489fd58b-4956-48d2-b479-ee9aed511050.html

<h4>Activity (15 minutes)</h4>
<p>In this activity, students represent a situation using a table of values, a graph, and an equation. From the
  exponential equation, it is a short step to thinking of the relationship between the quantities as a function.</p>
<p>Note that it is possible and acceptable to think of time as a function of area, but expressing this using an equation
  is out of the scope of this course. Students could, however, represent such a function with a graph, table, or
  description.</p>
<p>Making graphing technology available gives students an opportunity to choose appropriate tools strategically.</p>
<h4>Launch</h4>
<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>Represent the same information through different modalities by drawing a diagram. Encourage students who are unsure
    where to begin to sketch a diagram of a slice of bread on graph paper and to shade the area that is covered in mold
    after 1 day, then 2 days . . . until they reach the day when the slice of bread is completely covered in mold.</p>
    <p class="os-raise-text-italicize">Supports accessibility for: Conceptual processing; Visual-spatial processing</p>
</div>
</div>
<br>
<h4>Student Activity</h4>
<p>Clare noticed mold on the last slice of bread in a plastic bag. The area covered by the mold was about 1 square
  millimeter. She left the bread alone to see how the mold would grow. The next day, the area covered by the mold had
  doubled, and it doubled again the day after that.</p>
<ol class="os-raise-noindent">
  <li> If the doubling pattern continues, how many square millimeters will the mold cover 4 days after she noticed the
    mold? Show your reasoning. <br>
    <br>
    <strong>Answer:</strong> 16 square millimeters. \(2 \cdot 2 \cdot 2 \cdot 2\) or \(2^4\) is 16. <br>
    <br>
  </li>
  <li> Represent the relationship between the area \(A\), in square millimeters, covered by the mold and the number of
    days \(d\)&nbsp;since the mold was spotted using: </li>
  <ol class="os-raise-noindent" start="a">
    <li> A table of values, showing the values from the day the mold was spotted through 5 days later. </li>
    <li> An equation </li>
    <li> A graph </li>
  </ol>
  <p><br>
    <strong>Answer:</strong>
  </p>
  <ol class="os-raise-noindent" type="a">
    <li><br></li>
  </ol>
</ol>
<table class="os-raise-midsizetable">
  <thead>
    <tr>
      <th scope="col">Number of days, \(d\)</th>
      <th scope="col">Area, \(A\), in sq mm</th>
    </tr>
  </thead>
  <tbody>
    <tr>
      <td>
        0
      </td>
      <td>
       1
      </td>
    </tr>
    <tr>
      <td>
    1
      </td>
      <td>
        2
      </td>
    </tr>
    <tr>
      <td>
       2
      </td>
      <td>
        4
      </td>
    </tr>
    <tr>
      <td>
       3
      </td>
      <td>
       8
      </td>
    </tr>
    <tr>
      <td>
      4
      </td>
      <td>
      16
      </td>
    </tr>
    <tr>
      <td>
        5
      </td>
      <td>
       32
      </td>
    </tr>
  </tbody>
</table>
<br>
<ol class="os-raise-noindent">
  <ol class="os-raise-noindent" start="2" type="a">
    <li> Equation: \(A=1 \cdot 2^d\) or equivalent (like \(A=2^d\)). </li>
    <li>(Students may show a discrete or a continuous graph.)<br>
      <img src="https://k12.openstax.org/contents/raise/resources/12fcc0a764a3422882fd1d726e1ca3ad71b343f0" width="500">
    </li>
  </ol>
</ol>
<ol class="os-raise-noindent" start="3">
  <li> Discuss with your partner: Is the relationship between the area covered by mold and the number of days a
    function? If so, write ____ is a function of ____. If not, explain why it is not. <br>
    <br>
    <strong>Answer:</strong> Yes. Compare your answer: <br>
    <br>
    The area covered by mold is a function of the number of days that have passed. At any given time since mold was
    spotted, there is a certain area of the bread that is covered in mold.<br>
    <br>
    The number of days that have passed is a function of the area covered by the mold up to a certain point (e.g., when
    the entire bread is already completely covered, the area of mold would correspond to more than one day). We do not
    have a way to write this relationship using an equation, but we could use a table or a graph to show the
    relationship.
