# Alternatives

## Probability That a Random Triangle is Acute

Some time ago we looked into the probability that a random set of sides (from, say, a broken stick) form a triangle. A recent question asked about the probability that a random triangle is acute (all angles acute) or obtuse (at least one angle obtuse), which led to more discussion of what it means for …

## How to Think About the Product and Quotient Rules

Last time, we considered the Chain Rule for derivatives. This time, we’ll look at the product and quotient rules, focusing on how to keep the formulas straight, and make them easier to apply. We’ll look primarily at the quotient rule to start with, and then examine the product rule at the end.

## How to Think About the Chain Rule

Having recently helped some students (in person) with the rules of differentiation, I’m reminded to do so here, starting with the chain rule. It is easy to make this topic look harder than it really is; the two main ways to state the rule are often confusing, and different approaches fit different problems. We’ll try …

## Finding an Unknown Angle: Trig or Geometry

I am always interested in problems that can be solved in different ways, particularly because this can give a student a chance to be creative, as well as learning from experience that you don’t have to do it “the teacher’s way”. Here we’ll use trigonometry, and two different ways to add lines to a figure …

## Slow and Fast Ways to Solve a Probability Problem

Last week’s discussion reminded me of another question, from July, about a probability problem that was solved in a hard (but educational) way and an easy way. This instance is more extreme, and, due to its length, requires extreme editing in order to fit here.

## Different Ways to Solve a Probability Problem

Sometimes there are several ways a problem could be solved, and we encourage students to pursue the path they’ve started on, rather than give them a method that may not be natural to them. (And we encourage them to start, if they haven’t yet, in part so we can see what might be most helpful …

## What is Adjusted Frequency in a Histogram?

Some time ago we looked into the meaning of histograms, on the way to the concept of the Probability Density Function. A recent question focused on the histogram itself, in a way that will add to that discussion. We’ll learn about frequency density, which was overlooked there, and discover an alternative way to label a …

## Anything to the Zero Power: Why 1?

We’ve been looking at oddities of zero. Because “nothing” behaves differently than “something”, operations with it can be surprising. Although students learn that $$x^0=1$$ for any non-zero number x, they often wonder, why?? I’ve selected a few out of at least a dozen such questions in our archive.

## Comparing Logarithms With Different Bases

Logarithms are not hard to work with when only one base is involved (as in most real-life problems); but they can be challenging when each log has a different base. Here, we’ll look at a few problems in which we have to compare logarithms with different bases, showing various strategies.