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<h4>Activity (5 minutes)</h4>
<p>In this activity, students are asked to interpret in context the slope and vertical intercept of a linear model given a scatter plot and the equation for a linear model that fits the data well. The linear model is also used to interpolate and extrapolate information about the data in context. </p>
<p>Monitor for students who:</p>
<ol class="os-raise-noindent">
<li>Estimate values based on the graph.</li>
<li>Use the equation for the linear model to find the values.</li>
</ol>
<h4>Launch</h4>
<p>Arrange students in groups of 2.</p>
<h4>Student Activity</h4>
<p>The scatter plot shows the sale price of several food items, \(y\), and the cost of the ingredients used to produce those items, \(x\), as well as a line that models the data. The line is also represented by the equation \(y = 3.48x + 0.76\).</p>
<p><img alt="Scatterplot and line of best fit." src="https://k12.openstax.org/contents/raise/resources/741915e23b3675724338e3edda9489176518a7e5"><br>
</p>
<ol class="os-raise-noindent">
<li>What is the predicted sale price of an item that has ingredients that cost $1.50? Be prepared to show your reasoning.</li>
</ol>
<p><strong>Answer:</strong> Compare your answer: Your answer may vary, but here are some samples. $5.98 from the equation \((3.48⋅1.5+0.76=5.98)\), or approximately $6 from the scatter plot graph</p>
<ol class="os-raise-noindent" start="2">
<li>What is the predicted ingredient cost of an item that has a sale price of $7? Be prepared to show your reasoning.</li>
</ol>
<p><strong>Answer:</strong> Compare your answer: Your answer may vary, but here are some samples. $1.79 from the equation \(\left(\frac{7-0.76}{3.48}=\;1.79\right)\), or approximately $1.75 from the scatter plot graph</p>
<ol class="os-raise-noindent" start="3">
<li>What is the slope of the linear model?</li>
</ol>
<p><strong>Answer:</strong> 3.48</p>
<ol class="os-raise-noindent" start="4">
<li>What does the slope value mean in this situation?</li>
</ol>
<p><strong>Answer:</strong> Compare your answer: Your answer may vary, but here is a sample. This means that for every extra dollar spent on ingredients, the sale price of the food item is expected to increase by about $3.48
</p>
<ol class="os-raise-noindent" start="5">
<li>What is the \(y\)-intercept of the linear model?</li>
</ol>
<p><strong>Answer:</strong> 0.76</p>
<ol class="os-raise-noindent" start="6">
<li>What does the value of the \(y\)-intercept mean in this situation? </li>
</ol>
<p><strong>Answer:</strong> Compare your answer: Your answer may vary, but here is a sample. This means that a food item that is made with free ingredients would still be sold for approximately 76 cents.</p>
<ol class="os-raise-noindent" start="7">
<li> Does the value of the \(y\)-intercept make sense in this situation?</li>
</ol>
<p><strong>Answer:</strong> Compare your answer: Your answer may vary, but here are some samples.</p>
<ul>
<li>This doesn’t seem to make sense because something that is free to make should not cost anything for the customer.
</li>
<li>This might make sense since it would still cost some money to either run the machines or pay the people to make the item.
