BMath_Figure_11_04_011.png\" alt=\"The figure shows a grid of evenly spaced dots. There are 5 rows and 5 columns. There is a rubber band style triangle connecting three of the three points at column 1 row 3, column 1 row 4,and column 4 row 4.\" \/>\r\n[latex]\\begin{array}{cccc}\\text{The rise is }-1.\\hfill &amp; &amp; &amp; m={\\Large\\frac{-1}{\\text{run}}}\\hfill \\\\ \\text{The run is}3.\\hfill &amp; &amp; &amp; m={\\Large\\frac{-1}{3}}\\hfill \\\\ &amp; &amp; &amp; m=-{\\Large\\frac{1}{3}}\\hfill \\\\ &amp; &amp; &amp; \\text{The slope is }-{\\Large\\frac{1}{3}}\\hfill \\end{array}[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\n[ohm_question]147013[\/ohm_question]\r\n\r\n<\/div>\r\nNotice that in the first example, the slope is positive and in the second example the slope is negative. Do you notice any difference in the two lines shown in the images below.\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/25224547\/CNX_BMath_Figure_11_04_059.png\" alt=\"...\" \/>\r\nAs you read from left to right, the line in Figure A, is going up; it has positive slope. The line Figure B is going down; it has negative slope.\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/25224549\/CNX_BMath_Figure_11_04_060_img.png\" alt=\"...\" \/>\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nUse a geoboard to model a line with slope [latex]{\\Large\\frac{1}{2}}[\/latex].\r\n[reveal-answer q=\"225850\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"225850\"]\r\n\r\nSolution\r\nTo model a line with a specific slope on a geoboard, we need to know the rise and the run.\r\n<table id=\"eip-id1172468198215\" class=\"unnumbered unstyled\" summary=\".\">\r\n<tbody>\r\n<tr>\r\n<td>Use the slope formula.<\/td>\r\n<td>[latex]m=\\Large\\frac{\\text{rise}}{\\text{run}}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Replace [latex]m[\/latex] with [latex]{\\Large\\frac{1}{2}}[\/latex] .<\/td>\r\n<td>[latex]\\Large\\frac{1}{2}\\normalsize =\\Large\\frac{\\text{rise}}{\\text{run}}[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nSo, the rise is [latex]1[\/latex] unit and the run is [latex]2[\/latex] units.\r\nStart at a peg in the lower left of the geoboard. Stretch the rubber band up [latex]1[\/latex] unit, and then right [latex]2[\/latex] units.\r\n\r\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/25224550\/CNX_BMath_Figure_11_04_014.png\" alt=\"The figure shows a grid of evenly spaced dots. There are 5 rows and 5 columns. There is a rubber band style triangle connecting three of the three points at column 1 row 3, column 1 row 4,and column 3 row 3.\" \/>\r\nThe hypotenuse of the right triangle formed by the rubber band represents a line with a slope of [latex]{\\Large\\frac{1}{2}}[\/latex].\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\nUse a geoboard to model a line with the given slope: [latex]m=\\Large\\frac{1}{3}[\/latex].\r\n[reveal-answer q=\"152695\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"152695\"]\r\n\r\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/25224551\/CNX_BMath_Figure_11_04_015_img.png\" alt=\"The figure shows a grid of evenly spaced dots. There are 5 rows and 5 columns. There is a rubber band style triangle connecting three of the three points at column 2 row 3, column 2 row 4,and column 5 row 3.\" \/>\r\n\r\n[\/hidden-answer]\r\n\r\n&nbsp;\r\n\r\nUse a geoboard to model a line with the given slope: [latex]m=\\Large\\frac{3}{2}[\/latex].\r\n[reveal-answer q=\"64394\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"64394\"]\r\n\r\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/25224552\/CNX_BMath_Figure_11_04_016_img.png\" alt=\"The figure shows a grid of evenly spaced dots. There are 5 rows and 5 columns. There is a rubber band style triangle connecting three of the three points at column 1 row 1, column 1 row 4,and column 3 row 1.\" \/>\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nUse a geoboard to model a line with slope [latex]{\\Large\\frac{-1}{4}}[\/latex]\r\n[reveal-answer q=\"953715\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"953715\"]\r\n\r\nSolution\r\n<table id=\"eip-id1172466948346\" class=\"unnumbered unstyled\" summary=\".\">\r\n<tbody>\r\n<tr>\r\n<td>Use the slope formula.<\/td>\r\n<td>[latex]m=\\Large\\frac{\\text{rise}}{\\text{run}}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Replace [latex]m[\/latex] with [latex]-{\\Large\\frac{1}{4}}[\/latex] .