{"id":135,"date":"2017-07-07T21:25:36","date_gmt":"2017-07-07T21:25:36","guid":{"rendered":"http:\/\/pressbooks-dev.oer.hawaii.edu\/math111\/?post_type=chapter&#038;p=135"},"modified":"2018-01-25T23:11:51","modified_gmt":"2018-01-25T23:11:51","slug":"dots-and-boxes","status":"publish","type":"chapter","link":"https:\/\/pressbooks-dev.oer.hawaii.edu\/math111\/chapter\/dots-and-boxes\/","title":{"raw":"Dots and Boxes","rendered":"Dots and Boxes"},"content":{"raw":"<p class=\"p1\">Here are some dots; in fact there are nine of them:<\/p>\r\n<p style=\"text-align: center\"><img src=\"http:\/\/pressbooks-dev.oer.hawaii.edu\/math111\/wp-content\/uploads\/sites\/17\/2017\/10\/9dots.png\" alt=\"\" class=\"wp-image-363 aligncenter\" width=\"110\" height=\"101\" \/><\/p>\r\n<p class=\"p1\">We're going to a play an \"exploding dots\" game. Here's the only rule for the game:<\/p>\r\n\r\n<div class=\"textbox learning-objectives\">\r\n<h3 itemprop=\"educationalUse\"><b>The 1<\/b>\u21902<b> Rule<\/b><\/h3>\r\n<p class=\"p2\">Whenever there are two dots in single box, they \u201cexplode,'' disappear, and become one dot in the box to the left.<\/p>\r\n\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox key-takeaways\">\r\n<h3 itemprop=\"educationalUse\">Example:\u00a0Nine dots 1\u21902 in the system<\/h3>\r\n<p class=\"p1\">We start by placing nine dots in the rightmost box.<\/p>\r\n<p style=\"text-align: center\"><img src=\"http:\/\/pressbooks-dev.oer.hawaii.edu\/math111\/wp-content\/uploads\/sites\/17\/2017\/10\/Explode1a-300x87.png\" alt=\"\" class=\"alignnone size-medium wp-image-380 aligncenter\" width=\"300\" height=\"87\" \/><\/p>\r\n<p class=\"p1\">Two dots in that box explode and become one dot in the box to the left.<\/p>\r\n<p style=\"text-align: center\"><img src=\"http:\/\/pressbooks-dev.oer.hawaii.edu\/math111\/wp-content\/uploads\/sites\/17\/2017\/10\/Explode1b-300x83.png\" alt=\"\" class=\"alignnone size-medium wp-image-379\" width=\"300\" height=\"83\" \/><\/p>\r\n<p style=\"text-align: center\"><img src=\"http:\/\/pressbooks-dev.oer.hawaii.edu\/math111\/wp-content\/uploads\/sites\/17\/2017\/10\/Explode1c-300x84.png\" alt=\"\" class=\"size-medium wp-image-378 aligncenter\" width=\"300\" height=\"84\" \/><\/p>\r\n<p class=\"p1\">Once again, two dots in that box explode and become one dot in the box to the left.<\/p>\r\n<p style=\"text-align: center\"><img src=\"http:\/\/pressbooks-dev.oer.hawaii.edu\/math111\/wp-content\/uploads\/sites\/17\/2017\/10\/Explode1d-300x81.png\" alt=\"\" class=\"alignnone size-medium wp-image-377\" width=\"300\" height=\"81\" \/><\/p>\r\n<p style=\"text-align: center\"><img src=\"http:\/\/pressbooks-dev.oer.hawaii.edu\/math111\/wp-content\/uploads\/sites\/17\/2017\/10\/Explode1e-300x82.png\" alt=\"\" class=\"size-medium wp-image-376 aligncenter\" width=\"300\" height=\"82\" \/><\/p>\r\nWe do it again!\r\n<p style=\"text-align: center\"><img src=\"http:\/\/pressbooks-dev.oer.hawaii.edu\/math111\/wp-content\/uploads\/sites\/17\/2017\/10\/Explode1f-300x86.png\" alt=\"\" class=\"alignnone size-medium wp-image-375\" width=\"300\" height=\"86\" \/><\/p>\r\n<p style=\"text-align: center\"><img src=\"http:\/\/pressbooks-dev.oer.hawaii.edu\/math111\/wp-content\/uploads\/sites\/17\/2017\/10\/Explode1g-300x89.png\" alt=\"\" class=\"size-medium wp-image-374 aligncenter\" width=\"300\" height=\"89\" \/><\/p>\r\n<p class=\"p1\">Hey, now we have more than two dots in the second box, so those can explode and move!<\/p>\r\n<p style=\"text-align: center\"><img src=\"http:\/\/pressbooks-dev.oer.hawaii.edu\/math111\/wp-content\/uploads\/sites\/17\/2017\/10\/Explode1h-300x82.png\" alt=\"\" class=\"alignnone size-medium wp-image-366\" width=\"300\" height=\"82\" \/><\/p>\r\n<p style=\"text-align: center\"><img src=\"http:\/\/pressbooks-dev.oer.hawaii.edu\/math111\/wp-content\/uploads\/sites\/17\/2017\/10\/Explode1i-300x85.png\" alt=\"\" class=\"size-medium wp-image-367 aligncenter\" width=\"300\" height=\"85\" \/><\/p>\r\n<p class=\"p1\">And the rightmost box still has more than two dots.