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<h4>Activity (20 minutes)</h4>
<p>In this activity, students are <a href="https://k12.openstax.org/contents/raise/resources/8f49de2c25e987a13acf006126bbede16912fd1b" target="_blank">given cards</a> displaying scatter plots of data that can be fit by linear models with varying accuracy. Cards show data that are random, poorly fit by a linear model, well fit by a linear model, and better fit by another type of function, such as quadratic or exponential. Students should begin to recognize these differences and the connection to the correlation coefficient.</p>
<p>A sorting task gives students opportunities to analyze representations, statements, and structures closely and make connections.</p>
<p>Monitor for different ways groups choose to categorize the scatter plots, but especially for categories that distinguish between plots that would be modeled well with a linear function and those that would not.</p>
<h4>Launch</h4>
<p>Arrange students in groups of 2. Tell them that in this activity, they will sort some cards into categories of their choosing. When they sort the scatter plots, they should work with their partner to come up with categories. Distribute one copy of the blackline master to each group.</p>
<!--BEGIN ELL AND SWD GRAY BOX -->
<div class="os-raise-extrasupport">
<div class="os-raise-extrasupport-header">
<p class="os-raise-extrasupport-title">Support for English Language Learners</p>
</div>
<div class="os-raise-extrasupport-body">
<p class="os-raise-extrasupport-name">MLR 8 Discussion Supports: Conversing</p>
<p>As students work in pairs, ask them to take turns finding a match or category and explaining their reasoning to their partner. Display the following sentence frames for all to see: “_____ and _____ are alike because . . . ,” “I disagree because . . . ,” and “These are similar because . . . .” Encourage students to challenge each other when they disagree. This will help students clarify their understanding of linear models.</p>
<p class="os-raise-text-italicize">Design Principle(s): Support sense-making; Maximize meta-awareness</p>
<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: Develop Expression and Communication</p>
</div>
<div class="os-raise-extrasupport-body">
<p>Maintain a display of important terms and vocabulary. Review the following terms from previous lessons that students may have used or wanted words for during this activity: fit, linear model, non-linear, increasing, decreasing, pattern, random. Correlation coefficient can be added to this display as a new term.</p>
<p class="os-raise-text-italicize">Supports accessibility for: Memory; Language</p>
</div>
</div>
<!--END ELL AND SWD GRAY BOX -->
<br>
<h4>Student Activity</h4>
<p>Your teacher will give you a set of cards that show scatter plots of data. Sort the cards into 2 categories of your choosing. Be prepared to explain the meaning of your categories. Then, sort the cards into 2 categories in a different way. Be prepared to explain the meaning of your new categories.</p>
<p><strong>Answer:</strong></p>
<ul>
<li>Scatter plots that look linear: C, D, F, G. Scatter plots that are not very linear: A, B, E, H, I, J</li>
<li>Generally increasing: E, F, G, I. Generally decreasing: A, C, D, J. Not really increasing or decreasing: B, H</li>
</ul>
<h4>Discussion</h4>
<p>Now, display the scatterplots with the lines of best fit and \(r\) values included (see images below). <br>
After displaying the graphs, give students 1 minute of quiet time to think, and then 1 minute to discuss the things they notice with their partner, followed by a whole-class discussion.</p>
<p>Among the things students should notice are:</p>
<li>The sign of \(r\) is the same as the sign of the slope of the line of best fit.</li>
<li>The values for \(r\) seem to go from –1 to 1.</li>
<li>The closer \(r\) is to 1 or –1, the stronger the linear relationship between the variables.</li>
<li>The closer \(r\) is to 0, the weaker the linear relationship between the variables.</li>
<br>
<h4>Display:</h4>
<br>
<p>A: \(y=-0.83x-1.85\), \(r=-0.61\)</p>
<p><img alt="Scatter plot with line of best fit. " src="https://k12.openstax.org/contents/raise/resources/48942edeeadddcbff69a879381d6c4253e8cb34d"><br></p>
<br>
<p>B: \(y=-0.07x+5.63\), \(r=-0.06\)</p>
<p><img alt="Scatter plot with line of best fit. " src="https://k12.openstax.org/contents/raise/resources/0b15153fe12c24f0a20cf28234c2aa941fe8b124"><br></p>
<br>
<p>C: \(y=-0.31x+5.76\), \(r=-0.88\)</p>
<p><img alt="Scatter plot with line of best fit. " src="https://k12.openstax.org/contents/raise/resources/ccdf26ced34d143c824775e5e399ac51da421262"><br></p>
<br>
