0dbff3c2-3791-4f01-98d6-5c9f8224179d.html

<h4>Activity (15 minutes)</h4>
<p>This lesson uses the data from the video of orange weights from the activity Orange You Glad
  We&rsquo;re Boxing Fruit. The mathematical purpose of this activity is to introduce the concept of residuals and to
  have students plot and analyze the residuals to informally assess the fit of a function. In this activity, students
  also fit a function to data, and they use the function to solve problems. Students learn that a residual is the
  difference between the actual \(y\)-value for a point and the expected \(y\)-value for the point on the linear model
  with the same associated \(x\)-value.</p>
<h4>Launch</h4>
<p>The digital version of this activity includes instructions for plotting the residuals of the data. If you
  will be using graphing technology other than Desmos for this activity, you may need to prepare alternate instructions.
</p>
<p>Display the data from the video about weighing oranges:</p>
<table class="os-raise-midsizetable">
  <thead>
    <tr>
      <th scope="col">Number of Oranges</th>
      <th scope="col">Weight in Kilograms</th>
    </tr>
  </thead>
  <tbody>
    <tr>
      <td>
        3
      </td>
      <td>
        1.027
      </td>
    </tr>
    <tr>
      <td>
        4
      </td>
      <td>
        1.162
      </td>
    </tr>
    <tr>
      <td>
        5
      </td>
      <td>
        1.502
      </td>
    </tr>
    <tr>
      <td>
        6
      </td>
      <td>
        1.617
      </td>
    </tr>
    <tr>
      <td>
        7
      </td>
      <td>
        1.761
      </td>
    </tr>
    <tr>
      <td>
        8
      </td>
      <td>
        2.115
      </td>
    </tr>
    <tr>
      <td>
        9
      </td>
      <td>
        2.233
      </td>
    </tr>
    <tr>
      <td>
        10
      </td>
      <td>
        2.569
      </td>
    </tr>
  </tbody>
</table>
<br>
<h4>Student Activity</h4>
<ol class="os-raise-noindent">
  <p>Here is the data table for orange weights from a previous lesson. Use this information to answer the questions.</p>
  <li>Use the graphing tool or technology outside the course. Find the scatter plot and a line of best fit. Students were provided access to Desmos. </li>
  <br>
  <table class="os-raise-midsizetable">
    <thead>
      <tr>
        <th scope="col">Number of Oranges</th>
        <th scope="col">Weight in Kilograms</th>
      </tr>
    </thead>
    <tbody>
      <tr>
        <td>
          3
        </td>
        <td>
          1.027
        </td>
      </tr>
      <tr>
        <td>
          4
        </td>
        <td>
          1.162
        </td>
      </tr>
      <tr>
        <td>
          5
        </td>
        <td>
          1.502
        </td>
      </tr>
      <tr>
        <td>
          6
        </td>
        <td>
          1.617
        </td>
      </tr>
      <tr>
        <td>
          7
        </td>
        <td>
          1.761
        </td>
      </tr>
      <tr>
        <td>
          8
        </td>
        <td>
          2.115
        </td>
      </tr>
      <tr>
        <td>
          9
        </td>
        <td>
          2.233
        </td>
      </tr>
      <tr>
        <td>
          10
        </td>
        <td>
          2.569
        </td>
      </tr>
    </tbody>
  </table><br>
  <p><strong>Answer: </strong>The equation of a line of best fit is \(y=0.216x+0.345\).</p>
  <li>What level of accuracy makes sense for the slope and intercept values? Be prepared to show your reasoning.
    <p><strong>Answer: </strong>Rounding the values to the thousandths place makes sense because that would be to an accuracy of 1 gram, which is
      what the original data used.</p>
  </li>
  <li>What does the linear model estimate for how much the weight increases for of each additional box of oranges?
    <p><strong>Answer: </strong>0.216 kg for each orange</p>
  </li>

  <li>Compare the weight of the actual box with 3 oranges in it to the estimated weight of the box with 3 oranges in it.
