23bf685f-48d0-458f-8889-070f5270088a.html

<h4>Activity (15 minutes)</h4>
<p>The goal of this activity is to allow students to explore two growth patterns and to notice that a doubling pattern
  eventually grows very, very large, even with a small starting value. Students may perform a repeated calculation, or
  they may generalize the process by writing expressions or equations.</p>
<p>Making spreadsheet technology available gives students an opportunity to choose appropriate tools strategically.</p>
<h4>Launch</h4>
<p>Provide access to calculators or spreadsheets. Present the situation (either by asking students to read it quietly or
  by performing a dramatic reading). Before students do any calculating or any other work, poll the class about which
  option they think is better. Display the results of the poll (the number of students who think Purse A is better and
  the number who think Purse B is better) for all to see.</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 6 Three Reads: Reading, Listening, Conversing</p>
  </div>
  <div class="os-raise-extrasupport-body">
      <p> Use this routine to support reading comprehension of this word problem. Use the first read to orient students to
    the situation. Ask students to describe what the situation is about without using numbers (as a reward, students
    make a choice between Purse A and Purse B). Use the second read to identify quantities and relationships. Ask
    students what can be counted or measured without focusing on the values. Listen for, and amplify, the important
    quantities that vary in relation to each other in this situation: the amount of money, in dollars, and the amount of
    time, in days. After the third read, invite students to brainstorm possible strategies to answer the questions. This
    helps students connect the language in the word problem and the reasoning needed to solve the problem.</p>
    <p class="os-raise-text-italicize">Design Principle(s): Support sense-making</p>
    <p class="os-raise-extrasupport-title">Learn more about this routine</p>
    <p>
          <a href="https://www.youtube.com/watch?v=Q2PGJThrG2Q;&rel=0" target="_blank">View the instructional video</a>
          and
          <a href="https://k12.openstax.org/contents/raise/resources/bf750b41e6483d334d575e1d950851bfa07cfd26" target="_blank">follow along with the materials</a>
          to assist you with learning this routine.
        </p>
        <p class="os-raise-extrasupport-title">Provide support for students</p>
        <p>
          <a href="https://k12.openstax.org/contents/raise/resources/06eafa198e5345452a0adbc81731e7383785aa42" target="_blank">Distribute graphic organizers</a>
          to the students to assist them with participating in this routine.
        </p>
</div>
<br>
</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>
    <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 using diagrams. Encourage students to sketch
      diagrams that show how the amount of money grows over the first few days. Students may benefit from this visual as
      they transition to the use of a table or other representation to track growth. </p>
      <p class="os-raise-text-italicize">Supports accessibility for: Conceptual processing; Visual-spatial processing</p>
    </div>
  </div>
<br>
<h4>Student Activity</h4>
<p>You are walking along a beach, and your toe hits something hard. You reach down, grab onto a handle, and pull out a
  lamp! It is sandy. You start to brush it off with your towel. Poof! A genie appears.</p>
<p>He tells you, &ldquo;Thank you for freeing me from that bottle! I was getting claustrophobic. You can choose one of
  these purses as a reward.&rdquo;</p>
<ul>
  <li> Purse A, which contains $1,000 today. If you leave it alone, it will contain $1,200 tomorrow (by magic). The next
    day, it will be $1,400. This pattern of an additional $200 per day will continue. </li>
  <li> Purse B, which contains 1 penny today. Leave that penny in there because tomorrow it will (magically) turn into 2
    pennies. The next day, there will be 4 pennies. The amount in the purse will continue to double each day. </li>
</ul>
<ol class="os-raise-noindent">
  <li> How much money will be in Purse A after a 7-day week?<br><br><strong>Answer:</strong> After 7 days, Purse A will
    contain $2,400 because it grows by $200 each day for 7 days.</li><br>
  <li> How much money will be in Purse B after a 7-day week?<br><br><strong>Answer:</strong> Purse B will contain $1.28.
