fff84583-1d24-41b9-8845-6c677dcfaf08.html

<h4>Activity (15 minutes)</h4>
<p>Up to this point in the unit, students have written and interpreted equations representing exponential change that
  were meaningful for non-negative input values. In this lesson, they interpret the meaning of a negative value in a
  context. The independent variable is time, \(t\), so negative values of \(t\) refer to times before
  \(t\) is 0.</p>
<p>For the last question, to find the time when the coral had the given volume, students may:</p>
<ul>
  <li> Calculate the values of \(y\) when \(t\) is &ndash;3, &ndash;4, &ndash;5, etc. (i.e., extend the table
    of values, if they previously created one) until they reach 37.5 or a value close to it. </li>
  <li> Repeatedly divide by 2 (or multiply by \(\frac12\)) the value of \(y\) when \(t\) is &ndash;2 until it
    approaches or hits 37.5. </li>
  <li> Use the equation \(1,200\cdot 2^t=37.5\) to find the value of \(2^t\) and then reason about \(t\) from
    there. </li>
</ul>
<p>Look for students using these or other strategies so they can share later.</p>
<p>Making graphing technology available gives students an opportunity to choose appropriate tools strategically.</p>
<h4>Launch</h4>
<p>Arrange students in groups of 2. Encourage them to think quietly about the questions before discussing with their
  partner. Creating a table or spreadsheet may help students organize the work in the second question. If needed,
  encourage students to do so.</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>
    </div>
    <div class="os-raise-extrasupport-body">
        <p>To help students make sense of the language of exponential situations, start by displaying the task statement:
          &ldquo;A marine biologist estimates that a structure of coral has a volume of 1,200 cubic centimeters and that its
          volume doubles each year.&rdquo; Give students 1&ndash;2 minutes to write their own mathematical questions about the
          situation. Invite students to share their questions with the class, and then reveal the activity&rsquo;s questions.
          Listen for and amplify any questions referring to the value of \(t\), especially those wondering about time in the
          past. This will build student understanding and interpretation of the language used to represent negative exponents.</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>
  </div>
  <div class="os-raise-extrasupport-body">
      <p> Begin the activity with concrete or familiar contexts. Encourage students to create a timeline next to their
    tables, selecting a starting point year for zero and then labeling the years correlated with the positive exponents
    prior to labeling years associated with negative exponents.</p>
</div>
</div>
<br>
<h4>Student Activity</h4>
<p>A marine biologist estimates that a structure of coral has a volume of 1,200 cubic centimeters and that its volume
  doubles each year.</p>
<p><img height="300" src="https://k12.openstax.org/contents/raise/resources/915e980323c039f033c1086572a5d1f032cef8c3"
    width="300"></p>
<ol class="os-raise-noindent">
  <li> Write an equation of the form \(y=a\cdot b^t\) representing the relationship, where \(t\) is time in years
    since the coral was measured and \(y\) is volume of coral in cubic centimeters. (You need to figure out what
    numbers \(a\) and \(b\) are in this situation.) <br><br>
    <strong>Answer:</strong> \(y=1,200\cdot 2^t\)
  </li>
  <br>
</ol>
<ol class="os-raise-noindent" start="2">
  <li> Find the volume of the coral when \(t\) is 5, 1, 0, &ndash;1, and &ndash;2. Note that when \(t = -1\), this
    represents one year before the marine biologist made the original estimate. </li>
</ol>
<table class="os-raise-horizontaltable">
  <thead></thead>
  <tbody>
    <tr>
      <th scope="row">Time (years)</th>
      <td>5</td>
      <td>1</td>
      <td>0</td>
      <td>-1</td>
      <td>-2</td>
    </tr>
    <tr>
      <th scope="row">Volume of Coral (cubic centimeters)</th>
      <td>a.____
      </td>
      <td>b.____
      </td>
      <td>c.____
      </td>
      <td>d.____
      </td>
      <td>e.____
      </td>
    </tr>
  </tbody>
</table>
<br>
<p class="os-raise-text-bold">Answer:</p>
<ol type="a">
  <li>38,400 </li>
</ol>
<ol type="a" start="2">
  <li>24,000 </li>
</ol>
<ol type="a" start="3">
  <li>12,000 </li>
</ol>
<ol type="a" start="4">
  <li>600 </li>
</ol>
<ol type="a" start="5">
  <li>300 </li>
</ol>
<br>
<ol class="os-raise-noindent" start="3">
  <li> What does it mean, in this situation, when \(t\) is &ndash;2? <br><br>
    <strong>Answer:</strong> This represents the time 2 years before the scientist made the estimate of 1,200 cubic
    centimeters.
  </li>
  <br>
</ol>
<ol class="os-raise-noindent" start="4">
  <li> In a certain year, the volume of the coral is 37.5 cubic centimeters. Which year is this? Be prepared to show
    your reasoning. <br><br>
    <strong>Answer:</strong> \(t\) is &ndash;5, or 5 years before the scientist made the estimate of 1,200 cubic
