842820ad-0369-46f6-8c41-e9bc983abb6d.html

<h4>Activity (15 minutes)</h4>
<p>In an earlier activity, students discerned the relationship between two quantities by analyzing and looking for
  patterns in tables of values. They then described each relationship in words and identified a corresponding equation.
  In that activity, all relationships were linear.</p>
<p>This activity offers new opportunities to identify and represent relationships between pairs of quantities. In each
  of the first two situations, the relationship is an inverse variation. (Students do not need to know the term or the
  concept to be able to reason about the relationships.) The last situation involves a proportional relationship.
  Because the given information involves three quantities (volume in gallons, in cups, and in fluid ounces), students
  need to reason carefully about how two of them (volume in gallons and in fluid ounces) are related.</p>
<p>Monitor for different ways students use to figure out and articulate how the quantities are related. For instance,
  students may:</p>
<ul>
  <li>Use broad and qualitative descriptions (&ldquo;As the base length increases, the height decreases&rdquo;).</li>
  <li>Give specific and quantitative descriptions (&ldquo;The product of the base length and the height is always
    48&rdquo;).</li>
  <li>Use diagrams, tables, graphs, or equations to make sense of the relationships and to illustrate them.</li>
</ul>
<p>Identify students using varying strategies and ask them to share during discussion later.</p>
<h4>Launch</h4>
<p>Tell students that they will now describe the relationship between two quantities in some new situations.</p>
<p>If time is limited, ask students to focus on the first two questions.</p>
<p>With more time, there are opportunities to make this problem set more interesting. Ideas to increase student
  engagement are listed below:</p>
<ol class="os-raise-noindent">
  <li>For students who finish the first activity early, provide graph paper to start drawing the parallelograms
    indicated in the length/height table. These copies can be handed out to different groups as needed.</li>
  <li>Have students actually guess the number of beans (or other objects) in a jar. While other students go on to the
    next activity, let someone who already understands the topic pretty well count the results.</li>
  <li>If possible, bringing measurement containers/liquid (and paper towels!) can add engagement for the jug problem.
  </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">Action and Expression: Develop Expression and Communication</p>
  </div>
  <div class="os-raise-extrasupport-body">
    <p>
      Provide options for communicating understanding.
      Invite students to describe the relationships between quantities in different ways, for example, using verbal (written
      or oral) descriptions, tables, diagrams, or other representations. </p>
    <p class="os-raise-text-italicize">Supports accessibility for: Language; Organization</p>
    </div>
  </div>
<br>

<h4>Student Activity</h4>
<p>1. The table represents the relationship between the base length and the height of some parallelograms. Both
  measurements are in inches.</p>

<table class="os-raise-skinnytable">
  <thead>
    <tr>
      <th scope="col">
        base length<br>(inches)
      </th>
      <th scope="col">
        height<br>(inches)
      </th>
    </tr>
  </thead>
  <tbody>
    <tr>
      <td>
        1
      </td>
      <td>
        48
      </td>
    </tr>
    <tr>
      <td>
        2
      </td>
      <td>
        24
      </td>
    </tr>
    <tr>
      <td>
        3
      </td>
      <td>
        16
      </td>
    </tr>
    <tr>
      <td>
        4
      </td>
      <td>
        12
      </td>
    </tr>
    <tr>
      <td>
        6
      </td>
      <td>
        8
      </td>
    </tr>
  </tbody>
</table>
<br>
<p>What is the relationship between the base length and the height of these parallelograms?</p>
<p><strong>Answer:</strong></p>
<ul>
  <li>As the base length increases by 1, the height decreases, but not by a steady amount.</li>
  <li>Multiplying the base length and the height gives 48. The area of each parallelogram is 48 square inches.</li>
  <li>\( b*h=48 \) or&nbsp;\( \frac{48}{b}=h \) (or equivalent).</li>
  <li>The height varies inversely as \( b \).</li>
</ul>
