So the time of the year has come where we have to teach how to compare fractions (or generally, rational inequalities). You’ve got a standard approach, but it’s not working as well you want or as well as it did last year. Or maybe you’re looking for alternative ways of teaching this. Well, let’s take a look!

### Some Reference Problems

I’ll use \(R\) for “relationship” which can be one of \(>\), \(<\), or \(=\). We are interested in identifying \(R\) for problems like these:

Determine the relationship, \(R\) for $$\frac{a}{b}\mbox{ } R \mbox{ } \frac{c}{d}$$

Now, of course, if you’re teaching a unit on comparing fractions, you’re not going to state the problem like that. Instead, you may pose the question as

Determine which is greater

$$\frac{3}{8} \mbox{ or } \frac{4}{7}$$

followed by instructions to use the correct symbol (\(>, <, =\)).
One of the hardest parts about teaching (mathematics) is being able to anticipate student mistakes. There is no perfect delivery. And any deliver is going to *induce* mistakes. Yes, our very teaching of the subject will create mistakes in the minds of students. Students will create their own mistakes even if we aren’t teaching them. But the point is, don’t think that you’ve come up with a foolproof content delivery method. It may be very good, but odds are some students won’t be reached. And before we point the finger at them, let’s try a few other methods and let’s see if we can anticipate the mistakes that students will make as a consequence of how we delivered the material.

So before we start, let’s talk about student mistakes that inevitably occur. By the time we are teaching a unit on fraction comparisons, it’s assumed that students know how to compare integers. What is *natural* for students to do is to extend a concept they have understood on to the unfamiliar. Sometimes, the instructional pathways reinforce previous notions. This is a double-edged sword! But generally, there’s no way around it.

Let’s start with a basic introductory problem. In an attempt to ease students in, we might begin with something like this

Which is larger?

$$\frac{3}{8} \mbox{ or } \frac{4}{8}$$

The didactic goal is to not introduce too many moving parts. So we believe that showing a fraction with the same denominator will help to demonstrate how fraction comparison works. Maybe you don’t do this and that’s fine. But bear with me for a few more sentences. We point out that the denominators are the same, therefore \(\frac{4}{8}\) is greater than \(\frac{3}{8}\) since \(4\) is greater than \(3\). We write this as \(\frac{3}{8} < \frac{4}{8}\) and then there might be some confusion in that we used "the less than symbol" but said that "four-eighths is greater than three-eighths" --- so that's something that has to get sorted out (confusion #1!). To get around this, I recommend pausing and discussing how mathematics is read. It is not unusual to read from left to right OR from right to left in mathematics (or even from bottom to top and then left to right (and maybe a little bit of inside then out) --- \(\int_{a}^{b}f(x)\ dx\)). So, "three-eighths is less than four-eighths" is just as correct as "four-eighths is greater than three-eighths" AND both of those statements are a correct way to phrase \(\frac{3}{8} < \frac{4}{8}\). Ye be warned, many a teacher I have interacted with are militant about this and insist that there is only one way to read the statement. Please don't be! The symbol is called "less than", but that doesn't mean we must only read from left to right.
I do this on purpose because as much as we want to progress towards comparing fractions, the conversation about *reading* inequalities (math in general) is often lost up and down the curriculum. In effect, this confusion begins right here! (Or maybe just a wee bit earlier when comparing integers, but the confusion persists and then lingers on into Algebra where this knowledge is assumed.)

Anyhow, let’s move on to comparing fractions, but keep the symbolic confusion in mind because it’s a source of mistakes that’s easy to miss when diagnosing student confusion.

So, back to $$\frac{3}{8} \mbox{ vs } \frac{4}{8}$$ One of the other things that happens here is that in our appeal to smoothly transition and motivate the concepts by leveraging existing knowledge (comparing integers), is that we can artificially induce a false method in students’ minds. Then we have to spend more time addressing that. So my first suggestion, don’t give \(\frac{3}{8} \mbox{ vs } \frac{4}{8}\) as a first problem (or any in which the denominators are identical).

