{"id":71222,"date":"2020-04-24T07:00:54","date_gmt":"2020-04-24T11:00:54","guid":{"rendered":"http:\/\/schools.wrdsb.ca\/athome\/?page_id=71222"},"modified":"2020-05-25T13:09:14","modified_gmt":"2020-05-25T17:09:14","slug":"math-art-challenge-sierpinski-triangles","status":"publish","type":"page","link":"https:\/\/schools.wrdsb.ca\/athome\/learn\/elementary-home\/elementary-2\/math\/math-grades-4-6\/explorations-and-games\/math-art-challenge-sierpinski-triangles\/","title":{"rendered":"Math Art Challenge: Sierpinski Triangles"},"content":{"rendered":"<h5>Explorations and Games<\/h5>\n<h2>Learning Goals:<\/h2>\n<ul>\n<li>Construct triangles using a variety of tools<\/li>\n<li>Create and analyse symmetrical designs by translating, reflecting, or rotating a shape<\/li>\n<\/ul>\n<h2>The Challenge:<\/h2>\n<p>Creating a Sierpinski Triangle is simple, but results in a beautiful, symmetrical work of art!<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-71223\" src=\"http:\/\/schools.wrdsb.ca\/athome\/files\/2020\/04\/Triangles-GIF.gif\" alt=\"\" width=\"400\" height=\"347\" \/><\/p>\n<p>1. Start by creating a large triangle &#8211; either on paper or digitally in <a href=\"https:\/\/mathclips.ca\/swfPlayer.html?swfURL=tools\/Notepad1.swf&amp;title=Notepad\" target=\"_blank\" rel=\"noopener\">Mathies Notepad (dot paper option)<\/a>:<br \/>\n(it can be any type of triangle, but most people prefer equilateral)<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-71225\" src=\"http:\/\/schools.wrdsb.ca\/athome\/files\/2020\/04\/Triangle-1.jpg\" alt=\"Outline sketch of an equilateral triangle.\" width=\"400\" height=\"403\" srcset=\"https:\/\/schools.wrdsb.ca\/athome\/files\/2020\/04\/Triangle-1.jpg 1178w, https:\/\/schools.wrdsb.ca\/athome\/files\/2020\/04\/Triangle-1-150x150.jpg 150w, https:\/\/schools.wrdsb.ca\/athome\/files\/2020\/04\/Triangle-1-298x300.jpg 298w, https:\/\/schools.wrdsb.ca\/athome\/files\/2020\/04\/Triangle-1-768x774.jpg 768w, https:\/\/schools.wrdsb.ca\/athome\/files\/2020\/04\/Triangle-1-1016x1024.jpg 1016w, https:\/\/schools.wrdsb.ca\/athome\/files\/2020\/04\/Triangle-1-50x50.jpg 50w, https:\/\/schools.wrdsb.ca\/athome\/files\/2020\/04\/Triangle-1-100x100.jpg 100w, https:\/\/schools.wrdsb.ca\/athome\/files\/2020\/04\/Triangle-1-165x166.jpg 165w, https:\/\/schools.wrdsb.ca\/athome\/files\/2020\/04\/Triangle-1-1140x1149.jpg 1140w, https:\/\/schools.wrdsb.ca\/athome\/files\/2020\/04\/Triangle-1-945x952.jpg 945w, https:\/\/schools.wrdsb.ca\/athome\/files\/2020\/04\/Triangle-1-750x756.jpg 750w\" sizes=\"auto, (max-width: 400px) 100vw, 400px\" \/><\/p>\n<p>2. Mark the middle of each side of your triangle with a dot or tick mark:<br \/>\n(You can measure this or estimate &#8211; it\u2019s up to you!)<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-71226\" src=\"http:\/\/schools.wrdsb.ca\/athome\/files\/2020\/04\/Triangle-2.jpg\" alt=\"Outline sketch of an equilateral triangle with tick marks indicating the midpoint of each side.