Category: Mathematics
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Congruence [mathjax]As I discussed in an earlier post, there are two basic definitions of geometric congruence that are presented to students. The first is based on measurement: Definition 1. Two objects are congruent if all of their measurements are the same, and in the same order. That is, two segments are congruent if they’re the…
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I’ve been struggling for a while now to find a way to frame and articulate the answer to what seems like a simple question: “What is mathematics?” At the same time, I’ve been seeking to layout the similarities and differences between the concepts listed in the title: Computer programming, mathematics, and natural language. Recently, I…
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[mathjax]A poster on the Google Plus Mathematics community commented that one feature of the Golden Ratio ฯ is that adding one to ฯ yields the same value as squaring ฯ does. That is, \[\phi^2 = \phi + 1\] He was surprised that there would be such a number. While this is indeed an interesting attribute…
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And now, for something silly. In general, number bases are expected to be positive integers greater than one. The most widely used are decimal (because we have ten fingers and ten toes), binary (how computer data is stored), hexadecimal (a more convenient way of writing binary), and octal (base eight), but, mathematically speaking, there’s no…
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Puzzle I found this puzzle in the G+ Mathematics community, courtesy of Paul Cooper. Solve the final addition: 50 + 60 + 90 = 380 30 + 40 + 60 = 330 90 + 60 + 70 = 350 50 + 90 + 30 = 10 70 + 30 + 20 = 370 40 +…
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[mathjax]I’ve recently come upon two probability problems with counterintuitive solutions. One I’d seen before and dismissed because I didn’t understand the write-up (mea culpa); the other is new to me. Born on a Sunday Puzzle: You are introduced to a randomly selected family that happens to have two children. If one is a girl that…
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A standard high school geometry textbook talks about congruence in terms of three types of objects: Line segments, angles, and polygons. Congruence is then defined in terms of measurable parameters: “Two figures are congruent if they have the same size and the same shape” (Carnegie’s Bridge to Algebra Student Text, 2008, p. G-9). Math Open Reference…
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Introduction [mathjax]In my previous post, I included this image, which I’d made in GeoGebra. The image satisfies the conditions of the problem: \(AD\) is tangent to \(\odot P\) and \(\overline{BC} \cong \overline{AD}\). In order to create this image, I created a dynamic GeoGebra image where A, B, P and the radius of P can be…
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The Golden Ratio [mathjax]By definition, the Golden Ratio is a ratio involving overlapping line segments. Given collinear points A, B, and C, such that B is between A and C, if the ratio between the two subsegments is the same as the ratio between the entire segment and the longer segment, then that ratio is…
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[mathjax]The Monty Hall problem persists in Internet mathematics discussions, as if its results are somehow spectacularly unique or mystifying. Here is the problem: You are on a game show and are presented with three doors. Behind one door is some wonderful prize, and behind the other two is a goat (or something else of negligible…