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<h4>Activity</h4>
<br>
<div class="os-raise-ib-cta" data-button-text="Solution" data-fire-event="Reveal1" data-schema-version="1.0">
<div class="os-raise-ib-cta-content">
<ol class="os-raise-noindent">
<li>Here is a diagram of a rectangle with side lengths \(x+1\) and \(x+3\). Use this diagram to show that \((x+1)(x+3)\) and \(x^2+4x+3\) are equivalent expressions.</li>
</ol>
<p><img alt="Large rectangle, divided into 4 smaller rectangles. Left rectangle on top labeled x, Right rectangle on top labeled 3. Top rectangle on left side labeled x, bottom rectangle on left side labeled 1." height="168" src="https://k12.openstax.org/contents/raise/resources/d46e7dac037c0d1d77f1d1e4bb75d9727476ca83" width="254"></p>
</div>
<div class="os-raise-ib-cta-prompt">
<p>Write down your answer, then select the <strong>solution</strong> button to compare your work. </p>
</div>
</div>
<div class="os-raise-ib-content" data-schema-version="1.0" data-wait-for-event="Reveal1">
<p>Compare your answer:</p>
<p><img alt="Area model multiplying x+3 and x+1" height="168" src="https://k12.openstax.org/contents/raise/resources/f7fc78e091f6ece205f3e5d15d21e2a30472cc68" width="254"></p>
<p>\(x \cdot x =x^2\)<br>
\(1 \cdot x=x\)<br>
\(x \cdot 3=3x\)<br>
\(1 \cdot 3=3\)</p>
<p>Adding these together gives \(x^2+4x+3\).</p>
</div>
<br>
<div class="os-raise-ib-cta" data-button-text="Solution" data-fire-event="Reveal2" data-schema-version="1.0">
<div class="os-raise-ib-cta-content">
<ol class="os-raise-noindent" start="2">
<li>Draw diagrams to help you write an equivalent expression for each of the following:</li>
</ol>
<ol class="os-raise-noindent">
<ol class="os-raise-noindent" type="a">
<li>\((x+5)^2\) </li>
</ol>
</ol>
</div>
<div class="os-raise-ib-cta-prompt">
<p>Write down your answer, then select the <strong>solution</strong> button to compare your work. </p>
</div>
</div>
<div class="os-raise-ib-content" data-schema-version="1.0" data-wait-for-event="Reveal2">
<p>Compare your answer:</p>
<p>The diagram should show partial products that add up to \(x^2+10x+25\).</p>
</div>
<br>
<div class="os-raise-ib-cta" data-button-text="Solution" data-fire-event="Reveal3" data-schema-version="1.0">
<div class="os-raise-ib-cta-content">
<ol class="os-raise-noindent">
<ol class="os-raise-noindent" start="2" type="a">
<li> \(2x(x+4)\) </li>
</ol>
</ol>
</div>
<div class="os-raise-ib-cta-prompt">
<p>Write down your answer, then select the <strong>solution</strong> button to compare your work. </p>
</div>
</div>
<div class="os-raise-ib-content" data-schema-version="1.0" data-wait-for-event="Reveal3">
<p>Compare your answer:</p>
<p>The diagram should show partial products that add up to \(2x^2+8x\).</p>
</div>
<br>
<div class="os-raise-ib-cta" data-button-text="Solution" data-fire-event="Reveal4" data-schema-version="1.0">
<div class="os-raise-ib-cta-content">
<ol class="os-raise-noindent">
<ol class="os-raise-noindent" start="3" type="a">
<li> \((2x+1)(x+3)\) </li>
</ol>
</ol>
</div>
<div class="os-raise-ib-cta-prompt">
<p>Write down your answer, then select the <strong>solution</strong> button to compare your work. </p>
</div>
</div>
<div class="os-raise-ib-content" data-schema-version="1.0" data-wait-for-event="Reveal4">
<p>Compare your answer:</p>
<p>The diagram should show partial products that add up to \(2x^2+7x+3\).</p>
</div>
<br>
<div class="os-raise-ib-cta" data-button-text="Solution" data-fire-event="Reveal5" data-schema-version="1.0">
<div class="os-raise-ib-cta-content">
<ol class="os-raise-noindent">
<ol class="os-raise-noindent" start="4" type="a">
<li>\((x+m)(x+n)\) </li>
</ol>
</ol>
</div>
<div class="os-raise-ib-cta-prompt">
<p>Write down your answer, then select the <strong>solution</strong> button to compare your work. </p>
</div>
</div>
<div class="os-raise-ib-content" data-schema-version="1.0" data-wait-for-event="Reveal5">
<p>Compare your answer:</p>
<p>The diagram should show partial products that add up to \(x^2+xn+mx+xn\) or \(x^2+(m+n)x+mn\).</p>
</div>
<br>
<div class="os-raise-ib-cta" data-button-text="Solution" data-fire-event="Reveal6" data-schema-version="1.0">
<div class="os-raise-ib-cta-content">
<ol class="os-raise-noindent" start="3">
<li>Write an equivalent expression for each expression without drawing a diagram. Use the distributive property to help. Remember FOIL for multiplying binomials: Multiply the First terms, the Outer terms, the Inner terms, and the Last terms.</li>
</ol>
<ol class="os-raise-noindent">
<ol class="os-raise-noindent" type="a">
<li> \((x+2)(x+6)\) </li>
</ol>
</ol>
</div>
<div class="os-raise-ib-cta-prompt">
<p>Write down your answer, then select the <strong>solution</strong> button to compare your work. </p>
