What is a Ratio, Really?
A recent question reminded me I hadn’t yet written about the complexity surrounding the definition of ratio (and related terms, like rate and fraction). Here are four questions about the words.
Ambiguity
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.
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 …
Implicit Differentiation: What to Do When It’s “Wrong”
(A new question of the week) Having just discussed the Chain Rule and the Product and Quotient Rules, a recent question about implicit differentiation (which we covered in depth two years ago) fits in nicely. This raises an important issue: when you get an apparently wrong answer, you may just have done something wise that …
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Why Are There Different Definitions of Range?
A recent question about two interpretations of the range of a data set in statistics leads us into some older questions and some mysteries. Is “range” defined as the interval containing the data, or the difference between largest and smallest values, or 1 more than that? Yes! All three are used, and are useful.
Powers of Roots and Roots of Powers
Last time, we looked at two recent questions about combining squares and roots, and implications for the properties of exponents. We didn’t have space for some older questions that we referred to. Here, we will go there.
Types of Data: Discrete, Continuous, Nominal, Ordinal, …
Last time, we looked at some ideas about appropriate graph types, and the references I found put this in the context of identifying types of data. Here we’ll look at questions about two such classifications: nominal/ordinal/cardinal (with variants), and continuous/discrete. We’ll see that classifications can become distorted as they filter down from higher levels to …
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Inverse Trig Notation: What Do sin^-1 and arcsin Mean?
Since we’ve been looking at an example of ambiguity in notation, let’s look at a very different one. There is a lot to be confused by in inverse trigonometry! We’ll try to untangle the notations of \(\sin^{-1}\) and \(\arcsin\).
Implied Multiplication 2: Is There a Standard?
This is part 2 of a series of extracts from discussions we have had on whether multiplication implied by juxtaposition is to be done before division (which I call IMF, for Implied Multiplication First). Some people write to us claiming that there is one official correct answer. Are they right?
Implied Multiplication 1: Not as Bad as You Think
We keep getting new questions related to Order of Operations: Implicit Multiplication?, where we discussed expressions like 6/2(1+2) that keep showing up in social media arguments. Since I closed comments on that page some time ago, because of the toxicity of some of them, further questions have come through our Ask a Question page (as …
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