b7c7c2fb-046f-4c1c-b82e-3e7dfae05a21.html

<h4> Lesson Narrative&nbsp;</h4>
<p>In this lesson, students analyze the graph of an exponential function \(f\) given by \(f(x)=a \cdot b^x\). In
  particular, they study the effect of \(b\)&nbsp;on the shape of the graph. To observe how \(b\)&nbsp;impacts the shape
  of the graph when \(b&gt;1\) and when \( 0&lt;b&lt;1\), they simultaneously examine several functions of this form,
  all of which have the same value of \(a\).&nbsp;</p>
<p>Narrow contexts are used in these lessons to encourage greater attention to the growth factors of the different
  functions, rather than on interpreting the quantities in context. Since students are focusing mainly on understanding
  how varying \(b\)&nbsp;impacts the graph, they are looking at the structure of an exponential function and its graph.
  Connecting exponential functions to real-world situations helps students understand why it is important to learn about
  this type of function.</p>
<p>One of the activities in this lesson works best when each student has access to devices that can run a Desmos applet
  because students will benefit from seeing the relationship in a dynamic way.</p>
<h4>Learning Goals (Teacher Facing)&nbsp;</h4>
<p>Students will be able to:</p>
<ul>
  <li> Describe (orally and in writing) the effect of changing \(a\) and \(b\) on a graph that represents \(f(x)=a \cdot
    b^x\). </li>
  <li> Use equations and graphs to compare exponential functions. </li>
</ul>
<h4> Learning Targets (Student Facing)&nbsp;</h4>
<ul>
  <li> Describe the effect of changing \(a\) and \(b\) on a graph that represents \(f(x)=a \cdot b^x\). </li>
  <li> Use equations and graphs to compare exponential functions. </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(C) select tools, including real objects, manipulatives, paper and pencil, and technology as appropriate, and
        techniques, including mental math, estimation, and number sense as appropriate, to solve problems
      </td>
      <td>
        Math process coverage: Lesson provides content that supports this TEKS.
      </td>
    </tr>
    <tr>
      <td>
        A1(E) create and use representations to organize, record, and communicate mathematical ideas
      </td>
      <td>
        Math process coverage: Lesson provides content that supports this TEKS.
      </td>
    </tr>
    <tr>
      <td>
        A1(F) analyze mathematical relationships to connect and communicate mathematical ideas
      </td>
      <td>
        Math process coverage: Lesson provides content that supports this TEKS.
      </td>
    </tr>
    <tr>
      <td>
        A3(E) determine the effects on the graph of the parent function
        \(f(x) = x\) when \(f(x)\) is replaced by \(af(x), f(x) + d, f(x - c)\), \(f(bx)\) for specific values of \(a\), \(b\), \(c\), and \(d\)
      </td>
      <td>
        Full coverage: Lesson provides content that covers all parts of this TEKS.
      </td>
    </tr>
    <tr>
      <td>
        A9(C) <u>write exponential functions in the form</u>
        \(f(x)=ab^x\) <u>(where \(b\) is a rational number) to describe problems arising from</u> mathematical <u>and real-world
        situations, including</u> growth and <u>decay</u>
      </td>
      <td>
        Partial coverage: Lesson provides content that covers part of this TEKS. The parts that are covered have been
        underlined.
      </td>
    </tr>
    <tr>
      <td>
        A9(D) graph exponential functions that model growth and decay and identify key features, including
        \(y\)-intercept and asymptote, in mathematical and real-world problems
      </td>
      <td>
        Full coverage: Lesson provides content that covers all parts of this TEKS.
      </td>
    </tr>
  </tbody>
</table>
<br>
<h4> Lesson Activities</h4>
<p>Here are the instructional activities for the lesson:</p>
<ul>
  <li> 5.12.1: Modeling Exponential Decay </li>
  <li> 5.12.2: Equations and Their Graphs <ul>
      <li> 5.12.2: Self Check</li>
    </ul>
    <ul>
      <li> 5.12.2: Additional Resources</li>
    </ul>
  </li>
  <li> 5.12.3: Graphs Representing Exponential Decay <ul>
      <li> 5.12.3: Self Check</li>
    </ul>
    <ul>
      <li> 5.12.2: Additional Resources</li>
    </ul>
  </li>
  <li> 5.12.4: Possible Equation for a Function on a Graph </li>
</ul>
<p>Students will also complete a series of problems in 5.12.5: Practice.</p>
<h4>Required Materials</h4>
<p>Graphing technology</p>
<h4>Required Preparation</h4>
<p>Students should have access to Desmos (preferred) or other graphing technology.</p>
<h4>Lesson Vocabulary</h4>
<p>During this lesson, it is important to:</p>
<ul>
  <li> Familiarize students with the vocabulary words they will see throughout the lesson. Encourage students to look
    for these words and notice their use and meanings. </li>
  <li> Encourage students to use key vocabulary words in &ldquo;math talk&rdquo; 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 include:</p>
<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> exponential function </li>
          <li> exponential decay </li>
          <li> exponential growth </li>
          <li> asymptote </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/881785671/raise-unit-5-spn-vocabulary-flash-cards/?i=5eauv9&x=1jqt" target="_blank">Unit 5 Spanish Vocabulary</a></li>
  <li><a href="https://quizlet.com/881507407/raise-unit-5-vocabulary-flash-cards/?i=5eauv9&x=1jqt" target="_blank">Unit 5 Vocabulary</a></li>
</ul>
<br>
<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 1(E) internalize new basic and academic language by using and reusing it in meaningful ways in speaking and
    writing activities that build concept and language attainment</li>
  <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(F) ask and give information ranging from using a very limited bank of high-frequency, high-need, concrete
    vocabulary, including key words and expressions needed for basic communication in academic and social contexts, to
    using abstract and content-based vocabulary during extended speaking assignments</li>
  <li>ELPS 3(G) express opinions, ideas, and feelings ranging from communicating single words and short phrases to
    participating in extended discussions on a variety of social and grade-appropriate academic topics</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 <strong>grit</strong>.
</p>
<p>Here are some tips to try during this lesson:</p>
<ul>
  <li> <a href="https://characterlab.org/tips-of-the-week/what-matters/" target="_blank">What Matters</a> </li>
  <li> <a href="https://characterlab.org/tips-of-the-week/hard-fun/" target="_blank">Hard Fun</a> </li>
</ul>
<p>You can find other tips located here in the&nbsp;<a href="https://characterlab.org/playbooks/grit/"
    target="_blank">Playbook on Grit</a> from Character Lab.</p>