<\/td>\r\n<td>[latex]-\\Large\\frac{1}{4}\\normalsize =\\Large\\frac{\\text{rise}}{\\text{run}}[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nSo, the rise is [latex]-1[\/latex] and the run is [latex]4[\/latex].\r\nSince the rise is negative, we choose a starting peg on the upper left that will give us room to count down. We stretch the rubber band down [latex]1[\/latex] unit, then to the right [latex]4[\/latex] units.\r\n\r\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/25224553\/CNX_BMath_Figure_11_04_017_img.png\" alt=\"The figure shows a grid of evenly spaced dots. There are 5 rows and 5 columns. There is a rubber band style triangle connecting three of the three points at column 1 row 2, column 1 row 3,and column 5 row 3.\" \/>\r\nThe hypotenuse of the right triangle formed by the rubber band represents a line whose slope is [latex]-{\\Large\\frac{1}{4}}[\/latex].\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\nUse a geoboard to model a line with the given slope: [latex]m={\\Large\\frac{-3}{2}}[\/latex].\r\n[reveal-answer q=\"99402\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"99402\"]\r\n\r\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/25224555\/CNX_BMath_Figure_11_04_018_img.png\" alt=\"The figure shows a grid of evenly spaced dots. There are 5 rows and 5 columns. There is a rubber band style triangle connecting three of the three points at column 2 row 3, column 2 row 5,and column 3 row 5.\" \/>\r\n\r\n[\/hidden-answer]\r\n\r\n&nbsp;\r\n\r\nUse a geoboard to model a line with the given slope: [latex]m={\\Large\\frac{-1}{3}}[\/latex].\r\n[reveal-answer q=\"714001\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"714001\"]\r\n\r\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/25224556\/CNX_BMath_Figure_11_04_019_img.png\" alt=\"The figure shows a grid of evenly spaced dots. There are 5 rows and 5 columns. There is a rubber band style triangle connecting three of the three points at column 1 row 1, column 1 row 2,and column 4 row 2.\" \/>\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li>Use a model to determine slope<\/li>\n<li>Create a model of slope<\/li>\n<\/ul>\n<\/div>\n<p>In this section, we will explore the concepts of slope.<\/p>\n<p>Using rubber bands on a geoboard gives a concrete way to model lines on a coordinate grid. By stretching a rubber band between two pegs on a geoboard, we can discover how to find the slope of a line. And when you ride a bicycle, you <u>feel<\/u> the slope as you pump uphill or coast downhill.<\/p>\n<p>We\u2019ll start by stretching a rubber band between two pegs to make a line as shown in the image below.<\/p>\n<p><img decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/25224529\/CNX_BMath_Figure_11_04_001.png\" alt=\"The figure shows a grid of evenly spaced dots. There are 5 rows and 5 columns. There is a rubber band style loop connecting the point in column 1 row 4 and the point in column 4 row 2.\" \/><br \/>\nDoes it look like a line?<\/p>\n<p>Now we stretch one part of the rubber band straight up from the left peg and around a third peg to make the sides of a right triangle as shown in the image below. We carefully make a [latex]90^ \\circ[\/latex] angle around the third peg, so that one side is vertical and the other is horizontal.<\/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\/25224530\/CNX_BMath_Figure_11_04_002.png\" alt=\"The figure shows a grid of evenly spaced dots. There are 5 rows and 5 columns. There is a rubber band style triangle connecting three of the three points at column 1 row 2, column 1 row 4,and column 4 row 2.\" \/><br \/>\nTo find the slope of the line, we measure the distance along the vertical and horizontal legs of the triangle. The vertical distance is called the <em>rise<\/em> and the horizontal distance is called the <em>run<\/em>, as shown below.<\/p>\n<p><img decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/25224531\/CNX_BMath_Figure_11_04_003.png\" alt=\"This figure shows two arrows. The first arrow is vertical and is labeled\" \/><br \/>\nTo help remember the terms, it may help to think of the images shown below.<\/p>\n<p><img decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/25224533\/CNX_BMath_Figure_11_04_004.png\" alt=\"...