    Be prepared to show your reasoning.
    <p><strong>Answer: </strong>The actual weight is 1.027 kg, and the estimated weight is 0.993 kg. Therefore, the actual weight is 0.034
      kg greater than the estimate.</p>
  </li>
  <li>How many oranges are in the box when the linear model estimates the weight best? Be prepared to show your
    reasoning.
    <p><strong>Answer: </strong>6 oranges are in the box when the estimate is best. The difference between the estimated weight and the actual
      weight is only 0.023 kg, while the other points are farther from the line.</p>
  </li>
  <li>How many oranges are in the box when the linear model estimates the weight least well? Be prepared to show your
    reasoning.
    <p><strong>Answer: </strong>7 oranges are in the box when the estimate is worst. The difference between the estimated weight and the actual
      weight is 0.095, which is the most of any of the values.</p>
  </li>
  <p>Use the following information to answer questions 7 and 8.</p>
  <blockquote>
    <p>The difference between the actual value and the value estimated by a linear model is called the residual.</p>
    <ul>
      <li>If the actual value is greater than the estimated value, the residual is positive.&nbsp;</li>
      <li>If the actual value is less than the estimated value, the residual is negative.</li>
    </ul>
  </blockquote>
  <li> For the orange weight data set, what is the residual for the line of best fit when there are 3 oranges?

    <p><strong>Answer: </strong>The residual is \( 0.034 (1.027&minus;0.993=0.034)\).</p>
  </li>
  <li>On the same axes as the scatter plot, plot this residual at the point where \(x = 3\) and \(y\)
    has the value of the residual.
    <p><strong>Answer: </strong>The point is (8,0.043).</p>
  </li>
  <li>Find the residuals for each of the other points in the scatter plot and graph them.<br><br><img alt="Blank coordinate plane with grid. Horizontal axis from 0 to 11 by 1's. Vertical axis from negative 0 point 5 to 0 point 5, by 0 point 1's." src="https://k12.openstax.org/contents/raise/resources/ad8dea61f8ed8a9bdf59dc47c6446b93ba30332c">
    <br><br>
    <p><strong>Answer: </strong>
      <br><img alt="Discrete graph." src="https://k12.openstax.org/contents/raise/resources/b7243f82e7125491bd9a2a9ae3f52a9c03fd5bef">
    </p>
  </li>
  <li>Which point on the scatter plot has the residual closest to zero? What does this mean about the weight of the box
    with that many oranges in it?
    <p><strong>Answer: </strong>6 oranges. It means the estimated weight of the box with 6 oranges in it is closest to the actual value.</p>
  </li>
  <li>How can you use the residuals to decide how well a line fits the data?
    <p><strong>Answer: </strong>A line will result in more accurate estimates when the residuals are closer to zero.</p>
  </li>
</ol>
<h4>Video: Finding Residuals</h4>
<p>Watch the following video to learn more about finding residuals</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/23c4c5a71e02040cd50b142cc3fd9585b214bcaf">
      <track default="true" kind="captions" label="On" src="https://k12.openstax.org/contents/raise/resources/016997b431f1c5c96171aab3df5ced18ce0cc8f2" srclang="en_us">
      https://k12.openstax.org/contents/raise/resources/23c4c5a71e02040cd50b142cc3fd9585b214bcaf
    </video></div>
</div>
<br><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">Representation: Internalize Comprehension</p>
  </div>
  <div class="os-raise-extrasupport-body">
    <p> Use color coding and annotations to highlight connections
      between representations in a problem. For example, highlight the linear column in the table the same color as the
      line on the graph, and highlight the actual values on the table the same color as the corresponding points on the
      graph. To extend this further, the residual values can be displayed and highlighted in the same color as the
      residual points on the graph itself.</p>
    <p class="os-raise-text-italicize">Supports accessibility for: Visual-spatial processing</p>
  </div>
</div>
<br>
<h4>Anticipated Misconceptions</h4>
