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<h3>Activity</h3>
<p>In the previous activity, students matched the line of best fit best modeled a scatter plot of data. Now, given data in a table or graph, students will find a linear model that represents the data without technology. They will use the data to find the slope and \(y\)-intercept of an equation.</p>
<p>The discussion in the synthesis will prepare them for future activities when they use technology to find the line of best fit with technology and need to interpret the meaning of the slope and \(y\)-intercept for given scenarios.<br>
</p>
<h4>Launch</h4>
<p>Arrange students in pairs and briefly review slope-intercept form and the meaning of the variables in the form.</p>
<br>
<h4>Student Activity</h4>
<p>Linear models can also be found by hand in some situations when the data follows a linear pattern since the slope remains the same between data points. </p>
<p><strong>Linear Models from Tables</strong></p>
<br>
<blockquote>
<p>For questions 1-4, use the scenario and table below:</p>
<p>The table below shows the number of songs Marcus will have in his collection as he adds new songs each month.</p>
<table class="os-raise-horizontaltable">
<thead></thead>
<tbody>
<tr>
<th scope="row">Number of months, \(t\)</th>
<td>0</td>
<td>1</td>
<td>2</td>
<td>3</td>
</tr>
<tr>
<th scope="row">Number of songs, \(N\)</th>
<td>200</td>
<td>215</td>
<td>230</td>
<td>245</td>
</tr>
</tbody>
</table>
</blockquote>
<br>
<ol class="os-raise-noindent">
<li> What is the initial amount of songs, or the \(y\)-intercept of this situation? </li>
</ol>
<p><strong>Answer:</strong> 200</p>
<ol class="os-raise-noindent" start="2">
<li> What is the slope, or rate of change? </li>
</ol>
<p><strong>Answer:</strong> 15</p>
<ol class="os-raise-noindent" start="3">
<li> Write the equation of the linear model that represents the situation in slope-intercept form. </li>
</ol>
<p><strong>Answer:</strong> \(N = 15t + 200\)</p>
<ol class="os-raise-noindent" start="4">
<li> How many songs will Marcus have in 8 months? </li>
</ol>
<p><strong>Answer:</strong> 320<br>
Compare your answer: <br>
\(N = 15(8) + 200=320\)</p>
<br>
<blockquote>
<p>For questions 5-8, use the situation and table below:</p>
<p>A new plant food is introduced to a young tree to test its effect on the height of the tree. The table shows the height of the tree, H, in feet \(x\) months since measurements began.</p>
<table class="os-raise-horizontaltable">
<thead></thead>
<tbody>
<tr>
<th scope="row">Number of months, \(x\)</th>
<td>2</td>
<td>4</td>
<td>6</td>
<td>8</td>
</tr>
<tr>
<th scope="row">Height in feet, \(H\)</th>
<td>13.5</td>
<td>14.5</td>
<td>15.5</td>
<td>16.5</td>
</tr>
</tbody>
</table>
</blockquote>
<br>
<ol class="os-raise-noindent" start="5">
<li> What is the initial height of the tree, or the \(y\)-intercept, when measurements began? </li>
</ol>
<p>(Hint: Work each row backwards to where \(x= 0\).)</p>
<p><strong>Answer:</strong> 12.5</p>
<ol class="os-raise-noindent" start="6">
<li> What is the slope, or rate of change? </li>
</ol>
<p><strong>Answer:</strong> \(\frac12\) </p>
<ol class="os-raise-noindent" start="7">
<li> Write the equation of the linear model that represents the situation in slope-intercept form. </li>
</ol>
<p><strong>Answer:</strong> \(H= \frac12 x +12.5\)</p>
<ol class="os-raise-noindent" start="8">
<li> What will the height of the tree be in 14 months? </li>
</ol>
<p><strong>Answer:</strong> 19.5<br>
Compare your answer: <br>
