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<h4>Lesson Narrative </h4>
<p>This lesson continues to examine quantities that change exponentially, focusing on a quantity that decays or
decreases. Students examine more situations with quantities that decrease exponentially. They work from an equation to
a graph and from a graph to an equation. In both cases, they interpret the different parts of their equation in terms
of the situation and use the graph to answer questions.</p>
<p>Like many activities in this unit, the equations and graphs represent actual quantities (the area covered by algae
and the amount of insulin in a person’s body) and are to be interpreted in context. They also use discrete
graphs to depict measurements of quantities that vary continuously over time and answer questions using these graphs.
In the following lessons, we will represent situations where the domain is all real numbers with a continuous graph.
</p>
<p>One activity encourages students to view a quantity that decreases by a factor of itself using multiplication rather
than subtraction. If a computer costs $500 and loses \(\frac15\) of its value each year, after one year we could write
this, in dollars, as \(500−(\frac{1}{5}) \cdot 500\). But if we write this using multiplication, as \(500\cdot
(\frac{4}{5})\), then we are in a better position to see that after 2 years its value in dollars will be \(500 \cdot
(\frac{4}{5}) \cdot (\frac{4}{5})\), and after \(t\) years, the value will be \(500 \cdot (\frac{4}{5})^t\). In other
words, exponents are a particularly useful way to express repeated loss by a factor of the original amount. Students
will carry this understanding into future lessons that deal with repeated percentage change situations.</p>
<p>Technology isn’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>
<p>Students will be able to:</p>
<ul>
<li> Comprehend that the term “exponential growth” describes a quantity that changes by a growth factor
that is greater than 1, and the term “exponential decay” describes a quantity that changes by a growth
factor that is less than 1 but greater than 0. </li>
<li> Use only multiplication to represent “decreasing a quantity by some fraction of itself.”</li>
<li> Write an expression or an equation to represent a situation where a quantity decays exponentially. </li>
<li> Understand that the terms “growth factor” and “constant ratio” are interchangeable. </li>
<li> Calculate growth factor using points on a graph that represents exponential decay. </li>
<li> Graph equations that represent quantities that change by a growth factor between 0 and 1. </li>
<li> Interpret equations and graphs that represent exponential decay situations. </li>
</ul>
<h4>Learning Targets (Student Facing) </h4>
<ul>
<li> Use only multiplication to represent “decreasing a quantity by a fraction of itself.”</li>
<li> Write an expression or equation to represent a quantity that decays exponentially. </li>
<li> Know the meanings of “exponential growth” and “exponential decay.”</li>
<li>Explain the meanings of \(a\) and \(b\) in an equation that represents exponential decay and is written as \(y=a
\cdot b^x\).</li>
<li> Find a growth factor from a graph and write an equation to represent exponential decay. </li>
<li> Graph equations that represent quantities that change by a growth factor between 0 and 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>
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>
A1(G) display, explain, and justify mathematical ideas and arguments using precise mathematical language in written or oral communication
</td>
<td>
Math process coverage: Lesson provides content that supports this TEKS.
</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</u> growth and <u>decay 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> 5.5.1: Identifying Linear and Exponential Quantities </li>
<li> 5.5.2: Exponential Decay </li>
<ul>
<li> 5.5.2: Self Check </li>
<li> 5.5.2: Additional Resources </li>
</ul>
<li> 5.5.3: Using Graphs to Represent Exponential Decay </li>
<ul>
<li> 5.5.3: Self Check </li>
<li> 5.5.3: Additional Resources </li>
</ul>
<li> 5.5.4: Justifying Exponential Models </li>
<ul>
<li> 5.5.4: Self Check </li>
<li> 5.5.4: Additional Resources </li>
</ul>
<li> 5.5.5: Writing Equations to Represent Exponential Decay </li>
</ul>
<p>Students will also complete a series of problems in the 5.5.6: Practice.</p>
<h4>Required Materials</h4>
<p>None</p>
<h4>Required Preparation</h4>
<p>If possible, acquire devices that can run Desmos (recommended) or other graphing technology as an option for students
to select during the lesson.</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 “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>
<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>growth factor </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>
<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 audio tape, video, DVD, and CD ROM to build and reinforce concept and language attainment</li>
<li>ELPS 3(I) adapt spoken language appropriately for formal and informal purposes</li>
<li>ELPS 4(D) use prereading supports such as graphic organizers, illustrations, and pretaught topic- related vocabulary and other prereading activities to enhance comprehension of written text </li>
<li>ELPS 4(G) demonstrate comprehension of increasingly complex English by participating in shared reading, retelling or summarizing material, responding to questions, and taking notes commensurate with content area and grade level needs</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/success-story-tips/" target="_blank">Success Story</a>
</li>
<li> <a href="https://characterlab.org/tips-of-the-week/sample-then-specialize/" target="_blank">Sample, Then
Specialize</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>