<p>Students may not understand how to determine if the linear model estimates the weight of oranges well or poorly. Ask
  them to determine the weight that the model estimates. Then ask how close that estimate is to the actual weight.</p>
<h4>Activity Synthesis</h4>
<p>Compare student answers to the question about the point that the line estimates best to the answer for the question
  about the residual closest to zero.</p>
<p>Show a graph of the residuals.</p>
<p><img alt="Discrete graph." src="https://k12.openstax.org/contents/raise/resources/b7243f82e7125491bd9a2a9ae3f52a9c03fd5bef"><br></p>
<p>Ask students:</p>
<ul>
  <li>&ldquo;What does it mean for the residual to be positive? Negative?&rdquo; (The residual is positive when the
    actual data value is greater than what the model estimates for that \(x\)-value and negative when the actual data
    value is less than the estimate.)</li>
  <li>&ldquo;What does it mean when a residual is on or close to the horizontal axis?&rdquo; (It means that the line of
    best fit passes through or comes close to passing through that point on the graph.)</li>
  <li>&ldquo;Find the residual that has the farthest vertical distance from the horizontal axis. What does this mean in
    the context of the scatter plot and the line of best fit?&rdquo; (The residual that is farthest from the horizontal
    axis has the same \(x\)-coordinate as the point that is the greatest vertical distance away from the line of best
    fit in the scatter plot.)</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">MLR 2 Collect and Display: Conversing, Writing</p>
  </div>
  <div class="os-raise-extrasupport-body">
    <p>Before a whole-class discussion in the activity synthesis,
      invite students to discuss their understanding of residuals with a partner. Listen for and collect vocabulary and
      phrases students use to describe how to calculate residuals, what positive residual and negative residual mean, and
      what happens when the residual is 0. Display words and phrases, such as &ldquo;difference between actual data and
      estimates,&rdquo; &ldquo;estimate is greater/less than actual data,&rdquo; and &ldquo;linear estimate is close to
      actual data&rdquo; for all to see, and then encourage students to use this language in their written responses. This
      will help students read and use mathematical language during whole-group discussions.</p>
    <p class="os-raise-text-italicize">Design Principle(s): Support sense-making</p>
    <p class="os-raise-extrasupport-title">Provide support for students</p>
    <p>
      <a href="https://k12.openstax.org/contents/raise/resources/3765a3cff09304aea8d8db39295b1c00ffd0c12f" target="_blank">Distribute graphic organizers</a>
      to the students to assist them with participating in this routine.
    </p>
  </div>
</div>
<br>
<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>The equation for a line of best fit is \(y = 2.2x +3.4\). Find the residual for the point
  \((3, 9)\).</p>
<table class="os-raise-textheavytable">
  <thead>
    <tr>
      <th scope="col">Answers</th>
      <th scope="col">Feedback</th>
    </tr>
  </thead>
  <tbody>
    <tr>
      <td>
        \(0.4\)
      </td>
      <td>
        Incorrect. Let&rsquo;s try again a different way: Subtract the differences of the \(y\)-values of
        the actual value &ndash; estimated value. The answer is \(-1\).
      </td>
    </tr>
    <tr>
      <td>
        \(1\)
      </td>
      <td>
        Incorrect. Let&rsquo;s try again a different way: To find the residual, subtract the actual value
        &ndash; estimated value. The answer is \(-1\).
      </td>
    </tr>
    <tr>
      <td>
        \(-1\)
      </td>
      <td>
        That&rsquo;s correct! Check yourself. Substitute in \(x =3\) so that \(y=2.2(3)+3.4= 10\). This
        is the estimated value. The actual value at \(x=3\) is 9, so \(9-10 =-1\). The residual is \(-1\).
      </td>
    </tr>
    <tr>
      <td>
        \(14.2\)
      </td>
      <td>
        Incorrect. Let&rsquo;s try again a different way: Substitute 3 in for \(x\), not 9, in the
        equation. The answer is \(-1\).
