870e9ad7-c694-479c-885c-3e0bd62efdd0.html

<h3>Activity (5 minutes)</h3>
<p>This activity activates some familiar skills for writing and solving equations, which will be useful for specific tasks throughout the project.</p>
<p>It may have been a while since students thought about writing an equation for a line passing through two points. The two questions here are intentionally quite straightforward. Monitor for students taking different approaches, such as:</p>
<ul class="os-raise-noindent">
  <li> Plotting the points and determining the slope and \(y\)-intercept of a line passing through the points. </li>
  <li> Computing the slope by finding the quotient of the difference between the \(y\)-coordinates and difference between the \(x\)-coordinates. </li>
  <li> Considering what operation on each \(x\)-coordinate would produce its corresponding \(y\)-coordinate. </li>
</ul>
<p>Students have not yet solved a quadratic equation like the one in the second question, but they have learned and extensively practiced the skills needed to solve it. The two main anticipated approaches are:</p>
<ul class="os-raise-noindent">
  <li> Reasoning algebraically by performing the same operation to each side of the equation, applying the distributive property to expand factored expressions, combining like terms, rewriting an expression in factored form, and applying the zero product property. </li>
  <li> Graphing \(y=x+1\) and \(y=(x&minus;2)^2&minus;3\) and observing the \(x\)-coordinate of each point of intersection. </li>
</ul>
<h4>Launch</h4>
<ol><li>Encourage and support opportunities for peer interactions.</li> 
  <li>Invite students to talk about their ideas with a partner before writing them down.</li></ol>
<h4>Student Activity </h4>
<ol class="os-raise-noindent">
  <li>Write an equation representing the line that passes through each pair of points.</li>
</ol>
<ol class="os-raise-noindent" type="a">
  <li>\((3,3)\) and \((5,5)\)</li>
  <li>\((0,4)\) and \((-4,0)\)</li>
</ol>
<ol class="os-raise-noindent" start="2">
  <li>Solve this equation: \(x+1=(x&minus;2)^2&minus;3\). Show your reasoning.</li>
</ol>
<h4>Student Response</h4>
<ol class="os-raise-noindent">
  <li>Equations (or their equivalents):</li>
</ol>
<ol class="os-raise-noindent" type="a">
  <li>\(y=x\)</li>
  <li>\(y=x+4\)</li>
</ol>
<ol class="os-raise-noindent" start="2">
  <li>\(x=0\) and \(x=5\)</li>
</ol>
<p class="os-raise-noindent">\(\begin{array}{rcl}x+1\;&amp;=&amp;(x-2)^2-3\\x+1\;&amp;=&amp;x^2-4x+4-3\\x+1\;&amp;=&amp;x^2-4x+1\\x\;&amp;=&amp;x^2-4x\\0\;&amp;=&amp;x^2-5x\\0\;&amp;=&amp;x(x-5)\end{array}\\x=0\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;x=5\)</p>
<h4>Activity Synthesis</h4>
<p>Invite students taking different approaches to share their work. Ensure that students see more than one way to think about the equation representing a line for the first question. Reinforce that the second equation can be solved algebraically.</p>