# NQOTW

## A Challenging Geometry Problem

(A new problem of the week) We usually look here at problems or concepts that are relatively basic and generally applicable; that could give a wrong impression of the kinds of questions we get. Here I want to show a recent example of a discussion about a problem, related to a geometric figure called the …

## 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 …

## 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, …

## Proving an Identity in Different Ways

(A new question of the week) Having discussed trigonometric identities on Monday, let’s make this Trig Week, by looking at a discussion from two months ago in which we were asked about alternative routes to a proof.

## Two Solutions to an ODE

(New Question of the Week) This recent question involves an ordinary differential equation (ODE) and the relation between different solutions. It illustrates common difficulties in interpreting what a problem is asking for, as well as some communication problems involving language and notation.

## What Do Exponents Mean?

(New Question of the Week) We recently had a long discussion about a very common question from a somewhat different perspective: What do exponents (zero, negative, fractional, …) actually mean? The hard part, in the end, was to decide what “mean” means. What does it mean to define something in math? I will pick out the main thread of the …

## 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 …

## 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 …

## 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.

## Fun with a Quartic Equation

(New Question of the Week) Sometimes a problem leads to a very interesting discussion that brings out many good ideas – but then turns out to be something entirely different, which brings out even more (and simpler) ideas. This polynomial equation problem we helped with last week was like that. I will not be quoting …