145430d3-4236-4adb-a82e-0cad443ff6f4.html

<h4>Activity (10 minutes)</h4>
<p>Previously,
  students saw that an equation in two variables can have many solutions
  because there are many pairs of values that satisfy the equation. This
  activity illustrates that idea graphically. Students see that the
  coordinates of all points on the graphs are pairs of values that make
  the equation true, which means that they are all solutions to the
  equation.</p>
<p>They also see
  that because the given equation models the quantities and constraints in
  a situation, not all points on the graph are meaningful. For example,
  only positive \( x \)- or \( y \)-values
  on the graph (that is, only points in the first quadrant of the
  coordinate plane) have meanings in this context because almonds and figs
  cannot have negative values for their weight.</p>
<p>During the
  activity, look for students who perform numerical computations
  straightaway and those who first write a variable equation and then use
  it to answer the first two questions.</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 a
  partner.</p>
<br>
<br>
<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 (Clare purchased some snacks: salted almonds and dried figs). 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 cost per pound of each snack food, the amount of
      each snack food purchased, and the total amount of money spent before
      tax. After the third read, ask students to brainstorm possible
      strategies to determine the amount of one snack food purchased if the
      amount of the other snack food is known. 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>
</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>Provide appropriate reading accommodations and supports to ensure
      student access to written directions, word problems, and other
      text-based content. </p>
    <p class="os-raise-text-italicize">Supports accessibility for: Language; Conceptual processing
    </p>
  </div>
</div>
<br>
<h4>Student Activity</h4>
<p>To
  get snacks for a class trip, Clare went to the &ldquo;bulk&rdquo; section of the
  grocery store, where she could buy any quantity of a product and the
  prices are usually good.</p>
<p>Clare purchased some salted almonds at $6 a pound and some dried figs at $9 per pound. She spent $75 before tax.</p>
<img alt="Bowl of figs and almonds." height="300" src="https://k12.openstax.org/contents/raise/resources/5b214f5db89f6f9b4cdc2315a7a917f873009470" width="400">
<br><br>
<ol class="os-raise-noindent">
  <li>If she bought 2 pounds of almonds, how many pounds of figs did she buy?
  </li>
</ol>
<p><strong>Answer:</strong> 7 pounds</p>
<ol class="os-raise-noindent" start="2">
  <li>If she bought 1 pound of figs, how many pounds of almonds did she buy?</li>
</ol>
<p><strong>Answer:</strong> 11 pounds</p>
<ol class="os-raise-noindent" start="3">
  <li>Write an <span class="os-raise-ib-tooltip" data-schema-version="1.0" data-store="storename">equation</span> that
    describes the relationship between pounds of figs and pounds of almonds that Clare bought and the dollar amount that
    she paid. Be sure to specify what the <span class="os-raise-ib-tooltip" data-schema-version="1.0" data-store="storename">variables</span> represent.</li>
</ol>
<p><strong>Answer:</strong> \(6a+9f=75\)<br>
  \(a\): pounds of almonds, \(f\): pounds of figs</p>
<p>For questions 4 and 5, use the graph that represents the quantities in the situation.
</p>
<img alt="Graph of 2 intersecting lines and 5 points, origin O, with grid. Almonds (pounds) and dried figs (pounds)." height="196" src="https://k12.openstax.org/contents/raise/resources/6228bf48d50ee3aeb03d98c3cff3716dedb54f41" width="239">
<ol class="os-raise-noindent" start="4">
  <li>Choose any point on the line, state its coordinates, and explain what it tells us.
  </li>
</ol>
<p><strong>Answer:</strong> Your answer may vary, but here is a sample.<br>
  Point A or (2, 7). Clare bought 2 pounds of almonds and 7 pounds of figs for a total of $75.
</p>
<ol class="os-raise-noindent" start="5">
  <li>Choose any point that is not on the line, state its coordinates, and explain what it tells us.
  </li>
</ol>
<p><strong>Answer:</strong> Your answer may vary, but here is a sample.<br>
  Point D or \((1, 1)\). Clare bought 1 pound of almonds and 1 pound of figs, and the total was not $75.
