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a24a80af-f99c-40ec-908f-52adf8691223.html
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<h4>Activity</h4>
<p>Pattern 1</p>
<p><img alt="Pattern. Step 0,1 dot. Step 1, 3 dots, 2 by 2 with upper left dot removed. Step 2, 5 dots, 3 by 2 with upper left dot removed. Step 3, 7 dots, 4 by 2 with upper left dot removed." height="72" src="https://k12.openstax.org/contents/raise/resources/d40bf31a833606cb35897e28384bce0fad6c41e6" width="346"></p>
<p>Pattern 2</p>
<p><img alt="Pattern. Step 0, 0 dots. Step 1, 1 dot. Step 2, 4 dots, arranged 2 by 2. Step 3, 9 dots, arranged 3 by 3." height="83" src="https://k12.openstax.org/contents/raise/resources/32e3f0ae641cc7113cbd7ee68d29febdea28d604" width="346"></p>
<br>
<!--Text Entry Interaction Start -->
<div class="os-raise-ib-input" data-button-text="Solution" data-content-id="5e29ae67-e33c-4890-ab3d-f4cd5ce016d0" data-fire-event="eventShow" data-schema-version="1.0">
<div class="os-raise-ib-input-content">
<ol class="os-raise-noindent">
<li>Look at Pattern 1.</li>
</ol>
<ol class="os-raise-noindent">
<ol class="os-raise-noindent" type="a">
<li> How is the number of dots in each step changing?</li>
</ol>
</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>Each step has 2 rows of dots, with each row having the same number of dots as the step number, plus one extra
dot.</p>
</div>
</div>
<!--Interaction End -->
<br>
<br>
<!--Text Entry Interaction Start -->
<div class="os-raise-ib-input" data-button-text="Solution" data-content-id="bf1ac5f9-4bd2-419e-b7de-ae82c9e89092" data-fire-event="eventShow" data-schema-version="1.0">
<div class="os-raise-ib-input-content">
<ol class="os-raise-noindent">
<ol class="os-raise-noindent" start="2" type="a">
<li> How many dots will be in Step 5 in the pattern, and how will they be arranged?</li>
</ol>
</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>In Step 5, there will be 2 rows of 5 dots plus one additional dot.</p>
</div>
</div>
<!--Interaction End -->
<br>
<br>
<!--Text Entry Interaction Start -->
<div class="os-raise-ib-input" data-button-text="Solution" data-content-id="34b2502c-92ad-444c-afea-4d8160e1bd3d" data-fire-event="eventShow" data-schema-version="1.0">
<div class="os-raise-ib-input-content">
<ol class="os-raise-noindent" start="2">
<li>Look at Pattern 2.</li>
</ol>
<ol class="os-raise-noindent">
<ol class="os-raise-noindent" type="a">
<li> How is the number of dots in each step changing?</li>
</ol>
</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>Each step has as many rows and columns of dots as the step number.</p>
</div>
</div>
<!--Interaction End -->
<br>
<br>
<!--Text Entry Interaction Start -->
<div class="os-raise-ib-input" data-button-text="Solution" data-content-id="ff520f69-b918-4651-a899-14b0d8d67bfa" data-fire-event="eventShow" data-schema-version="1.0">
<div class="os-raise-ib-input-content">
<ol class="os-raise-noindent">
<ol class="os-raise-noindent" start="2" type="a">
<li> How many dots will be in Step 5 in the pattern, and how will they be arranged?</li>
</ol>
</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>Step 5 is a 5-by-5 array of dots.</p>
</div>
</div>
<!--Interaction End -->
<br>
<p>Consider the following table.</p>
<table class="os-raise-wideadjustedtable">
<thead>
<tr>
<th scope="col">
Step
</th>
<th scope="col">
