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9b47717f-226d-4a2a-8252-4eb69fc66a8e.html
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<h4>Activity </h4>
<p>Clare takes a piece of paper, cuts it in half, and then stacks the pieces. She takes the stack of two pieces and cuts
it in half again to form four pieces, stacking them. She keeps repeating the process.</p>
<ol class="os-raise-noindent">
<li>
The original piece of paper has length 8 inches and width 10 inches. Complete the table.
</li>
</ol>
<table class="os-raise-midsizetable">
<thead>
<tr>
<th scope="col">number of cuts</th>
<th scope="col">number of pieces</th>
<th scope="col">area in square inches of each piece</th>
</tr>
</thead>
<tbody>
<tr>
<td>
0
</td>
<td> </td>
<td> </td>
</tr>
<tr>
<td>
1
</td>
<td> </td>
<td> </td>
</tr>
<tr>
<td>
2
</td>
<td> </td>
<td> </td>
</tr>
<tr>
<td>
3
</td>
<td> </td>
<td> </td>
</tr>
<tr>
<td>
4
</td>
<td> </td>
<td> </td>
</tr>
<tr>
<td>
5
</td>
<td> </td>
<td> </td>
</tr>
</tbody>
</table>
<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">
</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">
<table class="os-raise-midsizetable">
<thead>
<tr>
<th scope="col">number of cuts</th>
<th scope="col">number of pieces</th>
<th scope="col">area in square inches of each piece</th>
</tr>
</thead>
<tbody>
<tr>
<td>
0
</td>
<td>1</td>
<td>80</td>
</tr>
<tr>
<td>
1
</td>
<td>2</td>
<td>40</td>
</tr>
<tr>
<td>
2
</td>
<td>4</td>
<td>20</td>
</tr>
<tr>
<td>
3
</td>
<td>8</td>
<td>10</td>
</tr>
<tr>
<td>
4
</td>
<td>16</td>
<td>5</td>
</tr>
<tr>
<td>
5
</td>
<td>32</td>
<td>\(\frac52\) or 2.5</td>
</tr>
</tbody>
</table>
<br>
</div>
<!--Text Entry Interaction Start -->
<div class="os-raise-ib-input" data-button-text="Solution" data-content-id="791a1f51-683a-4bcb-84bf-ee61fa960175"
data-fire-event="eventShow" data-schema-version="1.0">
<div class="os-raise-ib-input-content">
<ol class="os-raise-noindent" start="2">
<li>
Describe in words how you can use the results after 5 cuts to find the results after 6 cuts.
</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> Double the number of pieces, from 32 to 64. Halve the area of each piece, from \(\frac52\) to \(\frac54\).</p>
</div>
</div>
<!--Interaction End -->
<br>
<br>
<div class="os-raise-ib-desmos-gc" data-schema-version="1.0"> </div>
<br>
<br>
<!--Text Entry Interaction Start -->
<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" start="3">
<li>
<ol class="os-raise-noindent" type="a">
<li>Use the graphing tool above or technology outside the course, on the given axes, sketch a graph of the
number of pieces as a function of the number of cuts.
</li>
</ol>
</li>
</ol>
</div>
<div class="os-raise-ib-cta-prompt">
<p> 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><img height="186" src="https://k12.openstax.org/contents/raise/resources/640fe44d97e424cf3eefce9289f118c6bba06e3a"
width="315"></p>
</div>
<!--Interaction End -->
<br>
<br>
<!--Text Entry Interaction Start -->
<div class="os-raise-ib-input" data-button-text="Solution" data-content-id="e7be76b2-4a98-478b-89a7-7e62ecc4288f"
data-fire-event="eventShow" data-schema-version="1.0">
<div class="os-raise-ib-input-content">
<ol class="os-raise-noindent" start="2" type="a">
<li>How can you see on the graph how the number of pieces is changing with each cut?</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 height of each plotted point is twice the height of the previous plotted point.</p>
</div>
</div>
<!--Interaction End -->
<br>
<div class="os-raise-ib-desmos-gc" data-schema-version="1.0"></div>
<br>
<br>
<!--Text Entry Interaction Start -->
<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" start="4">
<li>
<ol class="os-raise-noindent" type="a">
<li>Use the graphing tool above or technology outside the course, on the given axes, sketch a graph of the
area of each piece as a function of the number of cuts.
</li>
</ol>
</li>
</ol>
</div>
<div class="os-raise-ib-cta-prompt">
<p> 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><img height="378" src="https://k12.openstax.org/contents/raise/resources/34702f050443b6b744a80e597b878040eca08b03"
width="322"></p>
</div>
<br>
<!--Interaction End -->
<br>
<br>
<!--Text Entry Interaction Start -->
<div class="os-raise-ib-input" data-button-text="Solution" data-content-id="c38aa21d-3b1c-4b86-b399-1c0b7dbb55d0"
data-fire-event="eventShow" data-schema-version="1.0">
<div class="os-raise-ib-input-content">
<ol class="os-raise-noindent" start="2" type="a">
<li>How can you see how the area of each piece is changing with each cut?</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 height of each plotted point is half the height of the previous plotted point.</p>
</div>
</div>
<!--Interaction End -->
<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="a21c6724-742a-438f-a6b3-61789d2eab59"
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>Clare has a piece of paper that is 8 inches by 10 inches.
<ol class="os-raise-noindent" type="a">
<li>How many pieces of paper will Clare have if she cuts the paper in half \(n\) times? </li>
</ol>
</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>\(2^n\) pieces</p>
</div>
</div>
<!--End interaction-->
<br>
<div class="os-raise-ib-input" data-button-text="Solution" data-content-id="32fee282-5ef0-44e9-b14e-9b2bf672b401"
data-schema-version="1.0" data-wait-for-event="RFM1">
<div class="os-raise-ib-input-content">
<ol class="os-raise-noindent">
<li>
<ol class="os-raise-noindent" start="2" type="a">
<li>What will the area of each piece be?</li>
</ol>
</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>Each piece has an area of \(\frac{80}{2^n}\) square inches.</p>
</div>
</div>
<!--End interaction-->
<br>
<div class="os-raise-ib-input" data-button-text="Solution" data-content-id="c7210faf-bfd4-4337-b89a-d67fc0d7d4bc"
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>Why is the product of the number of pieces and the area of each piece always the same? Explain how you know.
</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>It’s always the same because adding up the area of the little pieces of paper gives the area of the
original sheet of paper.</p>
</div>
</div>
<!--End interaction-->