<p>2. Visitors to a carnival are invited to guess the number of beans in a jar. The person who guesses the correct
  number wins $300. If multiple people guess correctly, the prize will be divided evenly among them.</p>
<p>What is the relationship between the number of people who guess correctly and the amount of money each person will
  receive?</p>
<p><strong>Answer:</strong></p>
<table class="os-raise-midsizetable">
  <thead>
    <tr>
      <th scope="col">
        Number of winners
      </th>
      <th scope="col">
        Amount in dollars<br>each winner receives
      </th>
    </tr>
  </thead>
  <tbody>
    <tr>
      <td>
        1
      </td>
      <td>
        300
      </td>
    </tr>
    <tr>
      <td>
        2
      </td>
      <td>
        150
      </td>
    </tr>
    <tr>
      <td>
        3
      </td>
      <td>
        100
      </td>
    </tr>
    <tr>
      <td>
        5
      </td>
      <td>
        60
      </td>
    </tr>
    <tr>
      <td>
        15</p>
      </td>
      <td>
        20</p>
      </td>
    </tr>
  </tbody>
</table>
<br>
<ul>
  <li>The amount in dollars received by each winner is 300 divided by the number of winners.</li>
  <li>\( a=\frac{300}{n} \) or \( an=300 \) (or equivalent), where \(a\) is the dollar amount a winner receives and
    \(n\) is the number of winners.</li>
  <li>The amount in dollars,&nbsp;\( a \), varies inversely as&nbsp;\( n \).</li>
</ul>
<p>3. A \( \frac{1}{2} \)-gallon jug of milk can fill 8 cups, while 32 fluid ounces of milk can fill 4 cups.</p>
<p>What is the relationship between the number of gallons and ounces? If you get stuck, try creating a table.</p>
<p><strong>Answer:</strong></p>
<ul>
  <li>If 4 cups contain 32 fluid ounces, then 8 cups must contain 64 fluid ounces. Because there are 16 cups in 1 gallon
    and 16 is twice as much as 8, there must be 128 fluid ounces in a gallon.</li>
</ul>
<table class="os-raise-midsizetable">
  <thead>
    <tr>
      <th scope="col">
        gallons
      </th>
      <th scope="col">
        cups
      </th>
      <th scope="col">
        fluid ounces
      </th>
    </tr>
  </thead>
  <tbody>
    <tr>
      <td>
        \( \frac{1}{2} \)
      </td>
      <td>
        8
      </td>
      <td>
        64
      </td>
    </tr>
    <tr>
      <td>
        1
      </td>
      <td>
        16
      </td>
      <td>
        128
      </td>
    </tr>
  </tbody>
</table><br>
<ul>
  <li>Multiplying the number of gallons by 128 gives the number of fluid ounces.</li>
  <li>\( f=8*8*2*g \) or \( f=128g \) (or equivalent), where \( f \) is the number of fluid ounces and \( g \) is the
    number of gallons.</li>
  <li>The number of fluid ounces,&nbsp;\( f \), varies directly as the number of gallons,&nbsp;\( g \).</li>
</ul>
<h4>Anticipated Misconceptions</h4>
<p>If students assume that the relationships must be expressed as equations and they get stuck, clarify that verbal
  descriptions, tables, or other representations are just as welcome.</p>
<p>When answering the first question, students may look only at the relationships between the values in the bottom few
  rows of the table and say that the \(y\)-values are decreasing by 4 each time, not noticing that this is not always
  the case. Encourage them to look at the upper rows in the table and to also look at the relationships between the
  values in the columns.</p>
<p>Some students may have trouble getting started on the question about the volume of milk or setting up a table. Ask
  students which units are given in the problem, or suggest the headings &ldquo;gallons,&rdquo; &ldquo;cups,&rdquo; and
  &ldquo;fluid ounces.&rdquo; Then, ask them to use the given information to complete a row in the table. This might
  involve trying a different unit to start with. (For example, if they start with \( \frac{1}{2} \) gallon and struggle
  to find the equivalent amount in cups and fluid ounces, try starting with &ldquo;4 cups&rdquo; or &ldquo;8
  cups.&rdquo;) A more direct hint is to suggest finding the number of fluid ounces in 8 cups.</p>
<h3>Activity Synthesis</h3>
<p>Select previously identified students to share their responses and thinking. Sequence the presentation in the order
  of precision, starting with the broader descriptions or illustrations and ending with equations. (Remind students who
  use equations to specify what the variables represent.) If students write equations in different forms to describe the