\" \/><br \/>\nOn our geoboard, the rise is [latex]2[\/latex] units because the rubber band goes up [latex]2[\/latex] spaces on the vertical leg. See the image below.<\/p>\n<p>What is the run? Be sure to count the spaces between the pegs rather than the pegs themselves! The rubber band goes across [latex]3[\/latex] spaces on the horizontal leg, so the run is [latex]3[\/latex] units.<\/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\/25224536\/CNX_BMath_Figure_11_04_005.png\" alt=\"The figure shows a grid of evenly spaced dots. There are 5 rows and 5 columns. There is a rubber band style triangle connecting three of the three points at column 1 row 2, column 1 row 4, and column 4 row 2. The triangle has a rise of 2 units and a run of 3 units.\" \/><br \/>\nThe slope of a line is the ratio of the rise to the run. So the slope of our line is [latex]{\\Large\\frac{2}{3}}[\/latex]. In mathematics, the slope is always represented by the letter [latex]m[\/latex].<\/p>\n<div class=\"textbox shaded\">\n<h3>Slope of a line<\/h3>\n<p>The slope of a line is [latex]m={\\Large\\frac{\\text{rise}}{\\text{run}}}[\/latex].<br \/>\nThe rise measures the vertical change and the run measures the horizontal change.<\/p>\n<\/div>\n<p>What is the slope of the line on the geoboard in the image above?<\/p>\n<p style=\"text-align: center\">[latex]m={\\Large\\frac{\\text{rise}}{\\text{run}}}[\/latex]<br \/>\n[latex]m={\\Large\\frac{2}{3}}[\/latex]<\/p>\n<p style=\"text-align: center\">The line has slope [latex]{\\Large\\frac{2}{3}}[\/latex]<\/p>\n<p>When we work with geoboards, it is a good idea to get in the habit of starting at a peg on the left and connecting to a peg to the right. Then we stretch the rubber band to form a right triangle.<\/p>\n<p>If we start by going up the rise is positive, and if we stretch it down the rise is negative. We will count the run from left to right, just like you read this paragraph, so the run will be positive.<\/p>\n<p>Since the slope formula has rise over run, it may be easier to always count out the rise first and then the run.<\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>What is the slope of the line on the geoboard shown?<\/p>\n<p><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/25224537\/CNX_BMath_Figure_11_04_006.png\" alt=\"The figure shows a grid of evenly spaced dots. There are 5 rows and 5 columns. There is a rubber band style loop connecting the point in column 1 row 5 and the point in column 5 row 2.\" \/><\/p>\n<p>Solution<br \/>\nUse the definition of slope.<br \/>\n[latex]m={\\Large\\frac{\\text{rise}}{\\text{run}}}[\/latex]<\/p>\n<p>Start at the left peg and make a right triangle by stretching the rubber band up and to the right to reach the second peg.<br \/>\nCount the rise and the run as shown.<\/p>\n<p><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/25224538\/CNX_BMath_Figure_11_04_007.png\" alt=\"The figure shows a grid of evenly spaced dots. There are 5 rows and 5 columns. There is a rubber band style triangle connecting three of the three points at column 1 row 2, column 1 row 5,and column 5 row 2.\" \/><br \/>\n[latex]\\begin{array}{cccc}\\text{The rise is }3\\text{ units}.\\hfill & & & m={\\Large\\frac{3}{\\text{run}}}\\hfill \\\\ \\text{The run is}4\\text{ units}.\\hfill & & & m={\\Large\\frac{3}{4}}\\hfill \\\\ & & & \\text{The slope is }{\\Large\\frac{3}{4}}\\hfill \\end{array}[\/latex]<\/p>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>What is the slope of the line on the geoboard shown?<\/p>\n<p><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/25224542\/CNX_BMath_Figure_11_04_010.png\" alt=\"The figure shows a grid of evenly spaced dots. There are 5 rows and 5 columns. There is a rubber band style loop connecting the point in column 1 row 3 and the point in column 4 row 4.\" \/><\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q698295\">Show Solution<\/span><\/p>\n<div id=\"q698295\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution<br \/>\nUse the definition of slope.<br \/>\n[latex]m={\\Large\\frac{\\text{rise}}{\\text{run}}}[\/latex]<\/p>\n<p>Start at the left peg and make a right triangle by stretching the rubber band to the peg on the right. This time we need to stretch the rubber band down to make the vertical leg, so the rise is negative.