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<h4>Activity (15 minutes)</h4>
<p>In this activity, students take turns with a partner, matching graphs of residuals to scatter plots that
display linear models. Students trade roles, explaining their thinking and listening, providing opportunities to
explain their reasoning and critique the reasoning of others. They should begin to recognize that a plot of the
residuals for data that are fit well by a linear model shows residuals that are close to the \(x\)-axis and do not
show a noticeable trend.</p>
<h4>Launch</h4>
<p>Arrange students in groups of 2. Demonstrate how to set up and find matches. Choose a student to be your
partner. Mix up the cards and place them face up. Point out that the cards contain either a scatter plot with a linear
model or a graph of the residuals. Select one of each style of card and then explain to your partner why you think the
cards do or do not match. Demonstrate productive ways to collaborate, by agreeing or respectfully disagreeing. Start
out by explaining your own mathematical thinking and asking clarifying questions. Give each group a set of cut-up
cards for matching.</p>
<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">MLR 1 Stronger and Clearer Each Time: Writing, Conversing</p>
</div>
<div class="os-raise-extrasupport-body">
<p>Use this routine to help students
improve their writing by providing them with multiple opportunities to clarify their explanations through
conversation. Give students time to meet with 2–3 partners to share their response to the last question.
Students should first check to see if they agree with each other about who had the better estimate of the line of
best fit. Provide listeners with prompts for feedback that will help their partner add detail to strengthen and
clarify their ideas. For example, students can ask their partner: “How did you use the graph of the residuals
to make your decision?” or ”How do you know . . . ?” Next, provide students with 3–4 minutes
to revise their initial draft based on feedback from their peers. This will help students produce a written
justification for determining best estimates of lines of best fit.<br><em>Design Principle(s): Support sense-making; Optimize output (for explanation</em>)</p>
<p class="os-raise-extrasupport-title">Provide support for students</p>
<p>
<a href="https://k12.openstax.org/contents/raise/resources/a5ae5bd09b27a5f53239a539c6009c19c92f7db7" 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 English Language Learners</p>
<p class="os-raise-extrasupport-name">Representation: Internalize Comprehension</p>
</div>
<div class="os-raise-extrasupport-body">
<p>Represent the same information through different modalities by showing the relationship between the original graph and the graph of the residuals. Provide students with tracing paper (to use directly over the card) or graph paper (to recreate the card) so they can draw in the distance between the point and the line to see the residuals.</p>
<p class="os-raise-text-italicize">Supports accessibility for: Conceptual processing; Visual-spatial processing</p>
</div>
</div>
<br>
<h4>Student Activity</h4>
<p><a href="https://k12.openstax.org/contents/raise/resources/9f51285a3b3686a226e9726c92548ddb5c735ff6" target="_blank">Matching Activity</a></p>
<ol class="os-raise-noindent">
<li>Match the scatter plots and given linear models to the graph of the residuals.
<p><strong>Answer: </strong></p>
<ul>
<li>A matches with K</li>
<li>B matches with G</li>
<li>C matches with L</li>
<li>D matches with H</li>
<li>E matches with J</li>
<li>F matches with I</li>
</ul>
</li>
<br>
<li>Turn over the scatter plots so that only the residuals are visible. Based on the residuals, which line would
produce the most accurate estimates? Which line fits its data worst?
<p><strong>Answer: </strong>B fits its data the best. C or E fits its data the worst.</p>
</li>
</ol>
<br>
<div class="os-raise-graybox">
<h5>What Do Residual Plots Tell Us?</h5>
<p>Residuals tell us how accurate a line of best fit really is.</p>
<p>Ideally, the residual from a line of best fit will be scattered randomly above and below the horizontal line \(y = 0\) on the residual plot. The residual plot would have points above and below the horizontal line \(y = 0\). This shows that the relationship between the two variables in the data set can be approximated by a linear model. </p>
<p>When the residuals are equal to 0, this indicates that the line of best fit goes through those points.</p>
<p>A curved pattern shows that a nonlinear model would be better to describe the relationships between points.</p>
</div>
<br>
<h4>Student Facing Extension</h4>
<h5>Are you ready for more?</h5>
<h5>Line of Best Fit</h5>
<p>Use the following informattion to answer questions 1 - 2.</p>
<blockquote><p>Tyler estimates a line of best fit for some linear data about the mass, in grams, of different numbers of apples.