Number of dots in Pattern 1
</th>
<th scope="col">
Number of dots in Pattern 2
</th>
</tr>
</thead>
<tbody>
<tr>
<td>
0
</td>
<td>
1
</td>
<td>
0
</td>
</tr>
<tr>
<td>
1
</td>
<td>
3
</td>
<td>
1
</td>
</tr>
<tr>
<td>
2
</td>
<td>
5
</td>
<td>
4
</td>
</tr>
<tr>
<td>
3
</td>
<td>
7
</td>
<td>
9
</td>
</tr>
<tr>
<td>
4
</td>
<td></td>
<td></td>
</tr>
<tr>
<td>
5
</td>
<td></td>
<td></td>
</tr>
<tr>
<td>
10
</td>
<td></td>
<td></td>
</tr>
<tr>
<td>
12
</td>
<td></td>
<td></td>
</tr>
<tr>
<td>
\(n\)
</td>
<td></td>
<td></td>
</tr>
</tbody>
</table>
<br>
<ol class="os-raise-noindent" start="3">
<li>Complete the table with the number of dots for Pattern 1.</li>
</ol>
<div class="os-raise-ib-pset" data-button-text="Check" data-content-id="5a588a3e-6b77-4f10-839e-fc7c8b6be633" data-fire-learning-opportunity-event="eventnameY" data-fire-success-event="eventnameX" data-retry-limit="0" data-schema-version="1.0">
<!--Q3a-->
<div class="os-raise-ib-pset-problem" data-content-id="8095bb6a-6ede-482f-978e-394579c2a588" data-problem-comparator="integer" data-problem-type="input" data-solution="9">
<div class="os-raise-ib-pset-problem-content">
<ol class="os-raise-noindent">
<ol class="os-raise-noindent" type="a">
<li> In Step 4, what is the number of dots in Pattern 1?</li>
</ol>
</ol>
<p>Enter your answer here: </p>
</div>
<div class="os-raise-ib-pset-correct-response">
<p>Correct! Step 4 has 9 dots in Pattern 1.</p>
</div>
<div class="os-raise-ib-pset-attempts-exhausted-response">
<p>The correct answer is 9. Step 4 has 9 dots in Pattern 1. </p>
</div>
</div>
<!--END QUESTION.-->
<!--Q3b-->
<div class="os-raise-ib-pset-problem" data-content-id="8139dd15-22b1-4be7-9a65-6a4492b88b72" data-problem-comparator="integer" data-problem-type="input" data-solution="11">
<div class="os-raise-ib-pset-problem-content">
<ol class="os-raise-noindent">
<ol class="os-raise-noindent" start="2" type="a">
<li> In Step 5, what is the number of dots in Pattern 1?</li>
</ol>
</ol>
<p>Enter your answer here: </p>
</div>
<div class="os-raise-ib-pset-correct-response">
<p>Correct! Step 5 has 11 dots in Pattern 1.</p>
</div>
<div class="os-raise-ib-pset-attempts-exhausted-response">
<p>The correct answer is 11. Step 5 has 11 dots in Pattern 1.</p>
</div>
</div>
<!--END QUESTION.-->
<!--Q3c-->
<div class="os-raise-ib-pset-problem" data-content-id="9330201a-1aa8-48b6-b4c5-f7beef96cfe7" data-problem-comparator="integer" data-problem-type="input" data-solution="21">
<div class="os-raise-ib-pset-problem-content">
<ol class="os-raise-noindent">
<ol class="os-raise-noindent" start="3" type="a">
<li> In Step 10, what is the number of dots in Pattern 1?</li>
</ol>
</ol>
<p>Enter your answer here: </p>
</div>
<div class="os-raise-ib-pset-correct-response">
<p>Correct! Step 10 has 21 dots in Pattern 1.</p>
</div>
<div class="os-raise-ib-pset-attempts-exhausted-response">
<p>The correct answer is 21. Step 10 has 21 dots in Pattern 1.</p>
</div>
</div>
<!--END QUESTION.-->
<!--Q3d-->
<div class="os-raise-ib-pset-problem" data-content-id="a74ee2de-72ef-4d03-bd65-da607a8f852e" data-problem-comparator="integer" data-problem-type="input" data-solution="25">
<div class="os-raise-ib-pset-problem-content">
<ol class="os-raise-noindent">
<ol class="os-raise-noindent" start="4" type="a">
<li> In Step 12, what is the number of dots in Pattern 1?</li>
</ol>
</ol>
<p>Enter your answer here: </p>
