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<h4>Activity (30 minutes)</h4>
<p>This activity is provided in case an additional activity is needed to reinforce the lesson goal. Alternatively, if
students are familiar with the Tower of Hanoi puzzle, use this activity instead, and use the structure of the launch
and discussion for the Tower of Hanoi activity for the discussion of this activity.</p>
<p>This activity works best when each student has access to manipulatives or devices that can run the GeoGebra applet
because students will benefit from seeing the relationship in a dynamic way. If students don’t have individual
access, projecting the applet during the launch is helpful.</p>
<p>Monitor for students writing clear explanations for Noah’s reasoning and for the number of moves needed for 7
checkers on each side to share during the discussion.</p>
<h4>Launch</h4>
<p>Students can use the embedded applet below.</p>
<p>Ask students to read the rules to the puzzle and then give them time to solve the puzzle when there are 3 spaces and
1 checker on each side. Remind them they need to keep track of the number of moves needed since the goal is to find
the smallest number of moves. Encourage groups to check in with those around them to see if anyone found a solution
with fewer moves. Before groups begin work on the rest of the questions, select a student to demonstrate, using the
applet, why it takes 3 moves to solve the puzzle when there are 3 spaces and 1 checker on each side as a check that
everyone understands the rules of the game.</p>
<div class="os-raise-extrasupport">
<div class="os-raise-extrasupport-header">
<p class="os-raise-extrasupport-title">Support for Students with Disabilities</p>
</div>
<div class="os-raise-extrasupport-body">
<p class="os-raise-extrasupport-name"> Representation: Develop Language and Symbols</p>
<p>Use virtual or concrete manipulatives to connect symbols to concrete objects. Provide black and red checkers if the
applet is not available. <p class="os-raise-text-italicize">Supports accessibility for: Conceptual processing</p>
</div>
</div>
<br>
<h4> Student Activity </h4>
<br>
<div class="os-raise-d-flex-nowrap os-raise-justify-content-center">
<div class="os-raise-video-container"><iframe scrolling="no" src="https://www.geogebra.org/material/iframe/id/qugvaudj/width/681/height/350/border/888888/sfsb/true/smb/false/stb/false/stbh/false/ai/false/asb/false/sri/true/rc/false/ld/false/sdz/false/ctl/false" title="Alg2 Moving Checkers"> </iframe></div>
</div>
<br>
<p>Some checkers are lined up, with blue on one side, red on the other, and 1 empty space between them. A move in this
checker game pushes any checker forward 1 space, or jumps over any 1 checker of the other color. Jumping the same
color is not allowed, moving backward is not allowed, and 2 checkers cannot occupy the same space.</p>
<p><img alt="THREE BLACK CHECKERS, A SPACE, AND THEN THREE RED CHECKERS.
" class="img-fluid atto_image_button_text-bottom" height="43" src="https://k12.openstax.org/contents/raise/resources/0830b28585b143bb3a3fa0c6a4ab68e6bda92a3b" width="298"></p>
<p>You complete the puzzle by switching the colors completely: ending up with black on the right, red on the left, and 1
empty space between them.</p>
<ol class="os-raise-noindent">
<li> Using 1 checker on each side, complete the puzzle. What is the smallest number of moves needed?</li>
<p> <strong>Answer: </strong> 3 moves </p>
<li> Using 3 checkers on each side, complete the puzzle. What is the smallest number of moves needed? </li>
<p> <strong>Answer: </strong> 15 moves </p>
<li> Estimate the number of moves needed if there are 2 or 4 checkers on each side. Then test your guesses. </li>
<p> <strong>Answer: </strong> 8 moves and 24 moves </p>
<li> Noah says he used the solution for 3 checkers on each side to help him solve the puzzle for 4 checkers. Describe
how this might happen. </li>
<p> <strong>Answer: </strong> For example: The moves that don’t involve the fourth checker on each side are the
same as before, so it is possible to decide the number of moves by running the first half of the three-checker game,
then moving the fourth checkers, and then running the second half. </p>
<li> How many moves do you think it will take to complete a puzzle with 7 checkers on each side? </li>
<p> <strong>Answer: </strong> 63 moves </p>
</ol>
<h4>Anticipated Misconceptions</h4>
<p>Students may incorrectly interpret the rules or presume that there is no better solution once they find one solution
that works. Encourage groups to regularly switch who is in charge of moving the checkers and to check with other
