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c85982d3-2861-4d59-8753-d6e7897cde40.html
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<h4>Activity </h4>
<p><img alt="Three steps of a growing pattern." height="168" src="https://k12.openstax.org/contents/raise/resources/cdcca2de2d8650c3f470bc20faffa7b1228fea96" width="422"></p>
<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> If the pattern continues, what will we see in Step 5 and Step 18? </li>
<ol class="os-raise-noindent" type="a">
<li> Sketch or describe the figure in each of these steps. </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="Reveal1">
<p>Compare your answer:</p>
<p>Step 5 is a collection of 25 small squares, in a 5-by-5 arrangement, surrounded by 4 other small squares at the corners. Step 18 is a collection of \(18^2\) small squares, in an 18 by 18 arrangement, surrounded by 4 other small squares at the corners.</p>
</div>
<br>
<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">
<ol class="os-raise-noindent" start="2" type="a">
<li> How many small squares are in each of these steps? Explain how you know. </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>Step 5: 29, because it is \(5^2+4\) or \((5 \cdot 5)+4\). Step 18: 328, because it is \(18^2+4\) or \((18 \cdot 18)+4\).</p>
</div>
<br>
<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" start="2">
<li> Write an equation to represent the relationship between the step number \(n\) and the number of squares \(y\). Be prepared to explain how each part of your equation relates to the pattern. (If you get stuck, try making a table.) </li>
</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>\(y=n^2+4\) or equivalent. The \(n^2\) term represents the number of small squares in the central \(n\)-by-\(n\) large square. The 4 represents the 4 small squares that are added to the large square at its 4 corners.</p>
</div>
<br>
<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" start="3">
<li> Sketch the first three steps of a pattern that can be represented by the equation \(y=n^2-1\). </li>
</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>An \(n\)-by-\(n\) array of squares with one square missing at the top right. The first step (for \(n=1\)) will have no squares.</p>
<p><img alt="DIAGRAM WITH 3 FIGURES SHOWING STEPS 1 THROUGH 3. STEP 1 DOES NOT SHOW ANYTHING. STEP 2 HAS A RECTANGLE WITH 1 ROW AND 2 COLUMNS. 1 SQUARE IS ADDED TO THE TOP LEFT. STEP 3 HAS A RECTANGLE WITH 2 ROWS AND 3 COLUMNS. 2 SQUARES ARE ADDED TO THE TOP LEFT." class="img-fluid atto_image_button_text-bottom" height="125" src="https://k12.openstax.org/contents/raise/resources/8965a03887be419785389417c8ac9bb33490ead8" width="384"></p>
</div>
<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="136e0c7f-9425-4cba-b248-e6526d42575c" data-schema-version="1.0" data-wait-for-event="RFM1">
<div class="os-raise-ib-input-content">
<h4>Extending Your Thinking</h4>
<ol class="os-raise-noindent">
<li> For the original step pattern in the statement, \(y=n^2+4\), write an equation to represent the relationship between the step number, \(n\), and the perimeter, \(P\). </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>\(P=4n+16\)</p>
</div>
</div>
<!--End interaction-->
<br>
<br>
<div class="os-raise-ib-input" data-button-text="Solution" data-content-id="f3f09518-440f-4ac1-b7fc-2c1288d5be27" 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> For the step pattern you created in Part 3 of the activity, \(y=n^2-1\), write an equation to represent the relationship between the step number, \(n\), and the perimeter, \(P\).</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>If the pattern was an \(n\)-by-\(n\) array of squares with a square missing from the corner, the perimeter could be given by \(P=0\) when \(n=1\), and by \(P=4n\) when \(n>1\).</p>
</div>
</div>
<!--End interaction-->
<br>
<br>
<div class="os-raise-ib-input" data-button-text="Solution" data-content-id="a76c69c0-6bf0-45bb-9dcb-05a995819002" 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> Are these linear functions?</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, in both cases, these are linear functions.</p>
</div>
</div>
<!--End interaction-->