{"id":1075,"date":"2011-01-17T10:17:51","date_gmt":"2011-01-17T17:17:51","guid":{"rendered":"http:\/\/www.tech4mathed.com\/blog\/?p=1075"},"modified":"2011-01-17T10:17:51","modified_gmt":"2011-01-17T17:17:51","slug":"japanese-multiplication-how-does-it-work","status":"publish","type":"post","link":"https:\/\/www.tech4mathed.com\/blog\/2011\/01\/17\/japanese-multiplication-how-does-it-work\/","title":{"rendered":"Japanese Multiplication &#8211; How Does it Work?"},"content":{"rendered":"<p>I have come across some fun math problems (yes, math CAN be fun!) through the internet and friends. \u00a0Here is a video of Japanese Multiplication. \u00a0First watch the video, and then see if you can figure out how they are multiplying \u00a0before you read my explanation below!<\/p>\n<p style=\"text-align: center;\"><object classid=\"clsid:d27cdb6e-ae6d-11cf-96b8-444553540000\" width=\"480\" height=\"385\" codebase=\"http:\/\/download.macromedia.com\/pub\/shockwave\/cabs\/flash\/swflash.cab#version=6,0,40,0\"><param name=\"allowFullScreen\" value=\"true\" \/><param name=\"allowscriptaccess\" value=\"always\" \/><param name=\"src\" value=\"http:\/\/www.youtube.com\/v\/e-P5RGdjICo?fs=1&amp;hl=en_US\" \/><param name=\"allowfullscreen\" value=\"true\" \/><embed type=\"application\/x-shockwave-flash\" width=\"480\" height=\"385\" src=\"http:\/\/www.youtube.com\/v\/e-P5RGdjICo?fs=1&amp;hl=en_US\" allowscriptaccess=\"always\" allowfullscreen=\"true\"><\/embed><\/object><\/p>\n<p>&#8212;&#8212;&#8212;&#8212;&#8212;&#8212;&#8212;&#8212;&#8212;&#8212;&#8212;&#8212;&#8212;&#8212;&#8212;&#8212;&#8212;&#8212;&#8212;&#8212;&#8212;-<\/p>\n<p>Well, did you figure it out? \u00a0Let&#8217;s take a closer look at the first example:<\/p>\n<p><a href=\"http:\/\/www.tech4mathed.com\/blog\/wp-content\/uploads\/2011\/01\/ImageFromVideo.png\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-1076\" title=\"ImageFromVideo\" src=\"http:\/\/www.tech4mathed.com\/blog\/wp-content\/uploads\/2011\/01\/ImageFromVideo.png\" alt=\"\" width=\"394\" height=\"315\" srcset=\"https:\/\/www.tech4mathed.com\/blog\/wp-content\/uploads\/2011\/01\/ImageFromVideo.png 394w, https:\/\/www.tech4mathed.com\/blog\/wp-content\/uploads\/2011\/01\/ImageFromVideo-300x239.png 300w\" sizes=\"auto, (max-width: 394px) 100vw, 394px\" \/><\/a>Now take it apart:<\/p>\n<ul>\n<li>The <span style=\"color: #00ff00;\">first set of lines, the green ones<\/span> in my image above, represent 2 sets of 10 , or 20<\/li>\n<li>The<span style=\"color: #ff9900;\"> second region with one line, which is orange in my image<\/span>, represents 1 set of 1<\/li>\n<li>Together these sets of lines, read top to bottom, \u00a0represent 2&#215;10 + 1&#215;1 \u00a0which is expanded notation for 21<\/li>\n<\/ul>\n<p>Let&#8217;s look at the next set of lines that were drawn perpendicular to these lines:<\/p>\n<ul>\n<li>The <span style=\"color: #0000ff;\">first region that contains the 1 blue line <\/span>represents 1 set of 10 or 1&#215;10<\/li>\n<li>The <span style=\"color: #ff0000;\">second set of lines that were drawn, the three red lines<\/span>, represent 3 sets of 1<\/li>\n<li>Together these 4 lines, read left to right, represent 1&#215;10+3&#215;1 = 13 in expanded notation.<\/li>\n<\/ul>\n<p>Now for the tricky part!<\/p>\n<p><a href=\"http:\/\/www.tech4mathed.com\/blog\/wp-content\/uploads\/2011\/01\/videoImage2.png\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-1077\" title=\"videoImage2\" src=\"http:\/\/www.tech4mathed.com\/blog\/wp-content\/uploads\/2011\/01\/videoImage2.png\" alt=\"\" width=\"420\" height=\"325\" srcset=\"https:\/\/www.tech4mathed.com\/blog\/wp-content\/uploads\/2011\/01\/videoImage2.png 420w, https:\/\/www.tech4mathed.com\/blog\/wp-content\/uploads\/2011\/01\/videoImage2-300x232.png 300w\" sizes=\"auto, (max-width: 420px) 100vw, 420px\" \/><\/a>Those of you who have ever FOIL-ed in Algebra will recognize the process of distributing the values by &#8220;First, Outer, Inner, Last&#8221;<\/p>\n<p><span style=\"color: #0000ff;\">Here is a quick Algebra example to remind you<br \/>\n(x+3)(2x+5) =<\/span><\/p>\n<p><span style=\"color: #0000ff;\">First = x * 2x = 2x^2<br \/>\nOuter= x*5 = 5x<br \/>\nInner = 3*2x = 6x<br \/>\nLast = 3*5 = 15<\/span><\/p>\n<p><span style=\"color: #0000ff;\">Then, \u00a02x^2 + 5x +6x +15 = 2x^2 + 11x +15 (the Outer and Inner were &#8220;like&#8221; terms, so could be added together)<\/span><\/p>\n<p>Now back to the arithmetic. \u00a0 If you look at the product 21&#215;13 by separating out each factor by its place values, you have:<\/p>\n<p>(20 + 1)(10 + 3) \u00a0and now you can FOIL out the values, just like in the Algebra problem!