ee14c5e7-3149-4182-99df-4ecd7476c832.html

<h3> Activity (10 minutes)</h3>
<br>
<h4> Launch</h4>
<p>Students study two stories characterized by different growth patterns and match expressions and tables with the two
  situations. The numbers in the two scenarios are identical, so students need to focus on the mathematical
  relationships and the operations in the expressions to successfully complete the activity. If students are evaluating
  the expressions, encourage them to reason about the operations without calculating anything.</p>
<h4>Student Activity</h4>
<p>Here are verbal descriptions of two situations, followed by tables and expressions that could help answer one of the
  questions in the situations.</p>
<ul>
  <li> Situation 1: A person has 80 followers on social media. The number of followers triples each year. How many
    followers will she have after 4 years? </li>
  <li> Situation 2: A tank contains 80 gallons of water and is getting filled at a rate of 3 gallons per minute. How
    many gallons of water will be in the tank after 4 minutes? </li>
</ul>
<p>Use the descriptions of Situation 1 and Situation 2 to help you answer questions 1&ndash;6. Be prepared to explain
  how the table or expression answers the question.</p>
<ol class="os-raise-noindent">
  <li> Which situation best describes \(80 \cdot 3 \cdot 3 \cdot 3 \cdot 3\)? <br><br>
    <strong>Answer:</strong> Situation 1
  </li><br>
  <li> Which situation best describes the following table? <br><br>
    <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>
          <td>
            4
          </td>
        </tr>
        <tr>
          <th scope="row">\(y\)</th>
          <td>
            80
          </td>
          <td>
            240
          </td>
          <td>
            720
          </td>
          <td>
            2,160
          </td>
          <td>
            6,480
          </td>
        </tr>
      </tbody>
    </table>
    <br>
    <strong>Answer:</strong> Situation 1. The table shows the growth of the followers multiplied by 3 each year.
  </li>
</ol>
<ol class="os-raise-noindent" start="3">
  <li> Which situation best describes \(80+3+3+3+3\)? <br>
    <br><strong>Answer:</strong> Situation 2
  </li><br>
  <li> Which situation best describes \(80+4 \cdot 3\)? <br><br>
    <strong>Answer:</strong> Situation 2. \(4\cdot 3\) is another way of expressing \(3+3+3+3\).
  </li><br>
  <li> Which situation best describes the following table? <br><br>
    <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>
          <td>
            4
          </td>
        </tr>
        <tr>
          <th scope="row">\(y\)</th>
          <td>
            80
          </td>
          <td>
            83
          </td>
          <td>
            86
          </td>
          <td>
            89
          </td>
          <td>
            92
          </td>
        </tr>
      </tbody>
    </table>
    <br>
    <strong>Answer:</strong> Situation 2. The table shows the gallons of water in the tank increasing by 3 gallons each
    minute.
  </li>
</ol>
<ol class="os-raise-noindent" start="6">
  <li> Which situation best describes \(80\cdot 81\)? <br>
    <br> <strong>Answer:</strong> Situation 1. The 81 comes from \(3 \cdot 3 \cdot 3 \cdot 3\).