<\/p>\r\n<p style=\"text-align: center\"><img src=\"http:\/\/pressbooks-dev.oer.hawaii.edu\/math111\/wp-content\/uploads\/sites\/17\/2017\/10\/Explode1j-300x89.png\" alt=\"\" class=\"alignnone size-medium wp-image-368\" width=\"300\" height=\"89\" \/><\/p>\r\n<p style=\"text-align: center\"><img src=\"http:\/\/pressbooks-dev.oer.hawaii.edu\/math111\/wp-content\/uploads\/sites\/17\/2017\/10\/Explode1k-300x84.png\" alt=\"\" class=\"alignnone size-medium wp-image-369 aligncenter\" width=\"300\" height=\"84\" \/><\/p>\r\n<p class=\"p1\">Keep going, until no box has two dots.<\/p>\r\n<p style=\"text-align: center\"><img src=\"http:\/\/pressbooks-dev.oer.hawaii.edu\/math111\/wp-content\/uploads\/sites\/17\/2017\/10\/Explode1l-300x86.png\" alt=\"\" class=\"alignnone size-medium wp-image-370\" width=\"300\" height=\"86\" \/><\/p>\r\n<p style=\"text-align: center\"><img src=\"http:\/\/pressbooks-dev.oer.hawaii.edu\/math111\/wp-content\/uploads\/sites\/17\/2017\/10\/Explode1m-300x83.png\" alt=\"\" class=\"alignnone size-medium wp-image-371\" width=\"300\" height=\"83\" \/><\/p>\r\n<p style=\"text-align: center\"><img src=\"http:\/\/pressbooks-dev.oer.hawaii.edu\/math111\/wp-content\/uploads\/sites\/17\/2017\/10\/Explode1n-300x86.png\" alt=\"\" class=\"alignnone size-medium wp-image-372\" width=\"300\" height=\"86\" \/><\/p>\r\n<p style=\"text-align: center\"><img src=\"http:\/\/pressbooks-dev.oer.hawaii.edu\/math111\/wp-content\/uploads\/sites\/17\/2017\/10\/Explode1o-300x88.png\" alt=\"\" class=\"alignnone size-medium wp-image-373 aligncenter\" width=\"300\" height=\"88\" \/><\/p>\r\n<p class=\"p1\">After all this, reading from left to right we are left with one dot, followed by zero dots, zero dots, and one final dot.<\/p>\r\n&nbsp;\r\n<p class=\"p1\"><b>Solution:\u00a0<\/b>The 1\u21902 code for nine dots is: 1001.<\/p>\r\n\r\n<\/div>\r\n&nbsp;\r\n<p class=\"p1\"><b>On Your Own. \u00a0<\/b>Here's a diagram showing what happens for seven dots in a 1\u21902 box. Trace through the diagram, and circle the pairs of dots that \u201cexploded'' at each step.<\/p>\r\n<p style=\"text-align: center\"><img src=\"http:\/\/pressbooks-dev.oer.hawaii.edu\/math111\/wp-content\/uploads\/sites\/17\/2017\/10\/Explode2a-300x85.png\" alt=\"\" class=\"alignnone size-medium wp-image-382\" width=\"300\" height=\"85\" \/><\/p>\r\n<p style=\"text-align: center\"><img src=\"http:\/\/pressbooks-dev.oer.hawaii.edu\/math111\/wp-content\/uploads\/sites\/17\/2017\/10\/Explode2b-300x81.png\" alt=\"\" class=\"alignnone size-medium wp-image-383\" width=\"300\" height=\"81\" \/><\/p>\r\n<p style=\"text-align: center\"><img src=\"http:\/\/pressbooks-dev.oer.hawaii.edu\/math111\/wp-content\/uploads\/sites\/17\/2017\/10\/Explode2c-300x83.png\" alt=\"\" class=\"alignnone size-medium wp-image-384\" width=\"300\" height=\"83\" \/><\/p>\r\n<p style=\"text-align: center\"><img src=\"http:\/\/pressbooks-dev.oer.hawaii.edu\/math111\/wp-content\/uploads\/sites\/17\/2017\/10\/Explode2d-300x84.png\" alt=\"\" class=\"alignnone size-medium wp-image-385\" width=\"300\" height=\"84\" \/><\/p>\r\n<p style=\"text-align: center\"><img src=\"http:\/\/pressbooks-dev.oer.hawaii.edu\/math111\/wp-content\/uploads\/sites\/17\/2017\/10\/Explode2e-300x87.png\" alt=\"\" class=\"size-medium wp-image-386 aligncenter\" width=\"300\" height=\"87\" \/><\/p>\r\n<p class=\"p1\"><b>Solution:\u00a0<\/b>The 1\u21902 code for seven dots is: 111.<\/p>\r\n&nbsp;\r\n<div class=\"textbox exercises\">\r\n<h3 itemprop=\"educationalUse\"><b>Problem 1<\/b><\/h3>\r\n<p class=\"p1\">Note: In solving this problem, you don't need to draw on paper; that can get tedious! Maybe you could use buttons or pennies for dots and do this by hand.<\/p>\r\n\r\n<ol>\r\n \t<li class=\"hanging-indent\">Draw 10 dots in the right-most box and perform the explosions. What is the 1\u21902 code for ten dots?