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<h4>Activity (10 minutes)</h4>
<p>Students have seen quadratic expressions in both standard form and factored form since the beginning of the unit. In
this activity, they learn to distinguish the expressions by their forms and to refer to each form by its formal name.
Refining their language about the different forms prepares students to be more precise in their thinking about the
graphs of quadratic functions in later lessons.</p>
<h4>Launch</h4>
<p>Arrange students in groups of two. Give students quiet work time and then time to share their work with a partner.
</p>
<br>
<div class="os-raise-extrasupport">
<div class="os-raise-extrasupport-header">
<p class="os-raise-extrasupport-title">Support for English Language Learners</p>
</div>
<div class="os-raise-extrasupport-body">
<p class="os-raise-extrasupport-name">MLR 1 Discussion Supports: Writing, Conversing</p>
<p>Use this routine to help students improve their writing by providing them with multiple opportunities to clarify their explanations through conversation. Give students time to meet with one to two partners to share their response to the final question, “Why do you think that form is called factored form?” Provide listeners with prompts for feedback that will help their partner add detail to strengthen and clarify their ideas. For example, students can ask their partner, “Can you say more about what each expression means?” or “I understand _____, but can you clarify . . .” Next, provide students with two to three minutes to revise their initial draft based on feedback from their peers.</p>
<p class="os-raise-text-italicize">Design Principle(s): Optimize output (for explanation)</p>
<p class="os-raise-extrasupport-title">Provide support for students</p>
<p>
<a href="https://k12.openstax.org/contents/raise/resources/a5ae5bd09b27a5f53239a539c6009c19c92f7db7" target="_blank">Distribute graphic organizers</a>
to the students to assist them with participating in this routine.
</p>
</div>
</div>
<br>
<div class="os-raise-extrasupport">
<div class="os-raise-extrasupport-header">
<p class="os-raise-extrasupport-title">Support for Students with Disabilities</p>
<p class="os-raise-extrasupport-name">Action and Expression: Internalize Executive Functions</p>
</div>
<div class="os-raise-extrasupport-body">
<p>Provide students with a graphic organizer that can be used to brainstorm the activity. Include one column for standard form and one column for factored form. Include a space at the bottom of the organizer for student-generated notes or reminders about standard form and factored form.</p>
<p class="os-raise-text-italicize">Supports accessibility for: Language; Organization</p>
</div>
</div>
<br>
<h4>Student Activity</h4>
<p>The quadratic expression \(x^2+4x+3\) is written in standard form.</p>
<p>Here are some other quadratic expressions. The expressions on the left are written in standard form and the
expressions on the right are not.</p>
<table class="os-raise-midsizetable">
<thead>
<tr>
<th scope="col">Written In Standard Form</th>
<th scope="col">Not Written In Standard Form</th>
</tr>
</thead>
<tbody>
<tr>
<td>
\(x^2-1\)
</td>
<td>
\((2x+3)x\)
</td>
</tr>
<tr>
<td>
\(x^2+9x\)
</td>
<td>
\((x+1)(x-1)\)
</td>
</tr>
<tr>
<td>
\(\frac {1}{2}x^2\)
</td>
<td>
\(3(x-2)^2+1\)
</td>
</tr>
<tr>
<td>
\(4x^2-2x+5\)
</td>
<td>
\(-4(x^2+x)+7\)
</td>
</tr>
<tr>
<td>
\(3x^2-x+6\)
</td>
<td>
\((x+8)(-x+5)\)
</td>
</tr>
</tbody>
</table>
<br>
<ol class="os-raise-noindent">
<li>What are some characteristics of expressions in standard form?</li>
</ol>
<p>Write down your answer, then select the <strong>solution</strong> button to compare your work.</p>
<p class="os-raise-text-bold">Answer:</p>
<ul>
<li> The standard form always starts with an \(x^2\) term and is in decreasing order of degree. </li>
<li> It does not need to have an \(x\) term or a numerical term. </li>
<li> Standard form is \(ax^2+bx+c\) where \(a\) is not zero. </li>
<li> The standard form shows a sum or difference of terms. </li>
</ul>
<ol class="os-raise-noindent" start="2">
<li>\((x+1)(x-1)\) and \((2x+3)x\) in the right column are quadratic expressions written in factored form. Why do you
think that form is called factored form?</li>
</ol>
<p><strong>Answer:</strong> It is a product of two factors. Each factor could be a variable, a number, or an expression
with a variable (with no exponent) and a number.</p>
<h4>Student Facing Extension</h4>
<h5>Are you ready for more?</h5>
<p>Which quadratic expression can be described as being both standard form and factored form? Explain how you know.</p>
<p><strong>Answer: </strong> \(x^2\) could be viewed as both standard and factored. It has an \(x^2\) term even though
there is not a constant or linear term. It can also be written as the product of two linear expressions because
\(x^2\) is equal to \(x \cdot x\).</p>
<h4>Activity Synthesis</h4>
<p>Display the two columns of expressions for all to see. Solicit students’ ideas on the features of each form.
