Infinitely Truncating a Triangle
(A new question of the week) Here is an intriguing question we got at the end of September from an adult whose name I’ll shorten to Arun.
Weighted Averages: Averaging Averages or Rates
In our series on averages, last week we introduced the idea of the weighted average (or weighted mean), where each item has a weight attached. The classic examples all involve grade averages in various ways. This time, we’ll look at how weighted averages arise when you need to average several averages together, something we touched …
A Mind-Stretching Exercise with a Stretched Cosine
(A new question of the week) A question in September, about graphing a horizontally-stretched cosine function, led to a long conversation. Between a typo in the problem and some inside-out thinking, this surprisingly non-routine problem led to some good mind-stretching! I have edited this down considerably by removing distractions from the main ideas, but it …
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Weighted Averages: Finding Your Grade
In our series on averages, we’ve looked at mean/median/mode, then at details of the (arithmetic) mean, and then at different kinds of mean (arithmetic, geometric, harmonic, quadratic). Next, I want to look at the weighted mean concept. In checking what we’ve said about this, I found a useful series of explanations of one application of …
Multiplying Fractions and Decimals
(A new question of the week) Let’s look at a quick question from mid-September, that had a number of different answers. In some ways, this is an easy question; but we’ll take it a little further, so keep reading to the end.
Four Kinds of “Mean”
Last week, we looked at exactly what the mean is, referring specifically to the arithmetic mean, the one we first learn as the “average”. But just as we previously saw that there are several things called “average” (mean, median, mode), there are in fact several different kinds of “mean”. We’ll look here at the arithmetic, …
A Limit: Getting the Algebra Right
(A new question of the week) I have often said that calculus class is where many students finally learn algebra, because now algebra is an essential tool, not just something to learn for an exam. This is especially true of a nontraditional student, who may not have taken math recently, or may even be learning …
Making the Mean More Meaningful
Last week, we started a series on averages, looking at a common list of three kinds of average: the mean, median, and mode. This time, we’ll focus in on the (arithmetic) mean, thinking about why it is appropriate for many applications; that will lead into next week’s discussion of when other kinds of mean are …
A Proof Problem: Chords and Tangents
One thing we enjoy doing is guiding a student through the process of problem-solving. Here is a problem from August that illustrates how to think through a complicated geometrical proof. In particular, this uses some circle theorems involving chords, secants, and tangents, together with a bit of algebra.
Three Kinds of “Average”
There are three different statistics that are commonly taught as “averages”, or “measures of central tendency”, of a set of numbers: mean, median, and mode. (There are others as well, which we will get to later.) What are they? How do they differ? How do you use them? We’ll look into questions like these as …