## Constructing a Tangent

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…

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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…

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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…

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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. This is a large…

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I was recently wondering why Euclid, the geometer, published a proof that there is an infinite number of primes. I should have known that his proof is geometric. It is: “Let A, B, and C be distinct lengths that cannot be…

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Algebra, Arenas of mathematical mastery, Education, Geometry, Mathematics

The first shoe: Multiplying Adding complex numbers is a straightforward task. Given two numbers, \(a + bi\) and \(c + di\), the sum is the sum of the real portion and the sum of the imaginary portion: \((a + c)…

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Abstract Unit, Algebra, Arenas of mathematical mastery, Geometry, Mathematics

I was reminded of the cylindrical wedge that casts shadows of a triangle, a square, and a circle, and it got me wondering: What if I wanted to create such a shape with an equilateral triangle as one of its…

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Algebra, Arenas of mathematical mastery, GeoGebra, Mathematics

Students often struggle with the concept of multiplying negative numbers, particularly with the notion that multiplying two negative numbers results in a positive. I’ve seen numerous attempts by teachers and teacher educators to explain why, conceptually, it is that the…

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Abstract Unit, Algebra, Arenas of mathematical mastery, Mathematics

This is an example of a common sort of story problem encountered in standardized tests: “1. A team of five professionals can do a certain job in nineteen days; a team of nine apprentices can do the same job in…

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Abstract Unit, Algebra, Arenas of mathematical mastery, Mathematics

Here’s an extension to the problem in my previous post. Time has run out, and a player is at the free throw line. If he makes the first shot, he gets a second try. If he makes both shots, his…

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At a recent workshop on collaboration, the other participants and I were presented with a version of this problem: Adam hits 60% of his free throws. He gets fouled just before the buzzer, and his team is down by one…

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Algebra, Arenas of mathematical mastery, GeoGebra, Geometry, Mathematics

I’ve seen variations of this one a few times, so I thought I’d give it a quick write-up. The simpler version is: Given two circles that are tangent and a line that is cotangent to them, what is the length…

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Most people are aware of two or three basic tests for divisibility by a prime number: A number n is divisible by 2 iff it ends in an even number (0, 2, 4, 6, or 8) by 5 iff it ends in a…

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