Here is the graph of the residuals. </p>
<p><img alt="Graph of residuals."
src="https://k12.openstax.org/contents/raise/resources/aaccd40ecc86ea94d64e6e642b851e5b58b11936"><br></p></blockquote>
<ol class="os-raise-noindent">
<li>What does Tyler’s line of best fit look like according to the graph of the residuals?<br><strong>Answer:</strong> The data are between 10 and 30 grams above the line of best fit.</li> <br>
<li>How well does Tyler’s line of best fit model the data? Be prepared to show your reasoning.<br><strong>Answer:</strong> Tyler’s line does not model the data well because all of the residuals are positive.</li> <br>
</ol>
<p>Use the following information to answer questions 3 and 4.</p>
<blockquote><p>Lin estimates a line of best fit for the same data. The graph shows the residuals.</p>
<p><img alt="Graph of residuals."
src="https://k12.openstax.org/contents/raise/resources/26815ec7f89ec7f0f4ecc054e36d103c253fba4b"><br></p></blockquote>
<ol class="os-raise-noindent" start="3">
<li>What does Lin’s line of best fit look like in comparison to the data?<br><strong>Answer:</strong> The line is below the lower half of the data and above the second half of the data.</li> <br>
<li>How well does Lin’s line of best fit model the data? Be prepared to show your reasoning.<br><strong>Answer:</strong> It is not a good fit because the first several residuals are all positive and the last several residuals are negative. They should not have a pattern if the line of best fit is good.</li> <br>
</ol>
<p>Use the following information to answer questions 5 - 6.</p>
<blockquote><p>Kiran also estimates a line of best fit for the same data. The graph shows the residuals.</p>
<p><img alt="Graph of residuals."
src="https://k12.openstax.org/contents/raise/resources/f23f4efdd3c00fad9ebdca9783c069ab1485b6af"><br></p></blockquote>
<ol class="os-raise-noindent" start="5">
<li>What does Kiran’s line of best fit look like in comparison to the data?<br><strong>Answer:</strong> Kiran’s line appears to go through the middle of the data with some points above and some points below the line.</li> <br>
<li>How well does Kiran’s line of best fit model the data? Be prepared to show your reasoning.<br><strong>Answer:</strong> Kiran’s line of best fit is a good one because it appears to pass through the middle of the data.</li> <br>
<li>Who has the best estimate of the line of best fit: Tyler, Lin, or Kiran? Be prepared to show your reasoning.<br><strong>Answer:</strong> Kiran’s line of best fit is the best because the graph of the residuals lets me know that some points are above the line of best fit and some are below. In addition, two of the residuals are close to 0, so that indicates that the line passes through or near those 2 points. Tylers line of best fit is not a good one because it is above all the data. Lin’s line of best fit is below all the lower data values and above all the high data values, so it is not a good fit.</li> <br>
</ol>
<h4>Activity Synthesis</h4>
<p>The goal is to make sure students understand the connections between a scatter plot displaying a linear model and a
graph of the residuals. A good linear model for the data will have residuals that are scattered on either side of the
\(x\)-axis without a clear pattern and close to the axis.</p>
<p>Much discussion takes place between partners. Invite students to share how they made the matches.</p>
<ul>
<li>“What were some ways you handled finding the matches for B and C? Recall that B and C used the same data but
different linear models.” (The line in C was not as good of a fit as line B, so I knew the graph of the
residuals would be farther away from the horizontal axis.)</li>
<li>“Look at the matches for E and J. How can you tell from the graph of the residuals that the linear model is
not a line of best fit?” (The first half of the residuals are positive, and the second half are negative. This
lets me know that the line does not pass through the middle of the data.)</li>
<li>“What do you notice about the residuals in graph K? Explain what you notice in the context of scatter plot
A.” (The residuals go in a u-shaped pattern. The data in the scatter plot do not appear linear and are curved,
so when you plot a line through them, you would expect the residuals to show the curvature.)</li>
<li>“Describe any difficulties you experienced and how you resolved them.” (It was tough figuring out how
to decide where to begin when finding matches. I used my partner’s strategy of looking at the values on the
\(x\)-axis to help narrow down the choices.)</li>
</ul>
<h3>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>Which residual plot shows the most accurate line of best fit for a scatter plot?
</p>
<table class="os-raise-textheavytable">
<thead>
<tr>
<th scope="col">Answers</th>
<th scope="col">Feedback</th>
</tr>
</thead>
<tbody>
<tr>
<td>
<img alt class="img-fluid atto_image_button_text-bottom" height="176" role="presentation"
src="https://k12.openstax.org/contents/raise/resources/59c32c1a7c427cbda9da367abf7eac41ecddc741"
width="215">
</td>
<td>
Incorrect. Let’s try again a different way: The residual shows a good line of best
fit if the points are scattered above and below the horizontal line \(y = 0\) without a pattern or curve. The
answer is <br>
<img alt height="176" role="presentation"
src="https://k12.openstax.org/contents/raise/resources/2ea7b5bc5457ce5ff8b45d136af951028a1d525a"
width="215">
</td>
</tr>
<tr>
<td>
<img alt class="img-fluid atto_image_button_text-bottom" height="176" role="presentation"
src="https://k12.openstax.org/contents/raise/resources/b41bf70fcf1fec0b55323d9f0376930008f37343"
width="215">
</td>
<td>
Incorrect. Let’s try again a different way: The residual shows a good line of best
fit if the points are scattered above and below the horizontal line \(y = 0\) without a pattern or curve. The
answer is <br><img alt height="176" role="presentation"
src="https://k12.openstax.org/contents/raise/resources/2ea7b5bc5457ce5ff8b45d136af951028a1d525a"
width="215">
</td>
</tr>
<tr>
<td>
<img alt class="img-fluid atto_image_button_text-bottom" height="176" role="presentation"
src="https://k12.openstax.org/contents/raise/resources/2ea7b5bc5457ce5ff8b45d136af951028a1d525a"
width="215">
</td>
<td>
That’s correct! Check yourself: The points in the residual are scattered randomly
above and below the horizontal line that is 0. There is no curve or pattern.