<\/p>\n<p>First = 20&#215;10 = 200<br \/>\nOuter = \u00a020&#215;3 = 60<br \/>\nInner = 1&#215;10 = 10<br \/>\nLast = \u00a01&#215;3 = 3<\/p>\n<p>The <span style=\"color: #ffff00;\"><span style=\"color: #000000;\">200 is represented by the<\/span> <span style=\"color: #000000;\">2 sets of crossing lines<span style=\"color: #ffff00;\"> circled<\/span> <\/span> in yellow<\/span> on the image above- that location on the paper represents the hundreds place value, so having a 2 in the hundreds location represents 2&#215;100 = 200. \u00a0In the video a 2 is placed as the first digit of the product, which will be the hundreds place.<\/p>\n<p>Next:<\/p>\n<p>The 60 is represented by the 6 sets of crossing lines <span style=\"color: #00ff00;\">in green<\/span> on the top right<br \/>\nThe 10 is represented by the 1 set of crossing lines<span style=\"color: #00ff00;\"> in green<\/span> on the bottom left<\/p>\n<p>Together the 60+10 gives 70. \u00a0In the video, the<span style=\"color: #00ff00;\"> areas circled in gree<\/span><span style=\"color: #00ff00;\">n <span style=\"color: #000000;\">on the image above<\/span><\/span> both represent the tens place value, so they are adding up the 6 crossed marks and the 1 crossed mark to get 7 sets in the tens place, or 7&#215;10=70.\u00a0\u00a0 \u00a0They then place a 7 to the right of the 2 in the product (placing it in the tens place)<\/p>\n<p>Finally:<\/p>\n<p>The 3 is represented by the 3 crossed marks in the lower right <span style=\"color: #ff0000;\"><span style=\"color: #000000;\">(<span style=\"color: #ff0000;\">circled<\/span> <\/span>in red <\/span>on the image above). \u00a0 This area of the paper represents the ones place, so we have 3&#215;1 = 3. \u00a0 They then place a 3 to the right of the 7 in the product, placing the 3 in the ones place.<\/p>\n<p><a href=\"http:\/\/www.tech4mathed.com\/blog\/wp-content\/uploads\/2011\/01\/videoImage3.png\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-1080\" title=\"videoImage3\" src=\"http:\/\/www.tech4mathed.com\/blog\/wp-content\/uploads\/2011\/01\/videoImage3.png\" alt=\"\" width=\"424\" height=\"328\" srcset=\"https:\/\/www.tech4mathed.com\/blog\/wp-content\/uploads\/2011\/01\/videoImage3.png 424w, https:\/\/www.tech4mathed.com\/blog\/wp-content\/uploads\/2011\/01\/videoImage3-300x232.png 300w\" sizes=\"auto, (max-width: 424px) 100vw, 424px\" \/><\/a><\/p>\n<p>This gives the final product of 200+60+10+3 or 200+70+3 = 273<\/p>\n<p>~Now look at the second product in the video and see if you can figure out how it works!<\/p>\n<div class=\"evernoteSiteMemory\"><a href=\"javascript:\" onclick=\"Evernote.doClip({title: 'Japanese Multiplication &amp;#8211; How Does it Work?',url: 'https:\/\/www.tech4mathed.com\/blog\/2011\/01\/17\/japanese-multiplication-how-does-it-work\/',contentID: 'post-1075',suggestTags: 'algebra,algorithm,Japanese,multiplication,teaching,youtube',providerName: 'Teaching Math and Music with Technology',styling: 'text' });return false\" class=\"evernoteSiteMemoryLink\"><img decoding=\"async\" src=\"http:\/\/static.evernote.com\/article-clipper.png\" class=\"evernoteSiteMemoryButton\" \/>\n\t\t\t\t<\/a>\t\t\t\t<div class=\"evernoteSiteMemoryClear\">&nbsp;<\/div>\n<\/div><!-- AddThis Advanced Settings generic via filter on the_content --><!-- AddThis Share Buttons generic via filter on the_content -->","protected":false},"excerpt":{"rendered":"<p>I have come across some fun math problems (yes, math CAN be fun!) through the internet and friends. \u00a0Here is a video of Japanese Multiplication. \u00a0First watch the video, and<!-- AddThis Advanced Settings generic via filter on wp_trim_excerpt --><!-- AddThis Share Buttons generic via filter on wp_trim_excerpt 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