4426cc2c-13ae-41b4-86fc-1dc631fb0df8.html

<h4>Activity (15 minutes)</h4>
<p>The goal of this activity is to illustrate that a perfect-square expression can take different forms, some of which may not look like \((something)\cdot(something)\) or \((something)^2\). The work here prompts students to recognize structure in perfect-square expressions, particularly when written in standard form, preparing them to complete the square in an upcoming lesson.</p>
<p>Students are given a series of expressions that are clearly perfect squares, for example, \((3x)^2\) or \((x+4)(x+4)\), and asked to rewrite them in standard form. As they repeatedly apply the distributive property to expand these expressions into standard form, students begin to recognize a pattern in how the two forms of expressions are related. They see that squaring an expression such as \((x+n)\) produces an expression in standard form in which the constant term is \(n^2\) and the linear term is \(2n\). Then, they use that insight to articulate why certain expressions in standard form (such as \(x^2&minus;16x+64\)) can be described as perfect squares.</p>
<h4>Launch</h4>
<p>Encourage and support opportunities for peer interactions. Invite students to talk about their ideas with a partner before writing them down.</p>
<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 3 Clarify, Critique, Correct: Reading, Writing, Speaking </p>
  </div>
  <div class="os-raise-extrasupport-body">
      <p>Before students share their explanations for the last question, present an incorrect answer and explanation. For example, “Some people may say they're perfect squares because they have a term that is a square.” Ask students to identify the error, critique the reasoning, and write a correct explanation. As students discuss with a partner, listen for students who identify and clarify the ambiguous language in the statement. For example, the author probably meant to say that each expression can be written as a squared linear expression. This will help students understand the language of “perfect squares” when referring to quadratic expressions and the relationship between the coefficients in standard and factored form.</p>
      <p class="os-raise-text-italicize">Design Principle(s): Optimize output (for explanation); Maximize meta-awareness</p>
  <p class="os-raise-extrasupport-title">Learn more about this routine</p>
      <p>
        <a href="https://www.youtube.com/watch?v=lozZJ21i3mQ;&rel=0" target="_blank">View the instructional video</a>
        and
        <a href="https://k12.openstax.org/contents/raise/resources/36e324584ae01e4eafc6674ac4ec02b4ae99002c" 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/76d1756428038b737f48dc515486cddceb99e2f8" 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>Activate or supply background knowledge. Represent the same information through different modalities by using a diagram to write each expression in expanded form. As students progress through the questions, invite them to notice how each side of their diagrams will be the same since the expressions are perfect squares. If students struggle on the last question, ask them to observe patterns of the entries in their diagrams (such as the lower left and upper right cells are the same) to help them diagram “in reverse,” starting from the standard form.</p>
      <p class="os-raise-text-italicize">Supports accessibility for: Conceptual processing; Visual-spatial processing</p>
</div>
</div>
<br>
<h4>Student Activity</h4>
<ol class="os-raise-noindent">
  <li>Each expression is written as the product of factors. Write an equivalent expression in standard form.</li>
</ol>
<ol class="os-raise-noindent" type="a">