    It doubles each day for 7 days, so it will have \(1 \cdot 2^7\) or 128 cents (the 1 here represents the 1 cent for
    when the genie appeared).</li><br>
  <li> How much money will be in Purse A after two weeks?<br><br> <strong>Answer:</strong> After 2 weeks, Purse A will
    contain an additional $1,400 for a total of $3,800.</li><br>
  <li> How much money will be in Purse B after two weeks?<br><br> <strong>Answer:</strong> After 2 weeks, Purse B will
    contain \(1 \cdot 2^{14}\) cents. This is 16,384 cents or $163.84. </li><br>
  <li> The genie later adds that he will let the money in each purse grow for three weeks.</li><br>
  <ol class="os-raise-noindent" type="a">
    <li> How much money will be in Purse A in three weeks? <br><br><strong>Answer:</strong> Purse A will have an
      additional $1,400 during the third week for a total of $5,200.</li><br>
    <li> How much money will be in Purse B in three weeks?<br><br> <strong>Answer:</strong> Purse B will have 128 times
      as much money (compared to the end of two weeks) for a total of $20,971.52.</li><br>
  </ol>
  <li> Which purse contains more money after 30 days?<br><br> <strong>Answer:</strong> Purse B will have a lot more
    money. Purse A will have \(1,000+30 \cdot 200\) or $7,000, which is much less than the doubling purse on day 21.
  </li><br>
</ol>
<h4>Anticipated Misconceptions</h4>
<p>Some students may confuse the units for the two purses. Remind them that Purse B contains pennies (and thus cents
  rather than dollars).</p>
<p>Vocabulary such as <em>claustrophobic</em> and <em>double</em> may need to be clarified so students are clear on what
  the question is asking.</p>
<p>Due to prior experience with genie stories, students may be suspicious of the offers. Ask them to show their
  mathematical reasoning, rather than basing their choice of purses on their suspicions or other considerations such as
  how to carry a purse containing over 2 million pennies.</p>
<h4> Activity Synthesis</h4>
<p>Focus the discussion on how students found the amounts of money in the two purses after many days. Invite students
  who performed recursive calculations and those who wrote expressions or equations to share. Record and display
  students&rsquo; reasoning for all to see. Seeing the repeated reasoning will support students&rsquo; generalization
  work later. For example, the amount in Purse A will be \(1,000+200\) for day 1, then \(1,000+200+200\) for day 2,
  \(1,000+200+200+200\) for day 3, and so on. Connect this to writing expressions like \(1,000+200 \cdot 3\) and
  \(1,000+200x\) where \(x\)&nbsp;represents the number of days since the genie appeared.</p>
<p>For Purse B, highlight the fact that after two days it contains \((1 \cdot 2) \cdot 2\) cents, which is \(2^2\)
  cents. After three days, it contains \((1 \cdot 2) \cdot 2 \cdot 2\) cents, which is \(2^3\) cents. (Here the day the
  genie appears is treated as day 0.) This is a good opportunity to emphasize the meaning of exponential notation.
  Exponential notation is particularly useful for expressing the amount in Purse B on day 30, which is \(2^{30}\)cents.
</p>
<p>If no students use a spreadsheet, consider demonstrating how it might be helpful in performing calculations like
  these.</p>
<h3>5.3.2: Self Check</h3>
<p class="os-raise-text-bold"><em>Following 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>Olivia has two different job offers for babysitting next Saturday.&nbsp;</p>
<ul>
  <li> Job 1: She will receive $5 for accepting the job, then $10 for every hour she works.&nbsp; </li>
  <li> Job 2: She will receive $1.50 for accepting the job, $3 after the first hour, $6 after the second hour,
    and for each additional hour her pay per hour will double. </li>
</ul>
<p>Olivia can only work 3 hours. How much would she make at each job?</p>
<table class="os-raise-textheavytable">
  <thead>
    <tr>
      <th scope="col">Answers</th>
      <th scope="col">Feedback</th>
    </tr>
  </thead>
  <tbody>
    <tr>
      <td>
        Olivia would make \(\$35\) at Job 1 and \(\$12\) at Job 2.
