91ba4f77-c324-40d7-80dc-ef2c16aa2181.html

<h4>Activity (15 minutes)</h4>
<p>In an earlier lesson, students encountered relationships where two quantities that could vary multiplied to a
  constant. The focus there was on studying the relationships and describing them using equations in any form. If \( x
  \) and \( y \) are quantities that vary and \( C \) is a constant, they might write \( xy=C \), \( y = \frac{C}{x} \),
or \( x = \frac {C}{y} \). </p>
<p>In this activity, students encounter a similar relationship. The goal here, however, is for students to
  recognize&mdash;by repeatedly calculating the value of one quantity and then the value of the other
  quantity&mdash;that a particular form of equation might be handy for finding one quantity but not so handy for finding
  the other. </p>
<p>When answering the first couple parts of the first question (finding the length of a section given the number of
  volunteers), students can use various strategies to efficiently reason about the answers. They might draw diagrams,
  use proportional reasoning, or think in terms of multiplication (asking, for example, &ldquo;8 times what number
  equals 2?&rdquo;). They might also write an equation such as \( n \cdot \ l = 2 \), substitute the number of
  volunteers for \( n \), and then solve the equation.</p>
<p>As students progress through the parts, they will likely notice that some strategies become less practical for
  finding the value of interest. One strategy, however, will remain efficient: dividing 2 by the number of volunteers,
  or evaluating \( 2 \div n \) or \( \frac2n \) at the given values of \( n \).</p>
<p>Likewise, when answering the third question (finding the number of volunteers given the length of a section),
  students could start out with a variety of strategies and fairly easily find the number of volunteers. Later, however,
  the number of volunteers becomes a bit more cumbersome to find except when using division (that is, dividing 2 by the
  given length, or evaluating \( 2 \div \ l \) or \( \frac {2}{\ l} \) at the given values of \( \ l \)).</p>
<p>Identify students who use these or other approaches and select them to share their strategies during discussion. </p>
<h4>Launch</h4>
<p>Arrange students in groups of 2 and provide access to calculators. Give students a few minutes of quiet work time and
  then time to share their responses with their partner. Follow with a whole-class discussion.</p>
<h4>Student Activity</h4>

<img alt="Mummers' Parade in Philly." class="img-fluid atto_image_button_text-bottom" height="394"
  src="https://k12.openstax.org/contents/raise/resources/dfb7838fc9d86de04e46523624735a8692051722" width="550">
<br>
<br>
<p>For numbers 1 - 4, use the situation below:
<p>
<blockquote>
  <p>
    After a parade, a group of volunteers is helping to pick up the trash along a 2-mile stretch of a road. The group
    decides to divide the length of the road so that each volunteer is responsible for cleaning up equal-length
    sections.
  </p>
  <p>Find the length of a road section for each volunteer if there are the following numbers of volunteers. Be prepared
    to
    show your reasoning.
  </p>
</blockquote>
<ol class="os-raise-noindent">
  <li>8 volunteers</li>
</ol>
<p><strong>Answer:</strong> \( \frac14 \) mile</p>
<ol class="os-raise-noindent" start="2">
  <li>10 volunteers</li>
</ol>
<p><strong>Answer:</strong> \( \frac15 \) mile</p>
<ol class="os-raise-noindent" start="3">
  <li>25 volunteers</li>
</ol>
<p><strong>Answer:</strong> \( \frac {2}{25} \) mile or 0.08 mile</p>
<ol class="os-raise-noindent" start="4">
  <li>36 volunteers</li>
</ol>
<p><strong>Answer:</strong> \( \frac{2}{36} \) or \( \frac{1}{18} \) mile</p>
<ol class="os-raise-noindent" start="5">
  <li>Write an equation that would make it easy to find \( l \), the length of a road section in miles for each
    volunteer, if there are \( n \) volunteers.
  </li>
</ol>
<p><strong>Answer:</strong> \( l = \frac{2}{n} \)</p>
<p>For numbers 6 - 9, find the number of volunteers in the group if each volunteer cleans up a section of the following
  lengths. Be prepared to show your reasoning.
