7cce28be-76bf-4030-a44a-839e68db4f5e.html

<h3>Activity (20 minutes)</h3>
<p>Previously, students analyzed a set of data on the power levels of a battery as it was being charged. They looked for a model that enabled them to predict when the battery would be 100% charged. In this activity, they engage in a similar, yet more challenging, modeling exercise to determine when a battery would be completely out of power.</p>
<p>Here, students are asked to define a function and then use their function to make some predictions. Then, they evaluate their predictions and adjust the function to account for new information. Data about the situation are presented in a way that may be unfamiliar, so students need to persevere in making sense of quantities and the relationships among them.</p>
<p>As students work, monitor for those who use different reasoning strategies and representations as they find a model that describes how the battery power was decreasing over time. Select them to share their work later.</p>
<p>No time estimate is specified for this activity as the length would depend on decisions about how students&rsquo; models are discussed, revised, tested, and presented. If students are to collect data using their own devices, or if they are to prepare a presentation that illustrates their results and their modeling process, additional time will be needed.</p>
<p>Making graphing and statistical technology available gives students an opportunity to choose appropriate tools strategically.</p>
<h4>Launch</h4>
<ol>
  <li>Ask students to close their books or devices. Then, display the graph for all to see.</li>
  <p><img alt="graph of battery usage" src="https://k12.openstax.org/contents/raise/resources/878a59b47b031b568820dd2669e635e32855b6a1" width="300"></p>
  <li>Tell students that the image shows the battery usage for a cell phone for 9 hours after it was fully charged. Give students a moment to observe the image and to make a prediction for how much longer the battery will last.</li>
  <li>Poll the class on their predictions and display the predictions for all to see. Then, ask students to work on the activity.</li>
  <li>Consider arranging students in groups of 2&ndash;4 and asking students to pause for a whole-class discussion after the first set of questions.</li>
</ol>
<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">Engagement: Develop Effort and Persistence</p>
  </div>
  <div class="os-raise-extrasupport-body">
    <p>Provide students with a blank two-column table. Label the columns time since full charge and percent remaining. Students can complete the table as they make sense of the given information, and then use the completed table to reason about the questions, to compare their interpretations of the data, and to discuss their reasoning.</p>
    <p class="os-raise-text-italicize">Supports accessibility for: Attention; Social-emotional skills</p>
    </div>
  </div>
<br>
<h4>Student Activity </h4>
<p>For questions 1-2, use the following prompt and image.</p>
<blockquote>The image shows the battery usage of a cell phone 9 hours since it was fully charged. It also shows a prediction that the battery would last 8 more hours.</blockquote>

<p><img alt="Advanced battery usage, 52 percent. Graph from 100 percent to 0 percent. 9 hours ago, 8 hours left." class="img-fluid atto_image_button_text-bottom" height="329" src="https://k12.openstax.org/contents/raise/resources/993366c891577df6fe2d656e62e6e51a0db8e9d7" width="300"></p><br>
  <ol class="os-raise-noindent">
    <li> Write an equation for a model that fits the data in the image and gives the percent of battery power as a function of time since the phone was fully charged. Be prepared to show your reasoning.
      <p>If you get stuck, consider creating a table of values or a scatter plot of the data.</p>
    </li><br>
    <li> Based on your function, what percentage of power would the battery have 4 hours after this image was taken? What about 5 hours after the image was taken? Be prepared to show your reasoning.<br>
      <br>
    </li>
</ol>


<p>For questions 3-5, use the following prompt and images.</p>

<blockquote>Here are two more images showing the battery usage at two later times, before the battery was charged again.</blockquote>

<p><img alt="Two images of advanced battery usage, 29 percent and 7 percent." src="https://k12.openstax.org/contents/raise/resources/e759b18e61a284924951bc44fec6d6c6471b4fe9" width="550"></p>
<br>
<ol class="os-raise-noindent" start="3">
  <li> How well did the function you wrote predict the battery power 4 and 5 hours after the first image was taken (that is, 13 and 14 hours after the battery was fully charged)? Be prepared to show your reasoning. </li><br>
  <li> What do you notice about the change in the prediction between \(t=13\) and \(t=14\)? </li><br>
  <li> Write a new equation for a function that would better fit the data shown in the last image. </li>
</ol>