</p>
<br>
<h4>Video: Using a Graph to Solve an Equation</h4>
<p>Watch
  the following video to see how this is done. You&rsquo;ll need to answer the
  question(s) that appear during the video to proceed to the next portion
  of the lesson.</p>
<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/a8f9efa6b344ccb6783e457a5769429b5d2ee9d6">
      <track default="true" kind="captions" label="On" src="https://k12.openstax.org/contents/raise/resources/596e67e418a392716595c779516e0d04f6ef8d1f " srclang="en_us">
      https://k12.openstax.org/contents/raise/resources/a8f9efa6b344ccb6783e457a5769429b5d2ee9d6
    </video></div>
</div>
</p>
<br>
<h4>Anticipated Misconceptions</h4>
<p>Some students
  may say that the points not on the line are impossible given that Clare
  spent $75. Encourage these students to think about what those points
  would mean if we didn&rsquo;t know how much money Clare spent.</p>
<h4>Activity Synthesis</h4>
<p>Display the graph for all to see. Invite students to
  share their equation for the situation and their interpretations of the
  points on and off the graph. Make sure students understand that a point
  on the graph of an equation in two variables is a solution to the
  equation. Discuss questions such as:</p>
<ul>
  <li>&ldquo;What does the point \( (10, 3) \) mean in this situation?&rdquo; (Clare purchased 10 pounds of almonds and
    3 pounds of figs.)</li>
  <li>&ldquo;Is that a possible combination of pounds of
    figs and almonds? Why or why not?&rdquo; (No. It doesn&rsquo;t lie on the graph.
    Also, if Clare bought 10 pounds of almonds and 3 pounds of figs, it
    would cost her $87, not $75.)</li>
  <li>&ldquo;From the graph, it looks like \( (7, 3.5) \)
    might be a solution, but it is hard to know for sure. Is there a way to
    verify?&rdquo; (Substitute the values into the equation and see if they make
    the equation true.)</li>
  <li>&ldquo;Suppose we extend the two ends of the graph beyond the first quadrant. Would a point on those parts of the
    line&mdash;say, \( (-1, 9) \)&mdash;be a solution to the equation \( 6a+9f=75 \)?
    Why or why not?&rdquo; (It would still be a solution to the equation, but it
    wouldn&rsquo;t make sense in this context. The weight of almonds or figs
    cannot be negative.)</li>
</ul>
<h3>1.5.3: 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>Which point is a solution to the line graphed below?</p>
<table class="os-raise-textheavytable">
  <thead>
    <tr>
      <th scope="col">Answers</th>
      <th scope="col">Feedback</th>
    </tr>
  </thead>
  <tbody>
    <tr>
      <td>\( (3, 1) \)</td>
      <td>That&rsquo;s correct! Check yourself: When \( x=3 \), \(y=1\), so this point is a solution on the graph. <br>
      </td>
    </tr>
    <tr>
      <td>\( (0, 1) \)</td>
      <td>Incorrect. Let&rsquo;s try again a different way: This point is located above the line that is graphed. When a
        point is a solution to an equation, it must be located on the graph of the equation. The answer is \( (3, 1) \).
      </td>
    </tr>
    <tr>
      <td>\( (1, 3) \)</td>
      <td>Incorrect.