<\/li>\r\n \t<li class=\"hanging-indent\">Find the 1\u21902 code for eighteen dots.<\/li>\r\n \t<li class=\"hanging-indent\">What number of dots has 1\u21902 code\u00a0101?<\/li>\r\n<\/ol>\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox examples\">\r\n<h3 itemprop=\"educationalUse\"><b>Think \/ Pair \/ Share<\/b><\/h3>\r\n<p class=\"p1\">After you worked on the problem, compare your answer with a partner.\u00a0 Did you both get the same code?\u00a0 Did you have the same process?<\/p>\r\n\r\n<\/div>","rendered":"<p class=\"p1\">Here are some dots; in fact there are nine of them:<\/p>\n<p style=\"text-align: center\"><img loading=\"lazy\" decoding=\"async\" src=\"\/\/pressbooks-dev.oer.hawaii.edu\/math111\/wp-content\/uploads\/sites\/17\/2017\/10\/9dots.png\" alt=\"\" class=\"wp-image-363 aligncenter\" width=\"110\" height=\"101\" srcset=\"https:\/\/pressbooks-dev.oer.hawaii.edu\/math111\/wp-content\/uploads\/sites\/17\/2017\/10\/9dots.png 194w, https:\/\/pressbooks-dev.oer.hawaii.edu\/math111\/wp-content\/uploads\/sites\/17\/2017\/10\/9dots-65x60.png 65w\" sizes=\"auto, (max-width: 110px) 100vw, 110px\" \/><\/p>\n<p class=\"p1\">We&#8217;re going to a play an &#8220;exploding dots&#8221; game. Here&#8217;s the only rule for the game:<\/p>\n<div class=\"textbox learning-objectives\">\n<h3 itemprop=\"educationalUse\"><b>The 1<\/b>\u21902<b> Rule<\/b><\/h3>\n<p class=\"p2\">Whenever there are two dots in single box, they \u201cexplode,&#8221; disappear, and become one dot in the box to the left.<\/p>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox key-takeaways\">\n<h3 itemprop=\"educationalUse\">Example:\u00a0Nine dots 1\u21902 in the system<\/h3>\n<p class=\"p1\">We start by placing nine dots in the rightmost box.<\/p>\n<p style=\"text-align: center\"><img loading=\"lazy\" decoding=\"async\" src=\"\/\/pressbooks-dev.oer.hawaii.edu\/math111\/wp-content\/uploads\/sites\/17\/2017\/10\/Explode1a-300x87.png\" alt=\"\" class=\"alignnone size-medium wp-image-380 aligncenter\" width=\"300\" height=\"87\" srcset=\"https:\/\/pressbooks-dev.oer.hawaii.edu\/math111\/wp-content\/uploads\/sites\/17\/2017\/10\/Explode1a-300x87.png 300w, https:\/\/pressbooks-dev.oer.hawaii.edu\/math111\/wp-content\/uploads\/sites\/17\/2017\/10\/Explode1a-65x19.png 65w, https:\/\/pressbooks-dev.oer.hawaii.edu\/math111\/wp-content\/uploads\/sites\/17\/2017\/10\/Explode1a-225x65.png 225w, https:\/\/pressbooks-dev.oer.hawaii.edu\/math111\/wp-content\/uploads\/sites\/17\/2017\/10\/Explode1a-350x101.png 350w, https:\/\/pressbooks-dev.oer.hawaii.edu\/math111\/wp-content\/uploads\/sites\/17\/2017\/10\/Explode1a.png 742w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/p>\n<p class=\"p1\">Two dots in that box explode and become one dot in the box to the left.<\/p>\n<p style=\"text-align: center\"><img loading=\"lazy\" decoding=\"async\" src=\"\/\/pressbooks-dev.oer.hawaii.edu\/math111\/wp-content\/uploads\/sites\/17\/2017\/10\/Explode1b-300x83.png\" alt=\"\" class=\"alignnone size-medium wp-image-379\" width=\"300\" height=\"83\" srcset=\"https:\/\/pressbooks-dev.oer.hawaii.edu\/math111\/wp-content\/uploads\/sites\/17\/2017\/10\/Explode1b-300x83.png 300w, https:\/\/pressbooks-dev.oer.hawaii.edu\/math111\/wp-content\/uploads\/sites\/17\/2017\/10\/Explode1b-65x18.png 65w, https:\/\/pressbooks-dev.oer.hawaii.edu\/math111\/wp-content\/uploads\/sites\/17\/2017\/10\/Explode1b-225x63.png 225w, https:\/\/pressbooks-dev.oer.hawaii.edu\/math111\/wp-content\/uploads\/sites\/17\/2017\/10\/Explode1b-350x97.png 350w, https:\/\/pressbooks-dev.oer.hawaii.edu\/math111\/wp-content\/uploads\/sites\/17\/2017\/10\/Explode1b.png 734w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/p>\n<p style=\"text-align: center\"><img loading=\"lazy\" decoding=\"async\" src=\"\/\/pressbooks-dev.oer.hawaii.edu\/math111\/wp-content\/uploads\/sites\/17\/2017\/10\/Explode1c-300x84.png\" alt=\"\" class=\"size-medium wp-image-378 