771e2fc0-b7d3-4c84-8ba9-8c20a6e66e53.html

<h3>Warm Up (5 minutes)</h3>
<p>The activities in this lesson require students to observe tables of values, look for patterns, and generalize their
  observations into equations. This warm up prompts students to think about how they could go about analyzing the values
  in the table and to articulate their reasoning.</p>
<h4>Launch</h4>
<p>Arrange students in groups of 2. Display the table for all to see. Explain that the quantities in each column are
  related.</p>
<p>Ask groups to try to find a relationship and pay attention to how they go about doing so. Emphasize that the goal is
  <em>not</em> to successfully find a relationship. It is to notice the strategies they use when attempting to figure
  out what the relationship might be.</p>
<h4>Student Activity</h4>
<p>Here is a table of values. The two quantities, \( x \)&nbsp;and \(y \), are related.</p>
<table class="os-raise-skinnytable">
  <thead>
    <tr>
      <th scope="col">
        \(x\)
      </th>
      <th scope="col">
        \(y\)
      </th>
    </tr>
  </thead>
  <tbody>
    <tr>
      <td>
        1
      </td>
      <td>
        0
      </td>
    </tr>
    <tr>
      <td>
        3
      </td>
      <td>
        8
      </td>
    </tr>
    <tr>
      <td>
        5
      </td>
      <td>
        24
      </td>
    </tr>
    <tr>
      <td>
        7
      </td>
      <td>
        48
      </td>
    </tr>
  </tbody>
</table>
<p><br>What are some strategies you could use to find a relationship between \( x \)&nbsp;and \(y \)? Brainstorm as many
  ways as possible.</p>
<h4>Student Responses</h4>
<ul>
  <li>See if there is an operation that could be done to \( x\) that would produce \(y \).</li>
  <li>Compare how the \( x\)-values change and the \( y\)-values change, and see if there is a pattern.</li>
  <li>See if the numbers in one column follow a special pattern and if that pattern could be connected to the numbers in
    the other column.</li>
  <li>Plot the \( x\)- and \( y\)-values on a coordinate plane and see if the graph looks like a familiar relationship.
  </li>
</ul>
<h4>Activity Synthesis</h4>
<p>Invite groups to share their strategies and record them for all to see. If not already described by students, apply
  each strategy using the values in the table, or ask students to give an example of how it could be applied.</p>
<p>Some students may notice that each time \( x \) increases by 2, \(y \) increases by 8 more than the previous time.
  Others may notice that the \(y \)-values are 1 less than the square numbers 1, 4, 9, 25, and 49, and that these
  numbers are the squares of the listed \( x\)-values, and from there concluded that the relationship is along the lines
  of: &ldquo;square \( x\) and subtract 1 to get \(y \).&rdquo; Neither of these observations is essential, but consider
  asking if they see any special patterns in either column that could help determine the relationship.</p>
<p>If no one mentions plotting the pairs of values as a way to understand the relationship between \( x\) and \( y\),
  and if time permits, consider displaying a graph such as the one shown (or displaying a blank coordinate plane and
  plotting the points together).</p>
<p><img alt="A scatter plot with plotted points (1, 0), (3, 8), (5, 24), and (7, 48)."
    class="img-fluid atto_image_button_text-bottom" height="222"
    src="https://k12.openstax.org/contents/raise/resources/46a63ce8c51d3b6a788e6e015b8defeef1c0a2a9" width="232"><br>
</p>
<p>Ask students to keep in mind the different strategies as they work on the activities in the lesson.</p>