dee206e7-812e-41cf-a3cc-9ac7797c32b7.html

<h3>Activity (20 minutes)</h3>
<p>This activity is an extension of the expectations in the TEKS.</p>
<p>This info gap activity gives students an opportunity to determine and request the information needed to write expressions that define quadratic functions with certain graphical features. To do so, students need to consider what they learned about the structure of quadratic expressions in various forms.</p>
<p>The info gap structure requires students to make sense of problems by determining what information is necessary, and then to ask for information they need to solve it. This may take several rounds of discussion if their first requests do not yield the information they need. It also allows them to refine the language they use and ask increasingly more precise questions until they get the information they need.</p>
<p>Because students are expected to make use of structure and construct logical arguments about how the structure helps them write expressions, technology is not an appropriate tool.</p>
<p>Here is the text of the cards for reference and planning. Note that the questions on Card 1 have many possible correct answers but no possible expressions, for \(f\) and \(g\) can define the same function.</p>
<h5>Problem Card 1</h5>
<ol class="os-raise-noindent">
  <li> Write an expression in vertex form that could define a quadratic function, \(f\). </li>
  <li> Write an expression in factored form that could define a quadratic function, \(g\). </li>
  <li> Show that \(f\) and \(g\) do not define the same function. </li>
</ol>
<h5>Problem Card 2</h5>
<p>Functions \(a\) and \(b\) are quadratic functions.</p>
<ol class="os-raise-noindent">
  <li> What are the zeros of function \(a\)? </li>
  <li> What is the vertex of the graph representing function \(b\)? </li>
  <li> Show that \(a\) and \(b\) do not define the same function. </li>
</ol>
<h5>Data Card 1</h5>
<ul class="os-raise-noindent">
  <li> The vertex of the graph of function \(f\) is \((6, -9)\). </li>
  <li> The \(x\)-intercepts of the graph of function \(g\) are \((-7, 0)\) and \((-5, 0)\). </li>
</ul>
<h5>Data Card 2</h5>
<ul class="os-raise-noindent">
  <li> Function \(a\) is defined by \((x-5)^2-4\). </li>
  <li> Function \(b\) is defined by \((x+1)(x-5)\). </li>
</ul>
<h4>Launch</h4>
<p>Tell students they will continue to write expressions in different forms that define quadratic functions. Explain the info gap structure, and consider demonstrating the protocol if students are unfamiliar with it.</p>
<p>Arrange students in groups of two. In each group, distribute a <a href="https://k12.openstax.org/contents/raise/resources/a9244c5fe407ac2ae6ca0ebc306bc8eb6260e3ca" target="_blank">problem</a> card to one student and a data card to the other student. After you review their work on the first problem, give them the cards for a second problem and instruct them to switch roles.</p>
<p>Since this activity was designed to be completed without technology, ask students to put away any devices.</p>
<div class="os-raise-extrasupport">
  <div class="os-raise-extrasupport-header">
    <p class="os-raise-extrasupport-title">Support for English Language Learners</p>
    <p class="os-raise-extrasupport-name">MLR 4 Information Gap: Conversing</p>
  </div>
  <div class="os-raise-extrasupport-body">
    <p>This activity gives students a purpose for discussing information necessary to solve problems involving features of quadratic functions. Display questions or question starters for students who need a starting point such as: “Can you tell me . . . (specific piece of information)”, and “Why do you need to know . . . (that piece of information)?"</p>
    <p class="os-raise-text-italicize">Design Principle(s): Cultivate Conversation</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">Engagement: Develop Effort and Persistence</p>
  </div>
  <div class="os-raise-extrasupport-body">
    <p>Display or provide students with a physical copy of the written directions. Check for understanding by inviting students to rephrase directions in their own words. Keep the display of directions visible throughout the activity.</p>
    <p class="os-raise-text-italicize">Supports accessibility for: Memory; Organization</p>
  </div>
</div>
<p>&nbsp;</p>
<h4>Student Activity</h4>
<p>Your teacher will give you either a problem card or a data card. Do not show or read your card to your partner.</p>
<p>If your teacher gives you the data card:</p>
<ol class="os-raise-noindent">
  <li> Silently read the information on your card. </li>
  <li> Ask your partner, "What specific information do you need?" and wait for your partner to ask for information. Only give information that is on your card. (Do not figure out anything for your partner!) </li>
  <li> Before telling your partner the information, ask, "Why do you need to know (that piece of information)?" </li>
  <li> Read the problem card and solve the problem independently. </li>
  <li> Share the data card and discuss your reasoning. </li>
</ol>
<p>If your teacher gives you the problem card:</p>
<ol class="os-raise-noindent">
  <li> Silently read your card and think about what information you need to answer the question. </li>
  <li> Ask your partner for the specific information that you need. </li>
  <li> Explain to your partner how you are using the information to solve the problem. </li>
  <li> When you have enough information, share the problem card with your partner, and solve the problem independently. </li>
  <li> Read the data card and discuss your reasoning. </li>
</ol>
<div class="os-raise-usermessage-lightbulb">
<p>Pause here so your teacher can review your work. Ask your teacher for a new set of cards and repeat the activity, trading roles with your partner.</p>
</div>
<ol class="os-raise-noindent">
  <li>Use the information about functions \(f\) and \(g\) to answer the questions.</li>
</ol>
<ul class="os-raise-noindent">
  <li> Function \(f\) is defined by \((x+3)(x+9)\). </li>
  <li> Function \(g\) is defined by \((x+6)^2-1\). </li>
</ul>
<ol class="os-raise-noindent" type="a">
  <li>Write an expression in vertex form that could define a quadratic function, \(f\).</li>
</ol>