aligncenter\" width=\"300\" height=\"84\" srcset=\"https:\/\/pressbooks-dev.oer.hawaii.edu\/math111\/wp-content\/uploads\/sites\/17\/2017\/10\/Explode1c-300x84.png 300w, https:\/\/pressbooks-dev.oer.hawaii.edu\/math111\/wp-content\/uploads\/sites\/17\/2017\/10\/Explode1c-65x18.png 65w, https:\/\/pressbooks-dev.oer.hawaii.edu\/math111\/wp-content\/uploads\/sites\/17\/2017\/10\/Explode1c-225x63.png 225w, https:\/\/pressbooks-dev.oer.hawaii.edu\/math111\/wp-content\/uploads\/sites\/17\/2017\/10\/Explode1c-350x98.png 350w, https:\/\/pressbooks-dev.oer.hawaii.edu\/math111\/wp-content\/uploads\/sites\/17\/2017\/10\/Explode1c.png 726w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/p>\n<p class=\"p1\">Once again, two dots in that box explode and become one dot in the box to the left.<\/p>\n<p style=\"text-align: center\"><img loading=\"lazy\" decoding=\"async\" src=\"\/\/pressbooks-dev.oer.hawaii.edu\/math111\/wp-content\/uploads\/sites\/17\/2017\/10\/Explode1d-300x81.png\" alt=\"\" class=\"alignnone size-medium wp-image-377\" width=\"300\" height=\"81\" srcset=\"https:\/\/pressbooks-dev.oer.hawaii.edu\/math111\/wp-content\/uploads\/sites\/17\/2017\/10\/Explode1d-300x81.png 300w, https:\/\/pressbooks-dev.oer.hawaii.edu\/math111\/wp-content\/uploads\/sites\/17\/2017\/10\/Explode1d-65x18.png 65w, https:\/\/pressbooks-dev.oer.hawaii.edu\/math111\/wp-content\/uploads\/sites\/17\/2017\/10\/Explode1d-225x61.png 225w, https:\/\/pressbooks-dev.oer.hawaii.edu\/math111\/wp-content\/uploads\/sites\/17\/2017\/10\/Explode1d-350x95.png 350w, https:\/\/pressbooks-dev.oer.hawaii.edu\/math111\/wp-content\/uploads\/sites\/17\/2017\/10\/Explode1d.png 724w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/p>\n<p style=\"text-align: center\"><img loading=\"lazy\" decoding=\"async\" src=\"\/\/pressbooks-dev.oer.hawaii.edu\/math111\/wp-content\/uploads\/sites\/17\/2017\/10\/Explode1e-300x82.png\" alt=\"\" class=\"size-medium wp-image-376 aligncenter\" width=\"300\" height=\"82\" srcset=\"https:\/\/pressbooks-dev.oer.hawaii.edu\/math111\/wp-content\/uploads\/sites\/17\/2017\/10\/Explode1e-300x82.png 300w, https:\/\/pressbooks-dev.oer.hawaii.edu\/math111\/wp-content\/uploads\/sites\/17\/2017\/10\/Explode1e-65x18.png 65w, https:\/\/pressbooks-dev.oer.hawaii.edu\/math111\/wp-content\/uploads\/sites\/17\/2017\/10\/Explode1e-225x61.png 225w, https:\/\/pressbooks-dev.oer.hawaii.edu\/math111\/wp-content\/uploads\/sites\/17\/2017\/10\/Explode1e-350x96.png 350w, https:\/\/pressbooks-dev.oer.hawaii.edu\/math111\/wp-content\/uploads\/sites\/17\/2017\/10\/Explode1e.png 718w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/p>\n<p>We do it again!<\/p>\n<p style=\"text-align: center\"><img loading=\"lazy\" decoding=\"async\" src=\"\/\/pressbooks-dev.oer.hawaii.edu\/math111\/wp-content\/uploads\/sites\/17\/2017\/10\/Explode1f-300x86.png\" alt=\"\" class=\"alignnone size-medium wp-image-375\" width=\"300\" height=\"86\" srcset=\"https:\/\/pressbooks-dev.oer.hawaii.edu\/math111\/wp-content\/uploads\/sites\/17\/2017\/10\/Explode1f-300x86.png 300w, https:\/\/pressbooks-dev.oer.hawaii.edu\/math111\/wp-content\/uploads\/sites\/17\/2017\/10\/Explode1f-65x19.png 65w, https:\/\/pressbooks-dev.oer.hawaii.edu\/math111\/wp-content\/uploads\/sites\/17\/2017\/10\/Explode1f-225x65.png 225w, https:\/\/pressbooks-dev.oer.hawaii.edu\/math111\/wp-content\/uploads\/sites\/17\/2017\/10\/Explode1f-350x101.png 350w, https:\/\/pressbooks-dev.oer.hawaii.edu\/math111\/wp-content\/uploads\/sites\/17\/2017\/10\/Explode1f.png 730w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/p>\n<p style=\"text-align: center\"><img loading=\"lazy\" decoding=\"async\" src=\"\/\/pressbooks-dev.oer.hawaii.edu\/math111\/wp-content\/uploads\/sites\/17\/2017\/10\/Explode1g-300x89.png\" alt=\"\" class=\"size-medium wp-image-374 aligncenter\" width=\"300\" height=\"89\" srcset=\"https:\/\/pressbooks-dev.oer.hawaii.edu\/math111\/wp-content\/uploads\/sites\/17\/2017\/10\/Explode1g-300x89.png 300w, https:\/\/pressbooks-dev.oer.hawaii.edu\/math111\/wp-content\/uploads\/sites\/17\/2017\/10\/Explode1g-65x19.png 65w, https:\/\/pressbooks-dev.oer.hawaii.edu\/math111\/wp-content\/uploads\/sites\/17\/2017\/10\/Explode1g-225x67.png 225w, https:\/\/pressbooks-dev.oer.hawaii.edu\/math111\/wp-content\/uploads\/sites\/17\/2017\/10\/Explode1g-350x104.png 350w, https:\/\/pressbooks-dev.oer.hawaii.edu\/math111\/wp-content\/uploads\/sites\/17\/2017\/10\/Explode1g.png 722w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/p>\n<p class=\"p1\">Hey, now we have more than two dots in the second box, so those can explode and move!