By Art Duval, Contributing Editor, University of Texas at El Paso
Why are fractions hard to learn for so many people? There are many reasons for this, but I like to think about one in particular, a mathematical idea hiding in plain sight, from elementary school to college: equivalence relations. Consider the fraction sum 2/3 + 1/5, which we of course compute by using 2/3=10/15 and 1/5=3/15, arriving at an answer of 13/15. This raises a whole host of fundamental questions about equality: If 2/3 equals 10/15, why can we use one but not the other in evaluating the sum? Does this mean something is wrong with our idea of “equals”? Could we have used something else besides 10/15; or, in the other extreme, should we always use 10/15? This shows that often when we say “equals”, what we really mean is “equivalent”. Equivalence introduces a number of useful mathematical connections, but we must be careful in how we handle it with our students who just want to know, for instance, how to add two fractions.
We’ll see in a bit how these same ideas arise in mathematical topics from elementary school through college, and what potential difficulties they may cause. I’ll conclude by sharing some specific examples of what I do in my classes to confront these issues.
But first, a very quick primer for readers who haven’t seen how to define rational numbers with mathematical rigor: Declare two fractions a/b and c/d (a,b,c,d integers, b, d not 0) to be equivalent if ad=bc. We then verify this relation really is an equivalence relation (a relation satisfying the familiar reflexive, symmetric, and transitive properties of equality), which also means this relation partitions all the fractions into equivalence classes. It’s those equivalence classes that are the rational numbers. So the rational number 2/3 is actually the equivalence class of all fractions that are equivalent to the fraction 2/3, by our usual method of reducing to lowest common denominator.
We should also verify that this relation behaves nicely with respect to operations such as addition (defined by the formula we get from using common denominators: a/b + c/d = (ad + bc)/bd) so that anything equivalent to 2/3 plus anything equivalent to 1/5 gives something equivalent to 13/15.
All of this is more meaningful (and the proofs are much more straightforward) if we define two fractions to be equivalent if they correspond to the same length on a number line. Indeed, this more natural way to define the equivalence is why we care about it in the first place. But that doesn’t help when we want to do exact calculations, or when our fractions contain algebraic expressions. Then we really are confronted with issues of equivalence:
- Although rational numbers are defined to be equivalence classes with infinitely many elements, we can’t add two infinite sets of equivalent fractions.
- Therefore we have to pick representatives from each equivalence class and add, but certain representatives (common denominators) make this job easier.
- And this works with adding fractions only because addition is an operation that respects the equivalence.
One more issue with some equivalence relations doesn’t show up with fractions, but does in geometry: there may be different equivalence relations on the same set. Sometimes, we want to consider geometric objects equivalent if they are congruent, or sometimes even just similar. But other times we need a finer partition that distinguishes between congruent objects. For instance, a left-hand glove is congruent to a right-hand glove, but not equivalent in most real-world settings, because they differ by a reflection; similarly, a triangle pointing up may be congruent to a triangle pointing to the left, but they convey different information, because they differ by a rotation.
Once you start looking, you can find equivalence relations in many mathematical ideas. Here is a sampling, showing that this notion of equivalence spans a wide variety of mathematical topics, from very elementary to advanced ideas. The four issues described above with fractions and geometry appear in various combinations throughout.
- Regrouping in multidigit addition and subtraction: Usually we think of 436 = 400 + 30 + 6 = 400 + 20 + 16 as being completely identical ways to write the same number. But changing from one of these representations to another is one way to think about regrouping, so in this setting I consider those three representations as merely equivalent.
- Factorization: Similarly, we usually think of 6 x 10 = 4 x 15, and in the context of equality, this is true. But in the context of factorization, these are now considered different factorizations of 60; factorization has a finer partition (more equivalence classes) than the set of non-negative integers.
- 0.999…: I surprise students by telling them 0.999… does not equal 1. After all, 0.999… isn’t even a number, it’s an infinite process. On the other hand, this infinite process gets arbitrarily close to 1, and “getting arbitrarily close” is an equivalence relation which respects the important operations and relations we have on numbers. So 0.999… is merely equivalent to 1, but this equivalence builds the real numbers.
- Algebra: Algebraic expressions that look different (e.g., (x^2)-1 and (x-1)(x+1) ) are still equivalent when they take the same value for all inputs. Algebraic equations are equivalent when they have the same solution set. In both cases, an important procedural skill is to chain together a sequence of easy equivalences by transitivity to show a complicated expression or equation is equivalent to a much simpler one.
- Combinatorics: One way to describe the difference between permutations (order matters) and combinations (order does not matter) is that combinations are equivalence classes on permutations.
- Vectors: To actually draw a vector, we need to pick a starting and ending point of a particular arrow. But vectors don’t change upon translation, so we can think of vectors as equivalence classes on arrows we can draw.
- Modular arithmetic: You probably thought of this example already.
- +C in anti-differentiaton: We put “+C” at the end of the solution to every indefinite integral because we know the answer to such problems only up to an equivalence relation.
- Cardinality: When we say the cardinality of an infinite set, we really mean the equivalence class of sets with the same cardinality.
- Etc.: There are a few more on the slides for a talk I gave on this topic. Also feel free to add your own to the comments section below.
Well, finding equivalence relations is a fun game for mathematics people, but so what? What can we do differently after finding them? I now do two things differently in my classes based on my thinking about equivalence:
- When a definition based on equivalence relations arises, I pay more attention to helping my students navigate the resulting potential trouble spots from the four issues described above. For instance, recently in Linear Algebra, we were talking about representing a linear transformation by a matrix, and some students had issues with how you must first pick a basis before constructing the matrix. I realized this was an equivalence relation issue (different matrices that represent the same transformation are similar, an equivalence relation), and spent a little more effort on the difficulty of this arbitrary, but necessary, choice of basis.
- I explicitly tell this story about equivalence relations to students planning to become teachers, so they can be aware of these difficulties for their own students. For some of my students (especially future high school teachers, who are math majors), the fairly careful development of equivalence relations, partitions, and equivalence classes is fine. But this rigor (and some of the above examples) may be too much for others (many future elementary school teachers). Fortunately, there are also a few simple “real-world” examples that illustrate the ideas well. Here are two:
- Money: Generally, a dollar bill is an equivalent sum of money as four quarters, but they are not equal: Some parking meters take quarters, but not dollar bills, whereas dollar bills fit more nicely in my wallet than quarters. Of course, this equivalence relation respects addition and subtraction.
- Pie: Even though we say 1/3 “equals” 2/6, if we’re talking about pie (isn’t it a rule that if there are fractions, you have to mention pie?), I’d rather have a 1/3 than 2/6, because cutting a 1/3 piece into two 1/6 pieces loses crumbs. On the other hand, if there are two hungry kids and no knife handy, everyone will be much happier with two 1/6 pieces than a single 1/3 piece.
To conclude, no matter what level you teach (or learn) at, keep looking for those hidden equivalence relations lurking throughout the mathematics curriculum. With a little awareness, we can help students avoid the traps and make more sense of this powerful tool.