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9e749b17-0286-48dd-b1b8-50d6c898bfea.html
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<h4>Lesson Narrative</h4>
<p>In a previous lesson, students recalled that a quadratic expression in vertex form can help us identify the vertex of a graph of a quadratic function. They then used completing the square to rewrite expressions from both standard and factored forms into vertex form. In this lesson, they use the vertex form to determine the maximum or minimum value of a function and to solve problems.</p>
<p>This is not the first time that students find a maximum or minimum value of a quadratic function. In an earlier unit on quadratic functions, students had a brief encounter with this idea, including the use of the vertex form to determine a maximum or minimum.</p>
<p>At that time, however, students did not yet know how to rewrite expressions in vertex form, so they could only use an expression to determine maximum or minimum if the given expression is already in vertex form. (Otherwise, students would have had to graph the expression or analyze a table of values.) Now that they can rewrite a given expression into vertex form, students can find a maximum or minimum of a function regardless of form and solve new kinds of problems.</p>
<p>An increased emphasis on using the structure of the vertex form to explain maximums and minimums also distinguishes the work in this lesson from earlier work. Previously, students may have relied on their observation of graphs, or recalling that the graph of an equation \(y=a(x−h)^2+k\) opens upward when \(a\) is positive and downward when \(a\) is negative. Here, they reason that:</p>
<ul class="os-raise-noindent">
<li> \((x−h)^2\) is always zero or positive. </li>
<li> When \(x=h\), the expression \((x−h)^2\) is 0 because \(0^2=0\). When \(x\) is any other value, the expression has a value greater than 0. </li>
<li> When \(a\) is positive, \(a(x−h)^2\) is positive except when \(x=h\) (at which point it is 0). This means 0 is the lowest possible value. </li>
<li> When \(a\) is negative, \(a(x−h)^2\) is negative except when \(x=h\) (at which point it is 0). This means 0 is the highest possible value. </li>
</ul>
<p>As students reason about and explain why a vertex is a maximum or a minimum, they practice constructing logical arguments and being precise in their communication.</p>
<h4>Learning Goals (Teacher Facing)</h4>
<ul class="os-raise-noindent">
<li> Explain (orally and in writing) how an expression in vertex form can show whether the vertex of a graph represents the maximum or minimum of a quadratic function. </li>
<li> Rewrite a quadratic expression in vertex form to identify the maximum or minimum value of the function the expression defines. </li>
<li> Use the structure of a quadratic expression in vertex form to determine whether the vertex of its graph represents the maximum or minimum of the quadratic function. </li>
</ul>
<h4>Learning Targets (Student Facing)</h4>
<ul class="os-raise-noindent">
<li> Find the maximum or minimum of a function by writing the quadratic expression that defines it in vertex form. </li>
<li> Explain why the vertex is a maximum or minimum when given a quadratic function in vertex form. </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><p>A1(G) display, explain, and justify mathematical ideas and arguments using precise mathematical language in written or oral communication</p></td>
<td><p>Math process coverage: Lesson provides content that supports this TEKS.</p></td>
</tr>
<tr>
<td><p>A7(A) graph quadratic functions on the coordinate plane and <u>use the graph to identify key attributes, if possible, including</u> \(x\)-intercept, \(y\)-intercept, zeros, <u>maximum value, minimum values, vertex</u>, and the equation of the axis of symmetry.</p></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>
<tr>
<td><p>A8(A) <u>solve quadratic equations having real solutions by</u> factoring, taking square roots, completing the square, and <u>applying the quadratic formula.</u></p></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> 9.11.1: Maximum and Minimum Value of a Function </li>
<li> 9.11.2: Does the Vertex Represent the Minimum or Maximum Value? </li>
<ul class="os-raise-noindent">
<li> 9.11.2: Self Check </li>
<li> 9.11.2: Additional Resources </li>
</ul>
<li> 9.11.3: Comparing Maximums between Quadratics </li>
<ul class="os-raise-noindent">
<li> 9.11.3: Self Check </li>
<li> 9.11.3: Additional Resources </li>
</ul>
<li> 9.11.4: Maximum, Minimum, and Analyzing the Vertex </li>
</ul>
<p>Students will also complete a series of problems in the 9.11.5: Practice.</p>
<h4>Required Materials</h4>
<p> None </p>
<h4>Required Preparation</h4>
<p> None </p>
<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>
<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> maximum </li>
<li> minimum </li>
<li> parabola </li>
<li> vertex (of a graph) </li>
<li> vertex form (of a quadratic expression) </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/881784766/raise-unit-9-spn-vocabulary-flash-cards/?i=5eauv9&x=1jqt" target="_blank">Unit 9 Spanish Vocabulary</a></li>
<li><a href="https://quizlet.com/881567564/raise-unit-9-vocabulary-flash-cards/?i=5eauv9&x=1jqt" target="_blank">Unit 9 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 1(B) monitor oral and written language production and employ self-corrective techniques or other resources </li>
<li> ELPS 2(A) distinguish sounds and intonation patterns of English with increasing ease </li>
<li> ELPS 2(B) recognize elements of the English sound system in newly acquired vocabulary such as long and short vowels, silent letters, and consonant clusters </li>
<li> ELPS 3(A) practice producing sounds of newly acquired vocabulary such as long and short vowels, silent letters, and consonant clusters to pronounce English words in a manner that is increasingly comprehensible </li>
<li> ELPS 4(A) learn relationships between sounds and letters of the English language and decode (sound out) words using a combination of skills such as recognizing sound-letter relationships and identifying cognates, affixes, roots, and base words </li>
<li> ELPS 5(A) learn relationships between sounds and letters of the English language to represent sounds when writing in English </li>
</ul>
<h4>Support for Building Character</h4>
<p>Throughout this unit, find ways to encourage and support students to work on cultivating their <strong>intellectual humility</strong>.</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/take-notice/" target="_blank">Take Notice</a> </li>
<li> <a href="https://characterlab.org/tips-of-the-week/admitting-mistakes/" target="_blank">Admitting Mistakes</a> </li>
</ul>
<p>You can find other tips located here in the <a href="https://characterlab.org/playbooks/intellectual-humility/" target="_blank">Playbook on Intellectual Humility</a> from Character Lab.</p>