<\/p>\n<p style=\"text-align: center\"><img loading=\"lazy\" decoding=\"async\" src=\"\/\/pressbooks-dev.oer.hawaii.edu\/math111\/wp-content\/uploads\/sites\/17\/2017\/10\/Explode1h-300x82.png\" alt=\"\" class=\"alignnone size-medium wp-image-366\" width=\"300\" height=\"82\" srcset=\"https:\/\/pressbooks-dev.oer.hawaii.edu\/math111\/wp-content\/uploads\/sites\/17\/2017\/10\/Explode1h-300x82.png 300w, https:\/\/pressbooks-dev.oer.hawaii.edu\/math111\/wp-content\/uploads\/sites\/17\/2017\/10\/Explode1h-65x18.png 65w, https:\/\/pressbooks-dev.oer.hawaii.edu\/math111\/wp-content\/uploads\/sites\/17\/2017\/10\/Explode1h-225x61.png 225w, https:\/\/pressbooks-dev.oer.hawaii.edu\/math111\/wp-content\/uploads\/sites\/17\/2017\/10\/Explode1h-350x96.png 350w, https:\/\/pressbooks-dev.oer.hawaii.edu\/math111\/wp-content\/uploads\/sites\/17\/2017\/10\/Explode1h.png 718w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/p>\n<p style=\"text-align: center\"><img loading=\"lazy\" decoding=\"async\" src=\"\/\/pressbooks-dev.oer.hawaii.edu\/math111\/wp-content\/uploads\/sites\/17\/2017\/10\/Explode1i-300x85.png\" alt=\"\" class=\"size-medium wp-image-367 aligncenter\" width=\"300\" height=\"85\" srcset=\"https:\/\/pressbooks-dev.oer.hawaii.edu\/math111\/wp-content\/uploads\/sites\/17\/2017\/10\/Explode1i-300x85.png 300w, https:\/\/pressbooks-dev.oer.hawaii.edu\/math111\/wp-content\/uploads\/sites\/17\/2017\/10\/Explode1i-65x18.png 65w, https:\/\/pressbooks-dev.oer.hawaii.edu\/math111\/wp-content\/uploads\/sites\/17\/2017\/10\/Explode1i-225x64.png 225w, https:\/\/pressbooks-dev.oer.hawaii.edu\/math111\/wp-content\/uploads\/sites\/17\/2017\/10\/Explode1i-350x99.png 350w, https:\/\/pressbooks-dev.oer.hawaii.edu\/math111\/wp-content\/uploads\/sites\/17\/2017\/10\/Explode1i.png 726w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/p>\n<p class=\"p1\">And the rightmost box still has more than two dots.<\/p>\n<p style=\"text-align: center\"><img loading=\"lazy\" decoding=\"async\" src=\"\/\/pressbooks-dev.oer.hawaii.edu\/math111\/wp-content\/uploads\/sites\/17\/2017\/10\/Explode1j-300x89.png\" alt=\"\" class=\"alignnone size-medium wp-image-368\" width=\"300\" height=\"89\" srcset=\"https:\/\/pressbooks-dev.oer.hawaii.edu\/math111\/wp-content\/uploads\/sites\/17\/2017\/10\/Explode1j-300x89.png 300w, https:\/\/pressbooks-dev.oer.hawaii.edu\/math111\/wp-content\/uploads\/sites\/17\/2017\/10\/Explode1j-65x19.png 65w, https:\/\/pressbooks-dev.oer.hawaii.edu\/math111\/wp-content\/uploads\/sites\/17\/2017\/10\/Explode1j-225x67.png 225w, https:\/\/pressbooks-dev.oer.hawaii.edu\/math111\/wp-content\/uploads\/sites\/17\/2017\/10\/Explode1j-350x104.png 350w, https:\/\/pressbooks-dev.oer.hawaii.edu\/math111\/wp-content\/uploads\/sites\/17\/2017\/10\/Explode1j.png 738w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/p>\n<p style=\"text-align: center\"><img loading=\"lazy\" decoding=\"async\" src=\"\/\/pressbooks-dev.oer.hawaii.edu\/math111\/wp-content\/uploads\/sites\/17\/2017\/10\/Explode1k-300x84.png\" alt=\"\" class=\"alignnone size-medium wp-image-369 aligncenter\" width=\"300\" height=\"84\" srcset=\"https:\/\/pressbooks-dev.oer.hawaii.edu\/math111\/wp-content\/uploads\/sites\/17\/2017\/10\/Explode1k-300x84.png 300w, https:\/\/pressbooks-dev.oer.hawaii.edu\/math111\/wp-content\/uploads\/sites\/17\/2017\/10\/Explode1k-65x18.png 65w, https:\/\/pressbooks-dev.oer.hawaii.edu\/math111\/wp-content\/uploads\/sites\/17\/2017\/10\/Explode1k-225x63.png 225w, https:\/\/pressbooks-dev.oer.hawaii.edu\/math111\/wp-content\/uploads\/sites\/17\/2017\/10\/Explode1k-350x98.png 350w, https:\/\/pressbooks-dev.oer.hawaii.edu\/math111\/wp-content\/uploads\/sites\/17\/2017\/10\/Explode1k.png 722w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/p>\n<p class=\"p1\">Keep going, until no box has two dots.