</li>
</ul>
<br>
<h4>Video: Linear Models</h4>
<p>Watch the following video to learn more about how to use linear models.</p>
<div class="os-raise-d-flex-nowrap os-raise-justify-content-center">
<div class="os-raise-video-container"><video controls="true" crossorigin="anonymous">
<source src="https://k12.openstax.org/contents/raise/resources/c7081f288d0cb5e03052a1f936334b0b53cd61ce">
<track default="true" kind="captions" label="On" src="https://k12.openstax.org/contents/raise/resources/f5db05987fcf3a598013376eb6a4f418e0d1a30e" srclang="en_us">
https://k12.openstax.org/contents/raise/resources/c7081f288d0cb5e03052a1f936334b0b53cd61ce
</video></div>
</div>
<br>
<h4>Anticipated Misconceptions</h4>
<p>Although the grid in the picture does not look like squares, all the lines in both the horizontal and vertical directions are 0.5 apart. Tell students who are confused to look at the values listed on the axes and identify the values attached to a few of the grid lines.</p>
<h4>Activity Synthesis</h4>
<p>The purpose of this discussion is for students to describe where to find the slope and vertical intercept from a scatter plot and interpret the values in terms of the context.</p>
<p>Select previously identified students to share in the order listed in the narrative. Ask students:</p>
<ul>
<li>“Are the values obtained from looking at the graph close to the values calculated using the equation?” (Yes, the values are close.)</li>
<li>“Which method will you use to predict values based on the model?” (Maybe a mixture of both methods. Using the equation produces a more precise value from the model, but it is also important to look at the graph to get a sense of how well the model fits the data where I am predicting.)</li>
<li>“Why do you think the \(y\)-intercept for the model is not \((0,0)\)?” (Companies may still charge money for items that cost nothing to produce. They need to pay for the salaries of their workers, research and development, as well as other things that require the company to make money more than just the cost of making the item. It may also be the case that it does not make sense to interpret the \(y\)-intercept since we have no evidence to believe that the same linear relationship will hold there.)</li>
</ul>
<br>
<div class="os-raise-extrasupport">
<div class="os-raise-extrasupport-header">
<p class="os-raise-extrasupport-title">Support for English Language Learners</p>
<p class="os-raise-extrasupport-name">MLR3 Clarify, Critique, Correct: Reading, Writing, Speaking</p>
</div>
<div class="os-raise-extrasupport-body">
<p>Before students share their interpretations of the slope, present an ambiguous explanation. For example: “The slope means that as sales increase, the cost of ingredients increases.” Ask students to identify inaccuracies, critique the reasoning, and write a correct explanation. As students discuss with a partner, listen for students who identify and clarify the ambiguous language in the statement. Invite students to share their critiques and corrected explanations with the class. Listen for and amplify the language students use to describe what happens to the sale price when the cost of ingredients increases by $1. This helps students evaluate and improve on the written mathematical arguments of others as they clarify the meaning of slope in context.</p>
<p class="os-raise-text-italicize">Design Principle(s): Optimize output (for explanation); Maximize meta-awareness</p>
<p class="os-raise-extrasupport-title">Learn more about this routine</p>
<p>
<a href="https://www.youtube.com/watch?v=lozZJ21i3mQ;&rel=0" target="_blank">View the instructional video</a>
and
<a href="https://k12.openstax.org/contents/raise/resources/36e324584ae01e4eafc6674ac4ec02b4ae99002c" target="_blank">follow along with the materials</a>
to assist you with learning this routine.
</p>
<p class="os-raise-extrasupport-title">Provide support for students</p>
<p>
<a href="https://k12.openstax.org/contents/raise/resources/76d1756428038b737f48dc515486cddceb99e2f8" target="_blank">Distribute graphic organizers</a>
to the students to assist them with participating in this routine.
</p>
</div>
</div>
<br>
<div class="os-raise-extrasupport">
<div class="os-raise-extrasupport-header">
<p class="os-raise-extrasupport-title">Support for Students with Disabilities</p>
<p class="os-raise-extrasupport-name">Action and Expression: Internalize Executive Functions</p>
</div>
<div class="os-raise-extrasupport-body">
<p>Provide students with a two-column graphic organizer. Label one column “graph” and the other column “equation.” Ask students to transfer their answers into the corresponding column on their organizer. When the whole group shares, they can continue to use the organizer to capture the answer if anyone used the alternative method for each question.</p>
<p class="os-raise-text-italicize">Supports accessibility for: Language; Organization</p>
</div>
</div>
<br>
