# Algorithms

## Long Division with Zero, Revisited

(A new question of the week) One of our first posts, in 2018, was about zeros in long division. But we still get many questions about this issue, and it’s time to dig in deeper. We’ll look here at two of them, answering the twin questions, “When do you put a zero in the quotient …

## Mean and Standard Deviation of Grouped Data

Two of our most-viewed posts deal with Mode and Median of Grouped Data: how to calculate these statistics for data that is supplied in the form of frequencies for classes of data (bins), rather than the individual data values. Here we’ll complete that topic with a look at the less troublesome cases of Mean and …

## Too Much Guessing?

Today we’ll look at a question from a student who was troubled by the amount of guessing needed to solve certain problems. This leads to an interesting survey of different kinds of guessing, and ways to develop that skill. When do we need to guess in math? The question is from 2017: More Methodical Than …

## Evaluating Square Roots by Hand

Square roots commonly are irrational numbers, so that it is necessary to estimate them. Usually today, we use a calculator to find them. Many students, however, are curious about how they could do it without a calculator. Here I want to reverse the usual format of this blog, presenting a summary I have written that …

## The Method of False Position: Old and New

(An archive question of the week) Last time, as part of our series on estimation, we looked at some numerical methods for solving equations approximately. I mentioned the Method of False Position, but when I looked for more detailed expositions in our archive, I realized that in a sense it is really two different things, …

## Partial Fractions: How and Why

I have often noted that calculus class is where you really learn algebra. Certain techniques in calculus demand algebraic skills that either were not taught in algebra classes (because they are not needed until you get to calculus), or have been forgotten. Chief among these is the method of partial fractions. I have here put …

## Rounding to the Nearest: The Basics

We frequently get questions about how to round; so many different issues arise that I won’t try to fit them all into one post. Children have trouble learning how to do it, and sometimes their parents are surprised to find that they are being taught a different way than they learned. There are several common …

## Long Division: When Zero Gets in the Way

I was going to move on from arithmetic to algebra, but the discussion of long division led me to think about some of the more ordinary difficulties students have asked about in that area. Here I will show several questions about the process of long division in which zero caused trouble. Zero in the quotient …

## Dividing Right to Left, Adding Left to Right

I now want to start looking more deeply at some specific questions at various levels, starting with arithmetic, then algebra and geometry. Students learning arithmetic (and their parents) tend to think in terms of following some rote procedure, just because “that’s the way it’s done”. Modern educators try to focus more on deep understanding, so …