I have long been of the opinion that ratios and fractions are effectively the same thing, but I’ve recently changed that belief. A ratio is a relationship between two values that states the relative size. These can generally be expressed in fractional form, but they need not be. How much sense the units make depends…
Category: Mathematics
“Answer”
I see or hear some variation of this often from mathematics teachers: “The answer to 4+5 is 9.” “So if we’re given (5+3)/2, that means we add five and three, then divide the result by two, and that gives us the answer.” In regular plain English, answers are coupled with questions. Expressions, though, aren’t questions,…
Squaring Two-Digit Numbers (Redux)
If you want to square a two-digit number, you could just use a calculator, or you could use the traditional algorithm. I’m going to talk about a different method here, but not because I think this is a particularly useful method. The point of this discussion is to look at how numbers are interrelated; if…
Squaring Two Digit Numbers
Math fun: If you want to square any two digit number more quickly than the traditional algorithm, here’s a strategy. It does require you to know the squares of all the single-digit numbers, as well as a bit of mental juggling. First, if the number ends in a zero, square the first digit and put…
Mathematics and Mnemonics
I’m currently reading “Why Don’t Students Like School?” (second edition) by Daniel T. Willingham. While there is a lot of good stuff in this book and I’m feeling fired out about setting my educational train back on its tracks, I winced at his cheerleading for mnemonics. And then: I reframed. He suggests the use of…
Operators
Moderate level definition: An operator is a mathematical symbol takes some defined number of inputs (also called arguments) and returns an output. The first operator that students learn about is \(+\). This operator takes two values and gives their sum. For instance, \(5 + 4 = 9\). The arguments of addition are called the addends;…
FTOC (Informal)
Let \(f_0\) and \(f_1\) be two functions such that \(f_1\) represents the rate at which \(f_0\) is changing with regards to some independent variable \(t\). (If you prefer to read an example first, jump down to that section.) This relationship is typically represented by saying that \(f_1(t)\) is the derivative of \(f_0(t)\), that is: \[f_1(t)=f_0^\prime(t)\]…
An Introduction to Sets
In casual parlance, we often distinguish collections from sets: My comic book collection consists of all the comic books I own, which may include duplicates. It is not an exhaustive collection of every comic book ever produced. However, if I happen to have a set of Groo comics, that means I have every title produced…
What is Algebra, Anyway?
Ask a general person on the street to define mathematics, and they’re likely to say something about manipulating or combining numbers. Ask a mathematician to define mathematics, and they’re likely to talk more about patterns. Here, for instance, is the first sentence of the definition provided by MathWorld: “Mathematics is a broad-ranging field of study…
Locus
A locus (plural loci) is a set of all the points that satisfy a particularly mathematical statement or rule. Such statements or rules can contain any number of distinct variables, but we can only easily graphically represent statements with two or three distinct variables. If a statement contains only one distinct variable, the graph of…