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<h3>Activity (20 minutes)</h3>
<p>In the activity, students will use an area model to find the product of two two-digit numbers. We want students to notice the total area of their models is consistent, and notice that partitioning by place value is "easier" to add up the parts than partitioning somewhere random. Another goal here is to have students see that they can group like-terms (in this case, all of the integers), to support them later with algebra tiles. Finally, students can test their knowledge of factors, partial products, and total area by playing an area model game. </p>

<h3>Launch</h3>
<ol>
  <li>Allow students to work independently to answer questions on their activity sheet, stopping where indicated to have small-group or whole-class discussions.</li>
  <li>Circulate as students work on Part 3 of the activity sheet, noting interesting responses. After you notice that groups have discussed #1c, pause and bring the class together for a whole-class discussion. </li>
  <li>Call on groups to share out what is the same and what is different in their various area models.</li> 
  <li>Ask if any partitions were noticeably "easier" to add than others.</li>
  <li>Encourage students to test their area models using the Generic screen in the simulation.</li>
</ol>
<p>Note: A common error might be partitioning 17 into 1 and 7 instead of 10 and 7. If you notice students making this error, ask them what the total side length of their rectangle is to push them to see their error. </p>
<h3>Student Activity</h3>


<div class="os-raise-familysupport">
  <p><a href="https://phet.colorado.edu/sims/html/area-model-algebra/latest/area-model-algebra_en.html?screens=1,2,3" target="_blank">Access the interactive simulator </a>to begin this project.</p>
</div>


<p>In this activity, whenever you see &#128172;, stop and share your responses with your partner. If you have different responses, try to come to a consensus.</p>
<p>For questions 1-4, use the prompt and instructions below.</p>
<blockquote>Suppose we want to find the product of 17 and 13 using an area model. Use the simulation screens labeled “Explore” and “Generic”. Note you can switch between using the toolbar located below the area model.</blockquote>

<ol class="os-raise-noindent">
  <li>Draw and label your partitions on the scaled area model below.</li> 
  <img height="300" src="https://k12.openstax.org/contents/raise/resources/ec6b7f372f58389772d9461682de2781c3f4d397">
<br>
  <strong>Answer:</strong>
  
  <img height="300" src="https://k12.openstax.org/contents/raise/resources/89512f5560068d2cfcdcd274beac150c43cdf2cf"></li><br>
 <li>
    Draw and label your partitions on the generic model below. <br>
    <img height="300" src="https://k12.openstax.org/contents/raise/resources/5d7078d0d035592651c72ffe93732201515db380"><br>
    <strong>Answer:</strong> 
    <img height="300" src="https://k12.openstax.org/contents/raise/resources/d756c1147a6bbe3feb1f58de7fa4b44991a0a6f4">
    
 </li><br>



  <li>
     Discuss with your group: How does your area model compare to those in your group? What is the same? What is different? &#128172; <br>
    <strong>Answer:</strong> Some of us used different partitions or broke it up differently. But no matter how we broke it up, the total area was the same and so were the dimensions. It was the partial products that were different. <br>
  </li><br>

<li>Justify how you know your model represents 17x13. Does your area model represent 17✕13? How do you know? &#128172;<br></li>
<p><strong>Answer:</strong> I know my area model represents 17x13 because I broke it into 10, 10, 3, and 7. 10 and 3 were across the top dimension with 10 and 7 along the side dimension. Then, I found the area - or number of unit tiles - in each section. For instance, 10x10 was 100 square units. 3x10 was 30, 7x10 was 70. So far, that gives me 200. But I still needed to multiply the 3 and 7 which gave me 21. So, in total, the area is 221. And if I multiply it out by hand on my paper, I get 221, too. </p><br>


  <li>
    Challenge yourself to work through levels 1-2 of the <a target="_blank" href="https://phet.colorado.edu/sims/html/area-model-algebra/latest/area-model-algebra_en.html?screens=4ß">Area Model Numbers Game</a>!
    <br><br>
    Note: In this challenge, students are given different rectangular scenarios and asked to determine partial products, dimensions, etc. Students may choose their level of difficulty/challenge.
  </li>

<br>
<li>  What are three different ways you could partition 17? </li>
  <strong>Answer:</strong>
<ul>
    <li>\(15 + 2\)</li>
    <li>\(10 + 7\)</li>
    <li>\(8 + 9\)</li>
    <li>Etc.</li>
</ul><br>

  <li>
    Write your own 2-digit times 2-digit multiplication problem that uses an area model, and find the total area. 
    <strong>Answer:</strong> Ask students to share their problems with their classmates.
  </li> 
  <br>

<p>
  <li>What is a convenient way to break up a multiplication problem into an area model, and why is it convenient for you?</li>
  <strong>Answer:</strong> If a problem is broken up into its place values, then it is “y” to multiply by sets of 10 and then add those pieces together.
</p> 
</ol>


<h3>Activity Synthesis</h3>
<p>After you notice that individuals have finished questions through the summary, pause and bring the class together for a whole-class discussion. Allow multiple students to share their responses for #4 and especially #5. Help the class notice that partitioning by place value means we'll get some “easier” partial products to add, since one dimension will end up being 10. This will also prepare students for area models that use binomial dimensions where constant terms are often placed “at the end.” If other students have other ideas about efficiency with an area model, allow them to share. </p>