d44dd0b4-d09b-4315-b171-4ab2505a3687.html

<h3>Activity (20 minutes)</h3>
<br>
<p>In this activity, students get a chance to practice applying their skills at connecting multiple representations of the same linear relationship.</p>
<p>They can use the structure of the card-matching task to check their thinking and make sense of the concepts they are practicing, as wrong matches will make some piles uneven or a representation that seems to match one representation in a pile may not match another.</p>
<h4>Launch</h4>
<p>Arrange students in groups of two and distribute a set of cut-up slips to each group. Ask students to take turns: the first partner identifies a graph to match the already matched equation, table, and situation and explains why they think the graph belongs, while the other listens and works to understand. When both partners agree on the match, they switch roles.</p>
<p>Use the Illustrative Mathematics Card Sort <a href="https://k12.openstax.org/contents/raise/resources/3be908da742d5d6de02d0d8af33cf276e5d18b25" target="_blank">Blackline Master.</a></p>
<h4>Student Activity</h4>
<p>Take turns with your partner to match a graph with each set of matching cards. Eventually all the cards will be sorted into groups of four cards (an equation, situation, table, and graph).</p>
<p>For each match that you find, explain to your partner how you know it&rsquo;s a match. Ask your partner if they agree with your thinking.</p>
<p>For each match that your partner finds, listen carefully to their explanation. If you disagree, discuss your thinking and work to reach an agreement.</p>
<ol class="os-raise-noindent">
  <li> What did you and your partner agree are the cards that match the equation on card 1? </li>
</ol>
<p><strong>Answer:</strong> Cards 1, 11, 20, 32</p>
<ol class="os-raise-noindent" start="2">
  <li> What did you and your partner agree are the cards that match the equation on card 2? </li>
</ol>
<p><strong>Answer:</strong> Cards 2, 16, 18, 25</p>
<ol class="os-raise-noindent" start="3">
  <li> What did you and your partner agree are the cards that match the equation on card 3? </li>
</ol>
<p><strong>Answer:</strong> Cards 3, 10, 21, 29</p>
<ol class="os-raise-noindent" start="4">
  <li> What did you and your partner agree are the cards that match the equation on card 4? </li>
</ol>
<p><strong>Answer:</strong> Cards 4, 9, 23, 30</p>
<ol class="os-raise-noindent" start="5">
  <li> What did you and your partner agree are the cards that match the equation on card 5? </li>
</ol>
<p><strong>Answer:</strong> Cards 5, 12, 22, 31</p>
<ol class="os-raise-noindent" start="6">
  <li> What did you and your partner agree are the cards that match the equation on card 6? </li>
</ol>
<p><strong>Answer:</strong> Cards 6, 15, 24, 26</p>
<ol class="os-raise-noindent" start="7">
  <li> What did you and your partner agree are the cards that match the equation on card 7? </li>
</ol>
<p><strong>Answer:</strong> Cards 7, 14, 19, 27</p>
<ol class="os-raise-noindent" start="8">
  <li> What did you and your partner agree are the cards that match the equation on card 8? </li>
</ol>
<p><strong>Answer:</strong> Cards 8, 13, 17, 28</p>
<br>
<h4>Anticipated Misconceptions</h4>
<p>Students may need reminders about using inverse operations to solve for \(y\).</p>
<h4>Activity Synthesis</h4>
<p>Ask pairs to reflect in a whole group discussion:</p>
<p>Which set of cards were the most difficult for you and your partner to match?</p>
<p>Why did they cause you difficulty?</p>
<p>Were there any sets that were easier to match and why?</p>
<p>The goal of this discussion is to help students understand that rows in tables and points on lines represent solutions to the related linear equation.</p>
<p>Display the graph, table, and equation for Mai&rsquo;s running speeds for students to see.</p>
<div class="os-raise-d-flex os-raise-justify-content-between"><p><img src="https://k12.openstax.org/contents/raise/resources/246bbdffbaab1d7f2b1e95b13f69db521ff2a527" width="300" height="282" /></p>
<table class="os-raise-skinnytable">
  <thead>
    <tr>
      <th scope="col">Minutes</th>
      <th scope="col">Miles</th>
    </tr>
  </thead>
  <tbody>
    <tr>
      <td>10</td>
      <td>0.4</td>
    </tr>
    <tr>
      <td>25</td>
      <td>1</td>
    </tr>
    <tr>
      <td>60</td>
      <td>2.4</td>
    </tr>
    <tr>
      <td>90</td>
      <td>3.6</td>
    </tr>
  </tbody>
</table></div>
<p>\(y = 0.04x\)</p>
<p>Possible questions for discussion:</p>
<p>If \(x\) is 10, what does \(y\) equal? How do you know? (0.4 because \(10 \cdot 0.04=0.4\).)</p>
<p>What does the point \((10, 0.4)\) mean in the story? (Mai can run 0.4 miles in 10 minutes.)</p>
<p>How can you see from the graph what \(y\) is if \(x\) is 10? (I can find the ordered pair on the graph and see that when \(x\) is 10, \(y\) is 0.4.)</p>
<p>How can you see from the table what \(y\) is if \(x\) is 10? (I can see that 10 minutes is paired with 0.4 miles.)</p>
<p>How can you use the equation to find out what \(y\) is if \(x\) is 10? (I can substitute 10 for \(x\) in the equation \(y = 0.04x\) and solve for y.)</p>
<p>When would you choose to use a graph to tell you an \(x\)-value if you know the \(y\)-value? When would you use a table? An equation? (A graph is useful if the coordinate is easy to determine. A table is useful if the values are present in the table. An equation can be used anytime but is best when the \(x\)-value is known and can be substituted in to find the \(y\)-value.)