</div>
<div class="os-raise-ib-pset-correct-response">
<p>Correct! Step 12 has 25 dots in Pattern 1.</p>
</div>
<div class="os-raise-ib-pset-attempts-exhausted-response">
<p>The correct answer is 25. Step 12 has 25 dots in Pattern 1.</p>
</div>
</div>
<!--END QUESTION.-->
<!--END QUESTION.-->
<!--Do not edit below line.-->
<div class="os-raise-ib-pset-correct-response">
<!-- INSERT ANY VALID HTML HERE -->
</div>
<div class="os-raise-ib-pset-encourage-response">
<!-- INSERT ANY VALID HTML HERE -->
</div>
</div>
<br>
<!--Text Entry Interaction Start -->
<div class="os-raise-ib-input" data-button-text="Solution" data-content-id="83c45254-9652-410e-b3e4-fdab3b80c988" data-fire-event="eventShow" data-schema-version="1.0">
<div class="os-raise-ib-input-content">
<ol class="os-raise-noindent">
<ol class="os-raise-noindent" start="5" type="a">
<li> In Step \(n\), what is the number of dots in Pattern 1?</li>
</ol>
</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>\(2n+1\)</p>
</div>
</div>
<!--Interaction End -->
<br>
<ol class="os-raise-noindent" start="4">
<li>Complete the table with the number of dots for Pattern 2.</li>
</ol>
<div class="os-raise-ib-pset" data-button-text="Check" data-content-id="ddf6eb48-dbac-454f-bdf2-060e3f6d2ea5" data-fire-learning-opportunity-event="eventnameY" data-fire-success-event="eventnameX" data-retry-limit="0" data-schema-version="1.0">
<!--Q4a-->
<div class="os-raise-ib-pset-problem" data-content-id="2b787d9d-44ad-4bb3-a433-1acdd39ddf49" data-problem-comparator="integer" data-problem-type="input" data-solution="16">
<div class="os-raise-ib-pset-problem-content">
<ol class="os-raise-noindent">
<ol class="os-raise-noindent" type="a">
<li> In Step 4, what is the number of dots in Pattern 2?</li>
</ol>
</ol>
<p>Enter your answer here: </p>
</div>
<div class="os-raise-ib-pset-correct-response">
<p>Correct! Step 4 has 16 dots in Pattern 2.</p>
</div>
<div class="os-raise-ib-pset-attempts-exhausted-response">
<p>The correct answer is 16. Step 4 has 16 dots in Pattern 2.</p>
</div>
</div>
<!--END QUESTION.-->
<!--Q4b-->
<div class="os-raise-ib-pset-problem" data-content-id="277c86eb-6bf2-4c19-824e-49ea954daf66" data-problem-comparator="integer" data-problem-type="input" data-solution="25">
<div class="os-raise-ib-pset-problem-content">
<ol class="os-raise-noindent">
<ol class="os-raise-noindent" start="2" type="a">
<li> In Step 5, what is the number of dots in Pattern 2?</li>
</ol>
</ol>
<p>Enter your answer here: </p>
</div>
<div class="os-raise-ib-pset-correct-response">
<p>Correct! Step 5 has 25 dots in Pattern 2.</p>
</div>
<div class="os-raise-ib-pset-attempts-exhausted-response">
<p>The correct answer is 25. Step 5 has 25 dots in Pattern 2.</p>
</div>
</div>
<!--END QUESTION.-->
<!--Q4c-->
<div class="os-raise-ib-pset-problem" data-content-id="95eaf6ac-0fbb-4732-9a7e-d258e1c80258" data-problem-comparator="integer" data-problem-type="input" data-solution="100">
<div class="os-raise-ib-pset-problem-content">
<ol class="os-raise-noindent">
<ol class="os-raise-noindent" start="3" type="a">
<li> In Step 10, what is the number of dots in Pattern 2?</li>
</ol>
</ol>
<p>Enter your answer here: </p>
</div>
<div class="os-raise-ib-pset-correct-response">
<p>Correct! Step 10 has 100 dots in Pattern 2.</p>
</div>
<div class="os-raise-ib-pset-attempts-exhausted-response">
