Category: Mathematics
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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 down by its rigid formalism. However, the nature of the exercise highlights that there are…
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[mathjax]The Fundamental Theorem of Arithmetic says that all integers greater than one can be written uniquely as the product of prime numbers. Another way of stating this is that, if \(P = (p_1, p_2, p_3, …)\) is the (infinite) set of all primes, in order from least to greatest, and \(K = (k_1, k_2, k_3,…
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Inspired by Math with Bad Drawings, here a trio of my own limerick creations: Two circles surrounding a square Was more than the poor thing could bear. It made itself fetal ‘Til planar was hedral. Cylindrical nets are a snare! Isometry! Great celebration! But tragedy followed elation, When off of the grid The image got…
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[mathjax]Most high school geometry textbooks will say 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 (blue is the pre-image): The first example is a simple translation, which can be written algebraically…
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Using Publisher, I’ve created a slide rule for multiplication tables (up to 10×10). To use it: — Print it out and cut along the dotted line. — Move the 1 on the bottom part to any single digit on the top strip. — Each number on the bottom strip lines up to its multiple on…
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[mathjax]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, and given that \(\angle BAD\) and \(\angle ADB\) are right. \(\Delta ADC \sim \Delta BDAย \simย \Delta BAC\). Due to the properties…
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[mathjax]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 \(DE = 3\), \(EC = 8\), and \(\angle AEB\) is right. Determine \(AB\). We can solve this by placing points…
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[mathjax]I was recently asked for an elegant proof of the following problem. It’s based on a construction challenge from Euclidea. Given: Circles A, B, and C, such that point C is on circle A, point B is on circles A and C, point E is on circles B and C, and point D is on…
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[mathjax]What is the equation of a line that is secant to a circle with radius \(r\) and center \((0,0)\)? This question started as a challenge with a student. She wanted to draw a pentagram on a graphing calculator, and while she could do the five lines freehand, she needed the equation of a circle. So…
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A friend of mine, a father, recently posted this item on his Facebook feed. It’s from Pearson, and he was struggling figuring it out. I also had to read it several times to figure it out. (Edit 6/17/23: I lost the original graphic; I located a version that has the answers on it, so ignore…