Why

Dividing Fractions: How and Why

Fractions have always given students trouble, and we have had many questions about working with them. Even looking only at division of fractions, I have had to restrict my attention to a few sample answers. These show the reasons for the standard method, presented in a variety of ways, together with some alternative methods.

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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Partial Fractions: How and Why

I have often noted that calculus class is where you really learn algebra. Certain techniques in calculus demand algebraic skills that either were not taught in algebra classes (because they are not needed until you get to calculus), or have been forgotten. Chief among these is the method of partial fractions. I have here put …

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Is Area of a Square a Circular Argument?

(New Question of the Week) I love it when students want to know why something has to be the way it is, and are not satisfied just being told to use a formula. Last month, Shunya asked this kind of question, which gave me a chance to refer to our archive and go beyond it.

Why Isn’t Slope Run Over Rise?

Definitions are interesting in several ways. Sometimes they are essentially arbitrary; other times there is a very good reason for them, and understanding that reason can be helpful in understanding and using them. But they are usually taught just as something to memorize. Let’s think about why slope is defined as it is, and not …

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Empty Sets and Vacuous Truth

I’m going to start this post with a simple question about the empty set, and gradually dive deeper. There will be connections here to previous discussions of conditional statements in logic.

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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