Integration by Substitution

(An archive question of the week) Last time, we looked at a method of integration, namely partial fractions, so it seems appropriate to find something about another method of integration (this one more specifically part of calculus rather than algebra). We will look at a question about integration by substitution; as a bonus, I will …

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Subtleties in a Logic Puzzle

(Archive Question of the Week) Logic puzzles can exercise our ability to reason carefully. Interestingly, the use of formal logic in doing so can actually get in our way, because such puzzles often have subtleties in their wording that are hard to capture in formal logic. Examining our thinking carefully can help us see wrong …

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Mathematical Thinking Solves an Operation Puzzle (Or Not)

(Archive problem of the week) Having just written about sequence puzzles, which sometimes can be solved mathematically, and sometimes are just psychological tests, I want to show a different kind of puzzle that I ran across while searching for those. At first, it looks like mere guess-and-check; then we find it can be solved easily …

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What is a Diamond?

(Archive Question of the Week) Having discussed various issues involving categorizing shapes, let’s take a look at a very different shape question, which didn’t fit into the last post.

Ranges of Inverse Trig Functions

(Archive Question of the Week) We have had a number of questions over the years about inverse trig functions and their ranges. For today’s question, I have chosen one from 2011, which will link to a number of others that I will not quote in detail.

From Histograms to Probability Distribution Functions

(Archive Question of the Week) We occasionally got questions about Probability Distribution Functions (PDFs) from students who lacked a full picture of what they are; when I searched for references to give them, I never found one that explained the whole concept as I wanted to. When the following question came in, I took it …

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Averaging Angles

(Archive Question of the Week) An interesting question that has been referred to many times since it was written in 1999 deals with averaging angles. At first the question seems trivial; then almost impossible; and then we end up with a rather simple formula that is totally unlike what we started with. And further applications …

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How to Convert a Fraction to a Decimal – and Why

(Archive Question of the Week) Fractions are a frequent source of questions from elementary students. I plan to devote several posts to various aspects of this, from using a common denominator to add fractions, to flipping and multiplying to divide fractions, to converting between improper fractions and mixed numbers. Here, I want to look at …

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When Math Doesn’t Make Sense

(Archive Question of the Week) One of my favorite questions, from 2001, asked about how to convince a skeptical friend, when a clear mathematical result goes against their intuition. Why should they believe the math? It led me into thoughts about the relationship of intuition to math, whether (and when) math can be trusted, and …

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