2a0a7e9e-7222-48ae-beab-48fc3fa2b86a.html

<h4>Lesson Narrative</h4>
<p>In this lesson, students study a situation characterized by exponential change and learn the term <span><em>growth factor</em></span> or <span><em>constant ratio</em></span>. They represent this relationship using a table, an expression, and a graph. They also explain the meaning of the numbers \(a\) and \(b\)&nbsp;in an exponential expression \(a\cdot b^x\), identifying their meaning in terms of a context (\(a\)&nbsp;is the initial amount, and \(b\)&nbsp;is the multiplier or growth factor) and also in terms of a graph (where \(a\)&nbsp;is the vertical intercept and \(b\)&nbsp;determines how quickly the graph increases). Students interpret the different representations of growth in terms of a bacteria population.</p>
<p>In this and following lessons, students will often work with properties of exponents, a topic developed in grade 8. There is also an activity to emphasize the convention that \(a^0=1\) for a non-zero number \(a\).</p>
<p>Technology isn&rsquo;t required for this lesson, but there are opportunities for students to choose to use appropriate technology to solve problems. We recommend making technology available.</p>
<h4>Learning Goals (Teacher Facing)</h4>
<ul>
  <li> Explain (in writing) how to see \(a\) and \(b\) on the graph of an equation of the form \(y=a\cdot b^x\). </li>
  <li> Interpret \(a\) and \(b\) given equations of the form \(y=a\cdot b^x\) and a context of exponential growth. </li>
  <li> Write an equation of the form of \(y=a\cdot b^x\) to represent a quantity \(a\) that changes by a growth factor \(b\). </li>
</ul>
<h4>Learning Targets (Student Facing)</h4>
<ul>
  <li> Explain the connections between an equation and a graph that represents exponential growth. </li>
  <li> Write and interpret an equation that represents exponential growth. </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>A1(D) communicate mathematical ideas, reasoning, and their implications using multiple representations, including symbols, diagrams, graphs, and language as appropriate </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>A9(A) <u>determine the domain and range of exponential functions of the form</u> \(f(x)=ab^x\) <u>and represent the domain and range</u> using inequalities </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(B) interpret the meaning of the values of  \(a\) and \(b\) in exponential functions of the form 
        \(f(x)=ab^x\) in real-world problems </td>
      <td>Full coverage: Lesson provides content that covers all parts of this TEKS. </td>
    </tr>
    <tr>
      <td>A9(C) write exponential functions in the form  
        \(f(x)=ab^x\) (where \(b\)  is a rational number) to describe problems arising from mathematical and real-world situations, including growth and decay </td>
      <td>Full coverage: Lesson provides content that covers all parts of this TEKS. </td>
    </tr>
    <tr>
      <td>A9(D) <u>graph exponential functions that model growth</u> and decay <u>and identify key features, including \(y\)-intercept</u> and asymptote, <u>in mathematical and real-world problems</u></td>
      <td>Partial coverage: Lesson provides content that covers part of this TEKS. The parts that are covered have been underlined. </td>
    </tr>
  </tbody>
</table>
<br>
<h4>Lesson Activities</h4>
<p>Here are the instructional activities for the lesson:</p>
<ul>
  <li> 5.4.1: Exponent Rules </li>
  <li> 5.4.2: Zero Exponent Rule
    <ul>
      <li> 5.4.2: Self Check </li>
      <li> 5.4.2: Additional Resources </li>
    </ul>
  </li>
  <li> 5.4.3: Exponential Change: The Growth Factor
    <ul>
      <li> 5.4.3: Self Check </li>
      <li> 5.4.3: Additional Resources </li>
    </ul>
  </li>
  <li> 5.4.4: Graphing Exponential Expressions
    <ul>
      <li> 5.4.4: Self Check </li>
      <li> 5.4.4: Additional Resources </li>
    </ul>
  </li>
  <li> 5.4.5: Interpret the Growth Factor of the Equation </li>
</ul>
<p>Students will also complete a series of problems in the 5.4.6: Practice.</p>
<h4>Required Materials</h4>
<p>None</p>
<h4>Required Preparation</h4>
<p>Acquire devices that can run Desmos (recommended) or other graphing technology as an optional tool for students.</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> none </li>
        </ul></td>
      <td><ul>
          <li> growth ratio or growth factor </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 audio tape, video, DVD, and CD ROM to build and reinforce concept and language attainment </li>
    <li>ELPS 3(B) expand and internalize initial English vocabulary by learning and using high-frequency English words necessary for identifying and describing people, places, and objects, by retelling simple stories and basic information represented or supported by pictures, and by learning and using routine language needed for classroom communication </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/me-to-me/" target="_blank">Me to Me</a> </li>
  <li> <a href="https://characterlab.org/tips-of-the-week/practice-makes-perseverance/" target="_blank">Practice Makes Perseverance</a> </li>
</ul>
<p>You can find other tips located here in the <a href="https://characterlab.org/playbooks/grit/" target="_blank">Playbook on Grit</a> from Character Lab.</p>