Author Archives: daniel-mcquillan

Useful Generalizations, part I

Very often in mathematics we see a nice argument or proof and we realize that the same argument can prove more than what was originally intended. The purpose of this post is to do just that for two recent posts, … Continue reading

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It’s a Mean Value Theorem

“The average rate of change will be the same as the instantaneous rate of change somewhere in the interval” under reasonable assumptions, is the gist of the Mean Value Theorem. A pleasant funny review is in the slow-paced 1966 (yes, … Continue reading

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A cardinal rule of base-running is hilariously incorrect

Sometimes I watch Major League Baseball games. I’ve noticed that the following is never questioned: Unwritten Rule: With fewer than 2 outs, a base-runner on first base shall be certain that a ball hit in the air will fall in … Continue reading

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On the value of believing that you know the answer

A beautiful inequality, the arithmetic -mean-geometric-mean inequality (AM-GM) seems reasonable but hard to prove. It states that the usual average of n positive real numbers is at least as big as the n-th root of the product. We’ll experiment with … Continue reading

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On the unreasonable efficiency of mathematical writing, part II

The writing of mathematics is very efficient. Perhaps the teaching of mathematics can afford a little more redundancy. Here is a key example: Mathematical induction-the champion of efficiency. The beautiful idea: Induction is often used (and usually taught) to prove … Continue reading

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On the unreasonable efficiency of mathematical writing

In literary writing, we are taught to provide an introduction, a body and a conclusion. We are encouraged to—inefficiently—relate parts of an essay to a central idea. In mathematics, our writing is at times too efficient. Perhaps the teacher can … Continue reading

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On Teaching Loving Mathematics

I love math—not because I think I’m good at it, but because it’s powerful, it’s supremely beautiful and it’s everywhere. Mathematicians and many others agree! However, it happens all too often that people claim not to agree. Teachers can change … Continue reading

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On the Value of Making up Your Own Question

First, we consider an example of how one might make up a question. Here is a nice fact: (1/2)+(1/3)+(1/6)=1 Let’s have some fun with this fact by making it the answer of a question. Think of it as a mystery … Continue reading

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Squaring the Superhero

I teach many different courses in mathematics. All of these courses contain wonderful material. I love math. Last year, my son asked me why some of the students at his elementary school did not seem to share my view regarding … Continue reading

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On Trigonometric Nostalgia

When my son asks me how something works, I often don’t have a satisfactory answer. After all, I can’t rip open my digital device and show him the parts in action. In sharp contrast, we can see exactly how a … Continue reading

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A Perfect Square Discovers Pi

Pi is defined as the ratio of the circumference of a circle, to its diameter. It follows (how?) that the same constant pi is also the number of r x r squares that have the same total area as a … Continue reading

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