    centimeters.
  </li>
</ol>
<h4>Anticipated Misconceptions</h4>
<p>Students may struggle to think of how to start finding the values for \(a\) and \(b\). Suggest that they make a
  table of corresponding values of \(t\) and \(y\), and think about starting with \(t\) and calculating \(y\).</p>
<p>They may also struggle with understanding a negative value for \(t\) referring to years. Emphasize that a value of
  \(t = -1\) represents the time 1 year before the estimate was made. Ask what a value of \(t = -4\) represents.</p>
<h4>Activity Synthesis</h4>
<p>Review how students found an equation, \(y=1,200\cdot 2^t\), to represent the volume of coral. Make sure that they
  recall that the 1,200 represents the 1,200 cubic centimeters of coral when it was first measured, and the 2 indicates
  the doubling of its volume each year.</p>
<p>Select students to share their interpretations of negative values of \(t\) and how they found the value of \(t\)
  when given a volume. Ask questions such as:</p>
<ul>
  <li>&ldquo;In this situation, what does it mean to say that when \(t\) is &ndash;3, \(y\) is \(1,200\cdot
    2^{-3}\) or 150?&rdquo; (3 years before the coral was measured to be 1,200 cubic centimeters, its volume was 150
    cubic centimeters.)</li>
  <li>&ldquo;How did you estimate or find \(t\) when the volume of the coral was 37.5 cubic centimeters?&rdquo;</li>
</ul>
<p>It may be helpful to display a graph of \(y=1,200\cdot 2^t\), such as the one embedded in the digital version of
  these materials. Display the graph (either with or without the points from the table plotted) and ask students to
  identify the volume of coral when \(t\) is &ndash;3 and the time when the volume of coral is 37.5 cubic
  centimeters.</p>
<p>If using the digital applet, consider initially turning off the function and the point, and hiding the expression
  list. Ask students to gesture at points on the graph corresponding to questions in the activity. Then, turn on the
  function and the point for demonstration. Remember that if you click on a plotted point, its coordinates are revealed.
</p>
<h3>5.6.2: 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>Using the equation \(p=10,000\cdot 2^d\), find the value of \(p\) when \(d = -2\).</p>
        <table class="os-raise-textheavytable">
          <thead>
            <tr>
              <th scope="col">Answers</th>
              <th scope="col">Feedback</th>
            </tr>
          </thead>
          <tbody>
            <tr>
      <td>
        2,500
      </td>
      <td>
        That&rsquo;s correct! Check yourself: The expression \(p=10,000\cdot2^{-2}=2,500\).
      </td>
    </tr>
    <tr>
      <td>
       20,000
      </td>
      <td>
      Incorrect. Let&rsquo;s try again a different way: This is the value when \(d =1\). Evaluate the expression
          when \(d=-2\). \(p=10,000\cdot2^{-2}\). The answer is 2,500.
      </td>
    </tr>
    <tr>
      <td>
       5,000
      </td>
      <td>
       Incorrect. Let&rsquo;s try again a different way: This is the value when \(d=-1\). Find the value when
          \(d=-2\). Evaluate the expression \(p=10,000\cdot2^{-2}\). The answer is 2,500.
      </td>
    </tr>
    <tr>
      <td>
       40,000
      </td>
      <td>
        Incorrect. Let&rsquo;s try again a different way: This is the value when \(d=2\). Find the value when
          \(d=-2\). Evaluate the expression \(p=10,000\cdot2^{-2}\). The answer is 2,500.
      </td>
    </tr>
  </tbody>
</table>
<br>
<h3>5.6.2: 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>Interpreting Negative Exponents in Exponential Growth</h4>
<p>Let&rsquo;s look at a problem involving an exponential growth equation and negative exponents.</p>
<p>A forest fire has been burning for several days. The burned area, in acres, is given by the equation \(y=4,800\cdot
  2^d\), where \(d\) is the number of days since the area of the fire was first measured.</p>
<ol class="os-raise-noindent">
  <li>
    Let&rsquo;s complete the table.
  </li>
  <br>
  <table class="os-raise-wideequaltable">
    <thead>
      <tr>
        <th scope="col">\(d\), days since first measurement</th>
        <th scope="col">Process column</th>
        <th scope="col">\(y\), acres burned since fire started</th>
      </tr>
    </thead>
    <tbody>
      <tr>
        <td>
       0
        </td>
        <td>\(= 4800 \cdot 2^0 = 4800 \cdot 1 \)
        </td>
        <td>4800</td>
      </tr>
      <tr>
        <td>
          &ndash;1
        </td>
        <td>
          \(= 4800 \cdot 2^1 = 4800 \cdot \frac12 \)
        </td>
        <td>2400</td>
      </tr>
      <tr>
        <td>&ndash;2
        </td>
        <td>0
        </td>
        <td>c. _____</td>
      </tr>
      <tr>
        <td>
         &ndash;3
        </td>
        <td>
          0
        </td>
        <td>d. _____</td>
      </tr>
      <tr>
        <td>&ndash;5
        </td>
        </td>
        <td>0
        <td>e. _____</td>
      </tr>
    </tbody>
  </table>
  <br>