<p>D: \(y=-0.8x+6\), \(r=-1\)</p>
<p><img alt="Scatter plot with line of best fit. " src="https://k12.openstax.org/contents/raise/resources/48f5624b0642bd994bfc419e04fd49269efd07ed"><br></p>
<br>
<p>E: \(y=0.69x+4.49\), \(r=0.46\)</p>
<p><img alt="Scatter plot with line of best fit. " src="https://k12.openstax.org/contents/raise/resources/09f953219e6bc868342d4330344aee79c91b9d41"><br></p>
<br>
<p>F: \(y=1.43x+2.86\), \(r=0.94\)</p>
<p><img alt="Scatter plot with line of best fit. " src="https://k12.openstax.org/contents/raise/resources/06c8206e8d6569a08388d5f6fc6c596134af9a40"><br></p>
<br>
<p>G: \(y=3x+2\), \(r=1\)</p>
<p><img alt="Scatter plot with line of best fit. " src="https://k12.openstax.org/contents/raise/resources/359faa19640031dd407d23673d302b125cc2162d"><br></p>
<br>
<p>H: \(y=-0.02x+1.64\), \(r=-0.13\)</p>
<p><img alt="Scatter plot with line of best fit. " src="https://k12.openstax.org/contents/raise/resources/2707ae52b9bb9b8533c24617b1d217c994d369c3"><br></p>
<br>
<p>I: \(y=14.55x−36.18\), \(r=0.95\)</p>
<p><img alt="Scatter plot with line of best fit. " src="https://k12.openstax.org/contents/raise/resources/e79aa4d7b7032703f34d7bcbb3790cc11403a3b6"><br></p>
<br>
<p>J: \(y=-0.62x+6.52\), \(r=-0.96\)</p>
<p><img alt="Discrete graph with equation." src="https://k12.openstax.org/contents/raise/resources/f4078f530807b756b4c708b8d0f2d2fef2ba4fb8"><br></p>
<p>Emphasize with students that the sign of the correlation coefficient matches the sign of the slope of the line of best fit, but the value for \(r\) is not otherwise related to the slope. If \(r=0.8\), the line of best fit will have a positive slope, but whether the slope is 0.2 or 2,000 is not clear without examining the data.</p>
<h4>Activity Synthesis</h4>
<p>Select groups of students to share their categories and how they sorted their equations. You can choose as many different types of categories as time allows, but ensure that one set of categories distinguishes between plots that would be modeled well with a linear function and those that would not. Attend to the language that students use to describe their categories and equations, giving them opportunities to describe their equations more precisely. Highlight the use of terms like linear model, fit, and non-linear. </p>
<p>During the second part of the activity, encourage partners or groups to share their observations from the scatter plots, line of best fit, and \(r\)-values. Use questioning about the signs of the slopes and the \(r\)-values and the \(r\)-value numbers and the closeness of points to the line of best fit to elicit the sample responses. Write the four properties on the board/screen.</p>
<h4>3.4.2: Self Check </h4>
<!--BEGIN SELF CHECK INTRO BEFORE Tables -->
<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>
<!--SELF CHECK QUESTION GOES BEFORE THE Table -->
<p class="os-raise-text-bold">QUESTION:</p>
<p><!--QUESTION GOES HERE FROM TABLE --></p>
<p>Which correlation coefficient best matches the scatter plot below?</p>
<p><img src="https://k12.openstax.org/contents/raise/resources/4a737224dfcdffdeacb21090c9a086106e38e830" width="300"/></p>
<br>
<!--SELF CHECK table-->
<table class="os-raise-textheavytable">
<thead>
<tr>
<th scope="col">Answers</th>
<th scope="col">Feedback</th>
</tr>
</thead>
<tbody>
<tr>
<td>
<p>\(r=0.95\)</p>
</td>
<td>
<p>That’s correct! Check yourself: The slope is positive, and there is a strong correlation because the points are close to each other. The correlation coefficient should be close to positive 1.</p>
</td>
</tr>
<tr>
<td>
<p>\(r=-0.89\)</p>
</td>
<td>
<p>Incorrect. Let’s try again a different way: The slope of the line of best fit would be positive, so the \(r\)-value should be positive. The answer is \(r=0.95\).</p>
</td>
</tr>
<tr>
<td>
<p>\(r=0.26\)</p>
</td>
<td>
<p>Incorrect. Let’s try again a different way: This indicates a weak positive correlation. The scatter plot has points that are very close to each other, so the \(r\)-value should be closer to 1. The answer is \(r=0.95\).</p>
</td>
</tr>
<tr>
<td>
<p>\(r=-0.39\)</p>
</td>
<td>
<p>Incorrect. Let’s try again a different way: This indicates a weak negative correlation. The scatter plot has points that are very close to each other, so the \(r\)-value should be closer to 1. The slope of the line of best fit would be positive, so the \(r\)-value should be positive. The answer is \(r=0.95\).</p><br>
</td>
</tr>
</tbody>
</table>
<br>
<!--END SELF CHECK INTRO BEFORE Tables -->
<br>
<h3>3.4.2: Additional Resources</h3>
<p><strong><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></strong></p>
<h4>Correlation Coefficient</h4>