<div class="os-raise-usermessage-lightbulb">
  <p>If you get stuck, consider creating a table of values or a scatter plot of the data. </p>
  </div>
<h4>Student Response</h4>
    <ol class="os-raise-noindent">
      <li>One equation could be: \( p=100 - \frac{100}{17}t \) or \( p=100 - 5.9t \). For example: In the table, the relationship between the quantities appears to be linear. The average rate of change between \( t=0 \) and \( t=17 \) is \( \frac{0-100}{17-0} \), which is \( \frac {-100}{17} \) or about &ndash;5.9, so the battery power was losing about 5.9% of power each hour. The rate of change in the first 9 hours is \( \frac {-48}{9} \) or about &ndash;5.3% per hour. The rate of change in the next 8 hours is \( \frac {-52}{8} \) or about &ndash;6.5% per hour. The mean of the two rates is \( \frac{-5.3 + (-6.5)}{2} \) or &ndash;5.9% per hour.</li>
      </ol>
          <br>
        <table class="os-raise-midsizetable">
          <thead>
            <tr>
              <th scope="col"> \( t \), time since fully charged (hours) </th>
              <th scope="col"> \( p \), percent of battery power </th>
            </tr>
          </thead>
          <tbody>
            <tr>
              <td>0</td>
              <td>100</td>
            </tr>
            <tr>
              <td>9</td>
              <td>52</td>
            </tr>
            <tr>
              <td>17</td>
              <td>0</td>
            </tr>
          </tbody>
        </table>
        <br>
    <ol class="os-raise-noindent" start="2">
      <li>Four hours since the image was taken means \( t=13 \). When \( t=13 \), \( p \) is about 23.5. The phone would have about 23.5% of battery power.Five hours since the image was taken means \( t=14 \). When \( t=14 \), \( p \) is about 17.6. The phone would have about 17.6% of battery power. </li>
      <br>
    <ol class="os-raise-noindent" start="3">
      <li>My prediction for \( t=13 \) is about 5% lower than what&rsquo;s shown in the image, and the one for \( t=14 \) is about 10.6% higher than what&rsquo;s shown in the last image.
      </li>
      <br>
      <li>At \( t=13 \), the phone software was still predicting that the battery would last 17 hours. At \( t=14 \), the prediction dropped drastically to 14.5 hours.
      </li>
      <br>
      <li>The average rate of change between \( t=0 \) and \( t=14.5 \) is \( \frac{0-100}{14.5-0} \), which is \( \frac {-100}{14.5} \) or about &ndash;6.9, so the battery was losing about 6.9% of power each hour. One equation could be: \( p=100 - 6.9t \).</li>
</ol>
<br>
<table class="os-raise-midsizetable">
  <thead>
    <tr>
      <th scope="col"> \( t \), time since fully charged (hours) </th>
      <th scope="col"> \( p \), percent of battery power </th>
    </tr>
  </thead>
  <tbody>
    <tr>
      <td>0</td>
      <td>100</td>
    </tr>
    <tr>
      <td>9</td>
      <td>52</td>
    </tr>
    <tr>
      <td>14</td>
      <td>7</td>
    </tr>
    <tr>
      <td>14.5</td>
      <td>0</td>
    </tr>
  </tbody>
</table><br>



<h4>Anticipated Misconceptions</h4>
<p>Because the images specify duration of time relative to the moment when they were taken, using phrases such as &ldquo;9 hours ago&rdquo; or &ldquo;8 hours left,&rdquo; students may think that time was measured from a different reference point each time. They may struggle to quantify the changes in time or to organize the values into a table. Ask students to consider when, or at what power level, we typically begin measuring the life of a battery. Clarify that all images referenced the time when the battery was at 100%, so they could use that as a starting point for measuring time.</p>
<h4>Project Synthesis</h4>
<p>Select students or groups to share the function they defined for the first set of questions and to explain their reasoning, including the assumptions they made. After each model is shared, ask if others had written the same equation but arrived at it a different way, or if they had reasoned the same way and made the same assumptions but arrived at a different model. Then, focus the discussion on how students refined their original models to account for the new information given in the second question. Showcase the variety of strategies and representations students created to make sense of the additional data or to represent their new function. Consider using graphing technology to graph (on the same coordinate plane) the different equations students wrote and display the graphs for analysis and comparison.</p>
<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 8 Discussion Supports: Speaking</p>
  </div>
  <div class="os-raise-extrasupport-body">
      <p>Use this routine to support small-group discussion. At the appropriate time, give students 2–3 minutes to prepare an explanation of their reasoning. Ask them to include an explanation about the assumptions they made and how these informed the function they wrote. Encourage students to consider what details are important to share and to think about how they will explain their reasoning using mathematical language. This will help students clarify their thinking which will improve the quality of explanations shared during the whole-class discussion.<br>
</p>
<p class="os-raise-text-italicize">Design Principle(s): Support sense-making; Maximize meta-awareness</p>
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