# Mistakes

## 1=0? Calculus Says So [or Not]

“False Proofs”, where seemingly good logic leads to nonsensical conclusions, can be a good way to learn the boundaries of reality — what to look out for when you are doing real math. We have a FAQ on the subject; there we discuss several well-known fallacies based in algebra, and have links to others. Today, …

## A Fraction Word Problem – Algebra or Not?

Sometime soon I will do a series of posts on word problems, which are a common point of difficulty with students. But here is one recent example from a high school student, where language was the main difficulty, but the algebra is worth discussing as well. We’ll look a little more deeply into the problem …

## Principles for Solving a Formula

(An archive question of the week) Last time I discussed issues that arise in solving a simple algebraic equation. In researching that, I found a discussion of solving a formula for a variable (which in some countries is called “making x the subject”, that is, changing an equation involving x into the form “x = …

## What is Cosine – With a Twist

(A new question of the week) Having written last week about the definitions of trigonometric functions, I want to look at a question from a few months ago that illustrates a rather common mistake students make in applying those definitions. It also demonstrates the patience required to find out what is in a student’s mind, …

## Integration: Partial Fractions and Substitution

(New Question of the Week) Many of the questions we answer are primarily about “how do you solve this problem?”, but at the same time ask deeper, more general questions: “How do you solve problems like this, and are there alternatives?” Today’s question is a good example of this, and raises an interesting point or …

## Avoiding “Stupid” Mistakes in Algebra

(New Question of the Week) I like working with a student who is willing to take chances, and also willing to be corrected. As I have often explained, just like a medical doctor, a Math Doctor wants you to “show me where it hurts” in order to diagnose the problem; so showing detailed work is …

## When Can a Function Equal Its Inverse?

(New Question of the Week) This week I answered a seemingly simple question that can be solved in several different ways when presented as multiple choice, but is rather difficult as a straightforward algebra problem. Trying to guess what the “patient” had done yielded an invalid method that gave the right answer — or was it really invalid? …

## How Can I Stop Making Careless Mistakes?

From time to time, a student will write to us asking for advice on studying, rather than on math itself. As either successful students, or teachers, or (quite often) as adults who recall overcoming difficulties in the past, we have some good advice to offer. Today, I want to look at three answers, by three …