      </td>
    </tr>
  </tbody>
</table>
<br>
<h3>3.3.2: 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>Finding Residuals</h4>
<p>The gestation time for an animal is the typical duration between conception and birth. The longevity of an
  animal is the typical lifespan for that animal. The gestation times, in days, and longevities, in years, for 13
  types of animals are shown in the table below.</p>
<table class="os-raise-midsizetable">
  <thead>
    <tr>
      <th scope="col">Animal </th>
      <th scope="col">Gestation Time
        (Days) </th>
      <th scope="col">Longevity
        (Years)</th>
    </tr>
  </thead>
  <tbody>
    <tr>
      <td>
        Baboon
      </td>
      <td>
        187
      </td>
      <td>
        20
      </td>
    </tr>
    <tr>
      <td>
        Black Bear
      </td>
      <td>
        219
      </td>
      <td>
        18
      </td>
    </tr>
    <tr>
      <td>
        Beaver
      </td>
      <td>
        105
      </td>
      <td>
        5
      </td>
    </tr>
    <tr>
      <td>
        Bison
      </td>
      <td>
        285
      </td>
      <td>
        15
      </td>
    </tr>
    <tr>
      <td>
        Cat
      </td>
      <td>
        63
      </td>
      <td>
        12
      </td>
    </tr>
    <tr>
      <td>
        Chimpanzee
      </td>
      <td>
        23
      </td>
      <td>
        20
      </td>
    </tr>
    <tr>
      <td>
        Cow
      </td>
      <td>
        284
      </td>
      <td>
        15
      </td>
    </tr>
    <tr>
      <td>
        Dog
      </td>
      <td>
        61
      </td>
      <td>
        12
      </td>
    </tr>
    <tr>
      <td>
        Fox (Red)
      </td>
      <td>
        52
      </td>
      <td>
        7
      </td>
    </tr>
    <tr>
      <td>
        Goat
      </td>
      <td>
        151
      </td>
      <td>
        8
      </td>
    </tr>
    <tr>
      <td>
        Lion
      </td>
      <td>
        100
      </td>
      <td>
        15
      </td>
    </tr>
    <tr>
      <td>
        Sheep
      </td>
      <td>
        154
      </td>
      <td>
        12
      </td>
    </tr>
    <tr>
      <td>
        Wolf
      </td>
      <td>
        63
      </td>
      <td>
        5
      </td>
    </tr>
  </tbody>
</table><br>
<p>The equation of a line of best fit is \(y = 9.875 + 0.02039x\), where \(x\) represents the gestational time, in days, and \(y\) represents longevity, in years.&nbsp;</p>
<p><img alt class="img-fluid atto_image_button_text-bottom" height="339" role="presentation" src="https://k12.openstax.org/contents/raise/resources/69b023d797f371f2dd27d011d5c66c2303d2ef6f" width="540"><br>
</p>
<p>A lion&rsquo;s gestation time is 100 days, and its longevity is 15 years. What does this line of best fit predict the
  lion&rsquo;s longevity to be?</p>
<p><strong>Solution</strong></p>
<p>The line of best fit is \(𝑦=9.875 + 0.02039(100)=11.9\)</p>
<p>The residual would be: actual value &ndash; estimated value or \(15&minus;11.9=3.1\).</p>
<p>The residual value is 3.1.</p>
<h4>Try It: Finding Residuals</h4>
<p>In the table above, a dog has a gestation time of 61 days and a longevity of 12 years.</p>
<p>Using the same equation, \(y = 9.875 + 0.02039x\), what would the residual value be?</p>
<p><strong>Answer:</strong></p>
<p>Substitute \(x =61\) into the equation, \(𝑦=9.875 + 0.02039(61)=11.1\).</p>
<p>So, the residual is the (actual value &ndash; estimated value):</p>
<p>\(12&minus;11.1=0.9\).</p>