\(H= \frac12 (14) +12.5 =19.5\)</p>
<br>
<p><strong>Linear Models from Graphs</strong></p>
<blockquote>
<p>For questions 9-12, use the situation and graph below:</p>
<p>The graph models the cost in dollars, \(C\), of renting a tent at a campground for n nights.<br>
<img src="https://k12.openstax.org/contents/raise/resources/dbc7eea714a05aa6c75d90c04a2cc22bd3aae58a" width="300"/></p>
</blockquote>
<ol class="os-raise-noindent" start="9">
<li> What is the \(y\)-intercept of the graph? </li>
</ol>
<p><strong>Answer:</strong> 30</p>
<ol class="os-raise-noindent" start="10">
<li> What is the slope, or rate of change? </li>
</ol>
<p><strong>Answer:</strong> 10</p>
<ol class="os-raise-noindent" start="11">
<li> Write the equation of the linear model that represents the situation in slope-intercept form. </li>
</ol>
<p><strong>Answer:</strong> \(C=10n + 30\)</p>
<ol class="os-raise-noindent" start="12">
<li> How much does it cost to rent a tent for 5 nights? </li>
</ol>
<p><strong>Answer:</strong> 80<br>
Compare your answer: <br>
\(C=10n + 30\)<br>
\(C=10(5) + 30=80\)</p>
<br>
<h3> 3.2.3: Self Check </h3>
<p class="os-raise-text-bold"><em>After the activity, students will answer the following question to check their understanding of the concepts explored in the activity.</em></p>
<p class="os-raise-text-bold">QUESTION:</p>
<p>The table below shows the revenue for the number of pizzas sold at a restaurant.</p>
<table class="os-raise-horizontaltable">
<thead>
</thead>
<tbody>
<tr>
<th scope="row">Number of Pizzas</th>
<td>20</td>
<td>30</td>
<td>40</td>
<td>50</td>
</tr>
<tr>
<th scope="row">Revenue</th>
<td>110</td>
<td>160</td>
<td>210</td>
<td>260</td>
</tr>
</tbody>
</table>
<br>
<p>Which linear model could be used to predict the revenue if 4000 pizzas were sold?</p>
<table class="os-raise-textheavytable">
<thead>
<tr>
<th scope="col">Answers</th>
<th scope="col">Feedback</th>
</tr>
</thead>
<tbody>
<tr>
<td>\(y=4000(10)\)</td>
<td>Incorrect. Let’s try again a different way: The change in y is 50 while the change in \(x\) is 10 so the slope is 5. Working the table back to 0 will give a \(y\)-intercept of 10. The answer is: \(y=5(4000)\). </td>
</tr>
<tr>
<td>\(y=80(4000)\)</td>
<td>Incorrect. Let’s try again a different way: The change in y is 50 while the change in \(x\) is 10 so the slope is 5. Working the table back to 0 will give a \(y\)-intercept of 10. The answer is: \(y = 3x - 5\). </td>
</tr>
<tr>
<td>\(y=5(4000)+10\)</td>
<td>That’s correct! Check yourself: The change in y is 50 while the change in \(x\) is 10 so the slope is 5. Working the table back to 0 will give a \(y\)-intercept of 10. Multiply this by 4000 to determine the revenue.</td>
</tr>
<tr>
<td>\(y= \frac15 (4000)+ 10\)</td>
<td>Incorrect. Let’s try again a different way: The change in y is 50 while the change in \(x\) is 10 so the slope is 5. The answer is: \(y=5(4000)\). </td>
</tr>
</tbody>
</table>
<br>
<h3>3.2.3: Additional Resources</h3>
<p class="os-raise-text-bold"><em>The following content is available to students who would like more support based on their experience with the self check. Students will not automatically have access to this content, so you may wish to share it with those who could benefit from it.</em></p>
<br>
<h4>Writing Linear Equations From Tables and Graphs</h4>
<p>Linear Equations from Tables</p>
<p>The number of texts a teen sends, \(T\), in days, \(d\), is shown in the table below.</p>
<table class="os-raise-horizontaltable">
<thead>
</thead>