Rather, let’s start with a problem to get as many bad habits out as possible. We want to get at two main things: (a) learn how to compare fractions mathematically and (b) understand what the mathematical process / method means both on paper and in a tactile way.

Now, really, let’s begin.

### Setting It Up

While there is no perfect delivery, some deliveries are better than others *and* some deliveries are as good as others if you know how to manage the sequence of lessons.

Here’s what I prefer

Which is larger?

$$\frac{6}{10} \mbox{ or } \frac{7}{8}$$

We know the myriad of mistakes and confusions that will happen. I want to try to catch a bunch of the major ones as early as possible with a single example. The problem above has a bunch of characteristics.

- All numbers are different and are greater than \(1\)
- The fraction with the larger numerator has a smaller denominator than its counterpart
- The denominators share a common factor greater than \(1\) (ie, they are not co-prime)
- One of the fractions is not in fully reduced form (another form of math militancy — we don’t
*have*to always reduce fractions!) - The numbers are large enough so as not to be trivial, but not so large that the fight changes to managing arithmetic errors

Can you see where I am going with this example? We’re going to (hopefully) catch these errors / misconceptions early or at the least head off some problem areas before they come to be.

- Numerators of \(1\) tend to oversimplify and obscure fraction comparison when we have to scale (ie, multiply). I avoid using a \(1\) early on. Also, keeping all the numbers different allows for a greater transparency in how the number crunching works.
- Students will want to simply compare numerators or denominators exclusively and deduce solely on this comparison. While this is perhaps where we want to be, if the pattern is identified too early, it’s an early habit that’s hard to correct or get its nuances sorted out. I prefer to showcase and remind students that fractions have two parts to them.
- Finding common denominators can be confusing and this is at the heart of the skill of comparing fractions. Some instructors enforce finding least common multiples. I don’t think it’s necessary the common multiple be least. It simply needs to be common. Having an 8 and a 10 allows for a few ways of scaling — either obtaining the least common multiple (40) or some other multiple (80, for example). Sometimes students see different things. Choosing, say, a 7 and an 8 directs students a little too much to one way.
- One fraction is fully-reduced another not. Again, students see different things.
- Too often we choose 1s, 2s, 3s, and 4s to keep things “simple” and to focus on the “concepts”. But we don’t have to have things be trivial.
- An extra point, the least common denominator is 40, which will require multiplications by 4 and 5 — two additional numbers different from those in the fraction. Transparency in computation!

### Standard #1: Find Common Denominators (aka Scaling)

This is probably where you want your students to be. In an ideal world you’d want students to get to $$\frac{24}{40} \mbox{ vs } \frac{35}{40}$$ at which point it will be clear that “35 out of 40” is greater than “24 out of 40” or “24 out of 40” is less then “35 out of 40”, whichever is your fancy.

The method works like so

$$

\begin{eqnarray}

\frac{6}{10} <>= & \frac{7}{8}\\

& \\

\frac{6 \times 4}{10 \times 4} <>= & \frac{7 \times 5}{8 \times 5}\\

\therefore & \\

\frac{24}{40} < & \frac{35}{40}
\end{eqnarray}
$$
This requires a few skills

- Finding common denominators which is identifying (least) common multiples
- Knowing what it means to “multiply by 1” both procedurally and conceptually — this is no easy task for students!!

If there is difficulty here, then first it’ll be a matter of identifying which of the two main skills are confusing. Is it about finding common denominators? Or is it about understanding how to multiply by 1?

If the problem is finding common denominators, then break this off as a separate lesson and take a step back remind students about finding common multiples.

If the problem is how and / or why to multiply by 1, start with some reasoning questions. Some of the confusion about fractions is about what they *can* represent (not what they *do* represent). Fractions can represent a ratio of one set of objects to another set of objects. Grab two sets of colored paper (or in general two sets of distinct objects). Pick 6 of one object and 10 of another. Now, we can make the statement “For every six of object two, we have 10 of object two”. From here, you can have a discussion about what it means to double, triple, quadruple the quantity of one of the objects and how it should affect the quantity of the other object in order to preserve the ratio.