Record their responses for all to see. Invite other students to express agreement or disagreement, or to clarify their
fellow students’ responses.</p>
<p>Define a quadratic expression in standard form explicitly as \(ax^2+bx+c\). Explain that we refer to \(a\) as the
coefficient of the squared term \(x^2\), \(b\) as the coefficient of the linear term \(x\), and \(c\) as the constant
term.</p>
<p>Ask students:</p>
<ul>
<li> “How would you write \((2x+3)x\) in standard form?” \((2x^2+3x)\) </li>
<li> “The expression \(2x^2+3x\) only has two terms. Is it still in standard form?” (Yes, there is no
constant term, which means \(c\) is \(0\).) </li>
<li> “How would you write \(-4(x^2+14x)+7\) in standard form?” \((-4x^2-56x+7)\) </li>
<li> “What are the values of the coefficients \(a\) and \(b\) in the expression \(-4x^2-x+7\)?” (\(–
4\) and \(– 1\)) </li>
<li> “What about the constant term?” (\(7\)) </li>
</ul>
<p>Then, clarify that a quadratic expression in factored form is a product of two factors that are each a linear
expression. For example, \((x+1)(x-1)\),\((2x+3)x\), and \(x(4x)\) all have two linear expressions for their factors.
An expression with two factors that are linear expressions and a third factor that is a constant, for example:
\(2(x+2)(x-1)\), is also in factored form.</p>
<p>Tell students that each form gives us interesting or useful information about the function that the expression
represents, and they will learn more about this in future lessons.</p>
<h3>7.9.3: Self Check</h3>
<p class="os-raise-text-bold"><em>After the activity, students will answer the following question to check their
understanding of the concepts
explored in the activity.</em></p>
<p class="os-raise-text-bold">QUESTION:</p>
<p>Which of the following expressions is a quadratic expression written in standard form?</p>
<table class="os-raise-textheavytable">
<thead>
<tr>
<th scope="col">Answers</th>
<th scope="col">Feedback</th>
</tr>
</thead>
<tbody>
<tr>
<td>
\(x-4\)
</td>
<td>
Incorrect. Let’s try again a different way: This expression does not have an \(x^2\) term. The answer
is \(x^2-9x+20\).
</td>
</tr>
<tr>
<td>
\(x^2-9x+20\)
<td>
That’s correct! Check yourself: This expression leads with an \(x^2\) term and is in standard form.
</td>
</tr>
<tr>
<td>
\(x(x-5)\)
</td>
<td>
Incorrect. Let’s try again a different way: This is in factored form. The answer is \(x^2-9x+20\).
</td>
</tr>
<tr>
<td>
\(7(x-6)^2+1\)
</td>
<td>
Incorrect. Let’s try again a different way: This is not in the form \(ax^2+bx+c\). The answer is
\(x^2-9x+20\).
</td>
</tr>
</tbody>
</table>
<br>
<h3>7.9.3: Additional Resources</h3>
<p class="os-raise-text-bold"><em>The following content is available to students who would like more support based on
their experience with the
self check. Students will not automatically have access to this content, so you may wish to share it with those
who could benefit from it.</em></p>
<h4>Standard Form of Quadratics</h4>
<p class="os-raise-text-bold">Examples of Standard Form Quadratics</p>
<p>\(x^2\)</p>
<p>\(x^2-9\)</p>
<p>\(x^2-3x\)</p>
<p>\(x^2+4x-45\)</p>
<p>These examples have an \(x^2\) and lead with it. Some have an \(x\) term or a constant term but it is not needed to
be in standard form. Standard form of a quadratic expression is of the form \(ax^2+bx+c\), where \(a \neq 0\).</p>
<p class="os-raise-text-bold">Non-Examples of Standard Form Quadratics</p>
<p>\(7-x\)</p>
<p>\(9(x-2)\)</p>
<p>\(x(x+10)\)</p>
<p>These examples do not have an \(x^2\) term.</p>
<h4>Try It: Standard Form of Quadratics</h4>
<p>Determine if \(4x(x-2)\) is a quadratic in standard form.</p>
<p>Write down your answer, then select the <strong>solution</strong> button to compare your work.</p>
<h5>Solution</h5>
<p>Here is how to determine if the expression is in standard form:</p>
<p>This expression is in factored form. If the \(4x\) is distributed, the expression becomes \(4x^2-8x\), which is in
standard form.</p>