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<h3>Activity (15 minutes)</h3>
<p>This activity will get students playing with a numerical area model while directing their attention toward the values on the outside and the values on the inside. It is important that students explain how the partial products are calculated, write the partial products as a sum, and discover that the sum of the partial products is equal to the total area of the rectangle.
<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 2 of the activity sheet, noting interesting responses. After you notice that groups have discussed #3, pause and bring the class together for a whole-class discussion. </li>
<li>Call on groups to share out what changes and what doesn't change when the partition slides are moved. </li>
<li>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 as a class notice that the partial products are different depending on where the partitions go, but the total area is always 150. </li>
</ol>
<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 💬, stop and share your responses with your partner. If you have different responses, try to come to a consensus.</p>
<p>For questions 1-8, use the following instructions.</p>
<blockquote>Use the simulation screen labeled “Explore”. Note you can switch between simulation screens using the toolbar located below the area model.</blockquote>
<ol class="os-raise-noindent">
<p><li>Explain what the red and blue sliders do to the outside of the rectangle. </li>
<strong>Answer:</strong> Split up the number, break up a number into two parts, make two numbers that add up to the total side </p>
<p><li>Explain what the red and blue sliders do to the inside of the rectangle.</li>
<strong>Answer:</strong> Make rectangles inside, make multiplication problems, make smaller areas, make a grid</p>
<p><li>Describe what changes and what doesn't change when the red and blue sliders are moved. 💬</li>
<br>
</p>
<strong>Answer:</strong>
</p>
<table class="os-raise-textheavytable">
<thead>
<tr>
<th scope="col">What changes</th>
<th scope="col">What doesn't change</th>
</tr>
</thead>
<tbody>
<tr>
<td>
<p>Side lengths have more numbers<br>
Number of inside rectangles<br>
Size/area of inside rectangles</p>
</td>
<td>
<p>Side lengths<br>
Total area<br>
Size of big rectangle</p>
</td>
</tr>
</tbody>
</table>
<p>
<li>
Multiply 10✕15 using an area model. Find two different ways to partition the 10x15 rectangle. Use the sim to support you in filling out this table. 💬<br>
<br>
<table class="os-raise-wideadjustedtable">
<thead>
<tr>
<th scope="col">Problem</th>
<th scope="col">Labeled Area Model with Partial Products</th>
<th scope="col">List partial products and write as a sum</th>
<th scope="col">Total area of rectangle</th>
</tr>
</thead>
<tbody>
<tr>
<td>
<p>10✕15</p>
</td>
<td>
<p><img src="https://k12.openstax.org/contents/raise/resources/98f8034b3c572d4c6d73853a245144f6ab3b43fa" height="200"></p>
</td>
<td>
<p> </p>
</td>
<td>
<p> </p>
</td>
</tr>
<tr>
<td>
<p>10✕15</p>
</td>
<td>
<p><img src="https://k12.openstax.org/contents/raise/resources/98f8034b3c572d4c6d73853a245144f6ab3b43fa" height="200"></p>
</td>
<td>
<p> </p>
</td>
<td>
<p> </p>
</td>
</tr>
</tbody>
</table>
<br>
<strong>Answer:</strong> Answers will vary, but here are some samples:
</li>
</p>
<table class="os-raise-wideadjustedtable">
<thead>
<tr>
<th scope="col">Problem</th>
<th scope="col">Labeled Area Model with Partial Products</th>
<th scope="col">List partial products and write as a sum</th>
<th scope="col">Total area of rectangle</th>
</tr>
</thead>
<tbody>
<tr>
<td>
<p>10✕15</p>
</td>
<td>
<p><img src="https://k12.openstax.org/contents/raise/resources/38c576f0ceaaa9759dc3df082e90b9b3fa522f83" height="200"></p>
</td>
<td>
<p>100+50</p>
</td>
<td>
<p>150</p>
</td>
</tr>
<tr>
<td>
<p>10✕15</p>
</td>
<td>
<p><img src="https://k12.openstax.org/contents/raise/resources/b60571da30d174b41c1bbb5d560b2b035130a1af" height="200"></p>
</td>
<td>
<p>40+40+35+35</p>
</td>
<td>
<p>150</p>
</td>
</tr>
</tbody>
</table>
<p><li>Use the sim to model 6✕18. What is the total area of this rectangle? Represent the total area in multiple ways.</li></p>
<img src="https://k12.openstax.org/contents/raise/resources/fbcd2c0d72399bc1a9f08f144315632f87b9fc23" height="200"><br>
<p>
<br>
<strong>Answer:</strong> 108. Using the area model, 6•18 = 6•10 + 6•8, which is 60+48.
<img src="https://k12.openstax.org/contents/raise/resources/53b13c4472c48e417696566d2d3c19aeb2c7236f" height="200"><br>
</p>
<p>
<li>In an area model, how do the partial products (interior numbers) get calculated?</li>
<strong>Answer:</strong> They are calculated by multiplying the side lengths after partitioning the sides of the big rectangle. Like in 6•18, the partial products were 6•10 and 6•8.
</p>
<p>
<li>In an area model, what are two different ways the total area could get calculated?</li>
<strong>Answer:</strong> One way is by partitioning the sides to get smaller rectangles, finding the areas of those, and adding them up. Another way is by adding up all of the square units of the large rectangle.
</p>
<p>
<li>How do the partial products relate to the total area of the area model? </li>
<strong>Answer:</strong> The partial products add up to the total area of the area model. Circulate around the room to hear similarities between groups. Write student ideas on the board to give value to contributions.
</p>
</ol>
<p><strong>Use the following questions to lead a class discussion.</strong></p>
<ol>
<li>Did anyone show the yellow boxes? What did they do? </li>
<li>If you were to add up the areas inside the rectangle, where is a useful place to break up the rectangle so you have “easier” numbers to add up? </li>
</ol>
<h3>Activity Synthesis</h3>
<p>Using an area model with numbers before variables allows students to connect their understanding of the area of a rectangle to algebraic thinking, namely the distributive property. When we multiply two numbers, we are essentially finding the area of a rectangle with a length and a width that are equivalent to those two factors. Since we will be talking about multiplying algebraic expressions, we could apply the same thinking as we do with the area of a rectangle. Before we can do that, students need to have a solid understanding of the area of a rectangle as the product of two factors. Starting with whole numbers is a jumping off point for students to see the distributive property in action with partial products that make sense.</p>
<p>If students are struggling at the start of this lesson, you could consider asking them to multiply 15✕18 by hand, and see what that process is like. Save their work and compare it with their area model. Which one makes more sense to them? </p>
</ol>