</p>
<p>If \(y\) is 3.6, what does \(x\) equal? How do you know? (90 because \(90 \cdot 0.04=3.6\).)</p>
<p>What does the point \((90, 3.6)\) mean in the story? (Jada can run 3.6 miles in 90 minutes.)</p>
<p>How can you see from the graph what \(x\) is if \(y\) is 3.6? (I can find the ordered pair on the graph and see that when \(y\) is 3.6, \(x\) is 90.)</p>
<p>How can you see from the table what \(x\) is if \(y\) is 3.6? (I can see that 3.6 miles is paired with 90 minutes.)</p>
<p>How can you use the equation to find out what \(x\) is if \(y\) is 3.6? (I can substitute 3.6 for \(y\) in the equation \(y = 0.04x\) and solve for \(x\).)</p>
<h3>1.13.3: 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 of the following is a representation of \(y = 30x\)?</p>
<table class="os-raise-textheavytable">
  <thead>
    <tr>
      <th scope="col">Answers</th>
      <th scope="col">Feedback</th>
    </tr>
  </thead>
  <tbody>
    <tr>
      <td><table class="os-raise-horizontalable">
          <thead>
          </thead>
          <tbody>
            <tr>
                <th scope="row">\(x\)</th>
              <td>0</td>
              <td>1</td>
              <td>2</td>
              <td>3</td>
            </tr>
            <tr>
                <th scope="row">\(y\)</th>
              <td>30</td>
              <td>35</td>
              <td>40</td>
              <td>45</td>
            </tr>
          </tbody>
        </table></td>
      <td>Incorrect. Let&rsquo;s try again a different way: In this table, 30 is the \(y\)-intercept, not the slope. In the equation, 30 is the slope.The answer is Josie paid $30 per pair of jeans.</td>
    </tr>
    <tr>
      <td><img src="https://k12.openstax.org/contents/raise/resources/d5cfdf81e3f7137f435e0fe319b4179523eaad71" width="300"/></td>
      <td>Incorrect. Let&rsquo;s try again a different way: In this graph, 30 is the \(y\)-intercept, not the slope. In the equation, 30 is the slope.The answer is Josie paid $30 per pair of jeans.</td>
    </tr>
    <tr>
      <td>Josie paid $30 per pair of jeans.</td>
      <td>That&rsquo;s correct! Check yourself: If Josie paid $30 per pair of jeans, that $30 represents the slope or rate of change in the equation.</td>
    </tr>
    <tr>
      <td>Eric was paid $30 to cut grass in July plus $5 each time he came to cut the grass.</td>
      <td>Incorrect. Let&rsquo;s try again a different way: Eric was paid $30 as a base rate. This is what the \(y\)-intercept would be. The slope would be 5. The answer is Josie paid $30 per pair of jeans.</td>
    </tr>
  </tbody>
</table>
<br>
<h3>1.13.3: 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>Linear Equations and Their Representations</h4>
<p>Linear equations can be written from situations, tables of values, and graphs.</p>
<p> Here is a linear situation:</p>
<p>Sadie pays $15 a month for ballet classes plus $5 each lesson she takes.</p>
<p>First, make a table of values to represent how much Sadie pays for the first 5 lessons.</p>
<table class="os-raise-horizontaltable">
  <thead>
  </thead>
  <tbody>
    <tr>
      <th scope="row">Number of Lessons</th>
      <td>0</td>
      <td>1</td>
      <td>2</td>
      <td>3</td>
      <td>4</td>
      <td>5</td>
    </tr>
    <tr>
      <th scope="row">Total Cost</th>
      <td>15</td>
      <td>20</td>
      <td>25</td>
      <td>30</td>
      <td>35</td>
      <td>40</td>
    </tr>
  </tbody>
</table>
<br>
<p>Next, graph all of the points:</p>
<p><img src="https://k12.openstax.org/contents/raise/resources/d54e1db9e68f90985c1ddefbc5827b27de708db4" width="300" height="301" /></p>
<p>Looking at the slope and \(y\)-intercepts of the table and graph, write an equation.</p>
<p><strong>Step 1</strong> - Identify the \(y\)-intercept.</p>
<p>When \(x = 0\), \(y = 15\) so the \(y\)-intercept is 15.</p>
<p><strong>Step 2</strong> - Identify the slope.</p>
<p>Slope is the change in \(y\) over the change in \(x\). For this table, the change in \(y\) is 43 and the change in \(x\) is 1. The slope is 3.</p>
<p><strong>Step 3</strong> - Write the equation in slope-intercept form.</p>
<p>In the slope-intercept form, \(y=mx+b\), \(m\) is the slope and \(b\) is the \(y\)-intercept.</p>
<p>The equation is \(y = 5x+15\).</p>
<h4>Try It: Linear Equations and Their Representations</h4>
<p>Write an equation from the table:</p>
<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>7</td>
      <td>11</td>
      <td>15</td>
      <td>19</td>
      <td>23</td>
    </tr>
  </tbody>
</table>
<br>
<p><strong>Answer:</strong> Compare your answer:</p>
<p>Here is how to write an equation from a table.</p>
<p><strong>Step 1</strong> - Identify the \(y\)-intercept.</p>
<p>When \(x = 0\), \(y = 7\) so the \(y\)-intercept is 7.</p>
<p><strong>Step 2</strong> - Identify the slope.</p>
<p>Slope is the change in \(y\) over the change in \(x\). For this table, the change in \(y\) is 4 and the change in \(x\) is 1. The slope is 4.</p>
<p><strong>Step 3</strong> - Write the equation in slope-intercept form.</p>
<p>In the slope-intercept form, \(y=mx+b\), \(m\) is the slope and \(b\) is the \(y\)-intercept.</p>
<p>The equation is \(y = 4x +7\).</p>