</p>
<ol class="os-raise-noindent" start="6">
  <li>0.4 mile</li>
</ol>
<p><strong>Answer:</strong> \(n=\frac{2}{0.4}=\) 5 volunteers</p>
<ol class="os-raise-noindent" start="7">
  <li>
    \( \frac {2}{7} \) mile
  </li>
</ol>
<p><strong>Answer:</strong> \(n=\frac2{\displaystyle\frac27}=\frac{2}{1}\times\frac{7}{2}=\) 7 volunteers</p>
<ol class="os-raise-noindent" start="8">
  <li>0.125 mile</li>
</ol>
<p><strong>Answer:</strong> \(n=\frac{2}{0.125}=\) 16 volunteers</p>
<ol class="os-raise-noindent" start="9">
  <li>
    \( \frac {6}{45} \) mile
  </li>
</ol>
<p><strong>Answer:</strong> \(n=\frac2{\displaystyle\frac{6}{45}}=\frac{2}{1} \times \frac{45}{6}=\) 15 volunteers</p>
<ol class="os-raise-noindent" start="10">
  <li>Write an equation that would make it easy to find the number of volunteers, \( n \), if each volunteer cleans up a
    section that is \( l \) miles. </li>
</ol>
<p><strong>Answer:</strong> \( n = \frac{2}{ l} \)</p>
<h4>Student Facing Extension</h4>
<h4>Are you ready for more?</h4>
<p>Let&rsquo;s think about the graph of the equation \( y = \frac{2}{x} \).</p>
<ol class="os-raie-noindent">
  <li>
    Create a list 
\((x,y)\) pairs that will help you graph the equation. Make sure to include some negative numbers for \(x\) and some numbers that are not integers.
  </li>
</ol>
<p><strong>Answer:</strong></p>
<p>Your answer may vary, but here is an example:</p>
<ul>
  <li>\((-2.5,-0.8)\)</li>
  <li>\((-1,-2)\)</li>
  <li>\((0.5,4)\)</li>
  <li>\((1.25,1.6)\)</li>
</ul>
<br>
<ol class="os-raie-noindent" start="2">
  <li>
    Use the graphing tool or technology outside the course. Graph the equation that represents this scenario using the
    Desmos tool below.&nbsp;
  </li>
</ol>
<p>(Students were provided access to Desmos.)</p>
<p><strong>Answer:</strong></p>
<div class="os-raise-d-flex os-raise-justify-content-between">
  <div class="os-raise-mx-auto">
    <img alt="Image of Demos screen of the plots of coordiantes for the equation of the fraction two over x." src="https://k12.openstax.org/contents/raise/resources/164b3020a1663ad321ecc53c4cbf4ab40f1d3d7f"> 
  </div>
  <div class="os-raise-mx-auto">
    <img alt="Graph of the equation of the fraction two over x." src="https://k12.openstax.org/contents/raise/resources/e1680e9f8278894aa656e8b20857a78c1dca29aa" width="400">
  </div>
</div>
<br>
<ol class="os-raie-noindent" start="3">
  <li>
    What do you think the graph looks like when \(x\) is between 0 and \(\frac12\)? Continue to use the graphing tool or
    technology outside the course to graph the function
    \(y=\frac{2}{x}\) with your points. Try some values of \(x\) to test your idea. (Students were provided access to Desmos.)
  
  </li>
</ol>
<p><strong>Answer:</strong></p>
<p>Your answer may vary, but here is an example:</p>
<p>The graph gets steeper and steeper as \(x\) gets closer to 0.</p>
<p><img src="https://k12.openstax.org/contents/raise/resources/bba34b4119919710e25d5a614d6d772cf523e636" width="300"/></p>
<ol class="os-raie-noindent" start="4">
  <li>
    What is the largest value \(y\) can ever be?
