# Strategies

## Order of Operations: The Basics

The order of operations in algebra (also called operator precedence) is a very common source of questions; I count at least 50 archived discussions explicitly about the topic (not just mentioning it in passing), in addition to the Ask Dr. Math FAQ on the subject.  I’ll devote the next few posts to looking at various aspects …

## Perimeter Magic Polygons

Last time we looked at the classic puzzle of magic squares. Many questions we get are about similar kinds of puzzles, and here I want to look at “magic polygons” (triangles, squares, pentagons) in which, unlike the traditional magic squares, only the edges count. These are a common subject of elementary-level questions. Four sides: corners …

## The Haybaler Problem: Several Ways to Solve

We often compare math problems to puzzles; and some puzzles are math problems. I want to devote a couple posts to interesting puzzles that can be attacked in various ways. Here, we are given the weights of every possible pair of hay bales, and have to work out the individual weights. This classic can be …

## 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: …

## A Rational Inequality with Huge Exponents

When a challenging type of problem is written with unexpectedly large numbers, it can look impossible. Today’s discussion illustrates how to get past the hurdles. The problem The problem came from Arsh in April: Q) [x((x+5)^2016)((x-3)^2017)((6-x) ^1231)]/((x-2)^10000)((x+1)^2015)((4-x)^242) ≥ 0 Since our site doesn’t yet allow LaTeX formatting, and Arsh chose not to insert the problem …

## Non-routine Algebra Problems

(A new problem of the week) Last week I mentioned “non-routine problems” in connection with the idea of “guessing” at a method. Let’s look at a recent discussion in which the same issues came up. How do you approach a problem when you have no idea where to start? We’ll consider some interesting implications for …

Last time we looked at applying Heron’s formula to problems about the area of a triangle, where knowing the side lengths is enough to determine the area; there was a passing mention of the fact that more is needed for quadrilaterals. We’ll start here with a repeat of that idea, and then look at several …

## Area of a Triangle: Heron’s Formula II

Last time we looked at a very useful formula for finding the area of any triangle, given only the lengths of its sides. Today I want to look at several problems in which the formula has been used, some of them surprising. Area of a plot of land We’ll start with a straightforward application in …

## Translating Logic Statements

The next few posts will examine aspects of logic, both symbolic logic, and how we talk about theorems in general. We’ll start here with issues in interpreting the wording of logic, and some of the semantic difficulties we face. English isn’t logical. (Well, I suppose humans in general aren’t logical.) Which kind of OR? We’ll …

## Graphing Transformed Sines

I’ll close out our look at transformations of functions with some trigonometric graphs. These are the best example of combined transformations, and involve some special tricks as well. We’ll start with an early question that gives an overview of the process, then focus in on important details. Overview First, a typical question from 1997, to …