\" width=\"400\" height=\"403\" srcset=\"https:\/\/schools.wrdsb.ca\/athome\/files\/2020\/04\/Triangle-2.jpg 1181w, https:\/\/schools.wrdsb.ca\/athome\/files\/2020\/04\/Triangle-2-150x150.jpg 150w, https:\/\/schools.wrdsb.ca\/athome\/files\/2020\/04\/Triangle-2-298x300.jpg 298w, https:\/\/schools.wrdsb.ca\/athome\/files\/2020\/04\/Triangle-2-768x774.jpg 768w, https:\/\/schools.wrdsb.ca\/athome\/files\/2020\/04\/Triangle-2-1016x1024.jpg 1016w, https:\/\/schools.wrdsb.ca\/athome\/files\/2020\/04\/Triangle-2-50x50.jpg 50w, https:\/\/schools.wrdsb.ca\/athome\/files\/2020\/04\/Triangle-2-100x100.jpg 100w, https:\/\/schools.wrdsb.ca\/athome\/files\/2020\/04\/Triangle-2-165x166.jpg 165w, https:\/\/schools.wrdsb.ca\/athome\/files\/2020\/04\/Triangle-2-1140x1149.jpg 1140w, https:\/\/schools.wrdsb.ca\/athome\/files\/2020\/04\/Triangle-2-945x952.jpg 945w, https:\/\/schools.wrdsb.ca\/athome\/files\/2020\/04\/Triangle-2-750x756.jpg 750w\" sizes=\"auto, (max-width: 400px) 100vw, 400px\" \/><\/p>\n<p>3. Connect those midpoint marks with straight lines:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-71227 size-full\" src=\"http:\/\/schools.wrdsb.ca\/athome\/files\/2020\/04\/Triangle-3.jpg\" alt=\"An equilateral triangle with three internal lines connecting at the midpoints.\" width=\"389\" height=\"346\" srcset=\"https:\/\/schools.wrdsb.ca\/athome\/files\/2020\/04\/Triangle-3.jpg 389w, https:\/\/schools.wrdsb.ca\/athome\/files\/2020\/04\/Triangle-3-300x267.jpg 300w, https:\/\/schools.wrdsb.ca\/athome\/files\/2020\/04\/Triangle-3-165x147.jpg 165w\" sizes=\"auto, (max-width: 389px) 100vw, 389px\" \/><\/p>\n<p>4. Now you should see that your original triangle is now made up of 4 similar triangles, 3 facing the same direction of the original, and 1 that is upside down.<\/p>\n<p>Colour in (or shade) the upside-down triangle:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-71228\" src=\"http:\/\/schools.wrdsb.ca\/athome\/files\/2020\/04\/Triangle-4.jpg\" alt=\"An equilateral triangle with a smaller, downward pointing green triangle contained inside.\" width=\"400\" height=\"354\" srcset=\"https:\/\/schools.wrdsb.ca\/athome\/files\/2020\/04\/Triangle-4.jpg 1213w, https:\/\/schools.wrdsb.ca\/athome\/files\/2020\/04\/Triangle-4-300x266.jpg 300w, https:\/\/schools.wrdsb.ca\/athome\/files\/2020\/04\/Triangle-4-768x680.jpg 768w, https:\/\/schools.wrdsb.ca\/athome\/files\/2020\/04\/Triangle-4-1024x907.jpg 1024w, https:\/\/schools.wrdsb.ca\/athome\/files\/2020\/04\/Triangle-4-165x146.jpg 165w, https:\/\/schools.wrdsb.ca\/athome\/files\/2020\/04\/Triangle-4-1140x1009.jpg 1140w, https:\/\/schools.wrdsb.ca\/athome\/files\/2020\/04\/Triangle-4-945x837.jpg 945w, https:\/\/schools.wrdsb.ca\/athome\/files\/2020\/04\/Triangle-4-750x664.jpg 750w\" sizes=\"auto, (max-width: 400px) 100vw, 400px\" \/><\/p>\n<p>Next, follow steps 2, 3, and 4 for your new smaller blank triangles, except choose a new colour for these new upside down triangles. You should end up with something like this:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-71229\" src=\"http:\/\/schools.wrdsb.ca\/athome\/files\/2020\/04\/Triangle-5.jpg\" alt=\"An equilateral triangle with a green triangle in the centre, surrounded by smaller, pink triangles.