      </td>
      <td>
        Incorrect. Let&rsquo;s try again a different way: \(\$12\) is the amount she would make after the third hour.
        To find the total, add: \(\$1.50+\$3+\$6+\$12\) or \($22.50\). The answer is Olivia would make \($35\) at Job
        1 and \(\$22.50\) at Job 2.
      </td>
    </tr>
    <tr>
      <td>
        Olivia would make \(\$35\) at Job 1 and \(\$22.50\) at Job 2.
      </td>
      <td>
        That&rsquo;s correct! Check yourself: Job 1 would be \(\$5 +\$10+\$10+\$10\) or \($35\). Job 2 would be
        \(\$1.50+\$3+\$6+\$12\) or \(\$22.50\).
      </td>
    </tr>
    <tr>
      <td>
        Olivia would make \(\$25\) at Job 1 and \(\$10.50\) at Job 2.
      </td>
      <td>
        Incorrect. Let&rsquo;s try again a different way: This would be the amount for 2 hours, not 3. For Job 1, add
        another \(\$10\). For Job 2, add \(\$12\). The answer is Olivia would make \(\$35\) at Job 1 and \(\$22.50\)
        at Job 2.
      </td>
    </tr>
    <tr>
      <td>
        Olivia would make \(\$30\) at Job 1 and \(\$21\) at Job 2.
      </td>
      <td>
        Incorrect. Let&rsquo;s try again a different way: Be sure to add the amount she makes for accepting the job,
        \(\$5\) for Job 1 and \(\$1.50\) for Job 2. The answer is Olivia would make \(\$35\) at Job 1 and \(\$22.50\)
        at Job 2.
      </td>
    </tr>
  </tbody>
</table>
<br>
<h3>5.3.2: 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>Exploring Linear and Exponential Growth</h4>
<p>The tables listed below are modeling four different functions, all showing growth. Which one does not have the same
  growth pattern as the others?</p>
<div class="os-raise-d-flex os-raise-justify-content-between">
  <table class="os-raise-skinnytable">
    <caption>Table A</caption>
    <thead>
      <tr>
        <th scope="col">\(x\)</th>
        <th scope="col">\(y\)</th>
      </tr>
    </thead>
    <tbody>
      <tr>
        <td>
          1
        </td>
        <td>
          8
        </td>
      </tr>
      <tr>
        <td>
          2
        </td>
        <td>
          16
        </td>
      </tr>
      <tr>
        <td>
          3
        </td>
        <td>
          24
        </td>
      </tr>
      <tr>
        <td>
          4
        </td>
        <td>
          32
        </td>
      </tr>
      <tr>
        <td>
          8
        </td>
        <td>
          64
        </td>
      </tr>
    </tbody>
  </table>
  <table class="os-raise-skinnytable">
    <caption>Table B</caption>
    <thead>
      <tr>
        <th scope="col">\(x\)</th>
        <th scope="col">\(y\)</th>
      </tr>
    </thead>
    <tbody>
      <tr>
        <td>
          0
        </td>
        <td>
          0
        </td>
      </tr>
      <tr>
        <td>
          2
        </td>
        <td>
          16
        </td>
      </tr>
      <tr>
        <td>
          4
        </td>
        <td>
          32
        </td>
      </tr>
      <tr>
        <td>
          6
        </td>
        <td>
          48
        </td>
      </tr>
      <tr>
        <td>
          8
        </td>
        <td>
          64
        </td>
      </tr>
    </tbody>
  </table>
</div>
<div class="os-raise-d-flex os-raise-justify-content-between">
  <table class="os-raise-skinnytable">
    <caption>Table C</caption>
    <thead>
      <tr>
        <th scope="col">\(x\)</th>
        <th scope="col">\(y\)</th>
      </tr>
    </thead>
    <tbody>
      <tr>
        <td>
          0
        </td>
        <td>
          1
        </td>
      </tr>
      <tr>
        <td>
          1
        </td>
        <td>
          4
        </td>
      </tr>
      <tr>
        <td>
          2