<p>The correlation coefficient is a number between \(-1\) and \(+1\) (including \(-1\) and \(+1\)) that measures the strength and direction of a linear relationship. The correlation coefficient is denoted by the letter \(r\). Several scatter plots are shown below. The value of the correlation coefficient for the data displayed in each plot is also given. </p>
<p><img alt class="img-fluid atto_image_button_text-bottom" height="440" role="presentation" src="https://k12.openstax.org/contents/raise/resources/145c40c22a66d767a839006e27971719b1f5ed40" width="350"><br></p>
<p><img alt class="img-fluid atto_image_button_text-bottom" height="340" role="presentation" src="https://k12.openstax.org/contents/raise/resources/6d12efe2b4803c74d0e500d8ded9bdfdc0c8c8ca" width="350"><br></p>
<p><img alt class="img-fluid atto_image_button_text-bottom" height="340" role="presentation" src="https://k12.openstax.org/contents/raise/resources/05ab55b9fcb372cf1f3d26385a2e6be01c33b3c8" width="350"><br></p>
<p><img alt class="img-fluid atto_image_button_text-bottom" height="340" role="presentation" src="https://k12.openstax.org/contents/raise/resources/92fb7cb25185ddad4146c7fd8a5b804d3ca1ff7f" width="350"><br></p>
<p><img alt class="img-fluid atto_image_button_text-bottom" height="340" role="presentation" src="https://k12.openstax.org/contents/raise/resources/c05dae989827fb0b0fc7ed78d260b9033167c3e1" width="350"><br></p>
<p><img alt class="img-fluid atto_image_button_text-bottom" height="340" role="presentation" src="https://k12.openstax.org/contents/raise/resources/a7ae5b629d8807a43abc7696ded9e1912ca72b90" width="350"><br></p>
<p><img alt class="img-fluid atto_image_button_text-bottom" height="340" role="presentation" src="https://k12.openstax.org/contents/raise/resources/3319a8d06f502242dc15ee91d771e05e409d8d05" width="350"><br></p>
<p>When is the value of the correlation coefficient positive?</p>
<ul>
<li>The correlation coefficient is positive when as the \(x\)-values increase, the \(y\)-values also tend to increase. </li>
</ul>
<p>When is the value of the correlation coefficient negative?</p>
<ul>
<li>The correlation coefficient is negative when as the \(x\)-values increase, the \(y\)-values tend to decrease. </li>
</ul>
<p>Is the linear relationship stronger when the correlation coefficient is closer to 0 or to 1 (or \(-1\))?</p>
<ul>
<li>As the points form a stronger negative or positive linear relationship, the correlation coefficient gets farther from 0. Students note that when all of the points are on a line with a positive slope, the correlation coefficient is \(+1\). The correlation coefficient is \(-1\) if all of the points are on a line with a negative slope. </li>
</ul>
<p>The table below shows how you can informally interpret the value of a correlation coefficient.</p>
<table class="os-raise-midsizetable">
<thead>
<tr>
<th scope="col">If the value of the correlation coefficient is . . .</th>
<th scope="col">You can say that . . .</th>
</tr>
</thead>
<tbody>
<tr>
<td>
<p>\(r = 1.0\)</p>
</td>
<td>
<p>There is a perfect positive linear relationship.</p>
</td>
</tr>
<tr>
<td>
<p>\( 0.7\;\leq\;r\;<\;1.0\)</p>
</td>
<td>
<p>There is a strong positive linear relationship.</p>
</td>
</tr>
<tr>
<td>
<p>\( 0.3\;\leq\;r\;<\;0.7\)</p>
</td>
<td>
<p>There is a moderate positive linear relationship.</p>
</td>
</tr>
<tr>
<td>
<p>\( 0\;<\;r\;<\;0.3\)</p>
</td>
<td>
<p>There is a weak positive linear relationship.</p>
</td>
</tr>
<tr>
<td>
<p>\(r = 0\)</p>
</td>
<td>
<p>There is no linear relationship.</p>
</td>
</tr>
<tr>
<td>
<p>\( -0.3\;<\;r\;<\;0\)</p>
</td>
<td>
<p>There is a weak negative linear relationship.</p>
</td>
</tr>
<tr>
<td>
<p>\( -0.7\;<\;r\;\leq\;-0.3\)</p>
</td>
<td>
<p>There is a moderate negative linear relationship.</p>
</td>
</tr>
<tr>
<td>
<p>\( -1.0\;<\;r\;\leq\;-0.7\)</p>
</td>
<td>
<p>There is a strong negative linear relationship.</p>
</td>
</tr>
<tr>
<td>
<p>\(r = -1.0\)</p>
</td>
<td>
<p>There is a perfect negative linear relationship.</p>
</td>
</tr>
</tbody>
</table>
<br>
<h4>Try It: Correlation Coefficient</h4>
<p>Which of the three scatter plots below shows the strongest linear relationship? Which shows the weakest linear relationship? Explain your answers.<br></p>
<p><img alt class="img-fluid atto_image_button_text-bottom" height="173" role="presentation" src="https://k12.openstax.org/contents/raise/resources/278eaa2d070c2003bcb24f90dabb39f48a33436b" width="549"><br></p>
<p><strong>Answer: </strong></p>
<p>The strongest linear relationship would be Scatter Plot 3. The points are closest to having a linear model. The weakest linear relationship would be Scatter Plot 2 because the points are spread apart the most and are less likely to have a strong linear relationship.</p><br>