## Distance from a Point to a Line in Space

(A new question of the week) Among many interesting recent questions we have one about vectors and equations in three dimensions. We’ll see four different ways to find the distance from a point to a line, proving two formulas and catching some of the errors one can make along the way. We’ll also see a …

## 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 …

## Sines Without Right Triangles

(A new question of the week) A recent question dealt with an apparent conflict between the right-triangle definition of sines and cosines, and the unit-circle definition, pertaining to multiples of 90° (angles on the axes). This provides an opportunity to look closely at the relationship between those two definitions. Two definitions Recall that the right-triangle …

## 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?

## Fractions and Felonies

(A new question of the week) A recent question involved a word problem about fractions, which will fit in nicely with the current series on fractions. We’ll explore several ways to solve a rather tricky fraction word problem, some avoiding fractions as much as possible, some focusing on the meaning of the fractions, and others …

## 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?

## One-sided limits of a composite function

(A new question of the week) A good way to develop a sense of what limits are and how they work comes from working with visual representations of them, in the form of graphs. In particular when the functions are defined by graphs rather than by equations, we have a lot more flexibility in creating …

## 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 …