{"id":274,"date":"2015-08-13T04:27:03","date_gmt":"2015-08-13T04:27:03","guid":{"rendered":"http:\/\/newblog.primefactorisation.com\/2015\/08\/13\/football-scores-problems\/"},"modified":"2024-11-02T15:09:15","modified_gmt":"2024-11-02T20:09:15","slug":"football-scores-problems","status":"publish","type":"post","link":"https:\/\/www.primefactorisation.com\/blog\/2015\/08\/13\/football-scores-problems\/","title":{"rendered":"Football scores problem"},"content":{"rendered":"<p>As a maths teacher, one of my aims is to get students to think about the world mathematically. So there aren&#8217;t many things more exciting than having a student come to me with a problem they noticed and are trying to solve themselves. Just for the fun of it. This is the story of one of those moments.<\/p>\n<p>The other day I had a student stay back after school and told me of a problem he was going to figure out. He had noticed a pattern in the football scores he&#8217;d seen over the weekend, and wanted to know how many different ways that pattern was possible.<\/p>\n<p>Now, unless you are from Australia, this going to take some explaining. In this part of the world, &#8220;football&#8221; refers to <a href=\"https:\/\/en.wikipedia.org\/wiki\/Australian_rules_football\">Australian rules football<\/a> (which is not rugby, despite the fact that I&#8217;ve <a href=\"https:\/\/www.primefactorisation.com\/blog\/2014\/10\/24\/kicking-goals\/\">blogged about that before<\/a>). <\/p>\n<p><img decoding=\"async\" src=\"https:\/\/upload.wikimedia.org\/wikipedia\/commons\/thumb\/a\/ad\/Stoppage_in_an_AFL_game.jpg\/640px-Stoppage_in_an_AFL_game.jpg\" alt=\"AFL\" \/><br \/>\n<em>Credit: <a href=\"http:\/\/www.flickr.com\/photos\/28135215@N00\">Tom Reynolds<\/a>. Sourced from Wikipedia.<\/em><\/p>\n<p>There are two ways to score in &#8220;Aussie Rules&#8221;:<\/p>\n<ul>\n<li>A &#8220;goal&#8221;, which is worth 6 points.<\/li>\n<li>A &#8220;behind&#8221;, which is worth 1 point.<\/li>\n<\/ul>\n<p>For example, a team with 3 goals and 4 behinds has 22 points, which is usually reported as &#8220;3.4.22&#8221;.<\/p>\n<p>My student had noticed that it is possible for the total score to be the product of the goals and behinds. For instance, 7.7.49 is a possible score, and 7&nbsp;&times;&nbsp;7&nbsp;=&nbsp;49.<\/p>\n<p>His question was: how many scores like this are possible?<\/p>\n<p>He&#8217;d already made some progress on the problem when he told me about it. He defined the problem as being the solution to 6a&nbsp;+&nbsp;b&nbsp;=&nbsp;ab, where a and b are both non-negative integers.<\/p>\n<p>How awesome is that?<\/p>\n<p>Now, this particular student is the type of kid who&#8217;ll go looking for problems like this, who just naturally love maths. But I&#8217;m wondering how I would go about using this in a whole class setting. How would I structure a lesson around this idea? What curriculum could it be fit into? This type of <a href=\"https:\/\/en.wikipedia.org\/wiki\/Diophantine_equation\">equation that only allows integer solutions<\/a> was something I studied as an undergrad, but this seems simple enough for high school kids to get &#8211; one of them did pose the problem, after all.<\/p>\n<p>We did manage to solve the problem. But I think I&#8217;m going to leave that for another blog post. I&#8217;ll give you a hint: 0.0.0 is also a solution. \ud83d\ude09<\/p>\n","protected":false},"excerpt":{"rendered":"<p>A student of mine noticed a pattern in the football scores over the last weekend. He came to me with a very interesting problem he was trying to solve, based on those scores.<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[1],"tags":[],"class_list":["post-274","post","type-post","status-publish","format-standard","hentry","category-uncategorized"],"_links":{"self":[{"href":"https:\/\/www.primefactorisation.com\/blog\/wp-json\/wp\/v2\/posts\/274","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.primefactorisation.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.primefactorisation.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.primefactorisation.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.primefactorisation.com\/blog\/wp-json\/wp\/v2\/comments?post=274"}],"version-history":[{"count":1,"href":"https:\/\/www.primefactorisation.com\/blog\/wp-json\/wp\/v2\/posts\/274\/revisions"}],"predecessor-version":[{"id":748,"href":"https:\/\/www.primefactorisation.com\/blog\/wp-json\/wp\/v2\/posts\/274\/revisions\/748"}],"wp:attachment":[{"href":"https:\/\/www.primefactorisation.com\/blog\/wp-json\/wp\/v2\/media?parent=274"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.primefactorisation.com\/blog\/wp-json\/wp\/v2\/categories?post=274"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.primefactorisation.com\/blog\/wp-json\/wp\/v2\/tags?post=274"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}