# Strategies

## Different Ways to Prove a Trigonometric Identity

Proving trigonometric identities can be a major challenge for students, as it is often very different from anything they have previously done. Often they confuse this concept with solving an equation. But also, they may be give overly rigorous standards to comply with. Here, I will look at several discussions we have had about different …

## Challenging Rate Questions

(New question of the week) A conversation last week went through a number of interesting questions, starting with a couple on percentages, and moving into some that I would call rate questions. I will extract these, which I think will be useful for others. (The rest could, too, but there was just too much there …

## 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 …

## Pattern and Sequence Puzzles

One of the harder types of question to answer effectively is a puzzle, which as I define it means that there is no routine way to solve it, so any hint would likely give away the answer. But sometimes these are only “puzzles” to us, because we don’t know the context that would have told …

## Constructing a Line in a Square

(New Question of the Week) Last month we had a question from a Czech student asking about a geometry problem. The discussion illustrates language issues that can arise, and how we try to guide a student to solve a problem himself. I will fill in some gaps as we examine how to approach an interesting …

## Finding the Range of a Function

Recall that the domain of a function is the set of all valid input values (x), and the range is the set of all possible output values (y). It is reasonably easy to find the domain: look for what could make it impossible to evaluate, such as dividing by zero or taking the square root …

## What Role Should a Figure Play in a Proof?

Questions about geometric proofs have often been handicapped by the inability to show us the associated figure (until we made that easier to do on this new site). In principle, that should not be a problem, because the statement to be proved should contain all the necessary information. It should never be necessary to refer …

## Minimizing an Angle

(New Question of the Week) An interesting trigonometry problem came through about a month ago, answered by Doctor Rick. It gives a nice example of how our process works at its best. It is also an interesting problem!

## 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 …

## How to Write a Proof: The Big Picture

Early in our history, we answered many questions about geometric proofs, particularly the “two-column” variety. Many of these were collected into a FAQ page. I want to briefly survey just some of what we have said about the big picture – an overall view of how to approach a proof, and how to work your …