A popular kind of question in combinatorics is to count the number of paths between two points in a grid (following simple constraints). This can be done by very different methods at different levels. We’ll look at several problems of this type, starting with the simplest.
(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 …
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 …
(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 …
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”.
(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.
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?
(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 …
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?
(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 …