<\/p>\n<p><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/25224543\/CNX_BMath_Figure_11_04_011.png\" alt=\"The figure shows a grid of evenly spaced dots. There are 5 rows and 5 columns. There is a rubber band style triangle connecting three of the three points at column 1 row 3, column 1 row 4,and column 4 row 4.\" \/><br \/>\n[latex]\\begin{array}{cccc}\\text{The rise is }-1.\\hfill & & & m={\\Large\\frac{-1}{\\text{run}}}\\hfill \\\\ \\text{The run is}3.\\hfill & & & m={\\Large\\frac{-1}{3}}\\hfill \\\\ & & & m=-{\\Large\\frac{1}{3}}\\hfill \\\\ & & & \\text{The slope is }-{\\Large\\frac{1}{3}}\\hfill \\end{array}[\/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><iframe loading=\"lazy\" id=\"ohm147013\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=147013&theme=oea&iframe_resize_id=ohm147013&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>Notice that in the first example, the slope is positive and in the second example the slope is negative. Do you notice any difference in the two lines shown in the images below.<\/p>\n<p><img decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/25224547\/CNX_BMath_Figure_11_04_059.png\" alt=\"...\" \/><br \/>\nAs you read from left to right, the line in Figure A, is going up; it has positive slope. The line Figure B is going down; it has negative slope.<\/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\/25224549\/CNX_BMath_Figure_11_04_060_img.png\" alt=\"...\" \/><\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Use a geoboard to model a line with slope [latex]{\\Large\\frac{1}{2}}[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q225850\">Show Solution<\/span><\/p>\n<div id=\"q225850\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution<br \/>\nTo model a line with a specific slope on a geoboard, we need to know the rise and the run.<\/p>\n<table id=\"eip-id1172468198215\" class=\"unnumbered unstyled\" summary=\".\">\n<tbody>\n<tr>\n<td>Use the slope formula.<\/td>\n<td>[latex]m=\\Large\\frac{\\text{rise}}{\\text{run}}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Replace [latex]m[\/latex] with [latex]{\\Large\\frac{1}{2}}[\/latex] .<\/td>\n<td>[latex]\\Large\\frac{1}{2}\\normalsize =\\Large\\frac{\\text{rise}}{\\text{run}}[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>So, the rise is [latex]1[\/latex] unit and the run is [latex]2[\/latex] units.<br \/>\nStart at a peg in the lower left of the geoboard. Stretch the rubber band up [latex]1[\/latex] unit, and then right [latex]2[\/latex] units.<\/p>\n<p><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/25224550\/CNX_BMath_Figure_11_04_014.png\" alt=\"The figure shows a grid of evenly spaced dots. There are 5 rows and 5 columns. There is a rubber band style triangle connecting three of the three points at column 1 row 3, column 1 row 4,and column 3 row 3.\" \/><br \/>\nThe hypotenuse of the right triangle formed by the rubber band represents a line with a slope of [latex]{\\Large\\frac{1}{2}}[\/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>Use a geoboard to model a line with the given slope: [latex]m=\\Large\\frac{1}{3}[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q152695\">Show Solution<\/span><\/p>\n<div id=\"q152695\" 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\/25224551\/CNX_BMath_Figure_11_04_015_img.png\" alt=\"The figure shows a grid of evenly spaced dots. There are 5 rows and 5 columns. There is a rubber band style triangle connecting three of the three points at column 2 row 3, column 2 row 4,and column 5 row 3.\" \/><\/p>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<p>Use a geoboard to model a line with the given slope: [latex]m=\\Large\\frac{3}{2}[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q64394\">Show Solution<\/span><\/p>\n<div id=\"q64394\" 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\/25224552\/CNX_BMath_Figure_11_04_016_img.png\" alt=\"The figure shows a grid of evenly spaced dots. There are 5 rows and 5 columns. There is a rubber band style triangle connecting three of the three points at column 1 row 1, column 1 row 4,and column 3 row 1.\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Use a geoboard to model a line with slope [latex]{\\Large\\frac{-1}{4}}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q953715\">Show Solution<\/span><\/p>\n<div id=\"q953715\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution<\/p>\n<table id=\"eip-id1172466948346\" class=\"unnumbered unstyled\" summary=\".