![\"post6pic1a\"](\"http:\/\/advancedmathyoungstudents.com\/blog\/wp-content\/uploads\/2014\/04\/post6pic1a.jpg\")
<\/a><\/p>\nAnd, yes, in Your First Book of Shapes<\/i> or its equivalent, that was the shape on the \u201cSquare\u201d page.\u00a0 But then in geometry class, you learned that the squares are a subset of the rectangles.\u00a0 They just happen to have equal length and width.\u00a0 Still, when I say \u201cpicture a rectangle\u201d,\u00a0 you probably don\u2019t picture a square.\u00a0 So I\u2019ll try again.<\/p>\n
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Here is a rectangle:<\/p>\n
OK, that is surely a rectangle and not a square.\u00a0 But is it what you picture when I say \u201crectangle\u201d?\u00a0 I\u2019m guessing that my rectangle is skinnier than the one in your mind.\u00a0 Some rectangles are more \u201crectangular-y\u201d than others.\u00a0 We could say they have higher \u201crectangularity\u201d.\u00a0 But is there a way to put a number on that?\u00a0 Actually, there is.\u00a0 We can use ratios.<\/p>\n Ratios are perfect for this job.\u00a0 A ratio is a comparison using division.\u00a0 To find the ratio of A to B, we calculate A\u00f7B.\u00a0 The answer tells us how many times bigger A is than B.<\/p>\n So let\u2019s define the \u201cRectangularity\u201d of a rectangle as the ratio of its length to its width.\u00a0 In that case, my first rectangle had rectangularity of 1 which made it a square.\u00a0 And my second rectangle had rectangularity of 12.\u00a0 Its length was 12 times as great as its width.<\/p>\n We can keep calling this ratio \u201crectangularity\u201d, a name that I made up, but in fact it already has a name: the length-to-width ratio of a rectangle is called its \u201caspect ratio\u201d.\u00a0\u00a0\u00a0 And in classical design, there is an aspect ratio that is considered ideal.\u00a0 It just looks right: not too square-y, not too rectangular-y\u00a0…just right.\u00a0\u00a0\u00a0 It is called the Golden Ratio.\u00a0 You can find tons of information about\u00a0 it on-line so I won\u2019t try to replicate it all here.\u00a0 I just want to look at one interesting feature of a golden rectangle and show how we can use that feature to calculate the Golden Ratio.<\/p>\n Start with a Golden Rectangle\u2026<\/p>\n It looks nice.\u00a0 But here\u2019s the interesting property.<\/p>\n Slice off a square on one end\u2026that little rectangle that remains is also a golden rectangle.\u00a0 And yes, if you slice off the end square of this little rectangle, you get another even littler golden rectangle.<\/p>\n Repeat as many times as you like to produce an endless stream of ever-smaller golden rectangles.<\/p>\n <\/p>\n Let\u2019s start again.\u00a0 Suppose the width of the rectangle = 1 and the length = x.<\/p>\n Since the ratio \u00a0x\/1=x, x must also be the value of the golden ratio.\u00a0 But what is that value?<\/p>\n Well, for the little rectangle, the length is 1 and the width is x-1.\u00a0 So that rectangle has an aspect ratio of 1\/(x-1).\u00a0 But that aspect ratio is the same as the aspect ratio of the bigger rectangle we started with.\u00a0 We can write:<\/p>\n To find the value of the golden ratio, we just have to solve that equation.\u00a0 When you cross-multiply, you get:<\/p>\n x(x-1) = 1 \u00a0\u00a0<\/i><\/p>\n You can distribute to get:<\/p>\n x2<\/sup> \u2013 x = 1<\/i><\/strong><\/p>\n But now what?\u00a0 If you use trial and error with a calculator, you will soon find that the golden ratio is a little more than 1.6.\u00a0 But if you want an exact solution, you need to know how to handle quadratic equations.<\/p>\n And that\u2019s what I really want to talk about.\u00a0 Quadratic equations pop up in physics.\u00a0 Not every day, but occasionally.\u00a0 And they usually can\u2019t be factored.\u00a0 (Neither can the one we have here.)\u00a0 My students deal with them a variety of ways.\u00a0 Some grab their TI89s and let the calculator do the work.\u00a0\u00a0 Some of them\u00a0 use the quadratic formula.\u00a0 Almost nobody \u201ccompletes the square\u201d.<\/p>\n I admit that I used to make fun of completing the square as a classic piece of cook-book math \u2013 all procedure, no understanding.\u00a0 But I think I was wrong about that.\u00a0 I wasn\u2019t picturing the procedure geometrically.\u00a0 And, as I have mentioned in a few other posts, it\u2019s better with pictures.\u00a0 I\u2019ll explain in the next post.<\/p>\n","protected":false},"excerpt":{"rendered":" These may all be digressions, but (as I reassure my students all the time), stay with me.\u00a0 I\u2019m heading somewhere with all of this.\u00a0 For now, let\u2019s just keep going. Here is a rectangle: And, yes, in Your First Book … Continue reading <\/a><\/p>\n<\/a><\/p>\n<\/a><\/p>\n<\/a><\/p>\n<\/a><\/p>\n<\/a><\/p>\n