\" width=\"400\" height=\"392\" srcset=\"https:\/\/schools.wrdsb.ca\/athome\/files\/2020\/04\/Triangle-5.jpg 1089w, https:\/\/schools.wrdsb.ca\/athome\/files\/2020\/04\/Triangle-5-300x294.jpg 300w, https:\/\/schools.wrdsb.ca\/athome\/files\/2020\/04\/Triangle-5-768x752.jpg 768w, https:\/\/schools.wrdsb.ca\/athome\/files\/2020\/04\/Triangle-5-1024x1002.jpg 1024w, https:\/\/schools.wrdsb.ca\/athome\/files\/2020\/04\/Triangle-5-50x50.jpg 50w, https:\/\/schools.wrdsb.ca\/athome\/files\/2020\/04\/Triangle-5-165x162.jpg 165w, https:\/\/schools.wrdsb.ca\/athome\/files\/2020\/04\/Triangle-5-945x925.jpg 945w, https:\/\/schools.wrdsb.ca\/athome\/files\/2020\/04\/Triangle-5-750x734.jpg 750w\" sizes=\"auto, (max-width: 400px) 100vw, 400px\" \/><br \/>\nNow each corner section matches what you created in step four! (a triangle divided into 4 smaller triangles) That\u2019s what is interesting about the Sierpinski triangle &#8211; every section looks like a miniature version of the whole thing. It\u2019s what\u2019s known as a fractal in Mathematics.<\/p>\n<p>How many more stages can you go from here?<br \/>\nThe larger your starting triangle, the farther you can get, and the more complex your finished work of art will be!<\/p>\n<div id=\"attachment_71230\" style=\"width: 410px\" class=\"wp-caption alignnone\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-71230\" class=\"wp-image-71230\" src=\"http:\/\/schools.wrdsb.ca\/athome\/files\/2020\/04\/Triangle-6.jpg\" alt=\"A completed Sierpinski Triangle containing a large green triangle, three medium pink triangles, nine small orange triangles, and twenty-seven tiny purple triangles.\" width=\"400\" height=\"300\" srcset=\"https:\/\/schools.wrdsb.ca\/athome\/files\/2020\/04\/Triangle-6.jpg 1600w, https:\/\/schools.wrdsb.ca\/athome\/files\/2020\/04\/Triangle-6-300x225.jpg 300w, https:\/\/schools.wrdsb.ca\/athome\/files\/2020\/04\/Triangle-6-768x576.jpg 768w, https:\/\/schools.wrdsb.ca\/athome\/files\/2020\/04\/Triangle-6-1024x768.jpg 1024w, https:\/\/schools.wrdsb.ca\/athome\/files\/2020\/04\/Triangle-6-165x124.jpg 165w, https:\/\/schools.wrdsb.ca\/athome\/files\/2020\/04\/Triangle-6-1140x855.jpg 1140w, https:\/\/schools.wrdsb.ca\/athome\/files\/2020\/04\/Triangle-6-945x709.jpg 945w, https:\/\/schools.wrdsb.ca\/athome\/files\/2020\/04\/Triangle-6-750x563.jpg 750w, https:\/\/schools.wrdsb.ca\/athome\/files\/2020\/04\/Triangle-6-100x75.jpg 100w\" sizes=\"auto, (max-width: 400px) 100vw, 400px\" \/><p id=\"caption-attachment-71230\" class=\"wp-caption-text\">Stage 4<\/p><\/div>\n<div id=\"attachment_71231\" style=\"width: 410px\" class=\"wp-caption alignnone\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-71231\" class=\"wp-image-71231\" src=\"http:\/\/schools.wrdsb.ca\/athome\/files\/2020\/04\/Triangle-7.png\" alt=\"A completed Sierpinski Triangle containing a large teal triangle, three medium blue triangles, and nine small orange triangles.