Category: Mathematics
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[mathjax]A recent comment from a colleague got me thinking about describing polygons using functions. His intent was that polygons (and all closed shapes) can be described as sets of functions; for instance, a triangle could be described by three linear functions with the domain of the triangle’s vertices. And, of course, any closed shape cannot…
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I have borrowed from a colleague a copy of G. A. Wentworth’s Plane and Solid Geometry, copyright 1899 and published 1902 by The Athenรฆum Press of Boston. I enjoy reading old textbooks because they either reinforce or give lie to certain claims about the longevity of mathematical concepts. This particular volume is attractive to me as…
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[mathjax]Here’s a quick one: All rational numbers except 0 can be expressed as \[(-1)^s \Pi p_i^{n_i}\] where \(s \in \{0, 1\}\), \(p_i\) is a prime number, and \(n_i\) is an integer. This reminds me of the restriction on the definition of rationals, i.e., that \(\frac{a}{b}\) is a rational number for all integers \(a\) and \(b\)…
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[mathjax]The version of Geometry most widely taught in high schools in the United States is an amalgam of the two most basic fields of geometry: Synthetic and analytic. The mixing of these two is done in such a way as to suggest that the fields are complementary, and so the points of differentiation between the…
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[mathjax]It is a persistently popular thing to do on social media to post challenges like this one. I used to be of a mind to be outraged at the abuse of the equal sign: Clearly these are not addition problems! This is not how math symbolism works! This is not math! However, I’ve since shifted…
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[mathjax]A question in this month’s Mathematics Teacher asks about the range of \(\sin(\sin(x))\). My initial concern about this was over the units of the input and output of the sine function. I’ll summarize those briefly, but this post is about the resolution of those concerns by clarifying what a “degree” is in the first place.…
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[mathjax]This is a quick proof based on an observation inspired by “Mathematical Lens” in the May 2016 Mathematics Teacher (“Fence Posts and Rails” by Roger Turton). A triangular number is the sum of all integers from 1 to n. The general formula for T(n), the nth triangular number, is \[T(n) = \frac{(n)(n + 1)}{2}\] Challenge:…
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The Internet is in a tizzy yet again about the evils of mathematics education. At least Common Core isn’t being demonized quite as front-and-center as in the recent past, but still. This time it’s about pizza. Which means every mathematics educator reading this will know it’s about fractions, because that’s why we ever mention pizza…
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[mathjax](Edit 6/18/23: The image has been lost, but I’ll leave the text in case I ever have the chance to reconstruct it.) Here’s a fun puzzle (via Brilliant.org): What is the area of the square \(ABCD\)? There may be a simpler approach; my solution wound up being more complicated than I expected. Since \(\Delta AEF\)…
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[mathjax]I was thinking about the third scenario described in al-Khwarizmi’s al-Jabr: \(x^2 = 3x + 4\). I was curious about the integer solutions of the general pattern, \(x^2 = ax + b\). It’s easy enough to demonstrate that this will hold if \(x = b = a + 1\), since that means \((a + 1)^2…