  </li>
</ol>
<p><strong>Answer:</strong> Your answer may vary, but here is an example:</p>
<p>There is no limit to the value of \(y\).</p>
<h4>Activity Synthesis</h4>
<p>Select students to present their strategies for solving either set (or both sets) of questions. Start with students
  using the least straightforward approach and end with those who wrote \( l = \frac{2}{n} \) for the first set of
  questions (or \( n = \frac{2}{\ l} \) for the second set of questions).</p>
<p>Emphasize that isolating the variable that we&rsquo;re interested in&mdash;before we substitute any known
  values&mdash;can be an efficient way of solving problems. Once we pin down the variable of interest and see what
  expression is equal to it, we can simply evaluate that expression and bypass some tedious steps.</p>
<p>Highlight that isolating a variable is called &ldquo;solving for a variable.&rdquo; In the road clean-up context, if
  we want to know the length of a road section for which each volunteer would be responsible, we can solve for \( \ l
  \). If we want to know how many volunteers would be needed, we can solve for \( n \).</p>
<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>
      Use color coding and annotations to highlight connections between
      representations. As students share their reasoning, scribe their thinking on a visible display. Invite students to
      describe connections they notice between approaches that use diagrams, proportional reasoning, or multiplication and
      those that use an equation. For example, ask, &ldquo;Where does \( l \) appear in this
      (diagram)?&rdquo; </p>
    <p class="os-raise-text-italicize">Supports accessibility for: Visual-spatial processing</p>
    </div>
  </div>
<br>
<h3>1.8.2: 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>Use the formula for the area of a triangle, \( A = \frac{1}{2} bh \), where \(b\) is the base of the triangle and
  \(h\) is the height of the triangle. Which equation would best be used to find the base of the triangle?</p>
<table class="os-raise-textheavytable">
  <thead>
    <tr>
      <th scope="col">Answers</th>
      <th scope="col">Feedback</th>
    </tr>
  </thead>
  <tbody>
    <tr>
      <td>\( b = \frac{2A}{h} \)</td>
      <td>That&rsquo;s correct! Check yourself: To find the base, solve for \(b\). Multiply both sides by 2, and then
        divide both sides by \(h\). Now, \( b = \frac{2A}{h} \).<br></td>
    </tr>
    <tr>
      <td>\( h = \frac{2A}{b} \)</td>
      <td>Incorrect. Let&rsquo;s try again a different way: Since \( b \) stands for the base in the formula, solve the
        formula for \( b \). The answer is \( b = \frac{2A}{h} \).</td>
    </tr>
    <tr>
      <td>\( 2A = bh \)</td>
      <td>Incorrect. Let&rsquo;s try again a different way: Since \( b \) stands for the base in the formula, solve the
        formula for \( b \). The answer is \( b = \frac{2A}{h} \).</td>
    </tr>
    <tr>
      <td>\( \frac{1}{2}b = \frac{A}{h} \)</td>
      <td> Incorrect. Let&rsquo;s try again a different way: Since \( b \) stands for the base in the formula, solve the
        formula for \( b \). The answer is \( b = \frac{2A}{h} \).</td>
    </tr>
  </tbody>
</table>
<br>
<h3>1.8.2 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>Solve a Formula for a Specific Variable</h4>
<p>We have all probably
  worked with some geometric formulas in our study of mathematics.
  Formulas are used in many fields, so it is important to recognize
  formulas and be able to manipulate them easily.</p>
<p>It is often helpful to
  solve a formula for a specific variable. If you need to put a formula
  in a spreadsheet, it is not unusual to have to solve it for a specific
  variable first. We isolate that variable on one side of the equal sign
  with a coefficient of 1, and all other variables and constants are on
  the other side of the equal sign.</p>
<p>Geometric formulas
