When I began teaching undergraduates, I mainly taught mathematics methods courses for elementary education majors. Put simply, I taught future elementary teachers how to teach mathematics. For many elementary teachers, fractions is one of their least favorite topics to teach. For many elementary students, fractions is one of their least favorite topics to learn. So why is it that fractions have become the-topic-which-must-not-be-named in school mathematics?
The short answer is because fractions are complicated! This is the first of a multipart series on the basics of fractions. In this first part, I’ll break down the two common ways of thinking about fractions.
m out of n
The first way that people think about fractions is what others have referred to as “m out of n.” Using this meaning of fractions, someone would think about 1/2 as 1 out of 2. So to put this in context, if I cut a piece an orange into two pieces, one of those two pieces is 1/2.
This way of thinking about fractions is not wrong, but it is a bit problematic. First of all, it doesn’t emphasize the size of the parts in relation to the whole very well (more on that later). Secondly, and super problematically, it makes understanding improper (when the fraction is greater than 1) fractions difficult. What does it mean to have 3 out of 2 parts of an orange? Nothing, actually. So this is not actually the preferred definition of fractions, but it is where many students and teachers start.
Consider the first reference to fractions in the Common Core.
Partition circles and rectangles into two and four equal shares, describe the shares using the words halves, fourths, and quarters, and use the phrases half of, fourth of, and quarter of. Describe the whole as two of, or four of the shares. Understand for these examples that decomposing into more equal shares creates smaller shares.
This standard seems similar to the m out of n definition. Students are expected to cut circles into two parts and describe the whole as two of the two parts (2/2) or one part as one of the two parts (1/2). Most people are surprised to learn that this is actually a first grade standard. That’s right, kids first encounter the dark lord, ahem, I mean fractions in first grade. Even more interesting is that this standard is in the geometry strand, not in number and operations as one might expect.
a times 1/b
The definition of fractions I prefer can also be found in the Common Core Standards (from the third grade number and operations strand).
Understand a fraction 1/b as the quantity formed by 1 part when a whole is partitioned into b equal parts; understand a fraction a/b as the quantity formed by a parts of size 1/b.
I’ve worked with a lot of people who’ve seen this and tuned out after the first sentence (you may have been one of those people!). However, if you are able to stick with me, you might come to love this definition as much as I do.
What this is saying is say you have something and you cut it into b equal parts. Each individual part will be 1/b of the original. So back to our orange, if you cut it into two equal parts, each part is 1/2 of the original orange. If you instead took that orange and cut it into three equal parts, each part would be 1/3 of the orange. With me so far?
Now, the next part of the definition says, if you take some number, let’s say a, of those 1/b parts, your total amount would be a/b. Back to the oranges, if I cut the orange into four pieces, each piece would be 1/4. Now, take two of those 1/4-sized pieces and what do you have? Two, 1/4-sized pieces or 2/4. That’s not too bad, right?
So why is this preferable to the other definition? Think about 5/4 of that orange. With the m out of n definition, I’d have to try to make sense of 5 out of 4 pieces of orange (nonsense!). With this definition, I’d take to think about a 1/4-sized piece and then think about five of those pieces to make 5/4 (sense!). So the real power of this definition lies in the ability to construct any fraction and the emphasis on the pieces in relation to the whole. If everyone learned this definition well, fractions would be a lot easier for everyone!
Stay tuned for part 2 of my fraction exposé where I delve into the whole and why it is so darn important.