<\/p>\n<p style=\"text-align: center\"><img loading=\"lazy\" decoding=\"async\" src=\"\/\/pressbooks-dev.oer.hawaii.edu\/math111\/wp-content\/uploads\/sites\/17\/2017\/10\/Explode1l-300x86.png\" alt=\"\" class=\"alignnone size-medium wp-image-370\" width=\"300\" height=\"86\" srcset=\"https:\/\/pressbooks-dev.oer.hawaii.edu\/math111\/wp-content\/uploads\/sites\/17\/2017\/10\/Explode1l-300x86.png 300w, https:\/\/pressbooks-dev.oer.hawaii.edu\/math111\/wp-content\/uploads\/sites\/17\/2017\/10\/Explode1l-65x19.png 65w, https:\/\/pressbooks-dev.oer.hawaii.edu\/math111\/wp-content\/uploads\/sites\/17\/2017\/10\/Explode1l-225x65.png 225w, https:\/\/pressbooks-dev.oer.hawaii.edu\/math111\/wp-content\/uploads\/sites\/17\/2017\/10\/Explode1l-350x101.png 350w, https:\/\/pressbooks-dev.oer.hawaii.edu\/math111\/wp-content\/uploads\/sites\/17\/2017\/10\/Explode1l.png 724w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/p>\n<p style=\"text-align: center\"><img loading=\"lazy\" decoding=\"async\" src=\"\/\/pressbooks-dev.oer.hawaii.edu\/math111\/wp-content\/uploads\/sites\/17\/2017\/10\/Explode1m-300x83.png\" alt=\"\" class=\"alignnone size-medium wp-image-371\" width=\"300\" height=\"83\" srcset=\"https:\/\/pressbooks-dev.oer.hawaii.edu\/math111\/wp-content\/uploads\/sites\/17\/2017\/10\/Explode1m-300x83.png 300w, https:\/\/pressbooks-dev.oer.hawaii.edu\/math111\/wp-content\/uploads\/sites\/17\/2017\/10\/Explode1m-65x18.png 65w, https:\/\/pressbooks-dev.oer.hawaii.edu\/math111\/wp-content\/uploads\/sites\/17\/2017\/10\/Explode1m-225x62.png 225w, https:\/\/pressbooks-dev.oer.hawaii.edu\/math111\/wp-content\/uploads\/sites\/17\/2017\/10\/Explode1m-350x97.png 350w, https:\/\/pressbooks-dev.oer.hawaii.edu\/math111\/wp-content\/uploads\/sites\/17\/2017\/10\/Explode1m.png 724w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/p>\n<p style=\"text-align: center\"><img loading=\"lazy\" decoding=\"async\" src=\"\/\/pressbooks-dev.oer.hawaii.edu\/math111\/wp-content\/uploads\/sites\/17\/2017\/10\/Explode1n-300x86.png\" alt=\"\" class=\"alignnone size-medium wp-image-372\" width=\"300\" height=\"86\" srcset=\"https:\/\/pressbooks-dev.oer.hawaii.edu\/math111\/wp-content\/uploads\/sites\/17\/2017\/10\/Explode1n-300x86.png 300w, https:\/\/pressbooks-dev.oer.hawaii.edu\/math111\/wp-content\/uploads\/sites\/17\/2017\/10\/Explode1n-65x19.png 65w, https:\/\/pressbooks-dev.oer.hawaii.edu\/math111\/wp-content\/uploads\/sites\/17\/2017\/10\/Explode1n-225x65.png 225w, https:\/\/pressbooks-dev.oer.hawaii.edu\/math111\/wp-content\/uploads\/sites\/17\/2017\/10\/Explode1n-350x101.png 350w, https:\/\/pressbooks-dev.oer.hawaii.edu\/math111\/wp-content\/uploads\/sites\/17\/2017\/10\/Explode1n.png 730w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/p>\n<p style=\"text-align: center\"><img loading=\"lazy\" decoding=\"async\" src=\"\/\/pressbooks-dev.oer.hawaii.edu\/math111\/wp-content\/uploads\/sites\/17\/2017\/10\/Explode1o-300x88.png\" alt=\"\" class=\"alignnone size-medium wp-image-373 aligncenter\" width=\"300\" height=\"88\" srcset=\"https:\/\/pressbooks-dev.oer.hawaii.edu\/math111\/wp-content\/uploads\/sites\/17\/2017\/10\/Explode1o-300x88.png 300w, https:\/\/pressbooks-dev.oer.hawaii.edu\/math111\/wp-content\/uploads\/sites\/17\/2017\/10\/Explode1o-65x19.png 65w, https:\/\/pressbooks-dev.oer.hawaii.edu\/math111\/wp-content\/uploads\/sites\/17\/2017\/10\/Explode1o-225x66.png 225w, https:\/\/pressbooks-dev.oer.hawaii.edu\/math111\/wp-content\/uploads\/sites\/17\/2017\/10\/Explode1o-350x102.png 350w, https:\/\/pressbooks-dev.oer.hawaii.edu\/math111\/wp-content\/uploads\/sites\/17\/2017\/10\/Explode1o.png 732w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/p>\n<p class=\"p1\">After all this, reading from left to right we are left with one dot, followed by zero dots, zero dots, and one final dot.