<h2>Understanding the Slope of a Line</h2>
<p>When you graph linear equations, you may notice that some lines tilt up as they go from left to right and some lines tilt down. Some lines are very steep and some lines are flatter. What determines whether a line tilts up or down or if it is steep or flat?</p>
<p>In mathematics, the ‘tilt’ of a line is called the <em>slope</em> of the line. The concept of slope has many applications in the real world. The pitch of a roof, grade of a highway, and a ramp for a wheelchair are some examples where you literally see slopes. And when you ride a bicycle, you feel the slope as you pump uphill or coast downhill.</p>
<p>In this section, we will explore the concept of slope.</p>
<h2>Using Geoboards to Model Slope</h2>
<p>A <strong>geoboard</strong> is a board with a grid of pegs on it. Using rubber bands on a geoboard gives us 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.</p>
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<p>We’ll start by stretching a rubber band between two pegs as shown in the figure below.</p>
<img alt="The figure shows a grid of evenly spaced pegs. There are 5 columns and 5 rows of pegs. A rubber band is stretched between the peg in column 1, row 4 and the peg in column 4, row 2, forming a line." src="https://data.ulearngo.com/assets/78a7617d-8dee-498e-b3e8-de9796bd0f20"/>
<p>Doesn’t it look like a line?</p>
<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 figure below</p>
<img alt="The figure shows a grid of evenly spaced pegs. There are 5 columns and 5 rows of pegs. A rubber band is stretched between the peg in column 1, row 2, the peg in column 1, row 4, and the peg in column 4, row 2, forming a right triangle. The 1, 2 peg is the vertex of the 90 degree angle, while the line between the 1, 4 and 4, 2 pegs forms the hypotenuse of the triangle." src="https://data.ulearngo.com/assets/85579653-87e3-42d8-b257-2f070ba74fd6"/>
<p>We carefully make a 90º angle around the third peg, so one of the newly formed lines is vertical and the other is horizontal.</p>
<p>To find the slope of the line, we measure the distance along the vertical and horizontal sides of the triangle. The vertical distance is called the <strong>rise</strong> and the horizontal distance is called the <strong>run</strong>, as shown in the figure below.</p>
<img alt="In this illustration, there are two perpendicular lines with arrows. The first line extends straight upward and is labeled “rise”. The second arrow extends straight rightward and is labeled “run”." src="https://data.ulearngo.com/assets/0c665109-3fcf-4541-ad61-482992ca4b96"/>
<p>If our <strong>geoboard</strong> and rubber band look just like the one shown in the figure below, the rise is 2. The rubber band goes up 2 units. (Each space is one unit.)</p>
<div class="wp-caption alignnone" style="width: 210px"><img alt="The figure shows a grid of evenly spaced pegs. There are 5 columns and 5 rows of pegs. A rubber band is stretched between the peg in column 1, row 2, the peg in column 1, row 4, and the peg in column 4, row 2, forming a right triangle where the 1, 2 peg is the vertex of the 90 degree angle and the line between the 1, 4 peg and the 4, 2 peg forms the hypotenuse. The line between the 1, 2 peg and the 1, 4 peg is labeled “2”. The line between the 1, 2 peg and the 4, 2 peg is labeled “3”." height="210" src="https://data.ulearngo.com/assets/e12a1218-d707-423b-8a66-394d12202b77" width="210"/><p class="wp-caption-text">The rise on this geoboard is 2, as the rubber band goes up two units.</p></div>
<p>What is the run?</p>
<p>The rubber band goes across 3 units. The run is 3 (see the figure above).</p>
<p>The slope of a line is the ratio of the rise to the run. In mathematics, it is always referred to with the letter \(m\).</p>
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<h3>Slope of a Line</h3>
<p>The <strong>slope of a line</strong> of a line is \(m=\frac{\text{rise}}{\text{run}}\).</p>
<p>The <strong>rise</strong> measures the vertical change and the <strong>run</strong> measures the horizontal change between two points on the line.</p>
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<p>What is the slope of the line on the geoboard in the figure above?</p>
<div>\(\begin{array}{ccc}\hfill m&amp; =\hfill &amp; \frac{\text{rise}}{\text{run}}\hfill \\ \hfill m&amp; =\hfill &amp; \frac{2}{3}\hfill \end{array}\)</div>
<p>The line has slope \(\frac{2}{3}\). This means that the line rises 2 units for every 3 units of run.</p>
