# Why

## Dividing Fractions: Why Invert and Multiply?

Last week, we looked at how to visualize division of fractions; in the process, we saw that you can multiply the first fraction (dividend) by the reciprocal of the second (divisor): “invert and multiply”. Here I want to look at a few of the many times we have been asked how to do it or …

## Dividing Fractions: Can You Picture It?

We’ve looked at what it means to multiply fractions, including whole and mixed numbers; now it’s time for division of fractions. We’ll start here with pictures, similar to what we did for multiplication, but a little more complicated. Then next time, we’ll see additional ways to understand why we “invert and multiply”.

## Simplifying Sums and Quotients of Radicals

(A new question of the week) A recent question asked about the reasons for differences in the work of simplifying different kinds of radical expressions. We’ll look at that general question, with two specific examples, and then consider an older problem of the same type.

## Multiplying Fractions by Whole or Mixed Numbers

Last week we looked at how to multiply fractions, and why we do it that way. But what do we do when one of the numbers is a whole number, or when one or both are mixed numbers? And do we have to do it the way we are taught?

## Multiplying Fractions

Last week we looked at some questions about multiplication that arise once students learn to multiply fractions or decimals. Let’s turn to the underlying question: How do you multiply fractions, and why do we do it that way? How does cancelling fit in?

## How Can Multiplication Make It Smaller?

A fairly common question arises when students learn to multiply or divide fractions and decimals: They discover that multiplication, which always used to make numbers larger (2, multiplied by 3, becomes 6), now can make them smaller (2, multiplied by 1/2, becomes 1). How can that be? Here we’ll look at a few answers we’ve …

## Decimals, Commas, Thousands, and Lakhs

I’ll finish this series on place value and writing numbers, with a question that’s not quite as simple as you might think: why we use commas and decimal points as we do. Americans may be surprised at some of the answers – and some of the questions. Commas are just a convenience We’ll start with …

## Why Can’t You Skip the Law of Sines?

(An archive question of the week) I’m in the middle of discussing the Law of Sines and the Law of Cosines, and in searching for questions about them, I ran across one that stands by itself. A student asks his teacher why his method without trig doesn’t work, and gets three answers from us. They …

## Volume of a Pyramid – Without Calculus

Last week we looked at ways to derive the formulas for volume and surface area of a sphere, without using calculus. Let’s do the same this time for a pyramid. We’ll be seeing one method that comes very close to calculus (slicing and infinite series), and another that is fully geometrical (dissection, which we’ll do …

## Fractions: What Are They, and Why?

We’ve looked at some specific ideas about fractions (their proper definition, their relationship to decimals, and how to divide them); it’s time to go through this topic from the beginning. Here we’ll look at how they are introduced to beginners, and how to keep them from hurting our brains! Parts of a whole Although, as …