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: General
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;…
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…
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…
In (Partial) Defense of Butterflies
Teaching students how to add fractions can be a real struggle. A big part of this is that we tend to get conceptually complicated about what fractions are. And a big part of this is because fractions can be conceptually complicated. I teach Algebra II. (For an elementary teachers reading this, though, hold on: I’m not…
“O Function! My Function!”
I’m going to build a simple set of functions. This set will wind up being very familiar when I’m done, but let’s pretend for a few minutes first. Function and Operators In case you’ve forgotten what a mathematical function is, there are a few quick ways to refresh your memory. A common way is to…
Operations on Fractions
A tale of carts and horses. My child is in fifth grade. Last Friday, we received the second volume of his mathematics workbook. It starts with multiplication of fractions; the next chapter is division of fractions. Addition (and subtraction) of fractions with different denominators is in the first volume. I believe this order persists partly…
Adding Fractions: The Common Error
This is one of the most common mistakes students make when adding fractions: \[\frac34 + \frac67=\frac9{11}\] For a long time, I thought that students did this solely based on confusion with multiplication. While we teach addition of integers before multiplication, addition of fractions is a more complicated process. To multiply fractions, multiply the numerators and…
Converting Between Bases
I was working through the November problems for NCTM’s Mathematics Teacher. There’s a problem on converting between bases, which led to me developing a new-to-me method. What I was taught I started by using the method I’d been taught by my computer programming teachers: Identify the value of each place Divide by the highest value,…
Is Zero a Factor of Zero?
Generally speaking, if \(a \times b = c\), then \(a\) and \(b\) are factors of \(c\). This concept appears at the secondary level in two contexts: The factors of positive integers, and the factors of a polynomial. If we limit the domain and range to be positive integers, for instance, the factors of 7 are…