<p>The correct answer is 100. Step 10 has 100 dots in Pattern 2.</p>
</div>
</div>
<!--END QUESTION.-->
<!--Q4d-->
<div class="os-raise-ib-pset-problem" data-content-id="af51617b-17bb-4e0d-9cd3-c26d38266205" data-problem-comparator="integer" data-problem-type="input" data-solution="144">
<div class="os-raise-ib-pset-problem-content">
<ol class="os-raise-noindent">
<ol class="os-raise-noindent" start="4" type="a">
<li> In Step 12, what is the number of dots in Pattern 2?</li>
</ol>
</ol>
<p>Enter your answer here: </p>
</div>
<div class="os-raise-ib-pset-correct-response">
<p>Correct! Step 12 has 144 dots in Pattern 2.</p>
</div>
<div class="os-raise-ib-pset-attempts-exhausted-response">
<p>The correct answer is 144. Step 12 has 144 dots in Pattern 2.</p>
</div>
</div>
<!--END QUESTION.-->
<!--Do not edit below line.-->
<div class="os-raise-ib-pset-correct-response">
<!-- INSERT ANY VALID HTML HERE -->
</div>
<div class="os-raise-ib-pset-encourage-response">
<!-- INSERT ANY VALID HTML HERE -->
</div>
</div>
<br>
<!--Text Entry Interaction Start -->
<div class="os-raise-ib-input" data-button-text="Solution" data-content-id="e4b3fed6-77fc-4f19-a30d-c2e5597de280" data-fire-event="eventShow" data-schema-version="1.0">
<div class="os-raise-ib-input-content">
<ol class="os-raise-noindent">
<ol class="os-raise-noindent" start="5" type="a">
<li> In Step \(n\), what is the number of dots in Pattern 2?</li>
</ol>
</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>\(n \cdot n\) or \(n^2\)</p>
</div>
</div>
<!--Interaction End -->
<br>
<br>
<div class="os-raise-ib-desmos-gc" data-schema-version="1.0"></div>
<br>
<ol class="os-raise-noindent" start="5">
<li>Use the graphing tool or technology outside the course. Plot the number of dots at each step number for Pattern 1
and Pattern 2.</li>
</ol>
<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">
<ol class="os-raise-noindent" type="a">
<li> Pattern 1 </li>
</ol>
</ol>
</div>
<div class="os-raise-ib-cta-prompt">
<p>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="Points plotted on a grid. Number of dots. Step number." height="157" src="https://k12.openstax.org/contents/raise/resources/cf515b348bca3de8f3b02eb7c8f2deca3dccc3e7" width="307"></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> Pattern 2 </li>
</ol>
</ol>
</div>
<div class="os-raise-ib-cta-prompt">
<p>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><img alt="Points plotted on a grid. Number of dots. Step number." height="204" src="https://k12.openstax.org/contents/raise/resources/bb98bf2467269ae301fca6f978cc21315c4ae42b" width="301"></p>
</div>
<br>
<br>
<!--Text Entry Interaction Start -->
<div class="os-raise-ib-input" data-button-text="Solution" data-content-id="9281ba85-e050-44c1-ab22-2941a92b4803" data-fire-event="eventShow" data-schema-version="1.0">
<div class="os-raise-ib-input-content">
<ol class="os-raise-noindent" start="6">
<li>Explain why the graphs of the two patterns look the way they do.</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>The points representing the first pattern lie on a line because getting from one step to the next involves adding
the same number of dots (2 dots). The points representing the second pattern do not lie on a line. They
“curve” upward because the number of dots added at each step increases from step to step.</p>
</div>
</div>
<!--Interaction End -->