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<p style="Normal" xid="2" props="text-align:left; line-height:1.5; dom-dir:ltr; margin-left:0.0mm"><c props="lang:en-US; font-size:12pt; text-decoration:underline; font-family:FreeSans; font-weight:bold">Slider Puzzle Lesson 7:</c></p>
<p style="Normal" xid="220" props="text-align:left; line-height:1.5; dom-dir:ltr; margin-left:0.0mm"><c props="lang:en-US; font-size:12pt; text-decoration:underline; font-family:FreeSans; font-weight:bold">The </c><c props="lang:en-US; font-size:12pt; text-decoration:underline; font-family:FreeSans; font-weight:bold; font-style:italic">n-puzzle</c><c props="lang:en-US; font-size:12pt; text-decoration:underline; font-family:FreeSans; font-weight:bold"> (Mathematical Patterns and Sequences)</c></p>
<p style="Normal (Web)" xid="170" props="text-align:left; margin-top:5pt; line-height:1.5; margin-bottom:5pt; dom-dir:ltr; margin-left:0.0mm"><c props="lang:en-US; font-size:12pt; font-family:FreeSans">The </c><c props="lang:en-US; font-weight:bold; font-family:FreeSans; font-size:12pt; font-style:italic">n</c><c props="lang:en-US; font-size:12pt; font-family:FreeSans; font-weight:bold">-puzzle</c><c props="lang:en-US; font-size:12pt; font-family:FreeSans"> is general name for a </c><a xlink:href="http://en.wikipedia.org/wiki/Sliding_puzzle" xid="171"><c props="lang:en-US; font-size:12pt; font-family:FreeSans">slider puzzle</c></a><c props="lang:en-US; font-size:12pt; font-family:FreeSans"> that </c><c props="lang:en-US; font-size:12pt; font-family:FreeSans; font-weight:bold">consists of a grid of numbered squares with one square missing, and the numbers out of sequence</c><c props="lang:en-US; font-size:12pt; font-family:FreeSans">. The 9-puzzle is a 3×3 numbered grid, the 16 puzzle is a 4×4 numbered grid. The goal of the n-puzzle is put the scrambled numbers back in sequence.</c></p>
<p style="Normal (Web)" xid="173" props="text-align:left; margin-top:5pt; line-height:1.5; margin-bottom:5pt; dom-dir:ltr; margin-left:0.0mm"><c props="lang:en-US; font-size:12pt; font-family:FreeSans; font-weight:bold">The n-puzzle is a classical problem for modeling repeatable steps (or “</c><a xlink:href="http://en.wikipedia.org/wiki/Algorithm" xid="174"><c props="lang:en-US; font-size:12pt; font-family:FreeSans; font-weight:bold">algorithms</c></a><c props="lang:en-US; font-size:12pt; font-family:FreeSans; font-weight:bold">”) involving mathematical patterns (or “</c><a xlink:href="http://en.wikipedia.org/wiki/Heuristic_(computer_science)" xid="176"><c props="lang:en-US; font-size:12pt; font-family:FreeSans; font-weight:bold">heuristics</c></a><c props="lang:en-US; font-size:12pt; font-family:FreeSans; font-weight:bold">.”)</c><c props="lang:en-US; font-size:12pt; font-family:FreeSans"> For example, you can begin by counting the number of misplaced tiles as you formulate a strategy for solving the puzzle. The n-puzzle also helps students to understand that some tile arrangements are unsolvable. </c></p>
<p style="Normal" xid="178" props="text-align:left; margin-top:0pt; line-height:1.5; margin-bottom:0pt; dom-dir:ltr; margin-left:0.0mm"><c props="lang:en-US; font-size:12pt; font-family:FreeSans; font-weight:bold">Skills</c><c props="lang:en-US; font-size:12pt; font-family:FreeSans">:</c></p>
<p level="1" listid="1000" parentid="0" style="Normal (Web)" xid="179" props="margin-top:0pt; list-style:Bullet List; margin-left:12.7mm; start-value:1; dom-dir:ltr; text-align:left; line-height:1.5; tabstops:0.6250in/L; margin-bottom:0pt; text-indent:-0.2500in; field-font:Symbol"><field type="list_label" xid="180" props="font-family:FreeSans; text-decoration:none"></field><c props="font-family:FreeSans; text-decoration:none"> </c><c props="lang:en-US; font-size:12pt; font-family:FreeSans">Recognize patterns.</c></p>
<p level="1" listid="1000" parentid="0" style="Normal (Web)" xid="181" props="margin-top:0pt; list-style:Bullet List; margin-left:12.7mm; start-value:1; dom-dir:ltr; text-align:left; line-height:1.5; tabstops:0.6250in/L; margin-bottom:0pt; text-indent:-0.2500in; field-font:Symbol"><field type="list_label" xid="182" props="font-family:FreeSans; text-decoration:none"></field><c props="font-family:FreeSans; text-decoration:none"> </c><c props="lang:en-US; font-size:12pt; font-family:FreeSans">Practice classification and grouping.</c></p>
