Why Do Logarithms Work That Way?
Last time, we introduced logarithms by way of their history. Here, we’ll look at their properties.
Where Do Logarithms Come From?
Having answered many questions recently about logarithms, I realized we haven’t yet covered the basics of that topic. Here we’ll introduce the concept by way of its history, and subsequently we’ll explore how they work.
Casting Out Nines: Why It Works
Last week we looked at how to “cast out nines” to check arithmetic, and touched only briefly on its relationship with modular arithmetic and remainders. Here we’ll look at several explanations of why it works, aimed at different levels of students, with varying levels of success..
Casting Out Nines: What and How
This old technique for checking arithmetic is both easy and hard to describe: easy to explain in advanced terms, but hard to explain in elementary terms. We’ll try to do it all here, but a fuller explanation of the “why” will come next week.
Law of Sines vs Law of Cosines: Which is Better?
Last month, four students from the same class wrote to us with the same question: Which is more accurate, the Law of Sines or the Law of Cosines? Those led to a couple deeper discussions, as we explored the context.
Permutation vs Combination: Clarifying Our Terms
A couple recent questions dealt with details in the way permutations and combinations are explained. What do we mean when we say that “order matters” for a permutation, and that there is “no repetition” or that the things being chosen are “different”? Teachers need to know how students hear such words.
Looks Like a Frustum, But …
Last time we looked at how to find the volume of a frustum of a pyramid or cone. But sometimes what looks at first like a rectangular frustum actually isn’t. This case turns out to have a more general formula almost as nice as what we have for an actual frustum. We’ll discover that the …
Frustums: Not Frustrating but Fascinating
We’ve looked in the past at volumes and surface areas of familiar geometric shapes like spheres, pyramids, and cones; but more can be done. If we cut parallel to the base of a pyramid or cone, the result is called a frustum (no, not a frustrum!). Let’s derive some formulas, which will be remarkably simple.
Algebra Word Problems: Learning from Mistakes
A recent series of questions from an insightful high school student about word problems, provided a number of opportunities to discuss how to find and correct your mistakes – or the book’s! We’ll look at five.
Proving the Chain Rule: Details Matter
When we recently looked at the Chain Rule, I considered including two questions about its proof, but decided they would be too much. However, when a recent question asked about a different version of the same proof, I decided to post all three. It is a nice illustration of how a mathematician’s view of a …