  <p>To determine the value of \(y\), given \(d\), follow these steps:</p>

  <h5>Example 1</h5>

  <p>Determine the value for \(y\) when \(d = -2\)</p>

  <p><strong>Step 1 </strong> -Substitute the given domain value for d into the expression.</p>

  <p>\(4800 \cdot 2^{-2}\)</p>

  <p><strong>Step 2 </strong> -Evaluate the exponential value. Recall that a negative exponent represents how many times
    the reciprocal of the base should be multiplied. </p>

  <p>\(= 4800 \cdot (\frac{1}{2})^2 = 4800 \cdot \frac14\)</p>


  <p><strong>Step 3 </strong> -Evaluate the remaining expression to determine \(y\).</p>

  <p>\(= 1200\)</p>

  <h5>Example 2</h5>

  <p>Determine the value for \(y\) when \(d = -3\)</p>

  <p><strong>Step 1 </strong> -Substitute the given domain value for d into the expression.</p>

  <p>\(4800 \cdot 2^{-3}\)</p>

  <p><strong>Step 2 </strong> -Evaluate the exponential value. Recall that a negative exponent represents how many times
    the reciprocal of the base should be multiplied. </p>

  <p>\(= 4800 \cdot (\frac{1}{2})^3 = 4800 \cdot \frac18\)</p>


  <p><strong>Step 3 </strong> -Evaluate the remaining expression to determine \(y\).</p>

  <p>\(= 600\)</p>

  <h5>Example 3</h5>

  <p>Determine the value for \(y\) when \(d = -5\)</p>

  <p><strong>Step 1 </strong> -Substitute the given domain value for d into the expression.</p>

  <p>\(4800 \cdot 2^{-5}\)</p>

  <p><strong>Step 2 </strong> -Evaluate the exponential value. Recall that a negative exponent represents how many times
    the reciprocal of the base should be multiplied. </p>