<\/p>\n<p>&nbsp;<\/p>\n<p class=\"p1\"><b>Solution:\u00a0<\/b>The 1\u21902 code for nine dots is: 1001.<\/p>\n<\/div>\n<p>&nbsp;<\/p>\n<p class=\"p1\"><b>On Your Own. \u00a0<\/b>Here&#8217;s a diagram showing what happens for seven dots in a 1\u21902 box. Trace through the diagram, and circle the pairs of dots that \u201cexploded&#8221; at each step.<\/p>\n<p style=\"text-align: center\"><img loading=\"lazy\" decoding=\"async\" src=\"\/\/pressbooks-dev.oer.hawaii.edu\/math111\/wp-content\/uploads\/sites\/17\/2017\/10\/Explode2a-300x85.png\" alt=\"\" class=\"alignnone size-medium wp-image-382\" width=\"300\" height=\"85\" srcset=\"https:\/\/pressbooks-dev.oer.hawaii.edu\/math111\/wp-content\/uploads\/sites\/17\/2017\/10\/Explode2a-300x85.png 300w, https:\/\/pressbooks-dev.oer.hawaii.edu\/math111\/wp-content\/uploads\/sites\/17\/2017\/10\/Explode2a-65x19.png 65w, https:\/\/pressbooks-dev.oer.hawaii.edu\/math111\/wp-content\/uploads\/sites\/17\/2017\/10\/Explode2a-225x64.png 225w, https:\/\/pressbooks-dev.oer.hawaii.edu\/math111\/wp-content\/uploads\/sites\/17\/2017\/10\/Explode2a-350x100.png 350w, https:\/\/pressbooks-dev.oer.hawaii.edu\/math111\/wp-content\/uploads\/sites\/17\/2017\/10\/Explode2a.png 730w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/p>\n<p style=\"text-align: center\"><img loading=\"lazy\" decoding=\"async\" src=\"\/\/pressbooks-dev.oer.hawaii.edu\/math111\/wp-content\/uploads\/sites\/17\/2017\/10\/Explode2b-300x81.png\" alt=\"\" class=\"alignnone size-medium wp-image-383\" width=\"300\" height=\"81\" srcset=\"https:\/\/pressbooks-dev.oer.hawaii.edu\/math111\/wp-content\/uploads\/sites\/17\/2017\/10\/Explode2b-300x81.png 300w, https:\/\/pressbooks-dev.oer.hawaii.edu\/math111\/wp-content\/uploads\/sites\/17\/2017\/10\/Explode2b-65x18.png 65w, https:\/\/pressbooks-dev.oer.hawaii.edu\/math111\/wp-content\/uploads\/sites\/17\/2017\/10\/Explode2b-225x61.png 225w, https:\/\/pressbooks-dev.oer.hawaii.edu\/math111\/wp-content\/uploads\/sites\/17\/2017\/10\/Explode2b-350x95.png 350w, https:\/\/pressbooks-dev.oer.hawaii.edu\/math111\/wp-content\/uploads\/sites\/17\/2017\/10\/Explode2b.png 732w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/p>\n<p style=\"text-align: center\"><img loading=\"lazy\" decoding=\"async\" src=\"\/\/pressbooks-dev.oer.hawaii.edu\/math111\/wp-content\/uploads\/sites\/17\/2017\/10\/Explode2c-300x83.png\" alt=\"\" class=\"alignnone size-medium wp-image-384\" width=\"300\" height=\"83\" srcset=\"https:\/\/pressbooks-dev.oer.hawaii.edu\/math111\/wp-content\/uploads\/sites\/17\/2017\/10\/Explode2c-300x83.png 300w, https:\/\/pressbooks-dev.oer.hawaii.edu\/math111\/wp-content\/uploads\/sites\/17\/2017\/10\/Explode2c-65x18.png 65w, https:\/\/pressbooks-dev.oer.hawaii.edu\/math111\/wp-content\/uploads\/sites\/17\/2017\/10\/Explode2c-225x62.png 225w, https:\/\/pressbooks-dev.oer.hawaii.edu\/math111\/wp-content\/uploads\/sites\/17\/2017\/10\/Explode2c-350x96.png 350w, https:\/\/pressbooks-dev.oer.hawaii.edu\/math111\/wp-content\/uploads\/sites\/17\/2017\/10\/Explode2c.png 726w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/p>\n<p style=\"text-align: center\"><img loading=\"lazy\" decoding=\"async\" src=\"\/\/pressbooks-dev.oer.hawaii.edu\/math111\/wp-content\/uploads\/sites\/17\/2017\/10\/Explode2d-300x84.png\" alt=\"\" class=\"alignnone size-medium wp-image-385\" width=\"300\" height=\"84\" srcset=\"https:\/\/pressbooks-dev.oer.hawaii.edu\/math111\/wp-content\/uploads\/sites\/17\/2017\/10\/Explode2d-300x84.png 300w, https:\/\/pressbooks-dev.oer.hawaii.edu\/math111\/wp-content\/uploads\/sites\/17\/2017\/10\/Explode2d-65x18.png 65w, https:\/\/pressbooks-dev.oer.hawaii.edu\/math111\/wp-content\/uploads\/sites\/17\/2017\/10\/Explode2d-225x63.png 