  </li>
</ol>
<h4>Student Facing Extension</h4>
<h5>Are you ready for more?</h5>
<p>What do you think is an appropriate domain for the mold area function \(A\)? Be prepared to show your reasoning.</p>
<h4>Extension Student Response</h4>
<p>Your answer may vary, but here are some samples: For example: Some negative values of \(t\) could be relevant, but we
  don&rsquo;t know how the mold was growing before Clare started watching. Positive values of \(t\) will not be valid
  indefinitely because the bread will soon be completely covered by mold. If a piece of bread is 10 cm by 10 cm,
  that&rsquo;s 10,000 square mm. An appropriate domain would be \(0 \leq d \leq 14\).</p>
<h4>Anticipated Misconceptions</h4>
<p>Students may have trouble understanding how to account for time in the first question. They may benefit from writing
  the area after 1 day has passed, after 2 days have passed, etc. A table is a convenient way to gather this
  information.</p>
<h4> Activity Synthesis</h4>
<p>Discuss why the area covered by mold is a function of the number of days that have passed. Attend explicitly to
  language students learned in the prior unit on functions: The area of the mold \(A\) is a function of the number of
  days \(d\) since the mold was spotted, \(A=f(d)\). The function \(f\)<em>&nbsp;</em>expressing the mold relationship
  can be written as \(f(d)=1 \cdot 2^d\), where \(d\)&nbsp;measures days since the mold was spotted and \(f(d)\) gives
  the area covered by the mold in square millimeters.</p>
<p>Discuss whether a discrete graph or a curve is more appropriate and what domain would be suitable in this context.
  Ask questions such as:</p>
<ul>
  <li>&ldquo;Can the independent variable be something like 1.5, a number that is not a whole number? Is there an area
    that is associated with 1.5 days?&rdquo; (Yes, some areas of the bread are covered by mold at any point in time. The
    mold doesn&rsquo;t disappear after being spotted and then reappear at exactly 1 full day, 2 full days, etc.) </li>
  <li>&ldquo;What would be the meaning of a point on the graph where the value of \(d\) is, for instance, between 2 and
    3?&rdquo; (It would mean the area covered by mold at some point longer than 2 days but less than 3 days after mold
    was spotted.) </li>
  <li>&ldquo;What domain would be appropriate for this function? Can the mold grow indefinitely?&rdquo; (Since the area
    of the bread (the range) is limited, the exponential growth cannot continue indefinitely. By the end of one week,
    more than 1 square cm will be covered and, by the end of the second week, the values of the function will be close
    to or will exceed the total area of the bread.) </li>
</ul>
<p>Students using paper and pencil may decide that it makes sense to connect the points on the graph, but they will not
  yet know how to do so. Consider stating that they are connected (in a very specific way), and their properties will be
  studied later.</p>
<p>Students using the digital version (or graphing technology along with the paper and pencil version) will see the
  continuous graph. If desired, you may want to demonstrate how using function notation to write the equation like
  \(A(d)=1 \cdot 2^d\) can be put to use. Try typing \(A(2)\) or \(A(-1)\).</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 2 Collect and Display: Conversing</p>
    </div>
    <div class="os-raise-extrasupport-body">
        <p>During the discussion, listen for and collect the language students use to describe the situation as a function.
          Call students&rsquo; attention to language such as &ldquo;independent or dependent variable&rdquo; or &ldquo;input
          or output value.&rdquo; Write the students&lsquo; words and phrases on a visual display and update it throughout the
          remainder of the lesson. Remind students to borrow language from the display as needed. This will help students use
          mathematical language for describing an exponential function and determining which variable is a function of the
          other.</p>
          <p class="os-raise-text-italicize">Design Principle(s): Maximize meta-awareness</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>5.8.2: 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>Which of the following equations represents the values in the table?</p>
        <table class="os-raise-horizontaltable">
          <thead></thead>
          <tbody>
            <tr>
              <th scope="row">\(x\)</th>
              <td>
                0
              </td>
              <td>
                1
              </td>
              <td>
               2
              </td>
              <td>
                3
              </td>
            </tr>
            <tr>
                <th scope="row">\(y\)</th>
              <td>
               1
              </td>
              <td>
                3
              </td>
              <td>
                9
              </td>
              <td>
                27
              </td>
            </tr>
          </tbody>
        </table>
        <br>
        <table class="os-raise-textheavytable">
          <thead>
            <tr>
              <th scope="col">Answers</th>
              <th scope="col">Feedback</th>
            </tr>
          </thead>
          <tbody>
            <tr>
      <td>
        \(y=3x\)
      </td>
      <td>
       Incorrect. Let&rsquo;s try again a different way: The values in the table are growing at an exponential rate,
          not a linear rate. The answer is \(y=1 \cdot 3^x\).
