## Secant to a Circle

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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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

The game of Set consists of 81 cards. Each card has one, two, or three identical symbols of one of three shapes (oval, diamond, or squiggle), in one of three colors (red, green, or purple) and one of three textures…

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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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