  same relationship, record and display the equivalent equations for all to see.</p>
<p>If no students considered using tables to make sense of the pairs of quantities, ask how these would help and show an
  example. A table, for instance, can be particularly helpful for reasoning about the last two relationships.</p>
<p>Display the equations that can represent the three situations. Highlight that writing equations is an efficient way
  to capture the constraints in a situation.</p>
<p>\( b*h = 48&nbsp;\) or \( h = \frac{48}{b} \)</p>
<p>\(&nbsp;a*n=300&nbsp;\) or \(&nbsp;a=\frac{300}{n} \)</p>
<p>\(&nbsp;f=128g \)<br></p>
<p>To help students connect the equations to prior work, ask students which quantities vary and which remain constant in
  each equation. Point out that these equations are also equations in two variables, but unlike the equations we saw in
  the previous activity, not all of these represent linear relationships.</p>
<h3>1.3.3: 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></strong></p>
<p class="os-raise-text-bold">QUESTION:</p>
<table class="os-raise-horizontaltable">
  <thead></thead>
  <tbody>
    <tr>
      <th scope="row">
        Number of guests, \( g \)
      </th>
      <td>
        <p>1
      </td>
      <td>
        <p>2
      </td>
      <td>
        <p>3
      </td>
      <td>
        <p>12
      </td>
    </tr>
    <tr>
      <th scope="row">
        Number of cupcakes per guest, \( n \)
      </th>
      <td>
        <p>24
      </td>
      <td>
        <p>12
      </td>
      <td>
        <p>8
      </td>
      <td>
        <p>2
      </td>
    </tr>
  </tbody>
</table>
<br>
<p>What is one way to write the relationship between the number of guests and the number of cupcakes per guest from the
  table above?</p>
<table class="os-raise-textheavytable">
  <thead>
    <tr>
      <th scope="col">Answers</th>
      <th scope="col">Feedback</th>
    </tr>
  </thead>
  <tbody>
    <tr>
      <td>\( n= 24g \)</td>
      <td>Incorrect. Let&rsquo;s try again a different way: Each guest does not get 24 cupcakes. The total number will
        always be 24 cupcakes. The correct answer is \( n=\frac{24}{g} \).</td>
    </tr>
    <tr>
      <td>\( n=\frac{24}{g}\)</td>
      <td>That&rsquo;s correct! Check yourself: The 24 cupcakes will be divided by the number of guests, so this is the
        correct answer. </td>
    </tr>
    <tr>
      <td>\( n=\frac{g}{24} \)</td>
      <td>Incorrect. Let&rsquo;s try again a different way: The number of cupcakes for each guest is found by dividing
        the total cupcakes by the number of guests. The correct answer is \( n=\frac{24}{g} \).</td>
    </tr>
    <tr>
      <td>\( n = 24 - g \)</td>
      <td>Incorrect. Let&rsquo;s try again a different way: This does not work for the values in the table. The number
        of cupcakes for each guest is found by dividing the total cupcakes by the number of guests. The correct answer
        is \( n=\frac{24}{g} \).</td>
    </tr>
  </tbody>
</table>
<br>
<h3>1.3.3: 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>Finding Relationships between Quantities</h4>
<p>There are times when the relationship between quantities may not be obvious. In some cases, the relationship between
  quantities might take a bit of work to figure out, by doing calculations several times or by looking for a pattern.
</p>
<p>Let's examine two examples.</p>
<p class="os-raise-text-bold">Example 1</p>
<p>A plane departed from New Orleans and is heading to San Diego. The table shows its distance from New Orleans, \( x
  \), and its distance from San Diego, \( y \), at some points along the way.</p>
<table class="os-raise-midsizetable">
  <thead>
    <tr>
      <th scope="col">miles from New Orleans</th>
      <th scope="col">miles from San Diego</th>
    </tr>
  </thead>
  <tbody>
    <tr>
      <td>100</td>
      <td>1,500</td>
    </tr>
    <tr>
      <td>300</td>
      <td>1,300</td>
    </tr>
    <tr>
      <td>500</td>
      <td>1,100</td>
    </tr>
    <tr>
      <td><br></td>
      <td>1,020</td>
    </tr>
    <tr>
      <td>900<br></td>
      <td>700</td>
    </tr>
    <tr>
      <td>1,450</td>
      <td></td>
    </tr>
    <tr>
      <td>\( x \)</td>
      <td>\( y \)</td>
    </tr>
  </tbody>
</table>
<br>
<ol class="os-raise-noindent">
  <li>What is the relationship between the two distances?