<h3>3.1.3: 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 scatter plot below shows the age (in years) and the price (in dollars) of used compact cars advertised in the local newspaper.</p>
<p><img alt="A SCATTER PLOT THAT SHOWS AGE IN YEARS ON THE X-AXIS AND PRICE IN DOLLARS ON THE Y-AXIS. THE LINE DRAWN DECREASES FROM LEFT TO RIGHT AND PASSES THROUGH THE POINTS (13, 6000) AND (7, 12000)." class="img-fluid atto_image_button_text-bottom" height="297" src="https://k12.openstax.org/contents/raise/resources/5f33547a78f8e26377fee2ffa0e7817c8d39546a" width="350"></p>
<table class="os-raise-textheavytable">
<thead>
<tr>
<th scope="col">Answers</th>
<th scope="col">Feedback</th>
</tr>
</thead>
<tbody>
<tr>
<td> \(-1000\), for every year older the car is, its price decreases 1000 dollars. </td>
<td> That’s correct! Check yourself: The slope is negative, and if the points \((13, 6000)\) and \((7, 12000)\) are used, the slope is \(\frac{12000-6000}{7-13}\;=\;-\;\frac{6000}6\;=\;-\;1000\). </td>
</tr>
<tr>
<td> \(1000\), for every year older the car is, its price decreases 1000 dollars. </td>
<td> Incorrect. Let’s try again a different way: The slope of the line is negative, and the price decreases as the cars get older. The answer is the slope is \(-1000\). For every year older the car is, its price decreases 1000 dollars. </td>
</tr>
<tr>
<td> \(1000\), for every year older the car is, its price increases 1000 dollars. </td>
<td> Incorrect. Let’s try again a different way: The slope of the line is negative, and the price decreases as the cars get older. The answer is the slope is \(-1000\). For every year older the car is, its price decreases 1000 dollars. </td>
</tr>
<tr>
<td> \(-\frac1{1000}\), for every year older the car is, it decreases \(\frac1{1000}\) of its value the year before. </td>
<td> Incorrect. Let’s try again a different way: To find slope, use the change of the \(y\)-values over the change of the \(x\)-values, or rise over run. For every year older the car is, its price decreases 1000 dollars. </td>
</tr>
</tbody>
</table>
<br>
<h3>3.1.3: 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>
<h4>Interpreting a Linear Model</h4>
<p>The scatter plot below shows the height and speed of some of the world’s fastest roller coasters. A line has been drawn to show a good fit for the data. <br>
<br>
</p>
<p><img alt="A SCATTER PLOT THAT SHOWS MAXIMUM HEIGHT IN FEET ON THE X-AXIS AND SPEED IN MILES PER HOUR ON THE Y-AXIS. THE LINE DRAWN INCREASES FROM LEFT TO RIGHT." class="img-fluid atto_image_button_text-bottom" height="309" src="https://k12.openstax.org/contents/raise/resources/0a1dcda7b995c59f5ccb90cf336b760df09a80bb" width="350"><br>
<br>
</p>
<p>The equation of the line drawn is \(S= 0.11h +60\).</p>
<p>What does this slope mean in the problem?</p>
<p>For every foot taller the roller coaster is, the speed increases 0.11 miles per hour.</p>
<h4>Try It: Interpreting a Linear Model</h4>
<p>Eight students were asked to estimate their score on a 10-point quiz. Their estimated and actual scores are given in the table below. Plot the points. Then sketch a line that fits the data. </p>
<table class="os-raise-midsizetable">
<thead>
<tr>
<th scope="col"> Predicted </th>
<th scope="col"> Actual </th>
</tr>
</thead>
<tbody>
<tr>
<td> 6 </td>
<td> 6 </td>
</tr>
<tr>
<td> 7 </td>
<td> 7 </td>
</tr>
<tr>
<td> 7 </td>
<td> 8 </td>
</tr>
<tr>
<td> 8 </td>
<td> 8 </td>
</tr>
<tr>
<td> 7 </td>
<td> 9 </td>
</tr>
<tr>
<td> 9 </td>
<td> 10 </td>
</tr>
<tr>
<td> 10 </td>
<td> 10 </td>
</tr>
<tr>
<td> 10 </td>
<td> 9 </td>
</tr>
</tbody>
</table>
<br>
<p>Here are the points from the table plotted on a scatter plot, with the \(x\)-axis being the predicted scores and the \(y\)-axis being the actual scores.</p>
<img alt="A SCATTER PLOT THAT SHOWS PREDICTED SCORES ON THE X-AXIS AND ACTUAL SCORES ON THE Y-AXIS. THE LINE OF BEST FIT DRAWN INCREASES FROM LEFT TO RIGHT." class="img-fluid atto_image_button_text-bottom" height="175" src="https://k12.openstax.org/contents/raise/resources/60e23f71d745d1c7c258d0a13366f857dbd01604" width="350"> <br>
<ol class="os-raise-noindent">
<li>Tell what you would expect someone’s grade to actually be if they predicted their grade to be a 5 out of 10.</li>
</ol>
<p>Write down your answer, then select the <strong>solution</strong> button to compare your work.</p>
<p><strong>Answer:</strong> Here is how to answer these questions using the linear model: Using the linear model above, if a student predicted a 5 on the quiz, they should have an actual score of 6.</p>
<ol class="os-raise-noindent" start="2">
<li>What does the slope mean in this problem?</li>
</ol>
<p>Write down your answer, then select the <strong>solution</strong> button to compare your work.</p>
<p><strong>Answer</strong>: Here is how to answer these questions using the linear model: The slope is the change in the actual grade point for each predicted grade point.</p>