<p><strong>Answer:</strong> \((x+6)^2&minus;9\)</p>
<ol class="os-raise-noindent" start="2" type="a">
  <li>Write an expression in factored form that could define a quadratic function, \(g\).</li>
</ol>
<p><strong>Answer:</strong> \((x+7)(x+5)\)</p>
<ol class="os-raise-noindent" start="3" type="a">
  <li>Show that \(f\) and \(g\) do not define the same function.</li>
</ol>

<p><strong>Answer:</strong> Rewriting \(g\) in standard form gives \(x^2+12x+35\), but \(f\) is \(x^2+12x+27\).</p>
<ol class="os-raise-noindent" start="2">
  <li>Functions \(a\) and \(b\) are quadratic functions.</li>
</ol>
<ul class="os-raise-noindent">
  <li> Function \(a\) is defined by \((x-5)^2-4\). </li>
  <li> Function \(b\) is defined by \((x+1)(x-5)\). </li>
</ul>
<ol class="os-raise-noindent" type="a">
  <li>What are the zeros of function \(a\)? <strong>Select two.</strong></li>
</ol>
<ul class="os-raise-noindent">
  <li> \(x = 1\) </li>
  <li> \(x = 3\) </li>
  <li> \(x = 7\) </li>
  <li> \(x = 9\) </li>
</ul>
<p><strong>Answer:</strong> \(x = 3\), \(x = 7\)</p>
<ol class="os-raise-noindent" start="2" type="a">
  <li>What is the \(x\)-coordinate of the vertex of the graph representing function \(b\)?</li>
</ol>
<p><strong>Answer:</strong> 2</p>
<ol class="os-raise-noindent" start="3" type="a">
  <li>What is the \(y\)-coordinate of the vertex of the graph representing function \(b\)?</li>
</ol>
<p><strong>Answer:</strong> -9</p>
<ol class="os-raise-noindent" start="4" type="a">
  <li>Show that \(a\) and \(b\) do not define the same function.</li>
</ol>

<p><strong>Answer:</strong> Rewriting \(a\) in standard form gives \(x^2&minus;10x+21\), and \(b\) in standard form is \(x^2&minus;4x&minus;5\).</p>
<h4>Activity Synthesis</h4>
<p>After students have completed their work, discuss the correct answers to the questions and any difficulties that come up.</p>
<p>Highlight for students that different forms of quadratic expressions are useful in different ways, so it helps to be able to move flexibly across forms. For example, a quadratic expression in factored form makes it straightforward to tell the zeros of the function that the expression defines and the \(x\)-intercepts of its graph. The vertex form makes it easy to identify the coordinates of the vertex of a graph of function.</p>
<p>To know whether two expressions define the same function, we can rewrite the expression in an equivalent form. There are many tools at our disposal. For instance, we can rewrite an expression into factored form, apply the distributive property to expand a factored expression, rearrange parts of an expression, combine like terms, or complete the square.</p>