      </td>
    </tr>
    <tr>
      <td>
        \(y=3 \cdot 1^x\)
      </td>
      <td>
        Incorrect. Let&rsquo;s try again a different way: This equation only works when \(x= 1\) and \(y=3\). A
          different exponential pattern is needed. Remember the base is the growth factor, 3. The answer is \(y=1 \cdot
          3^x\).
      </td>
    </tr>
    <tr>
      <td>
      \(y=1 \cdot 3^x\)
      </td>
      <td>
       That&rsquo;s correct! Check yourself: When the exponent, \(x\), is 3, the equation is equivalent to
          \(3^3=27\). All values in the table follow this exponential pattern.
      </td>
    </tr>
    <tr>
      <td>
        \(y=3x+1\)
      </td>
      <td>
        Incorrect. Let&rsquo;s try again a different way: This equation only works when \(x = 1\) and \(y=3\). The
          values in the table are growing at an exponential rate, not a linear rate. The answer is \(y=1 \cdot 3^x\).
      
      </td>
    </tr>
  </tbody>
</table>
<br>
<h3>5.8.2: 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> Different Exponential Representations</h4>
<p>A table of values is shown below for a function \(g(x)=(\frac{1}{2})^x\), which represents exponential decay.</p>
<table class="os-raise-horizontaltable">
  <thead></thead>
  <tbody>
    <tr>
      <th scope="row">\(x\)</th>
      <td>
    -3
      </td>
      <td>
      -2
      </td>
      <td>
       -1
      </td>
      <td>
        0
      </td>
      <td>
       1>
      </td>
      <td>
        2
      </td>
      <td>
       3
      </td>
    </tr>
    <tr>
        <th scope="row">\(g(x) = (\frac{1}{2})^x\)</th>
      <td>
       8
      </td>
      <td>
      4
      </td>
      <td>
       2
      </td>
      <td>
     1
      </td>
      <td>
       \(\frac12\)
      </td>
      <td>
      \(\frac14\)
      </td>
      <td>
   \(\frac18\)
      </td>
    </tr>
  </tbody>
</table>
<br>
<p>When these points are graphed, they show the decay because the function graphed decreases from left to right:</p>
<p><img src="https://k12.openstax.org/contents/raise/resources/640dc842516ed96046308a6b80d6a2df0ebdf749"
    width="5rm *.mbz00"></p>
<h4> Try It: Different Exponential Representations</h4>
<p>Make a table for the function \(f(x)=4^x\). Then graph the function.</p>
<p>Write down your answers. Then select the <strong>solution</strong> button to compare your work.</p>
<h5>Solution</h5>
<p>Here is how to make a table and then graph the exponential function:</p>
<p>First, substitute values in for \(x\) to find the values of \(f(x)\).</p>
<table class="os-raise-horizontaltable">
  <thead></thead>
  <tbody>
    <tr>
      <th scope="row">Time (years)</th>
      <td>
        0
      </td>
      <td>
       1
      </td>
      <td>
        2
      </td>
      <td>
       3
      </td>
      <td>
     4
      </td>
    </tr>
    <tr>
        <th scope="row"> \(f(x)=4^x\)</th>
      <td>
        \(4^0=1\)
      </td>
      <td>
        \(4^1=4\)
      </td>
      <td>
        \(4^2=16\)
      </td>
      <td>
        \(4^3=64\)
      </td>
      <td>
        \(4^4=256\)
      </td>
    </tr>
  </tbody>
</table>
<br>
<p><img src="https://k12.openstax.org/contents/raise/resources/c4e972074be9c587e82641a2054c2fead8f5a718" width="300">
</p>