47023f4c-88f9-4029-a99c-a98d17772b7b.html

<h3>Warm Up (5 minutes)</h3>
<p>In this warm up, students look for a relationship between two quantities by interpreting a verbal description and
  analyzing pairs of values in a table. They then use the observed relationship to find unknown values of one quantity
  given the other and to think about possible equations that could represent the relationship more generally (TEKS A.1(F)).</p>
<p>The work here reinforces the idea that the relationship between two quantities can be expressed in more than one way
  and that some forms might be more helpful than others, depending on what we want to know. In this context, for
  instance, if we know the area of a parallelogram and want to know its base length, the equation \( b = \frac{A}{3} \)
  is more helpful than \( A=3b \).</p>
<h4>Launch</h4>
<p><a href="https://teacher.desmos.com/activitybuilder/custom/646fb71b698f31b7e49f640a" target="_blank">Desmos Activity: RAISE 1.8.1
    Expressing Relationships Between Two Quantities</a></p>
<p>Give students access to four-function calculators, if requested.</p>
<p>Desmos activities are different from the graphing calculator application because they lead students through one or
  more interactive tasks. As students progress through the activity, teachers may use the Teacher Dashboard to monitor
  student responses and use them for whole group discussion. Activities may also be set to identify student responses
  that are correct or incorrect</p>
  <div class="os-raise-familysupport">
    <p>Access the <a href="https://k12.openstax.org/contents/raise/resources/7a719f4f974cdf749ead9dbffe099eb70fc7d296" target="_blank">Desmos guide PDF</a> to learn more about accessing and customizing activities.</p>
</div>
<h4>Student Activity</h4>
<p>In the image below, all the parallelograms have the same height. Base  length is measured in inches, and area is measured in square inches. </p>
<p id="yui_3_17_2_1_1689891442508_40"> A table can be used to show the relationship between the base length, b , and the area, A, of the parallelograms. </p>
<img alt="image shows three parallelograms with dimensions as shown in the table."
  class="img-fluid atto_image_button_text-bottom" height="283"
  src="https://k12.openstax.org/contents/raise/resources/5d4504dfc43395c726ce06afb55cd8a9d0122ec1" width="300"><br>
<br>
<br>
<table class="os-raise-skinnytable">
  <thead>
    <tr>
      <th scope="col">
        \( b \) (inches)
      </th>
      <th scope="col">
        \( A \) (square inches)
      </th>
    </tr>
  </thead>
  <tbody>
    <tr>
      <td>1</td>
      <td>3</td>
    </tr>
    <tr>
      <td>2</td>
      <td>6</td>
    </tr>
    <tr>
      <td>3</td>
      <td>9</td>
    </tr>
    <tr>
      <td>4.5</td>
      <td>a. ____ </td>
    </tr>
    <tr>
      <td>\( \frac{11}{2} \)</td>
      <td>b. ____ </td>
    </tr>
    <tr>
      <td>c. ____ </td>
      <td>36</td>
    </tr>
    <tr>
      <td>d. ____ </td>
      <td>46.5</td>
    </tr>
  </tbody>
</table>
<br>
<p>In the following activity, you will complete the table by entering the missing information. </p>
<p><div class="os-raise-usermessage-link">
  <p>Log into <a href="https://student.desmos.com" target="_blank">student.desmos.com</a> using the information provided by your teacher to complete the activity.</p>
</div>
<p><strong>Answer:</strong></p>
<table class="os-raise-skinnytable">
  <thead>
    <tr>
      <th scope="col">
        \( b \) (inches)
      </th>
      <th scope="col">
        \( A \) (square inches)
      </th>
    </tr>
  </thead>
  <tbody>
    <tr>
      <td>1</td>
      <td>3</td>
    </tr>
    <tr>
      <td>2</td>
      <td>6</td>
    </tr>
    <tr>
      <td>3</td>
      <td>9</td>
    </tr>
    <tr>
      <td>4.5</td>
      <td>a. <u>13.5</u></td>
    </tr>
    <tr>
      <td>\( \frac{11}{2} \)</td>
      <td>b. <u>\( \frac{33}{2} \)</u></td>
    </tr>
    <tr>
      <td>c. <u>12</u></td>
      <td>36</td>
    </tr>
    <tr>
      <td>d. <u>15.5</u></td>
      <td>46.5</td>
    </tr>
  </tbody>
</table>
<br>
<p>&nbsp;</p>

<h4>Anticipated Misconceptions</h4>
<p>Some students may think that the height must be known before they could find the missing area or base. Encourage them
  to look for a pattern in the table and to reason from there.</p>
<h4>Activity Synthesis</h4>
<p>Invite students to share their responses and explanations. Then, focus the whole-class discussion on the second
  question. Discuss with students:</p>
<ul>
  <li>&ldquo;Are the two equations we chose equivalent? How do you know?&rdquo; (Yes. There is an acceptable move that
    takes one to the other. If we divide each side of \( A=3b \) by 3, we have \( \frac{A}{3}=b \), which can also be
    written as \( b=\frac{A}{3} \). If we multiply each side of \( b=\frac{A}{3} \) by 3, we have \( 3b=A \) or \( A =
    3b \).)</li>
  <li>&ldquo;If we know the base, which equation would make it easier to find the area? Why?&rdquo; (\( A=3b \). The
    variable for area is already isolated. All we have to do is multiply the base by 3 to find the area.)</li>
  <li>&ldquo;If we know the area, which equation would make it easier to find the base? Why?&rdquo; (\( b=\frac{A}{3}
    \). The variable for the base is already isolated. We can just divide the area by 3 to find the base.)</li>
</ul>