<p level="1" listid="1000" parentid="0" style="Normal (Web)" xid="183" props="margin-top:0pt; list-style:Bullet List; margin-left:12.7mm; start-value:1; dom-dir:ltr; text-align:left; line-height:1.5; tabstops:0.6250in/L; margin-bottom:0pt; text-indent:-0.2500in; field-font:Symbol"><field type="list_label" xid="184" props="font-family:FreeSans; text-decoration:none"></field><c props="font-family:FreeSans; text-decoration:none"> </c><c props="lang:en-US; font-size:12pt; font-family:FreeSans">Practice sequencing.</c></p>
<p level="1" listid="1000" parentid="0" style="Normal (Web)" xid="185" props="margin-top:0pt; list-style:Bullet List; margin-left:12.7mm; start-value:1; dom-dir:ltr; text-align:left; line-height:1.5; tabstops:0.6250in/L; margin-bottom:0pt; text-indent:-0.2500in; field-font:Symbol"><field type="list_label" xid="186" props="font-family:FreeSans; text-decoration:none"></field><c props="font-family:FreeSans; text-decoration:none"> </c><c props="lang:en-US; font-size:12pt; font-family:FreeSans">Put numbers in order.</c></p>
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<p style="Normal (Web)" xid="244" props="text-align:left; line-height:1.5; dom-dir:ltr; margin-left:0.0mm"><c props="lang:en-US; font-size:12pt; font-family:FreeSans; font-weight:bold">Activity</c><c props="lang:en-US; font-size:12pt; font-family:FreeSans">:</c></p>
<p level="1" listid="1010" parentid="0" style="Normal" xid="189" props="list-style:Numbered List; start-value:1; dom-dir:ltr; tabstops:0.5000in/L; margin-left:0.5000in; text-indent:-0.3000in; line-height:1.5; text-align:left; field-font:NULL"><field type="list_label" xid="228" props="width:0in; font-family:FreeSans; display:inline; font-variant:normal; font-weight:normal; lang:en-US; font-style:normal; color:000000; homogeneous:1; bgcolor:transparent; font-size:12pt; text-decoration:none; list-style:None; height:0in; text-position:normal; font-stretch:normal"></field><c props="width:0in; font-family:FreeSans; display:inline; font-variant:normal; font-weight:normal; lang:en-US; font-style:normal; color:000000; homogeneous:1; bgcolor:transparent; font-size:12pt; text-decoration:none; list-style:None; height:0in; text-position:normal; font-stretch:normal"> </c><c props="lang:en-US; font-size:12pt; font-family:FreeSans; font-weight:bold">Start the Slider Puzzle</c><c props="lang:en-US; font-size:12pt; font-family:FreeSans"> activity and </c><c props="lang:en-US; font-size:12pt; font-family:FreeSans; font-weight:bold">choose the category labeled “Sequencing.”</c><c props="lang:en-US; font-size:12pt; font-family:FreeSans"> Sequencing is when you put things into a logical order. </c></p>
<p level="1" listid="1010" style="Normal" xid="191" props="list-delim:%L; list-decimal:.; list-style:Numbered List; start-value:1; margin-left:0.5000in; text-indent:-0.3000in; line-height:1.5; dom-dir:ltr; text-align:left; field-font:NULL"><field type="list_label" xid="229" props="width:0in; font-family:FreeSans; display:inline; color:000000; font-weight:normal; text-position:normal; lang:en-US; font-style:normal; font-variant:normal; text-decoration:none; bgcolor:transparent; list-style:Numbered List; homogeneous:1; height:0in; font-size:12pt; font-stretch:normal"></field><c props="width:0in; font-family:FreeSans; display:inline; color:000000; font-weight:normal; text-position:normal; lang:en-US; font-style:normal; font-variant:normal; text-decoration:none; bgcolor:transparent; list-style:Numbered List; homogeneous:1; height:0in; font-size:12pt; font-stretch:normal"> </c><c props="lang:en-US; font-size:12pt; font-family:FreeSans; font-weight:bold">When the first image comes up, think how this image is set up differently from the one on the Slider Puzzle</c><c props="lang:en-US; font-size:12pt; font-family:FreeSans">. The Slider Puzzle creates a large image, where all the squares are pieces of the larger image, while the Sequencing image contains a number on each tile. </c></p>
