On mathematics
Euler’s Identity has been called “the most beautiful equation” in mathematics. It neatly encapsulates five key values and three operators into a true equation: \[e^{\pi i} – 1 = 0\] But why is it true? In this entry, I’m going…
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Generally speaking, if \(a \times b = c\), then \(a\) and \(b\) are factors of \(c\). This concept appears at the secondary level in two contexts: The factors of positive integers, and the factors of a polynomial. If we limit…
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The Pythagorean Theorem states that, given a right triangle, the areas of squares placed along the two legs will have the same area as a square placed on the hypotenuse. This is normally written as \(a^2 + b^2 = c^2\),…
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Contemporary plane geometry of the sort taught in the standard American high school is most heavily informed by two books and a third mathematician. The first of these is Euclid’s Elements, which is so conceptually tied to planar geometry that…
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This morning, I’ve been watching YouTube videos. I started with Tarleen Kaur’s video on Middle Term Splitting. What I find interesting about Kaur’s Chapter to Chapter videos is that, because she’s a student in India, her methods are often different…
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They say that when you’re a hammer, everything looks like a nail. Since I’m currently thinking about conceptual vs procedural teaching, I’m noticing examples. Here’s a good definition of absolute value: “the magnitude of a real number without regard to…
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I have been thinking about procedural vs conceptual thinking, which Skemp’s seminal article refers to as relational vs instructional. One of the questions on this year’s geometry final asks: Given a triangle ABC with sides AB = 5, BC = 6,…
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A persistent topic in mathematics education is whether to focus on conceptual or procedural knowledge. After reading Kris Boulton’s recent post that argues, “It depends,” I found myself thinking about the disconnect between arithmetic and algebra. What is needed to understand…
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Dan Meyer’s latest post is on an exercise involving using a gridless coordinate plane to place fruit along two dimensions. The goal is a worthy one: To give students the opportunity to explore what the coordinate plane is without getting tied…
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Most high school geometry textbooks will proclaim that there are four basic transformations. Three of these (translations, reflections, and rotations) are rigid transformations; the resulting copy (image) is congruent to the original version (pre-image). Here are examples: The first example…
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Discussing the properties of similar triangles today, I derived a simple proof of the Pythagorean Theorem that uses ratios. (I do not claim this is original to me; I’m sure it isn’t.) Consider the diagram. \(\Delta ADC \sim \Delta BDA \sim \Delta…
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In this entry, I’m going to start with a concrete problem and develop an abstract generalization. The starting problem: Given isosceles trapezoid \(ABCD\) with an altitude of 6. Point \(E\) is on \(\overline{DC}\) such that \(DE = 3\), \(EC =…
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