  </li>
</ol>
<h4>Video: Matching Verbal Situations to Expressions and Tables</h4>
<p>Watch the following video to learn more about matching verbal representations of a function with tables and
  expressions.</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/184e8762df04ba9f7374004a7694c2581541f978">
      <track default="true" kind="captions" label="On" src="https://k12.openstax.org/contents/raise/resources/9185ecea01ae0e39e327cfd497e05ce0a9a3010d" srclang="en_us">
      https://k12.openstax.org/contents/raise/resources/184e8762df04ba9f7374004a7694c2581541f978
    </video></div>
</div>
<br>
<br>
<h4>Activity Synthesis</h4>
<p>Focus the discussion on how students went about matching the cards and the situations. Record students&rsquo;
  reasoning for all to see. In particular, highlight observations about common differences and common ratios. Ask
  questions such as:</p>
<ul>
  <li>&ldquo;Besides evaluating the expression, how could you tell (a certain expression) matched one of the
    situations?&rdquo;</li>
  <li>&ldquo;How could you tell (a certain table) matched one of the situations?&rdquo;</li>
  <li>&ldquo;How is the number of followers growing every 2 years?&rdquo;</li>
  <li>&ldquo;How is the amount of water in the tank increasing every 2 minutes?&rdquo;</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>
  </div>
  <div class="os-raise-extrasupport-body">
    <p>Use this routine to support whole-class discussion. For each observation that is shared, ask students to restate
      what they heard using precise mathematical language. Consider providing students time to restate what they hear to a
      partner before selecting one or two students to share with the class. Ask the original speaker if their peer was
      accurately able to restate their thinking. Call students&rsquo; attention to any words or phrases that helped
      clarify the original statement. This provides more students with an opportunity to produce language as they
      interpret the reasoning of others.</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>Use color coding and annotations to highlight important connections between the text of each situation and the
      corresponding tables and expressions. Begin by asking students what they noticed to help identify how common
      differences and common factors appear in each representation.</p>
  </div>
</div>
<br>
<h3>5.3.4: 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>Which of the following is <strong>NOT</strong> an example of a way to represent a chain of stores that
  currently has 4 stores but plans to double the number of stores they have each year for the next 5 years?</p>
<table class="os-raise-textheavytable">
  <thead>
    <tr>
      <th scope="col">Answers</th>
      <th scope="col">Feedback</th>
    </tr>
  </thead>
  <tbody>
    <tr>
      <td>
        \(4 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2\)
      </td>
      <td>
        Incorrect. Let&rsquo;s try again a different way: You are looking for an expression that does NOT represent
        the situation. This is an expression that represents having 4 stores and then multiplying by 2, or doubling
        the amount, 5 times to show 5 years. The answer is \(4 + 5 \cdot 2\).
      </td>
    </tr>
    <tr>
      <td>
        \(4 + 5 \cdot 2\)
      </td>
      <td>
        That&rsquo;s correct! Check yourself: They started with 4 stores, and doubling means multiplying by 2 for
        each additional year. You would need to multiply by 2 five times, NOT \(5 \cdot 2\).
      </td>
    </tr>
    <tr>
      <td>
        \(4 \cdot 32\)
      </td>
      <td>
        Incorrect. Let&rsquo;s try again a different way: You are looking for an expression that does NOT represent
        the situation. This is an expression that represents having 4 stores and doubling the amount each year for 5
        years, or multiplying by 2 five times, or \(2 \cdot 2 \cdot 2 \cdot 2 \cdot 2\), which is 32. The answer is
        \(4 + 5 \cdot 2\).
      </td>
    </tr>
    <tr>
      <td>
        <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>
              <td>
                4
              </td>
              <td>
                5
              </td>
            </tr>
            <tr>
              <th scope="row">\(y\)</th>
              <td>
                4
              </td>
              <td>
                8
              </td>
              <td>
                16
              </td>
              <td>
                32
              </td>
              <td>
                64
              </td>
              <td>
                128
              </td>
            </tr>
          </tbody>
        </table>
        <br>
      </td>
      <td>
        Incorrect. Let&rsquo;s try again a different way: You are looking for an expression that does NOT represent
        the situation. This is the table that represents starting with 4 stores and multiplying by 2, or doubling, for
        5 years. The answer is \(4 + 5 \cdot 2\).