groups around them once they think they have found a solution with the smallest number of moves.</p>
<h4>Activity Synthesis</h4>
<p>Select students to share the smallest number of moves they found for 2, 3, and 4 checkers on each side. Each time,
ask whether anyone in the class solved it with fewer moves. If no one finds the minimal solution, ask students to keep
looking if time allows, or share the minimum number of moves and challenge them to do it, or demonstrate the minimal
solution. </p>
<p>Invite previously identified students to share their explanation for Noah’s strategy and the number of moves
needed to complete a puzzle with 3 checkers on each side. The bottom row of the table should now have the numbers 3,
8, 15, and 24 filled in.</p>
<p>If this activity was used instead of the Tower of Hanoi, make sure to define “sequence” and
“term” for students before inviting students to describe a rule for the next term in the pattern. After a
brief quiet time to think individually, select 2–3 students to share their thinking and write down any notation
they come up with to describe the recursive rule, such as “first add 5, and then keep adding the next odd
number.” There is no need to introduce formal notation or discuss a specific rule for finding term \(n\) at this
time, but if students suggest these (time permitting), welcome their explanations.</p>
<h3>4.14.3: Self Check </h3>
<p class="os-raise-text-bold"><em>After the activity, students will answer the following question to check their
understanding of the concepts explored in the activity.</em></p>
<p class="os-raise-text-bold">QUESTION:</p>
<p>The table below shows the number of checkers and the number of moves needed to win a skipping game.</p>
<table class="os-raise-horizontaltable">
<thead></thead>
<tbody>
<tr>
<th scope="row">Number of Checkers</th>
<td>
1
</td>
<td>
2
</td>
<td>
3
</td>
<td>
4
</td>
</tr>
<tr>
<th scope="row">Number of Skips Needed</th>
<td>
2
</td>
<td>
8
</td>
<td>
18
</td>
<td>
32
</td>
</tr>
</tbody>
</table>
<br>
<p>How many skips are needed with 6 checkers on each side?</p>
<table class="os-raise-textheavytable">
<thead>
<tr>
<th scope="col"> Answers</th>
<th scope="col">Feedback</th>
</tr>
</thead>
<tbody>
<tr>
<td>
50
</td>
<td>
Incorrect. Let’s try again a different way: This is the number of skips needed with 5 checkers. The
answer is 72.
</td>
</tr>
<tr>
<td>
72
</td>
<td>
That’s correct! Check yourself: In every column, the number of checkers is multiplied by 2 more than
the number in the previous column. 6 will be multiplied by 12, which results in 72 moves.
</td>
</tr>
<tr>
<td>
128
</td>
<td>
Incorrect. Let’s try again a different way: This assumes the number of skips needed started to double,
which is not the case. The answer is 72.
</td>
</tr>
<tr>
<td>
48
</td>
<td>
Incorrect. Let’s try again a different way: This did not change the number being multiplied by each
time and kept multiplying the number of checkers by 8. The answer is 72.
</td>
</tr>
</tbody>
</table>
<br>
<h3>4.14.3: Additional Resources</h3>
<p class="os-raise-text-bold"><em>The following content is available to students who would like more support based on
their experience with the self check. Students will not automatically have access to this content, so you may wish
to share it with those who could benefit from it.</em></p>
<h4>Using a Table to Find Missing Values</h4>
<br>
<table class="os-raise-horizontaltable">
<thead></thead>
<tbody>
<tr>
<th scope="row">Number of Checkers</th>
<td>
1
</td>
<td>
2
</td>
<td>
3
</td>
<td>
4
</td>
<td>
7
</td>
<td>
8
</td>
</tr>
<tr>
<th scope="row">>Number of Moves Needed</th>
<td>
3
</td>
<td>
8
</td>
<td>
15
</td>
<td>
24
</td>
<td>
63
</td>
<td>
?
</td>
</tr>
</tbody>
</table>
<br>
<p>What is the pattern that you used in the previous activity to find the number of moves depending on the number of
checkers?</p>
<p>Each row is multiplied by 1 more, so:</p>
<p>\(1\cdot3=3\)</p>
<p>\(2\cdot4=8\)</p>
<p>\(3\cdot5=15\)</p>
<p>\(4\cdot6=24\)</p>
<p>How many moves are needed with 5 checkers?</p>
<p>\(5\cdot7=35\) moves</p>
<p>What about 6 checkers?</p>
<p>\(6\cdot8=48\) moves</p>
<h4>Try It: Using a Table to Find Missing Values</h4>
<p>Using the table and the pattern, how many moves are needed with 8 checkers?</p>
<p>Write down your answer. Then select the<strong> Solution</strong> button to compare your work.</p>
<h5>Solution</h5>
<p>Here is how to solve this problem using a general strategy:</p>
<p>The next column in the table had 7 checkers and then multiplied 7 by 9 to get 63 moves.</p>
<p>Next, the 8 would need to be multiplied by 10, so:</p>
<p>\(8\cdot10=80\)</p>
<p>80 moves are needed.</p>