f89e0577-e09a-4abd-a8ca-671d2df38749.html

<h4>Lesson Narrative</h4>
<p>Up to this point, most quadratic expressions that students have transformed from standard form to factored form had a
  leading coefficient of 1; that is, they were in the form of \(x^2+ bx + c\) because the squared term had a coefficient
  of 1. There were a few instances in which students rewrote expressions in standard form with a leading coefficient
  other than 1. Those expressions were differences of two squares, where there were no linear terms (for instance,
  \(9x^2&minus; 64\) or \(25x^2&minus; 9\)). Students learned to rewrite these as \((3x + 8)(3x &minus; 8)\) or \((5x +
  3)(5x &minus; 3)\), respectively.</p>
<p>In this lesson, students consider how to rewrite expressions in standard form where the leading coefficient is not 1
  and the expression is not a difference of two squares. They notice that the same structure used to rewrite \(x^2+ 5x +
  4\) as \((x + 4)(x + 1)\) can be used to rewrite expressions such as \(3x^2+ 8x + 4\), but the process is now a little
  more involved because the coefficient of \(x^2\) has to be taken into account when finding the right pair of factors.
</p>
<p>This lesson aims to give students a flavor of rewriting more complicated expressions in factored form, and to suggest
  that it is not always practical or possible. This experience motivates the need for other strategies for solving
  equations and prepares students to complete the square in a series of upcoming lessons.</p>
<h4>Learning Goals (Teacher Facing)</h4>
<ul class="os-raise-noindent">
  <li> Write an equivalent expression in factored form given a quadratic expression of the form \(ax^2+ bx + c\), where
    \(a\) is not 1. </li>
  <li> Write a quadratic equation that represents a context, consider different methods for solving it, and describe
    (orally) the limitations of each method. </li>
</ul>
<h4>Learning Targets (Student Facing)</h4>
<ul class="os-raise-noindent">
  <li> Use the factored form of a quadratic expression or a graph of a quadratic function to answer questions about a
    situation. </li>
  <li> Write equivalent expressions in factored form when given quadratic expressions of the form \(ax^2+ bx + c\) and
    \(a\) is not 1. </li>
</ul>
<table class="os-raise-textheavytable">
  <caption>Texas Essential Knowledge and Skills (TEKS)</caption>
  <thead>
    <tr>
      <th scope="col">
        TEKS
      </th>
      <th scope="col">
        Explanation of Coverage
      </th>
    </tr>
  </thead>
  <tbody>
    <tr>
      <td>A1(A) apply mathematics to problems arising in everyday life, society, and the workplace</td>
      <td>Math process coverage: Lesson provides content that supports this TEKS.</td>
    </tr>
    <tr>
      <td>
        A8(A) <u>solve quadratic equations having real solutions by factoring,</u> taking square roots, completing the
        square, and applying the quadratic formula.
      </td>
      <td>
        <p>Partial coverage: Lesson provides content that covers part of this TEKS. The parts that are covered have been
          underlined.</p>
      </td>
    </tr>
  </tbody>
</table>
<br>
<h4>Lesson Activities</h4>
<p>Here are the instructional activities for the lesson:</p>
<ul class="os-raise-noindent">
  <li> 8.10.1: Analyzing Various Quadratic Expressions </li>
  <li> 8.10.2: Working with Quadratic Factored Form with Leading Coefficient Other Than One </li>
  <ul class="os-raise-noindent">
    <li> 8.10.2: Self Check </li>
    <li> 8.10.2: Additional Resources </li>
  </ul>
  <li> 8.10.3: Using Technology to Find Rational Factors </li>
  <ul class="os-raise-noindent">
    <li> 8.10.3: Self Check </li>
    <li> 8.10.3: Additional Resources </li>
  </ul>
  <li> 8.10.4: Finding the Factors of Quadratic Expressions in Standard Form </li>
  <ul class="os-raise-noindent">
    <li> 8.10.4: Self Check </li>
    <li> 8.10.4: Additional Resources </li>
  </ul>
  <li> 8.10.5: Solving Quadratic Equations by Any Method </li>
</ul>
<p>Students will also complete a series of problems in the 8.10.6: Practice.</p>
<h4>Required Materials</h4>
<ul class="os-raise-noindent">
  <li> Paper and pencil </li>