  <p>\(= 4800 \cdot (\frac{1}{2})^5 = 4800 \cdot \frac{1}{32}\)</p>


  <p><strong>Step 3 </strong> -Evaluate the remaining expression to determine \(y\).</p>

  <p>\(= 150\)</p>

  <h5>Example 4</h5>

  <p>What does the value of \(y\) tell you about the area burned in the fire when \(d=-1\)?</p>

  <p>The \(y\) value at \(d= -1\) represents the fact that 2,400 acres had burned 1 day before the first measurement was
    taken. </p>

  <h5>Example 5</h5>

  <p>How much area had the fire burned a week before it measured 4,800 acres? Explain your reasoning. </p>

  <p>Since there are 7 days in a week, to find the area burned a week before the fire was measured, we need to determine
    the value of \(y\) when \(d = -7\). </p>

  <p><strong>Step 1 </strong> -Substitute the given domain value for d into the expression.</p>

  <p> \(4800 \cdot 2^{-7}\) </p>


  <p><strong>Step 2 </strong> -Evaluate the exponential value. Recall that a negative exponent represents how many times
    the reciprocal of the base should be multiplied.</p>

  <p> \(= 4800 \cdot \frac12 ^7 = 4800 \cdot \frac{1}{128}\) </p>


  <p><strong>Step 3 </strong> -Evaluate the remaining expression to determine \(y\).</p>

  <p> \(= 37.5\) </p>

  <p> \(37.5\) acres had burned a week before the fire measured \(4,800\) acres. The value is found by substituting
    \(-7\) for d since that represents \(7\) days before the first measurement of \(4,800\) acres. </p>




  <br>
  <li> Look at the value of \(y=4,800\cdot 2^d\) when \(d=-1\). What does it tell you about the area burned in the fire?
  </li>
  <p><strong>Answer: </strong>It tells you that 1 day before the first measurement, 2,400 acres had burned.</p>
  <li> How much area had the fire burned a week before it measured 4,800 acres? Be prepared to show your reasoning.
  </li>
  <p><strong>Answer: </strong>Substitute &ndash;7 for \(d\).<br>
    <br>
    \(y=4,800\cdot 2^d\) <br>
    \(y=4,800\cdot 2^{-7}\) <br>
    \(y=4,800\cdot\frac1{128}\) <br>
    \(y=37.5\)
  </p>
  <p> 37.5 acres had burned a week before the fire measured 4,800 acres. The solution is found by substituting &ndash;7
    for \(d\) since that represents 7 days before the first measurement of 4,800 acres.</p>
</ol>
<h4> Try It: Interpreting Negative Exponents in Exponential Growth</h4>
<p>The value of a home in 2015 was $400,000. Its value has doubled each decade.</p>


<ol class="os-raise-noindent">
  <li> If \(v\) is the value of the home, in dollars, write an equation for \(v\) in terms of \(d\), the number of
    decades since 2015. </li>
</ol>

<ol class="os-raise-noindent" start="2">
  <li> What is \(v\) when \(d=-1\)? </li>
  <p> <strong>Answer:</strong>\(v=200,000\)
    <br>\(v=400,000\cdot 2^{-1}\)
    <br>\(v=400,000\cdot\frac12\)
    <br>\(v=200,000\)
  </p>
</ol>
<ol class="os-raise-noindent" start="3">
  <li>What does this value mean? </li>
  <p> <strong>Answer:</strong>This means the home had a value of 200,000 in 2005, one decade before 2015. </p>
</ol>
<ol class="os-raise-noindent" start="4">
  <li> What is \(v\) when \(d=-3\)?</li>
  <p> <strong>Answer:</strong>\(v=50,000\)
    <br>\(v=400,000\cdot 2^{-3}\)
    <br>\(v=400,000\cdot\frac18\)
    <br>\(v=50,000\)
  </p>
</ol>
<ol class="os-raise-noindent" start="5">
  <li>What does this value mean? </li>
  <p> <strong>Answer:</strong>This means the home had a value of 200,000 in 2005, one decade before 2015. </p>
</ol>