  <li> \((3x)^2\) </li>
</ol>
<p class="os-raise-noindent">Write down your answer, then select the <strong>solution</strong> button to compare your work.</p>
<p class="os-raise-noindent"><strong>Answer: </strong>\(9x^2\)</p>
<ol class="os-raise-noindent" start="2" type="a">
  <li> \(7x \cdot 7x\) </li>
</ol>
<p class="os-raise-noindent">Write down your answer, then select the <strong>solution</strong> button to compare your work.</p>
<p class="os-raise-noindent"><strong>Answer:</strong> \(49x^2\)</p>
<ol class="os-raise-noindent" start="3" type="a">
  <li> \((x+4)(x+4)\) </li>
</ol>
<p class="os-raise-noindent">Write down your answer, then select the <strong>solution</strong> button to compare your work.</p>
<p class="os-raise-noindent"><strong>Answer:</strong> \(x^2+8x+16\)</p>
<ol class="os-raise-noindent" start="4" type="a">
  <li> \((x+1)^2\) </li>
</ol>
<p class="os-raise-noindent">Write down your answer, then select the <strong>solution</strong> button to compare your work.</p>
<p class="os-raise-noindent"><strong>Answer:</strong> \(x^2+2x+1\)</p>
<ol class="os-raise-noindent" start="5" type="a">
  <li> \((x&minus;7)^2\) </li>
</ol>
<p class="os-raise-noindent">Write down your answer, then select the <strong>solution</strong> button to compare your work.</p>
<p class="os-raise-noindent"><strong>Answer:</strong> \(x^2&minus;14x+49\)</p>
<ol class="os-raise-noindent" start="6" type="a">
  <li> \((x+n)^2\) </li>
</ol>
<p class="os-raise-noindent">Write down your answer, then select the <strong>solution</strong> button to compare your work.</p>
<p class="os-raise-noindent"><strong>Answer:</strong> \(x^2+2nx+n^2\)</p>
<ol class="os-raise-noindent" start="2">
  <li>Why do you think the following expressions can be described as perfect squares?</li>
</ol>
<p class="os-raise-noindent">\(x^2+6x+9\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;x^2&minus;16x+64\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;x^2+ \frac {1}{3}x+ \frac {1}{36}\)</p>
<p>Write down your answer, then select the <strong>solution</strong> button to compare your work.</p>
<p><strong>Answer:</strong> If these expressions are written in factored form, each one is the square of a linear expression:</p>
<p>\((x+3)^2\), \((x&minus;8)^2\), and \((x+\frac {1}{6})^2\).</p>
<h4>Student Facing Extension</h4>
<h5>Are you ready for more?</h5>
<p>Write each expression in factored form.</p>
<ol class="os-raise-noindent">
  <li>\(x^4&minus;30x^2+225\)</li>
</ol>
<p><strong>Answer:</strong> \((x^2&minus;15)^2\)</p>
<ol class="os-raise-noindent" start="2">
  <li>\(x+14\sqrt{x}+49\)</li>
</ol>
<p><strong>Answer:</strong> \((\sqrt{x}+7)^2\)</p>
<ol class="os-raise-noindent" start="3">
  <li>\(5^{2x}+6\cdot 5^x+9\)</li>
</ol>
<p><strong>Answer:</strong> \((5^x+3)^2\)</p>
<h4>Anticipated Misconceptions</h4>
<p>If students have trouble generalizing \((x+n)^2\) as \(x^2+2nx+n^2\) from working only with expressions, ask them to draw a rectangular diagram showing \(x\) and \(n\) along both sides of the rectangle and see if they can show on the diagram where the \(2n\) and \(n^2\) come from. (If some scaffolding is needed, consider starting with numbers, for example, \((10+4)^2\), then \((x+4)^2\), and then \((x+n)^2\).)</p>
<table class="os-raise-doubleheadertable">
  <thead>
    <tr>
      <th scope="col">&nbsp;</th>
      <th scope="col">
        \(x\)
      </th>
      <th scope="col">
        \(n\)
      </th>
    </tr>
  </thead>
  <tbody>
    <tr>
      <th scope="row">
        \(x\)
      </th>
      <td>
        \(x^2\)
      </td>
      <td>
        \(nx\)
      </td>
    </tr>
    <tr>
      <th scope="row">
        \(n\)
      </th>
      <td>
        \(nx\)
      </td>
      <td>