</td>
</tr>
<tr>
<td>
<img alt class="img-fluid atto_image_button_text-bottom" height="176" role="presentation"
src="https://k12.openstax.org/contents/raise/resources/3c933e9a588ba22651de23e9237dd72a998aecd4"
width="215">
</td>
<td>
Incorrect. Let’s try again a different way: The residual shows a good line of best
fit if the points are scattered above and below the horizontal line \(y = 0\) without a pattern or curve. The
answer is<br> <img alt height="176" role="presentation"
src="https://k12.openstax.org/contents/raise/resources/2ea7b5bc5457ce5ff8b45d136af951028a1d525a"
width="215">
</td>
</tr>
</tbody>
</table>
<br>
<h3>3.3.3: 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>The Meaning of Residuals</h4>
<p>Suppose that you have a scatter plot and that you have drawn the line of best fit on your plot. Remember that
the residual for a point in the scatter plot is the vertical distance of that point from the line of best fit. In
the previous lesson, you looked at a scatter plot showing how fuel efficiency was related to curb weight for five
compact cars. The scatter plot and line of best fit are shown below.</p>
<p><img
alt="A SCATTER PLOT THAT SHOWS CURB WEIGHT IN HUNDREDS OF POUNDS ON THE X-AXIS AND FUEL EFFICIENCY IN MILES PER GALLON ON THE Y-AXIS. A LINE OF BEST FIT IS DRAWN. FOUR POINTS ARE ABOVE THE LINE AND ONE POINT IS BELOW."
class="atto_image_button_text-bottom" height="398" role="presentation"
src="https://k12.openstax.org/contents/raise/resources/06f888b0d9412ba32049e5eeb65ac188265286eb" width="450">
</p>
<p>Consider the following questions:</p>
<ol class="os-raise-noindent">
<li>What kind of residual does Point A have? </li><br><p class="os-raise-indent">Point A has a large positive residual.</p>
<li>What kind of residual does Point B have?</li><br><p class="os-raise-indent">Point B has a small positive residual.</p>
<li>What kind of residual does Point C have?</li><br><p class="os-raise-indent">Point C has a very large negative residual.</p>
</ol><br>
<p><img
alt="A RESIDUAL PLOT THAT SHOWS CURB WEIGHT IN HUNDREDS OF POUNDS ON THE X-AXIS AND RESIDUAL ON THE Y-AXIS. FOUR POINTS ARE ABOVE THE HORIZONTAL AXIS AND ONE POINT IS BELOW. "
class="img-fluid atto_image_button_text-bottom" height="398"
src="https://k12.openstax.org/contents/raise/resources/8eef14cd5bf3fda2801d674db24c0da4f28f3d56" width="450">
</p>
<h4>Try It: The Meaning of Residuals</h4>
<p>Suppose you are given a scatter plot and a line of best fit that looks like this:</p>
<p><img
alt="A SCATTER PLOT WITH A LINE OF BEST FIT DRAWN. FROM LEFT TO RIGHT, POINTS ARE BELOW THE LINE, ABOVE THE LINE, AND THEN BELOW THE LINE."
class="img-fluid atto_image_button_text-bottom" height="411"
src="https://k12.openstax.org/contents/raise/resources/e98902ab87bf79960aac410b87f0fb244d26c87f" width="450">
</p>
<p>Describe what you think the residual plot would look like.</p>
<p><strong>Answer:</strong></p>
<p>Here is how to sketch the residual plot.</p>
<p>Moving from left to right, the points initially tend to be below the line of best fit, then move above it, and
then below it. The residuals are negative, then positive, and then negative again. This means that the points in the
residual plot will be below the horizontal axis, then above it, and then below it again.</p>
<p>The residual plot has an arch shape like this:</p>
<p><img alt="A RESIDUAL PLOT THAT SHOWS FROM LEFT TO RIGHT POINTS BELOW THE HORIZONTAL AXIS, ABOVE THE HORIZONTAL AXIS, AND THEN BELOW THE HORIZONTAL AXIS.
" class="img-fluid atto_image_button_text-bottom" height="369"
src="https://k12.openstax.org/contents/raise/resources/b41bf70fcf1fec0b55323d9f0376930008f37343" width="450">
</p>