  often need to be solved for another variable, too. The formula \( V=
  \frac{1}{3} &pi;r^2h \) is used to find the volume of a right circular
  cone when given the radius of the base and height. In the next example,
  we will solve this formula for the height.</p>
<p>Example 1</p>
<p>Solve \(V=\frac{1}{3}πr^2h\) for \(h\).</p>
<p><strong>STEP 1:</strong> Remove the fraction on the right.<br>
  Multiply both sides by 3.<br>
  <img src="https://k12.openstax.org/contents/raise/resources/9669f1c3605c13fca45ef756dc66969f482a50de" alt="." />
</p>
<p><strong>Step 2: </strong>Simplify.<br>
  <img src="https://k12.openstax.org/contents/raise/resources/80bab41ddb0d7d65cfc39fd39004be058362ce66" alt="." />
</p>
<p><strong>Step 3: </strong>Divide both sides by \( &pi;r^2 \).<br>
  <img src="https://k12.openstax.org/contents/raise/resources/c187db6fb8bfc8fc85701172ee191718658f7db5" alt="." />
</p>
<p>We could now use this formula
  to find the height of a right circular cone when we know the volume and
  the radius of the base, by using the formula \( h= \frac{3V}{&pi;r^2} \)</p>
<p>In the sciences, we often need to change temperature from Fahrenheit to Celsius or vice versa. If you travel in a
  foreign country, you may want to change the Celsius temperature to the more familiar Fahrenheit temperature.</p>
<p><strong>Example 2&nbsp;</strong></p>
<p>Solve&nbsp;</p>
<p>\( C = \frac{5}{9} ( F - 32 ) \) for \( F \).</p>
<p><strong>STEP 1:</strong> Remove the fraction on the right.<br>
  Multiply both sides by 9/5.<br>
  <img src="https://k12.openstax.org/contents/raise/resources/76bd338fd5f4648c7191063d5ffec4c07a187dc4" alt="." />
</p>
<p><strong>STEP 2:</strong> Simplify.<br>
  <img src="https://k12.openstax.org/contents/raise/resources/84124e3df6951b154bebf3186ba2680e3c02faed" alt="." />
</p>
<p><strong>STEP 3:</strong> Add 32 to both sides.<br>
  <img src="https://k12.openstax.org/contents/raise/resources/4b4f0d6febadd60c32ede7756462e9b521c73330" alt="." />
</p>
<p>Continue explanation as is</p>
<p><strong>Example 3</strong></p>
<p>Solve the formula \( 8x + 7y = 15 \) for \( y \).</p>
<p><strong>STEP 1:</strong> Subtract 8x from both sides to isolate the term with \(y\).<br>
  <img src="https://k12.openstax.org/contents/raise/resources/07c980d0f1e405555ee08341049660f63e5e1533" alt="." />
</p>
<p><strong>Step 2: </strong>Simplify.<br>
  <img src="https://k12.openstax.org/contents/raise/resources/d0cefdbcb454e78f881f4ca656102a1d3ed806bc" alt="." />
</p>
<p><strong>STEP 3:</strong> Divide both sides by 7 to make the coefficient of \(y\) equal to 1.<br>
  <img src="https://k12.openstax.org/contents/raise/resources/9998fc36167333f2c1d26e55334676e1ede8c127" alt="." />
</p>
<p><strong>STEP 4:</strong> Simplify.</p>
<p> \(\frac{2d}{t}=(v_0+ v)\)</p>
<p>Now, try to solve \(d =\frac{1}{2} (v_0 + v)^t\) for \(v_0\).</p>
<p><strong>Answer:</strong></p>
<p><strong>STEP 1:</strong> Remove the fraction on the right.<br>
  Multiply by 2.<br>
  \(2 \times d =2 \times \frac{1}{2} (v_0+ v)t\) </p>
<p><strong>STEP 2:</strong> Simplify.<br>
  \(2d =(v_0+ v)t\) </p>
<p><strong>Step 3: </strong>Divide both sides by \(t\).<br>
  \(\frac{2d}{t}=\frac{(v_0+ v)^t}{t}\) </p>
<p><strong>STEP 4:</strong> Simplify.<br>
  \(\frac{2d}{t}=(v_0+ v)\) </p>
<p><strong>STEP 5:</strong> Subtract \(v\) from both sides to solve for v0.<br>
  \(\frac{2d}{t}-v=v_0+ v-v\) <br>
  \(\frac{2d}{t}-v=v_0\) </p>
<p><strong>Answer:</strong></p>
<h4>Try It: Solve a Formula for a Specific Variable</h4>
<p>Solve \(8x+7y=15\) for \(x\)
</p>
<p>Here is how to solve this equation for \(x\).</p>
<p><strong>Step 1: </strong>Subtract \(7y\) from each side.<br>
  \( 8x+ 7y -7y =15 -7y \)</p>
<p><strong>Step 2: </strong>Simplify.<br>
  \( 8x =15-7y \)</p>
<p><strong>Step 3: </strong>Divide both sides by 8.<br>
  \( \frac{8x}{8}=\frac{15-7y}{8} \)</p>
  <p>The equation solved for \( x \) is \( x=\frac{15}{8}-\frac{7y}{8} \).</p>