  </li>
</ol>
<p><strong>Answer:</strong> Your answer may vary, but here are some samples.
</p>
<p>The distances represent one path—the distance from New Orleans to San Diego. The two distances sum to 1600 miles. \(x
  + y = 160\)
</p>
<ol class="os-raise-noindent" start="2">
  <li>Do you see any patterns in how each quantity is changing?
  </li>
</ol>
<p><strong>Answer:</strong> Your answer may vary, but here are some samples.
</p>
<p>As the \(x\)-values increase, the \(y\)-values decrease. As the \(x\)-values increase by 200 miles, the \(y\)-values
  decrease by 200 miles. The quantities, \(x\) and \(y\), sum to 1600 even after changing.
</p>
<ol class="os-raise-noindent" start="3">
  <li>What is the value of \(x\) when \(y\) = 1020?
  </li>
</ol>
<p><strong>Answer:</strong> 580 miles
</p>
<ol class="os-raise-noindent" start="4">
  <li>What is the value of \(y\) when \(x\) = 1450?
  </li>
</ol>
<p><strong>Answer:</strong> 150 miles
</p>
<p class="os-raise-text-bold">Example 2</p>
<p>A company decides to donate $50,000 to charity. It will select up to 20 charitable organizations, as nominated by its employees. Each selected organization will receive an equal donation amount.
</p>
<p>What is the relationship between the number of students, \(s\), and the dollar amount each student will receive,
  \(d\)?<br>
  To begin, let's examine some specific values to help uncover the pattern.
</p>
<ol class="os-raise-noindent" start="5">
  <li>If 5 organizations are selected, how much will each charity receive?</li>
</ol>
<p><strong>Answer:</strong> $10,000</p>
<ol class="os-raise-noindent" start="6">
  <li>If 10 organizations are selected, how much will each charity receive?
  </li>
</ol>
<p><strong>Answer:</strong> $5000
</p>
<ol class="os-raise-noindent" start="7">
  <li>If 20 organizations are selected, how much will each charity receive?
  </li>
</ol>
<p><strong>Answer:</strong> $2500</p>
<h4>Try It: Finding Relationships between Quantities</h4>
<p>A local business is going to hand out $20,000 in scholarships to students at local high schools.</p>
<p>What is the relationship between the number of students, \( s \), and the dollar amount each student will receive, \(
  d \)?</p>
<ol class="os-raise-noindent">
  <li>If 2 students are selected, what is the amount of the scholarship they will receive?
  </li>
</ol>
<p><strong>Answer:</strong> $10,000
</p>
<ol class="os-raise-noindent" start="2">
  <li>If there are 5 students selected, what is the amount of the scholarship they will receive?
  </li>
</ol>
<p><strong>Answer:</strong> $4000
</p>
<ol class="os-raise-noindent" start="3">
  <li>If there are 10 students selected, what is the amount of the scholarship they will receive?
  </li>
</ol>
<p><strong>Answer:</strong> $2000
</p>
<ol class="os-raise-noindent" start="4">
  <li>If there are 20 students selected, what is the amount of the scholarship they will receive?
  </li>
</ol>
<p><strong>Answer:</strong> $1000
</p>
<ol class="os-raise-noindent" start="5">
  <li>What equation can be used to model the relationship between the number of students, \(s\), receiving scholarships
    and the dollar amount, \(d\), they receive?</li>
</ol>
<p><strong>Answer:</strong> Your answer may vary, but here is a sample.<br>
  The amount each student receives is $20,000 divided by the number of students or, \(d=\frac{20000}{s}\). The dollar amount each student receives, \(d\), varies inversely as the number of students, \(s\).
</p>