<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. If the rise goes up it is positive and if it goes down it is negative. The run will go from left to right and be positive.</p>
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<h2>Example 1:</h2>
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<p>What is the slope of the line on the geoboard shown?</p>
<img alt="The figure shows a grid of evenly spaced pegs. There are 5 columns and 5 rows of pegs. A rubber band is stretched between the peg in column 1, row 5 and the peg in column 5, row 2, forming a line." src="https://data.ulearngo.com/assets/c6c74796-051c-4ad2-8f86-768dbeec935a"/></div>
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<h3>Solution</h3>
<p>Use the definition of slope: \(m=\frac{\text{rise}}{\text{run}}.\)</p>
<p>Start at the left peg and count the spaces up and to the right to reach the second peg.</p>
<img alt="The figure shows a grid of evenly spaced pegs. There are 5 columns and 5 rows of pegs. A rubber band is stretched between the peg in column 1, row 2, the peg in column 1, row 5 and the peg in column 5, row 2, forming a right triangle. The 1, 2 peg forms the vertex of the 90 degree angle and the line from the 1, 5 peg to the 5, 2 peg forms the hypotenuse of the triangle. The line from the 1, 2 peg to the 1, 5 peg is labeled “3”. The line from the 1, 2 peg to the 5, 2 peg is labeled “4”." src="https://data.ulearngo.com/assets/2bc347df-108f-472a-9ab0-7bab8749309e"/>
<p>\(\begin{array}{ccc}\text{The rise is 3.}\hfill &amp; &amp; \phantom{\rule{5em}{0ex}}m=\frac{3}{\text{run}}\hfill \\ \text{The run is 4.}\hfill &amp; &amp; \phantom{\rule{5em}{0ex}}m=\frac{3}{4}\hfill \\ &amp; &amp; \phantom{\rule{5em}{0ex}}\text{The slope is}\phantom{\rule{0.4em}{0ex}}\frac{3}{4}.\hfill \end{array}\)</p>
<p>This means that the line rises 3 units for every 4 units of run.</p>
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<h2>Example 2:</h2>
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<p>What is the slope of the line on the geoboard shown?</p>
<img alt="The figure shows a grid of evenly spaced pegs. There are 5 columns and 5 rows of pegs. A rubber band is stretched between the peg in column 1, row 3 and the peg in column 4, row 4, forming a line." src="https://data.ulearngo.com/assets/6dac7d61-db8c-42dc-9505-6ffafa83bb7f"/></div>
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<h3>Solution</h3>
<p>Use the definition of slope: \(m=\frac{\text{rise}}{\text{run}}.\)</p>
<p>Start at the left peg and count the units down and to the right to reach the second peg.</p>
<img alt="The figure shows a grid of evenly spaced pegs. There are 5 columns and 5 rows of pegs. A rubber band is stretched between the peg in column 1, row 3, the peg in column 1, row 4 and the peg in column 4, row 4, forming a right triangle. The 1, 3 peg forms the vertex of the 90 degree angle and the line from the 1, 4 peg to the 4, 4 peg forms the hypotenuse of the triangle. The line from the 1, 3 peg to the 1, 4 peg is labeled “negative 1”. The line from the 1, 4 peg to the 4, 4 peg is labeled “3”." src="https://data.ulearngo.com/assets/d564435a-6a2b-4dbf-a41f-e0203e284691"/>
<p>\(\begin{array}{cccc}\text{The rise is}\phantom{\rule{0.2em}{0ex}}-1.\hfill &amp; &amp; &amp; \phantom{\rule{1.5em}{0ex}}m=\frac{-1}{\text{run}}\hfill \\ \text{The run is 3.}\hfill &amp; &amp; &amp; \phantom{\rule{1.5em}{0ex}}m=\frac{-1}{3}\hfill \\ &amp; &amp; &amp; \phantom{\rule{1.5em}{0ex}}m=-\frac{1}{3}\hfill \\ &amp; &amp; &amp; \text{The slope is}\phantom{\rule{0.2em}{0ex}}-\frac{1}{3}.\hfill \end{array}\)</p>
<p>This means that the line drops 1 unit for every 3 units of run.</p>
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<p>Notice that in example 1 above the slope is positive and in example 2 above the slope is negative. Do you notice any difference in the two lines shown in figure (a) below and figure (b) below?</p>
<p><img alt="The figure shows two grids of evenly spaced pegs, one labeled (a) and one labeled (b). There are 5 columns and 5 rows of pegs in each grid. In the (a) grid, a rubber band is stretched between the peg in column 1, row 5 and the peg in column 5, row 2, forming a line. Below this grid is the slope of a line defined as m equals 3 fourths. In the (b) grid, a rubber band is stretched between the peg in column 1, row 3 and the peg in column 4, row 4, forming a line. Below this grid is the slope of a line defined as m equals negative 1 third." src="https://data.ulearngo.com/assets/54bc55e3-9621-4683-984f-d011afd31b4f"/></p>