<tbody>
<tr>
<th scope="row">Days, \(d\)</th>
<td>1</td>
<td>2</td>
<td>3</td>
<td>4</td>
</tr>
<tr>
<th scope="row">Number of texts, \(T|)</th>
<td>65</td>
<td>130</td>
<td>195</td>
<td>260</td>
</tr>
</tbody>
</table>
<br>
<p>Write an equation in slope-intercept form to represent the situation then predict the number of texts sent in 8 days.</p>
<p><strong>Step 1</strong> - Find the \(y\)-intercept.<br>
Since the value when \(d=0\) is not present, work backwards to \(d=0\). <br>
If \(d=0\) was present, then \(T=0\) since as each x-value increases by 1, the \(y\)-values increase by 65.</p>
<p><strong>Step 2</strong> - Find the slope of the situation.<br>
Find the rate of change. The change in y is 65 and the change in \(x\) is 1, so the slope is 65</p>
<p><strong>Step 3</strong> -Write the equation in slope-intercept form.<br>
\(y=mx+b\)<br>
\(T=65d +0\)<br>
\(T=65d\)</p>
<p><strong>Step 4 </strong>-Make a prediction.<br>
Substitute \(d=8\) into the equation.<br>
\(T=65(8)=520\)</p>
<p>After 8 days, the teen would have sent 520 texts.</p>
<br>
<p><strong>Linear Equations from Graphs</strong></p>
<p>The cost, C, for the number of days, \(d\), a dog spends at doggie daycare is shown in the graph below.<br>
<img src="https://k12.openstax.org/contents/raise/resources/3f66172ff755fe8a3f00243f3a79bb981754dab1" width="300"/></p>
<p>Write an equation in slope-intercept form to represent the situation then predict the cost of a dog staying at doggie daycare after 7 days.</p>
<p><strong>Step 1 </strong>- Find the \(y\)-intercept.<br>
Since the value when \(d=0\) is not present, work backwards to \(d=0\). <br>
If \(d=0\) was present, then \(C=0\) since as each \(x\)-value increases by 1, the \(y\)-values increase by 35.</p>
<p><strong>Step 2 </strong>- Find the slope of the situation.<br>
Find the rate of change. The change in \(y\) is 35 and the change in \(x\) is 1, so the slope is 35</p>
<p><strong>Step 3</strong> -Write the equation in slope-intercept form.<br>
\(y=mx+b\)<br>
\(C=35d +0\)<br>
\(C=35d\)</p>
<p><strong>Step 4</strong> -Make a prediction.<br>
Substitute \(d=7\) into the equation.<br>
\(C=35(7)=245\)</p>
<br>
<p>After 7 days, the cost of doggie daycare would be $245.</p>
<br>
<h4>Try It-Writing Linear Equations From Tables and Graphs</h4>
<p>Naomi is a professional painter. The table below shows how many square feet, \(F\), she can paint in \(h\) hours.</p>
<table class="os-raise-horizontaltable">
<thead>
</thead>
<tbody>
<tr>
<th scope="row">Time in hours, \(h\)</th>
<td>1</td>
<td>2</td>
<td>3</td>
<td>4</td>
</tr>
<tr>
<th scope="row">Number of square feet, \(F\)</th>
<td>120</td>
<td>240</td>
<td>360</td>
<td>480</td>
</tr>
</tbody>
</table>
<br>
<p>How many square feet can Naomi paint in 8 hours?</p>
<p><strong>Answer:</strong> 960</p>
<p><strong>Step 1</strong> - Find the \(y\)-intercept.<br>
Since the value when \(h=0\) is not present, work backwards to \(h=0\). <br>
If \(h=0\) was present, then \(F=0\) since as each \(x\)-value increases by 1, the \(y\)-values increase by 120.</p>
<p><strong>Step 2</strong> - Find the slope of the situation.<br>
Find the rate of change. The change in \(y\) is 120 and the change in \(x\) is 1, so the slope is 120</p>
<p><strong>Step 3</strong> -Write the equation in slope-intercept form.<br>
\(y=mx+b\)<br>
\(F=120h+0\)<br>
\(F=120h\)</p>
<p><strong>Step 4</strong> -Make a prediction.<br>
Substitute \(h=8\) into the equation.<br>
\(F=120(8)=960\)</p>
<p>Naomi can paint 960 square feet in 8 hours.</p>