225w, https:\/\/pressbooks-dev.oer.hawaii.edu\/math111\/wp-content\/uploads\/sites\/17\/2017\/10\/Explode2d-350x98.png 350w, https:\/\/pressbooks-dev.oer.hawaii.edu\/math111\/wp-content\/uploads\/sites\/17\/2017\/10\/Explode2d.png 734w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/p>\n<p style=\"text-align: center\"><img loading=\"lazy\" decoding=\"async\" src=\"\/\/pressbooks-dev.oer.hawaii.edu\/math111\/wp-content\/uploads\/sites\/17\/2017\/10\/Explode2e-300x87.png\" alt=\"\" class=\"size-medium wp-image-386 aligncenter\" width=\"300\" height=\"87\" srcset=\"https:\/\/pressbooks-dev.oer.hawaii.edu\/math111\/wp-content\/uploads\/sites\/17\/2017\/10\/Explode2e-300x87.png 300w, https:\/\/pressbooks-dev.oer.hawaii.edu\/math111\/wp-content\/uploads\/sites\/17\/2017\/10\/Explode2e-65x19.png 65w, https:\/\/pressbooks-dev.oer.hawaii.edu\/math111\/wp-content\/uploads\/sites\/17\/2017\/10\/Explode2e-225x65.png 225w, https:\/\/pressbooks-dev.oer.hawaii.edu\/math111\/wp-content\/uploads\/sites\/17\/2017\/10\/Explode2e-350x102.png 350w, https:\/\/pressbooks-dev.oer.hawaii.edu\/math111\/wp-content\/uploads\/sites\/17\/2017\/10\/Explode2e.png 724w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/p>\n<p class=\"p1\"><b>Solution:\u00a0<\/b>The 1\u21902 code for seven dots is: 111.<\/p>\n<p>&nbsp;<\/p>\n<div class=\"textbox exercises\">\n<h3 itemprop=\"educationalUse\"><b>Problem 1<\/b><\/h3>\n<p class=\"p1\">Note: In solving this problem, you don&#8217;t need to draw on paper; that can get tedious! Maybe you could use buttons or pennies for dots and do this by hand.<\/p>\n<ol>\n<li class=\"hanging-indent\">Draw 10 dots in the right-most box and perform the explosions. What is the 1\u21902 code for ten dots?<\/li>\n<li class=\"hanging-indent\">Find the 1\u21902 code for eighteen dots.<\/li>\n<li class=\"hanging-indent\">What number of dots has 1\u21902 code\u00a0101?<\/li>\n<\/ol>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox examples\">\n<h3 itemprop=\"educationalUse\"><b>Think \/ Pair \/ Share<\/b><\/h3>\n<p class=\"p1\">After you worked on the problem, compare your answer with a partner.\u00a0 Did you both get the same code?\u00a0 Did you have the same process?<\/p>\n<\/div>\n","protected":false},"author":21,"menu_order":1,"template":"","meta":{"pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-135","chapter","type-chapter","status-publish","hentry"],"part":134,"_links":{"self":[{"href":"https:\/\/pressbooks-dev.oer.hawaii.edu\/math111\/wp-json\/pressbooks\/v2\/chapters\/135","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/pressbooks-dev.oer.hawaii.edu\/math111\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/pressbooks-dev.oer.hawaii.edu\/math111\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/pressbooks-dev.oer.hawaii.edu\/math111\/wp-json\/wp\/v2\/users\/21"}],"version-history":[{"count":26,"href":"https:\/\/pressbooks-dev.oer.hawaii.edu\/math111\/wp-json\/pressbooks\/v2\/chapters\/135\/revisions"}],"predecessor-version":[{"id":2768,"href":"https:\/\/pressbooks-dev.oer.hawaii.edu\/math111\/wp-json\/pressbooks\/v2\/chapters\/135\/revisions\/2768"}],"part":[{"href":"https:\/\/pressbooks-dev.oer.hawaii.edu\/math111\/wp-json\/pressbooks\/v2\/parts\/134"}],"metadata":[{"href":"https:\/\/pressbooks-dev.oer.hawaii.edu\/math111\/wp-json\/pressbooks\/v2\/chapters\/135\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/pressbooks-dev.oer.hawaii.edu\/math111\/wp-json\/wp\/v2\/media?parent=135"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/pressbooks-dev.oer.hawaii.edu\/math111\/wp-json\/pressbooks\/v2\/chapter-type?post=135"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/pressbooks-dev.oer.hawaii.edu\/math111\/wp-json\/wp\/v2\/contributor?post=135"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/pressbooks-dev.oer.hawaii.edu\/math111\/wp-json\/wp\/v2\/license?post=135"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}