        Let&rsquo;s try again a different way: To get to \((1, 3)\), you move 1 right from the origin and 3 up, so this
        point is not on the line. When a point is a solution to an equation, it must be located on the graph of the
        equation. The answer is \( (3, 1) \).</td>
    </tr>
    <tr>
      <td>\( (-3, 3) \)</td>
      <td>Incorrect. Let&rsquo;s
        try again a different way: To get to \((-3, 3)\), move 3 left and then 3 up. This point is not a solution to the
        equation of the line graphed. When a point is a solution to an equation, it must be located on the graph of the
        equation. The answer is \( (3, 1) \).</td>
    </tr>
  </tbody>
</table><br>
<h3>1.5.3: 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>Determining the Meaning of Solutions of a Graphed Line</h4>
<p>Suppose
  we are buying beans and rice to feed a large gathering of people, and we
  plan to spend $120 on the two ingredients. Beans cost $2 a pound, and
  rice costs $0.50 a pound.<br></p>
<p>If \( x \) represents pounds of beans and \( y \) pounds of rice, the equation \( 2x + 0.50y = 120 \) can represent
  the constraints in this situation.</p>
<p>The graph of \( 2x + 0.50y = 120 \) shows a straight line.</p>
<img alt="Graph of a line. Vertical axis, pounds of rice. Horizontal axis, pounds of beans." height="232" src="https://k12.openstax.org/contents/raise/resources/586a7b603355fd20cb8f01e888ce5baf39f6a9e6" width="400"><br><br>
<p>Each point on the line is a pair of \( x \)-
  and \( y \)-values that make the equation true and is thus a solution.
  It is also a pair of values that satisfy the constraints in the
  situation.</p>
<ul>
  <li>The point \( (10, 200) \) is on the
    line. If we buy 10 pounds of beans and 200 pounds of rice, the cost will
    be \( 2(10) + 0.50(200) \), which equals 120.</li>
  <li>The points \( (60, 0) \) and \( (45, 60)
    \) are also on the line. If we buy only beans&mdash;60 pounds of them&mdash;and no
    rice, we will spend $120. If we buy 45 pounds of beans and 60 pounds of
    rice, we will also spend $120.</li>
</ul>
<p>What about points that are <em>not</em> on the line? They are not solutions because they don&rsquo;t satisfy the
  constraints, but they still have meaning in the situation.</p>
<ul>
  <li>The point \( (20, 80) \) is not on the
    line. Buying 20 pounds of beans and 80 pounds of rice costs \( 2(20) +
    0.50(80) \) or 80, which does not equal 120. This combination costs less
    than what we intend to spend.</li>
  <li>The point \( (70, 180) \) means that we
    buy 70 pounds of beans and 180 pounds of rice. It will cost \(
    2(70)+0.50(180) \) or 230, which is over our budget of 120.</li>
</ul>
<p>Let&rsquo;s look at an example.</p>
<p><strong>Example:</strong></p>
<ol class="os-raise-noindent">
  <li>Looking at the graph about purchasing beans and rice, is the point (30, 120) a solution?
  </li>
</ol>
<p><strong>Answer:</strong> Yes, because \(2(3)+0.50(12) = 60 + 60 =120\).</p>
<ol class="os-raise-noindent" start="2">
  <li>What does the point (30, 120) mean in this situation?
  </li>
</ol>
<p><strong>Answer:</strong> The point means that 30 pounds of beans and 120 pounds of rice were purchased. Since the
  point is on the line, the total spent was $120.</p>
<h4>Try It: Determining the Meaning of Solutions of a Graphed Line</h4>
<p>At a high school baseball game, the
  concession stand sold hot dogs and hamburgers. Hot dogs were $1.50, and
  hamburgers cost $3. The goal is to make $150 for the game.</p>
<p>What is the meaning of the point \((40, 30)\) on the graph?</p>
<p><img alt class="img-fluid atto_image_button_text-bottom" height="190" role="presentation" src="https://k12.openstax.org/contents/raise/resources/6b33b707e3132aa959eb31db9921beaeb279036c" width="300"></p>
<br><br>
<p><strong>Answer:</strong></p>
<p>Here is how to determine the meaning of the point \( (40, 30) \) on this
  graph. </p>
<p>The \( x \)-axis is labeled &ldquo;number of hot
  dogs,&rdquo; and the \( y \)-axis is labeled &ldquo;number of hamburgers.&rdquo; Since
  \( x = 40\), there were 40 hot dogs sold, and since \( y = 30\), there
  were 30 hamburgers sold. Since the point is on the line, the total
  amount from the hot dogs and hamburgers sold was $150.<br></p>