# AQOTW

## Order of Operations: Fractions, Evaluating, and Simplifying

(An archive problem of the week) Last time we looked at the subtle distinction between the order of operations, which defines the meaning of an expression, and properties that allow us to do something other than what an expression literally says. Here I want to look at one longer discussion that brings out these issues …

## Order of Operations: Trigonometric Functions

(An archive question of the week) Last time we looked at some details that are rarely mentioned in stating the conventions for interpreting algebraic expressions. I couldn’t fit a discussion of the most complicated case: trigonometric functions, which when written without parentheses, as they traditionally have been, can raise several issues. (Much of the same …

## Calculus: Is This All There Is?

(An archive question of the week) I’m going to do something unusual, and post a discussion that was never archived. I ran across it while searching for the original of an archived discussion to check something, and this one stood out as worth posting around the start of the school year. It’s a question from …

## An Introduction to Trigonometry

(An archive question of the week) While I’m showing some recent explanations of basic trigonometry techniques, this is a good time to look at an even more basic explanation of the essentials of the subject for a beginner. Right triangle trigonometry Here is the question, from 2001: Trigonometry in a Nutshell I’m in 8th grade …

## Equations with Fractions: Three Ways to Solve Them

Since we just looked at a complicated rational inequality, let’s look at some simpler rational equations, first a linear equation with fractions, and then truly rational equations, in which the variable(s) appear in the denominator. This discussion dealt with a common confusion I’ve seen in students. The problem The question came from Fairooz in 2017: …

## Chinese Remainders With and Without the Theorem

(An archive question of the week) My title is tongue-in-cheek, as we’ll be looking at the Chinese Remainder Theorem, which is really a Chinese theorem about remainders, not a theorem about “Chinese remainders”. But we’ll work on a problem that can be solved with or without knowledge of the theorem, and with various doses of …

## Interpreting and Solving a Counting Problem

(An archive question of the week) Combinatorics can be inherently tricky; making up your own problem is doubly so. Here we have a problem created by a teacher, who then is not entirely sure what it means. How can we figure out what meaning to give it? Combine that with working out how to solve …

## How Many Different Pizzas?

(An archive question of the week) We’ve been looking at examples of extended discussions with students about various kinds of problems. Here, we have one (not from a student) that led to some good thinking about combinatorics – the techniques of counting the ways something can happen. The problem: Triple toppings Here’s the question, from …

## Too Many Variables?

(An archive question of the week) Students often struggle with solving an equation with several variables, for one of those variables. This is also called “solving a formula”, or a “literal equation”; or “making one variable the subject”. Learning to use variables instead of just numbers (as we looked at last week) is the first …

## How Variables and Equations Work

(An archive question of the week) We’re looking at extended discussions of a single topic, which illustrate how we try to guide a student to a deeper understanding. Here, a student asks how to solve an equation, and Doctor Ian takes him through the whole process, clarifying what it means to solve an equation, and …