<p>We ‘read’ a line from left to right just like we read words in English. As you read from left to right, the line in figure (a) below is going up; it has <strong>positive slope</strong>. The line in figure (b) below is going down; it has <strong>negative slope</strong>.</p>
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<h3>Positive and Negative Slopes</h3>
<img alt="The figure shows two lines side-by-side. The line on the left is a diagonal line that rises from left to right. It is labeled “Positive slope”. The line on the right is a diagonal line that drops from left to right. It is labeled “Negative slope”." src="https://data.ulearngo.com/assets/5a908d1c-6cb6-4160-a354-abc167cc24ed"/></div>
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<h2>Example</h2>
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<p>Use a geoboard to model a line with slope \(\frac{1}{2}\).</p>
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<h3>Solution</h3>
<p>To model a line on a geoboard, we need the rise and the run.</p>
<p>\(\begin{array}{ccccc}\text{Use the slope formula.}\hfill &amp; &amp; \hfill \phantom{\rule{8em}{0ex}}m&amp; =\hfill &amp; \frac{\text{rise}}{\text{run}}\hfill \\ \text{Replace}\phantom{\rule{0.2em}{0ex}}m\phantom{\rule{0.2em}{0ex}}\text{with}\phantom{\rule{0.2em}{0ex}}\frac{1}{2}.\hfill &amp; &amp; \hfill \phantom{\rule{8em}{0ex}}\frac{1}{2}&amp; =\hfill &amp; \frac{\text{rise}}{\text{run}}\hfill \end{array}\)</p>
<p>So, the rise is 1 and the run is 2.</p>
<p>Start at a peg in the lower left of the geoboard.</p>
<p>Stretch the rubber band up 1 unit, and then right 2 units.</p>
<img alt="The figure shows a grid of evenly spaced pegs. There are 5 columns and 5 rows of pegs. A rubber band is stretched between the peg in column 1, row 3, the peg in column 1, row 4 and the peg in column 3, row 3, forming a right triangle. The 1, 3 peg forms the vertex of the 90 degree angle and the line from the 1, 4 peg to the 3, 3 peg forms the hypotenuse of the triangle. The line from the 1, 3 peg to the 1, 4 peg is labeled “1”. The line from the 1, 3 peg to the 3, 3 peg is labeled “2”." src="https://data.ulearngo.com/assets/589238ba-4e42-4f42-b730-337d6af4d02a"/>
<p>The hypotenuse of the right triangle formed by the rubber band represents a line whose slope is \(\frac{1}{2}\).</p>
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<h2>Example</h2>
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<p>Use a geoboard to model a line with slope \(\frac{-1}{4}.\)</p>
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<h3>Solution</h3>
<p>\(\begin{array}{ccccc}\text{Use the slope formula.}\hfill &amp; \phantom{\rule{9em}{0ex}}\hfill &amp; \hfill m&amp; =\hfill &amp; \frac{\text{rise}}{\text{run}}\hfill \\ \text{Replace}\phantom{\rule{0.2em}{0ex}}m\phantom{\rule{0.2em}{0ex}}\text{with}\phantom{\rule{0.5em}{0ex}}\frac{-1}{\phantom{\rule{0.4em}{0ex}}4}.\hfill &amp; \phantom{\rule{9em}{0ex}}\hfill &amp; \hfill \frac{-1}{\phantom{\rule{0.4em}{0ex}}4}&amp; =\hfill &amp; \frac{\text{rise}}{\text{run}}\hfill \end{array}\)</p>
<p>So, the rise is \(-1\) and the run is 4.</p>
<p>Since the rise is negative, we choose a starting peg on the upper left that will give us room to count down.</p>
<p>We stretch the rubber band down 1 unit, then go to the right 4 units, as shown.</p>
<div><img alt="The figure shows a grid of evenly spaced pegs. There are 5 columns and 5 rows of pegs. A rubber band is stretched between the peg in column 1, row 2, the peg in column 1, row 3 and the peg in column 5, row 3, forming a right triangle. The 1, 3 peg forms the vertex of the 90 degree angle and the line from the 1, 2 peg to the 5, 3 peg forms the hypotenuse of the triangle. The line from the 1, 2 peg to the 1, 3 peg is labeled “negative 1”. The line from the 1, 3 peg to the 5, 3 peg is labeled “4”." src="https://data.ulearngo.com/assets/b824b47c-3046-43b5-af66-48cbafeb44ca"/></div>
<p>The hypotenuse of the right triangle formed by the rubber band represents a line whose slope is \(\frac{-1}{4}\).</p>
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Using Geoboards to Model Slope

Mathematics studies measurement, relationships, and properties of quantities and sets, using numbers and symbols. Arithmetic, algebra, geometry, and calculus are popular topics in mathematics.

Mathematics

Explore how to graph linear equations, find solutions to equations and inequalities, identify intercepts on a graph, model slope using geoboards, and find equations of lines using slope and intercepts or given points.


Using Geoboards to Model Slope

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