      </td>
    </tr>
  </tbody>
</table>
<br>
<h4> 5.3.4: Additional Resources</h4>
<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.&nbsp;</em></p>
<h4>Multiple Representations Verbal Scenario</h4>
<p>Here are verbal descriptions of two situations, followed by tables and expressions that could help answer one of the
  questions in the situations.</p>
<p>A food company currently has 5 convenience stores. It is considering two plans for expanding its chain of stores.</p>
<p>Plan A: Open 20 new stores each year.</p>
<p>Plan B: Double the number of stores each year.</p>
<p>How many stores will there be after 5 years under each plan?</p>
<p>Match each representation (a table or an expression) with one situation. Be prepared to explain how the table or
  expression answers the question.</p>
<ol class="os-raise-noindent">
  <li> \(5+20+20+20+20+20\)</li>
  <li> \(5 \cdot 32\) </li>
  <li> \(5 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2\) </li>
  <li> </li>
</ol>
<table class="os-raise-horizontaltable">
  <thead></thead>
  <tbody>
    <tr>
      <th scope="row">years</th>
      <td>
        0
      </td>
      <td>
        1
      </td>
      <td>
        2
      </td>
      <td>
        3
      </td>
      <td>
        4
      </td>
      <td>
        5
      </td>
    </tr>
    <tr>
      <th scope="row">number of stores</th>
      <td>
        5
      </td>
      <td>
        10
      </td>
      <td>
        20
      </td>
      <td>
        40
      </td>
      <td>
        80
      </td>
      <td>
        160
      </td>
    </tr>
  </tbody>
</table>
<br>
<ol class="os-raise-noindent" start="5">
  <li> \(5+100\) </li>
  <li> </li>
</ol>
<table class="os-raise-horizontaltable">
  <thead></thead>
  <tbody>
    <tr>
      <th scope="row">years</th>
      <td>
        0
      </td>
      <td>
        1
      </td>
      <td>
        2
      </td>
      <td>
        3
      </td>
      <td>
        4
      </td>
      <td>
        5
      </td>
    </tr>
    <tr>
      <th scope="row">number of stores</th>
      <td>
        5
      </td>
      <td>
        25
      </td>
      <td>
        45
      </td>
      <td>
        65
      </td>
      <td>
        85
      </td>
      <td>
        105
      </td>
    </tr>
  </tbody>
</table>
<br>
<p>To determine which options go with which scenario, you first need to analyze each scenario.&nbsp;</p>
<p>Let&rsquo;s look at Plan A.&nbsp;</p>
<ul>
  <li> Plan A is to open 20 new stores each year for a food company that currently has 5 convenience stores. In both
    scenarios, the initial value or starting point is 5 stores, which is why in each table the first value is \((0,5)\).
    If they are opening 20 additional stores each year, they are adding 20 stores to the total from the previous year.
    So in year 1 there would be 25 total stores, in year 2 there would be 45 total stores, and so on. These values are
    represented in option 6, so option 6 would represent Plan A. </li>
  <li> The mathematical expressions contain either multiplication or addition. Plan A is ADDING 20 stores a year, so
    year 5 could be represented as option 1 because it would be the initial number of stores plus 20 per year for 5
    years, or \(5 +20+20+20+20+20\).&nbsp; </li>
  <li> Option 5 is also added, and it represents Plan A since \(5 +20+20+20+20+20\) is the same as \(5+100\). </li>
</ul>
<p>Let&rsquo;s look at Plan B.</p>
<ul>
  <li> Plan B would also start at 5, but it says to double the number of stores each year. Double means to MULTIPLY by
    2, so it would be \(5 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2\), which is option 3.&nbsp; </li>
  <li> Since \(5 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2\) can be simplified to \(5 \cdot 32\), option 2 also represents
    Plan B.</li>
  <li> Option 4 is the other table option, and as mentioned before, each table should have an initial value of 5 stores.
    Since Plan B is to double the number of stores, at the end of year 1 there should be 10 stores. When you check
    option 4, you can see that each term is double the term before. </li>
</ul>
<p>Therefore Plan A is options 1, 5, and 6, and Plan B is options 2, 3, and 4.</p>
<h4> Try It: Multiple Representations Verbal Scenario</h4>
<p>Represent the following scenario as a table and as two different mathematical expressions:</p>
<p>A tank contains 80 gallons of water and is getting filled at a rate of 3 gallons per minute. How many gallons of
  water will be in the tank after 4 minutes?</p>
<p>Write down your answer, then select the <strong>solution</strong> button to compare your work.</p>
<h5>Solution</h5>
<p>Here is how to represent the scenario:</p>
<p>The table would record time in minutes and total water in gallons. The initial value is 80 gallons, so the first
  entry would be \((0,80)\) and each additional entry would be 3 more than the previous entry.</p>
<table class="os-raise-horizontaltable">
  <thead></thead>
  <tbody>
    <tr>
      <th scope="row">Time (in minutes)</th>
      <td>
        0
      </td>
      <td>
        1
      </td>
      <td>
        2
      </td>
      <td>
        3
      </td>
      <td>
        4
      </td>
    </tr>
    <tr>
      <th scope="row">Total of water (in gallons)</th>
      <td>
        80
      </td>
      <td>
        83
      </td>
      <td>
        86
      </td>
      <td>
        89
      </td>
      <td>
        92
      </td>
    </tr>
  </tbody>
</table>
<br>
<p>One mathematical expression would be 80 plus 3 repeated 4 times for the 4 minutes that passed, or \(80+3+3+3+3\).
  This could also be simplified to \(80+12\) or 92 gallons.</p>