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<h4>Lesson Narrative</h4>
<p>The goal of this lesson is to encounter two different growth patterns—one pattern is linear, and the other is
exponential, though students don’t need to use those words yet. Students think about and compare the patterns by
performing calculations and using graphs. This lesson contains many opportunities for students to notice and make use
of structure, for example, noticing that \(1,000+200+200+200+200\) can be expressed as \(1,000+200 \cdot 4\), which is
useful when they must add on more and more 200s. There is also an opportunity to use appropriate tools strategically,
for example, if students choose to use a spreadsheet to perform many iterations of such calculations.</p>
<p>They see that the pattern that grows by repeatedly doubling starts off slowly but eventually overtakes the other
pattern, which increases by repeatedly adding the same amount. In fact, the first pattern eventually leaves the second
pattern far behind. It is important that the teacher makes the connections between linear and exponential growth and
the real world. Students need to understand how exponential growth can affect them financially through debts and
investments, and how it’s always important to compare different offers given.</p>
<p>Students continue to examine two types of patterns by looking at tables, noting that one type is characterized by
common differences and the other by common factors. They then match different representations with two contexts. This
is an opportunity for students to notice structure in expressions. Note we are still not expecting students to use the
words <em>linear</em> or <em>exponential</em> to describe these patterns, though they may naturally use the word
<em>linear</em> to describe linear growth since they have encountered these types of functions before.</p>
<p>Throughout the unit, students will study exponential patterns systematically (eventually viewing them as functions)
before returning to this comparison with linear functions toward the end of the unit.</p>
<p>Some technology is required for this lesson, but there are opportunities for students to select appropriate
technology in the lesson. We recommend making scientific calculators and spreadsheet technology available.</p>
<h4>Learning Goals (Teacher Facing)</h4>
<ul>
<li> Compare linear and exponential relationships by performing calculations and by interpreting graphs that show two
growth patterns. </li>
<li> Describe (using words and expressions) patterns in tables that represent linear and exponential relationships.
</li>
<li> Given descriptions of linear and exponential relationships, create tables of values and write expressions. </li>
</ul>
<h4> Learning Targets (Student Facing)</h4>
<ul>
<li> Compare growth patterns using calculations and graphs. </li>
<li> Use words and expressions to describe patterns in tables of values. </li>
<li> Write expressions and create tables of values to represent them using descriptions of linear and exponential
relationships. </li>
<li> Begin making connections between linear and exponential growth and the real world (especially financially).</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(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(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>
Foundational: Lesson provides foundational content necessary for students to be able to perform this TEKS in later lessons.
</td>
</tr>
</tbody>
</table>
<br>
<h4>Lesson Activities</h4>
<p>Here are the instructional activities for the lesson:</p>
<ul>
<li> 5.3.1: Creating a Pictorial Representation of Exponential Growth </li>
<li> 5.3.2: Exploring Different Growth Patterns </li>
<ul>
<li> 5.3.2: Self Check </li>
<li> 5.3.2: Additional Resources </li>
</ul>
<li> 5.3.3: Compare and Contrast Two Patterns </li>
<ul>
<li> 5.3.3: Self Check </li>
<li> 5.3.3: Additional Resources </li>
</ul>
<li> 5.3.4: Match Expressions and Tables with Situations </li>
<ul>
<li> 5.3.4: Self Check </li>
<li> 5.3.4: Additional Resources </li>
</ul>
<li> 5.3.5: Describe Mathematically Changes in Growth </li>
</ul>
<p>Students will also complete a series of problems in the 5.3.6: Practice.</p>
<h4> Required Materials</h4>
<ul>
<li> Scientific calculators </li>
<li> Spreadsheet technology </li>
</ul>
<h4> Required Preparation</h4>
<p>Acquire scientific calculators or devices that can run GeoGebra (recommended) or other spreadsheet technology. It is
ideal if each student has their own device. (GeoGebra Spreadsheet is available <a
href="https://www.geogebra.org/m/v4fk7gua" target="_blank">here</a>)</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 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> common difference </li>
<li> common ratio </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 audiotape, video, DVD, and CD-ROM to build and reinforce concept and language attainment
</li>
<li>ELPS 2(I) demonstrate listening comprehension of increasingly complex spoken English by following directions, retelling or summarizing spoken messages, responding to questions and requests, collaborating with peers, and taking notes commensurate with content and grade-level needs</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>
<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/v-for-vulnerability/" target="_blank">V Is for
Vulnerability</a> </li>
<li> <a href="https://characterlab.org/tips-of-the-week/resilience/" target="_blank">Resilience </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>