  <li> Graphing technology </li>
</ul>
<h4>Required Preparation</h4>
<ul class="os-raise-noindent">
  <li> Acquire devices that can run Desmos (recommended) or other graphing technology. It is ideal if each student has
    their own device. </li>
</ul>
<h4>Lesson Vocabulary</h4>
<p>During this lesson, it is important to:</p>
<ul class="os-raise-noindent">
  <li> Familiarize students with the vocabulary words they will see throughout the lesson. </li>
  <li> Encourage students to look for these words and notice their use and meanings. </li>
  <li> Encourage students to use key vocabulary words in "math talk" and their written and oral explanations. </li>
  <li> Utilize a word wall. Sample cards are located here: <a
      href="https://k12.openstax.org/contents/raise/resources/8ed3aa68763498713b44d3c2e537ffe61b4755f0"
      target="_blank">Mathematics Vocabulary Word Wall Cards</a>. </li>
</ul>
<p>Vocabulary words that are emphasized in this lesson:</p>
<br>
<table class="os-raise-textheavytable">
  <thead>
    <tr>
      <th scope="col">
        Previous Vocabulary
      </th>
      <th scope="col">
        New Vocabulary
      </th>
    </tr>
  </thead>
  <tbody>
    <tr>
      <td>
        <ul>
          <li> quadratic expression </li>
          <li> quadratic equation </li>
          <li> quadratic function </li>
          <li> factored form (of a quadratic expression) </li>
          <li> standard form (of a quadratic expression) </li>
          <li> variable </li>
          <li> coefficient </li>
          <li> linear term </li>
          <li> root </li>
          <li> zero product property </li>
          <li> zero of the function </li>
        </ul>
      </td>
      <td>
        <ul>
          <li> none </li>
        </ul>
      </td>
    </tr>
  </tbody>
</table>
<br>
<p>To support newcomers or students identified at the beginning level of language proficiency, share the following Quizlet links to help students gain an understanding of the academic vocabulary. Use the Spanish versions to anchor student understanding before bridging to the English versions. </p>
<ul>
  <li><a href="https://quizlet.com/881785079/raise-unit-8-spn-vocabulary-flash-cards/?i=5eauv9&x=1jqt" target="_blank">Unit 8 Spanish Vocabulary</a></li>
  <li><a href="https://quizlet.com/881565975/raise-unit-8-vocabulary-flash-cards/?i=5eauv9&x=1jqt" target="_blank">Unit 8 Vocabulary</a></li>
</ul>
<h4>Support for English Language Learners</h4>
    <p>Throughout this lesson, activities are incorporated that align to the following ELPS. The suggested activities are
  only a sampling of the types of support and scaffolding that can extend the learning for English language learners.
  Continue to find additional opportunities as you build your own set of ELL learning routines.</p>
<ul>
  <li>ELPS 2(F) listen to and derive meaning from a variety of media such as audiotape, video, DVD, and CD-ROM to build and
    reinforce concept and language attainment</li>
  <li>ELPS 3(E) share information in cooperative learning interactions
  </li>
  <li>ELPS 3(H) narrate, describe, and explain with increasing specificity and detail as more English is acquired
  </li>
  <li>ELPS 4(F) use visual and contextual support and support from peers and teachers to read grade-appropriate content area
    text, enhance and confirm understanding, and develop vocabulary, grasp of language structures, and background
    knowledge needed to comprehend increasingly challenging language
  </li>
</ul>
<br>
<h4>Support for Building Character</h4>
<p>Throughout this unit, find ways to encourage and support students to work on cultivating their curiosity.</p>
<p>Here are some tips to try during this lesson:</p>
<ul class="os-raise-noindent">
  <li> <a href="https://characterlab.org/tips-of-the-week/letting-kids-drive/" target="_blank">Letting Kids Drive</a>
  </li>
  <li> <a href="https://characterlab.org/tips-of-the-week/active-learning/" target="_blank">A Lesson in Active
      Learning</a> </li>
</ul>
<p>You can find other tips located here in the <a href="https://characterlab.org/playbooks/curiosity/"
    target="_blank">Playbook on Curiosity</a> from Character Lab.</p>