        \(n^2\)
      </td>
    </tr>
  </tbody>
</table>
<br>
<p>Additionally, many students will jump from \((x+n)^2\) to \(x^2+n^2\). It can be helpful to show students that this is not equivalent to the answer \(x^2+2nx+n^2\). Encourage students to always rewrite expressions of the form \((x+n)^2\) as \((x+n)(x+n)\).</p>
<h4>Activity Synthesis</h4>
<p>Invite students to share their responses and reasoning for the last question. Make sure students see the structure that relates the expression in standard form and its equivalent expression in factored form. Highlight that:</p>
<ul class="os-raise-noindent">
  <li> In each given example, the constant term is a number squared \((n^2)\), and the coefficient of the linear term is twice that number \((2n)\). </li>
  <li> This is also true for the expression that contains fractions: \(\frac {1}{36}\) is \((\frac {1}{6})^2\) and \(\frac {1}{3}\) is \(2\cdot \frac {1}{6}\). </li>
  <li> We call these expressions "perfect squares" because they can be written as something squared: \((x+3)^2\), \((x&minus;8)^2\), and \((x+ \frac {1}{6})^2\). </li>
  <li> In general, when \((x+n)\) is squared and expanded, we have: \(x^2+2nx+n^2\). </li>
</ul>
<p>Tell students that knowing about quadratic expressions that are perfect squares can help us solve all kinds of equations in upcoming lessons.</p>
<h3>9.1.2: 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>Which of the following is equivalent to \((x-4)^2\)?</p>
        <table class="os-raise-textheavytable">
<thead>
    <tr>
      <th scope="col">
        Answers
      </th>
      <th scope="col">
        Feedback
      </th>
    </tr>
  </thead>
  <tbody>
    <tr>
      <td>
        <p>\(x^2-16\)</p>
      </td>
      <td>
        <p>Incorrect. Let's try again a different way: Make sure to remember \((x-4)^2=(x-4)(x-4)\). The answer is \(x^2-8x+16\).</p>
      </td>
    </tr>
    <tr>
      <td>
        <p>\(x^2+16\)</p>
      </td>
      <td>
        <p>Incorrect. Let's try again a different way: Make sure to remember \((x-4)^2=(x-4)(x-4)\). The answer is \(x^2-8x+16\).</p>
      </td>
    </tr>
    <tr>
      <td>
        <p>\(x^2-8x+16\)</p>
      </td>
      <td>
        <p>That's correct! Check yourself: \((x-4)^2=(x-4)(x-4)\). Then use the distributive property (FOIL).</p>
      </td>
    </tr>
    <tr>
      <td>
        <p>\(x^2+8x+16\)</p>
      </td>
      <td>
        <p>Incorrect. Let's try again a different way: When multiplying \((x - 4)(x - 4)\), the product gives a \(-4x + -4x\). The answer is \(x^2-8x+16\).</p>
      </td>
    </tr>
  </tbody>
</table>
<br>
<h3>9.1.2: 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>Creating Perfect Square Trinomials</h4>
<p>A perfect square trinomial is created by multiplying a binomial by itself.</p>
<p class="os-raise-text-bold">Example</p>
<p>Write \((x-2)^2\) as an expanded polynomial in standard form:</p>
<p><strong>Step 1</strong> - Multiply the binomial by itself.<br>
  \((x-2)(x-2)\)</p>
<p><strong>Step 2</strong> - Use the distributive property to expand (FOIL).<br>
  \(x^2-2x-2x+4\)</p>
<p><strong>Step 3</strong> - Combine like terms.<br>
  \(x^2-4x+4\)</p>
<h4>Try It: Creating Perfect Square Trinomials</h4>
<p>Write \((x+5)^2\) as an expanded polynomial in standard form.</p>
<p>Write down your answer, then select the <strong>solution</strong> button to compare your work.</p>
<h5> Solution</h5>
<p>Here is how to expand the polynomial:</p>
<p><strong>Step 1</strong> - Multiply the binomial by itself.<br>
  \((x+5)(x+5)\)</p>
<p><strong>Step 2</strong> - Use the distributive property (FOIL).<br>
  \(x^2 +5x+5x+25\)</p>
<p><strong>Step 3</strong> - Combine like terms.<br>
  \(x^2+10x+25\)</p>