\">\n<tbody>\n<tr>\n<td>Use the slope formula.<\/td>\n<td>[latex]m=\\Large\\frac{\\text{rise}}{\\text{run}}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Replace [latex]m[\/latex] with [latex]-{\\Large\\frac{1}{4}}[\/latex] .<\/td>\n<td>[latex]-\\Large\\frac{1}{4}\\normalsize =\\Large\\frac{\\text{rise}}{\\text{run}}[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>So, the rise is [latex]-1[\/latex] and the run is [latex]4[\/latex].<br \/>\nSince the rise is negative, we choose a starting peg on the upper left that will give us room to count down. We stretch the rubber band down [latex]1[\/latex] unit, then to the right [latex]4[\/latex] units.<\/p>\n<p><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/25224553\/CNX_BMath_Figure_11_04_017_img.png\" alt=\"The figure shows a grid of evenly spaced dots. There are 5 rows and 5 columns. There is a rubber band style triangle connecting three of the three points at column 1 row 2, column 1 row 3,and column 5 row 3.\" \/><br \/>\nThe hypotenuse of the right triangle formed by the rubber band represents a line whose slope is [latex]-{\\Large\\frac{1}{4}}[\/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>Use a geoboard to model a line with the given slope: [latex]m={\\Large\\frac{-3}{2}}[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q99402\">Show Solution<\/span><\/p>\n<div id=\"q99402\" 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\/25224555\/CNX_BMath_Figure_11_04_018_img.png\" alt=\"The figure shows a grid of evenly spaced dots. There are 5 rows and 5 columns. There is a rubber band style triangle connecting three of the three points at column 2 row 3, column 2 row 5,and column 3 row 5.\" \/><\/p>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<p>Use a geoboard to model a line with the given slope: [latex]m={\\Large\\frac{-1}{3}}[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q714001\">Show Solution<\/span><\/p>\n<div id=\"q714001\" 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\/25224556\/CNX_BMath_Figure_11_04_019_img.png\" alt=\"The figure shows a grid of evenly spaced dots. There are 5 rows and 5 columns. There is a rubber band style triangle connecting three of the three points at column 1 row 1, column 1 row 2,and column 4 row 2.\" \/><\/p>\n<\/div>\n<\/div>\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-3791\">\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 147013. <strong>Authored by<\/strong>: Lumen Learning. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><\/ul><div class=\"license-attribution-dropdown-subheading\">CC licensed content, 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":25777,"menu_order":9,"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 147013\",\"author\":\"Lumen Learning\",\"organization\":\"\",\"url\":\"\",\"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-3791","chapter","type-chapter","status-web-only","hentry"],"part":1040,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/slcc-mathforliberalartscorequisite\/wp-json\/pressbooks\/v2\/chapters\/3791","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\/25777"}],"version-history":[{"count":1,"href":"https:\/\/courses.lumenlearning.com\/slcc-mathforliberalartscorequisite\/wp-json\/pressbooks\/v2\/chapters\/3791\/revisions"}],"predecessor-version":[{"id":3792,"href":"https:\/\/courses.lumenlearning.com\/slcc-mathforliberalartscorequisite\/wp-json\/pressbooks\/v2\/chapters\/3791\/revisions\/3792"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/slcc-mathforliberalartscorequisite\/wp-json\/pressbooks\/v2\/parts\/1040"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/slcc-mathforliberalartscorequisite\/wp-json\/pressbooks\/v2\/chapters\/3791\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/slcc-mathforliberalartscorequisite\/wp-json\/wp\/v2\/media?parent=3791"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/slcc-mathforliberalartscorequisite\/wp-json\/pressbooks\/v2\/chapter-type?post=3791"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/slcc-mathforliberalartscorequisite\/wp-json\/wp\/v2\/contributor?post=3791"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/slcc-mathforliberalartscorequisite\/wp-json\/wp\/v2\/license?post=3791"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}