<p level="1" listid="1010" style="Normal" xid="194" props="list-delim:%L; list-decimal:.; list-style:Numbered List; start-value:1; margin-left:0.5000in; text-indent:-0.3000in; line-height:1.5; dom-dir:ltr; text-align:left; field-font:NULL"><field type="list_label" xid="230" props="width:0in; font-family:FreeSans; display:inline; font-variant:normal; font-weight:normal; lang:en-US; font-style:normal; color:000000; homogeneous:1; bgcolor:transparent; font-size:12pt; text-decoration:none; list-style:None; height:0in; text-position:normal; font-stretch:normal"></field><c props="width:0in; font-family:FreeSans; display:inline; font-variant:normal; font-weight:normal; lang:en-US; font-style:normal; color:000000; homogeneous:1; bgcolor:transparent; font-size:12pt; text-decoration:none; list-style:None; height:0in; text-position:normal; font-stretch:normal"> </c><c props="lang:en-US; font-size:12pt; font-family:FreeSans; font-weight:bold">Try to solve the 9-tile puzzle first. Try to put the numbers in order from smallest to largest.</c><c props="lang:en-US; font-size:12pt; font-family:FreeSans"> Model for other students how to slide the keys and place them in order.</c></p>
<p level="1" listid="1010" style="Normal" xid="197" props="list-delim:%L; list-decimal:.; list-style:Numbered List; start-value:1; margin-left:0.5000in; text-indent:-0.3000in; line-height:1.5; dom-dir:ltr; text-align:left; field-font:NULL"><field type="list_label" xid="231" props="width:0in; font-family:FreeSans; display:inline; color:000000; font-weight:normal; text-position:normal; lang:en-US; font-style:normal; font-variant:normal; text-decoration:none; bgcolor:transparent; list-style:Numbered List; homogeneous:1; height:0in; font-size:12pt; font-stretch:normal"></field><c props="width:0in; font-family:FreeSans; display:inline; color:000000; font-weight:normal; text-position:normal; lang:en-US; font-style:normal; font-variant:normal; text-decoration:none; bgcolor:transparent; list-style:Numbered List; homogeneous:1; height:0in; font-size:12pt; font-stretch:normal"> </c><c props="lang:en-US; font-size:12pt; font-family:FreeSans; font-weight:bold">Once you are able to solve the 9-puzzle, share your strategies with the class.</c><c props="lang:en-US; font-size:12pt; font-family:FreeSans"> How is this different from the Slider Puzzles where you are assembling a picture? </c></p>
<p level="1" listid="1010" style="Normal" xid="200" props="list-delim:%L; list-decimal:.; list-style:Numbered List; start-value:1; margin-left:0.5000in; text-indent:-0.3000in; line-height:1.5; dom-dir:ltr; text-align:left; field-font:NULL"><field type="list_label" xid="232" props="width:0in; font-family:FreeSans; display:inline; font-variant:normal; font-weight:normal; lang:en-US; font-style:normal; color:000000; homogeneous:1; bgcolor:transparent; font-size:12pt; text-decoration:none; list-style:None; height:0in; text-position:normal; font-stretch:normal"></field><c props="width:0in; font-family:FreeSans; display:inline; font-variant:normal; font-weight:normal; lang:en-US; font-style:normal; color:000000; homogeneous:1; bgcolor:transparent; font-size:12pt; text-decoration:none; list-style:None; height:0in; text-position:normal; font-stretch:normal"> </c><c props="lang:en-US; font-size:12pt; font-family:FreeSans; font-weight:bold">Now try to solve the 16-puzzle. How is this harder or easier?</c></p>
<p level="1" listid="1010" style="Normal" xid="203" props="list-delim:%L; list-decimal:.; list-style:Numbered List; start-value:1; margin-left:0.5000in; text-indent:-0.3000in; line-height:1.5; dom-dir:ltr; text-align:left; field-font:NULL"><field type="list_label" xid="233" props="width:0in; font-family:FreeSans; display:inline; font-variant:normal; font-weight:normal; lang:en-US; font-style:normal; color:000000; homogeneous:1; bgcolor:transparent; font-size:12pt; text-decoration:none; list-style:Numbered List; height:0in; text-position:normal; font-stretch:normal"></field><c props="width:0in; font-family:FreeSans; display:inline; font-variant:normal; font-weight:normal; lang:en-US; font-style:normal; color:000000; homogeneous:1; bgcolor:transparent; font-size:12pt; text-decoration:none; list-style:Numbered List; height:0in; text-position:normal; font-stretch:normal"> </c><c props="lang:en-US; font-size:12pt; font-family:FreeSans">After completing the 16-puzzle,</c><c props="lang:en-US; font-size:12pt; font-family:FreeSans; font-weight:bold"> walk around and help those who are having trouble.</c></p>
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