\" width=\"400\" height=\"399\" srcset=\"https:\/\/schools.wrdsb.ca\/athome\/files\/2020\/04\/Triangle-7.png 1600w, https:\/\/schools.wrdsb.ca\/athome\/files\/2020\/04\/Triangle-7-150x150.png 150w, https:\/\/schools.wrdsb.ca\/athome\/files\/2020\/04\/Triangle-7-300x300.png 300w, https:\/\/schools.wrdsb.ca\/athome\/files\/2020\/04\/Triangle-7-768x765.png 768w, https:\/\/schools.wrdsb.ca\/athome\/files\/2020\/04\/Triangle-7-1024x1020.png 1024w, https:\/\/schools.wrdsb.ca\/athome\/files\/2020\/04\/Triangle-7-50x50.png 50w, https:\/\/schools.wrdsb.ca\/athome\/files\/2020\/04\/Triangle-7-100x100.png 100w, https:\/\/schools.wrdsb.ca\/athome\/files\/2020\/04\/Triangle-7-165x164.png 165w, https:\/\/schools.wrdsb.ca\/athome\/files\/2020\/04\/Triangle-7-1140x1136.png 1140w, https:\/\/schools.wrdsb.ca\/athome\/files\/2020\/04\/Triangle-7-945x941.png 945w, https:\/\/schools.wrdsb.ca\/athome\/files\/2020\/04\/Triangle-7-750x747.png 750w\" sizes=\"auto, (max-width: 400px) 100vw, 400px\" \/><p id=\"caption-attachment-71231\" class=\"wp-caption-text\">Stage 3<\/p><\/div>\n<h2>Extensions, Adaptations, and Math Conversations:<\/h2>\n<ul>\n<li>What changes when you change your original triangle? (right or isosceles)<\/li>\n<li>How many uncolored triangles are there after Stage 1? After Stage 2? Stage 3? How is this pattern growing? Can you make predictions about future stages?<\/li>\n<li>Check out <a href=\"https:\/\/mathigon.org\/course\/fractals\/introduction\" target=\"_blank\" rel=\"noopener\">Mathigon&#8217;s Introduction to Fractals<\/a> if you want to learn more!<\/li>\n<\/ul>\n<p>Source: Kent Haines &#8211; <a href=\"https:\/\/www.gamesforyoungminds.com\/blog\/2020\/3\/27\/sierpinski-triangles\" target=\"_blank\" rel=\"noopener\">Games for Young Minds<\/a><\/p>\n","protected":false},"excerpt":{"rendered":"<p>Creating a Sierpinski Triangle is simple, but results in a beautiful, symmetrical work of art!<\/p>\n","protected":false},"author":15985,"featured_media":0,"parent":72861,"menu_order":5,"comment_status":"closed","ping_status":"closed","template":"","meta":{"footnotes":""},"categories":[7],"tags":[],"class_list":["post-71222","page","type-page","status-publish","hentry","category-elementary"],"_links":{"self":[{"href":"https:\/\/schools.wrdsb.ca\/athome\/wp-json\/wp\/v2\/pages\/71222","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/schools.wrdsb.ca\/athome\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/schools.wrdsb.ca\/athome\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/schools.wrdsb.ca\/athome\/wp-json\/wp\/v2\/users\/15985"}],"replies":[{"embeddable":true,"href":"https:\/\/schools.wrdsb.ca\/athome\/wp-json\/wp\/v2\/comments?post=71222"}],"version-history":[{"count":0,"href":"https:\/\/schools.wrdsb.ca\/athome\/wp-json\/wp\/v2\/pages\/71222\/revisions"}],"up":[{"embeddable":true,"href":"https:\/\/schools.wrdsb.ca\/athome\/wp-json\/wp\/v2\/pages\/72861"}],"wp:attachment":[{"href":"https:\/\/schools.wrdsb.ca\/athome\/wp-json\/wp\/v2\/media?parent=71222"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/schools.wrdsb.ca\/athome\/wp-json\/wp\/v2\/categories?post=71222"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/schools.wrdsb.ca\/athome\/wp-json\/wp\/v2\/tags?post=71222"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}