</div>
</div>
<div class="os-raise-ib-content" data-schema-version="1.0" data-wait-for-event="Reveal6">
<p>Compare your answer:</p>
<p>\(x^2+8x+12\)</p>
</div>
<br>
<div class="os-raise-ib-cta" data-button-text="Solution" data-fire-event="Reveal7" data-schema-version="1.0">
<div class="os-raise-ib-cta-content">
<ol class="os-raise-noindent">
<ol class="os-raise-noindent" start="2" type="a">
<li>\((x+5)(2x+10)\)</li>
</ol>
</ol>
</div>
<div class="os-raise-ib-cta-prompt">
<p>Write down your answer, then select the <strong>solution</strong> button to compare your work. </p>
</div>
</div>
<div class="os-raise-ib-content" data-schema-version="1.0" data-wait-for-event="Reveal7">
<p>Compare your answer:</p>
<p>\(2x^2+20x+50\)</p>
</div>
<br>
<br>
<h4>Video: Using Diagrams to Find Equivalent Quadratic Expressions</h4>
<p>Watch the following video to learn more about using diagrams to find equivalent quadratic expressions.</p>
<div class="os-raise-d-flex-nowrap os-raise-justify-content-center">
<div class="os-raise-video-container"><video controls="true" crossorigin="anonymous">
<source src="https://k12.openstax.org/contents/raise/resources/4d74c6bb24c118c9ee654b330391716aa3d0144f">
<track default="true" kind="captions" label="On" src="https://k12.openstax.org/contents/raise/resources/e00d4277ae0ede45984559afc9789d08abd8c5df" srclang="en_us">https://k12.openstax.org/contents/raise/resources/4d74c6bb24c118c9ee654b330391716aa3d0144f
</video></div>
</div>
<br>
<br>
<br>
<!-- "Start "Are you Ready for More? click to reveal. If you have more than one on a page, you will need to change the Data-fire/data-wait-for-events for each set.-->
<div class="os-raise-ib-cta" data-button-text="Are you ready for more?" data-fire-event="RFM1" data-schema-version="1.0">
<div class="os-raise-ib-cta-content">
<!-- INSERT ANY VALID HTML HERE -->
</div>
<div class="os-raise-ib-cta-prompt">
<!-- INSERT ANY VALID HTML HERE -->
</div>
</div>
<!--Start interaction. If multiple interactions appear under "are you ready for more?, they should all have matching "data-wait-for-event", which should also match the "data-fire-event" for the button. Note in this sample, they are all RFM1-->
<div class="os-raise-ib-input" data-button-text="Solution" data-content-id="3a557210-e23e-4980-a3ac-24bb4aba1af1" data-schema-version="1.0" data-wait-for-event="RFM1">
<div class="os-raise-ib-input-content">
<h4>Extending Your Thinking</h4>
<p>Image 1</p>
<p><img alt="Square, side length = x. 5 rectangles, width = x, length = 1. 6 squares with side length = 1." height="159" src="https://k12.openstax.org/contents/raise/resources/c5b15617ab51a34141305b88d49a53bb6a7757a5" width="624"></p>
<p>Image 2</p>
<p><img alt="Rectangle decomposed into smaller squares and rectangles" height="264" src="https://k12.openstax.org/contents/raise/resources/b7cb24d8f86dc740773265298778e6c195ac0045" width="254"></p>
<ol class="os-raise-noindent">
<li>Are Image 1 and Image 2 equivalent? Explain or show your reasoning.</li>
</ol>
</div>
<div class="os-raise-ib-input-prompt">
<p>Enter your answer here:</p>
</div>
<div class="os-raise-ib-input-ack">
<p>Compare your answer:</p>
<p>Yes, because \(x \cdot x+5(1x)+6(1)=(x+2)(x+3)\).</p>
</div>
</div>
<!--End interaction-->
<br>
<div class="os-raise-ib-input" data-button-text="Solution" data-content-id="8c3d80dc-66e7-439d-9508-3d094e8d6600" data-schema-version="1.0" data-wait-for-event="RFM1">
<div class="os-raise-ib-input-content">
<ol class="os-raise-noindent" start="2">
<li>What does this tell you about an equivalent expression for \(x^2+5x+6\)?</li>
</ol>
</div>
<div class="os-raise-ib-input-prompt">
<p>Enter your answer here:</p>
</div>
<div class="os-raise-ib-input-ack">
<p>Compare your answer:</p>
<p>This shows that an equivalent expression is \((x+2)(x+3)\).</p>
</div>
</div>
<!--End interaction-->
<br>
<div class="os-raise-ib-input" data-button-text="Solution" data-content-id="cbf2cfc0-ea9f-48a4-ae9c-4bf5c8bb5114" data-schema-version="1.0" data-wait-for-event="RFM1">
<div class="os-raise-ib-input-content">
<ol class="os-raise-noindent" start="3">
<li>Is there a different non-zero number of 1-by-1 squares that we could use instead that would allow us to arrange the combined figures into a single large rectangle?</li>
</ol>
</div>
<div class="os-raise-ib-input-prompt">
<p>Enter your answer here:</p>
</div>
<div class="os-raise-ib-input-ack">
<p>Compare your answer:</p>
<p>We could use four 1-by-1 squares. This would give the expression \(x^2+5x+4\). The rectangle would be \((x + 4)(x + 1)\).<br>
<br>
<img alt height="346" role="presentation" src="https://k12.openstax.org/contents/raise/resources/5d64e5540c5f24105b1a4349cd0d352d8cfdc110" width="200">
</p>
</div>
</div>
<!--End interaction-->