        </td>
        <td>
          16
        </td>
      </tr>
      <tr>
        <td>

          3
        </td>
        <td>
          64
        </td>
      </tr>
      <tr>
        <td>
          4
        </td>
        <td>
          256
        </td>
      </tr>
    </tbody>
  </table>
  <table class="os-raise-skinnytable">
    <caption>Table D</caption>
    <thead>
      <tr>
        <th scope="col">\(x\)</th>
        <th scope="col">\(y\)</th>
      </tr>
    </thead>
    <tbody>
      <tr>
        <td>
          0
        </td>
        <td>
          4
        </td>
      </tr>
      <tr>
        <td>
          1
        </td>
        <td>
          8
        </td>
      </tr>
      <tr>
        <td>
          2
        </td>
        <td>
          12
        </td>
      </tr>
      <tr>
        <td>
          3
        </td>
        <td>
          16
        </td>
      </tr>
      <tr>
        <td>
          4
        </td>
        <td>
          20
        </td>
      </tr>
    </tbody>
  </table>
</div>
<br>
<p>When determining growth, there are a few ways to check. When given a table, you could graph each, or you could check
  the rate of change. Linear functions have a constant rate of change, where consecutive terms have a common difference.
  The common difference is a value that is added to or subtracted from consecutive terms to find the next term.
  Exponential functions have a constant ratio or common multiplier. The constant ratio is a value that is multiplied or
  divided from consecutive terms to find the next term.</p>
<ul>
  <li> Table A is linear and has a slope of 8 since the change in \(y\) is 8 and the change in \(x\) is 1.&nbsp;&nbsp;
  </li>
  <li> Table B is also linear because the change in \(y\) is 16 and the change in \(x\) is 2 for a slope of
    8.&nbsp;&nbsp; </li>
  <li> Table C is an example of an exponential function. There is a common ratio instead of a common difference. Notice
    that each \(y\)-term is 4 times the previous term. Therefore, the common ratio is 4. </li>
  <li> Table D is also linear with a common difference of 4. </li>
</ul>
<p>Table C is the only exponential function. Therefore, it has a different growth pattern since Tables A, B, and D are
  linear.</p>
<h4>Try It: Exploring Linear and Exponential Growth</h4>
<p>The table shows the height, in centimeters, of the water in a swimming pool at different times since the pool started
  to be filled.</p>

  <table class="os-raise-skinnytable">
    <thead>
      <tr>
        <th scope="col">minutes</th>
        <th scope="col">height</th>
      </tr>
    </thead>
    <tbody>
      <tr>
      <td>
        0
      </td>
      <td>
     150
      </td>
    </tr>
    <tr>
      <td>
        1
      </td>
      <td>
       150.5
      </td>
    </tr>
    <tr>
      <td>
       2
      </td>
      <td>
       151
      </td>
    </tr>
    <tr>
      <td>
       3
      </td>
      <td>
        151.5
      </td>
    </tr>
  </tbody>
</table>
<br>
<ol class="os-raise-noindent">
  <li> Does the height of the water have a common difference or a common ratio? </li>
  <p><strong>Answer:</strong> Common Difference</p>
  <li> Explain how you know. </li>
  <p><strong>Answer:</strong> When you look at the \(y\)-values in the table, you notice that they are all 0.5 more than
    the previous term. That means there is a common difference of 0.5. </p>
  <li> What type of function is represented by the table?</li>
  <p><strong>Answer:</strong> The function is linear.</p>
  <li> Explain how you know. </li>
  <p><strong>Answer:</strong> The function is linear because the common difference means the rate of change, or slope, is
      constant.
  </p>
</ol>