If you don’t want to deal with all the material, then use Xs and Os and have students write out 6 Xs and 10 Os. From there you can demonstrate reducing the fraction (by an integer) or scaling up.

From an arithmetic level, you can ask them what happens to a number if you multiply it by 1. Then have a discussion about how \(\frac{4}{4}\) is actually just one. Thus, scaling the numerator and denominator by 4 is just multiplying by 1 and as a consequence we haven’t changed the ratio. It may be more instructive to show what happens when you break this rule! Have students try and see what happens when they *add* 4 to both the numerator and the denominator.

### Standard #2: Clear Fractions

I don’t care for this method even as a scaffolding technique. In fact, I want to discourage you from doing this. Clearing fractions works like this:

$$\frac{6}{10} \mbox{ R } \frac{7}{8}$$

We “clear the fractions” by multiplying both sides by \(10 \times 8\) to arrive at

$$(10 \times 8)\frac{6}{10} \mbox{ R } \frac{7}{8}(10 \times 8)$$

yielding

$$6 \times 8 \mbox{ R } 7 \times 10$$ and we can see that \(48 < 70\). Therefore,
$$\frac{6}{10} < \frac{7}{8}$$
While this can feel generally clean and will enable students to move through the material more quickly, I don't care to introduce this at all. Here are a few reasons why.

- After the arithmetic maneuvering, it’s easy to lose sight of the original problem. We’ve lost the original object we were working with — the fraction!
- In the not too distant future, they’ll have to work out a problem like this \(\displaystyle \frac{4}{x} > -1\) and, yeah, let’s just say, I’ve seen too many students try to clear the fraction … Let’s aim for extendibility in methods, whereever we can
- As a general practice, multiplying across an inequality is tricky business. We have to be careful not to multiply by zero or actually, more fatally, multiply by a negative value and not switch the direction of the inequality. I think of “clearing fractions” as a bad quality control process. It’ll get students through, but we’re probably not helping them in the longer run
- Scaling is one thing we want them to understand, not this hack

### Standard #3: Subtraction!

I absolutely prefer the following method. But it isn’t for everyone. We want to determine the relationship, \(R\) in $$\frac{6}{10} \mbox{ R } \frac{7}{8}$$

Rather than work through inequality mumbo jumbo, have students fall back on something they already know with fractions! Subtraction! Have students work out

$$\frac{6}{10} – \frac{7}{8}$$

This will do a bunch of things all at once. It continues to reinforce fraction arithmetic (a huge sore point in math education, forever). Reinforces skills like finding least common multiples and their basic multiplication facts. But also, it gets them to understand how numbers (fractions are numbers!) relate to each other. If the result of the subtraction is negative, what does that mean? It means that seven-eighths was larger than six-tenths (or six-tenths was smaller than seven-eighths). If the result of the subtraction is positive, what does that mean? If the result is 0, then what?

I absolutely prefer this. Though, fraction inequalities aren’t often taught this way. From a “quality control” standpoint, we don’t run the risk of dubious multiplications across an inequality. Additionally, this works cleanly in Algebra as well: \(\displaystyle \frac{4}{x} > -1\) becomes \(\displaystyle \frac{4+x}{x} > 0\) and then it’s a matter of setting up critical points and testing what happens in \((-\infty,-4), (-4,0), \mbox{ and } (0,\infty)\). Notice that with Standard #1, students will be stuck in the mud with \(\frac{4}{x} > \frac{-x}{x}\) after which all sorts of hell breaks loose. And Standard #2 is just plain wrong.

I’m also of the habit of changing the order in which things are taught. First comes fraction arithmetic. Then inequalities. I don’t like it when curriculum has fraction inequalities coming before fraction arithmetic.

### Hacks (Do Not Use!)

Avoid at almost any cost, things like “cross-multiplying”, the “butterfly method”, or anything where the entire meaning of what’s going on has been obfuscated. You may have been taught that way and it’s worked out fine (I was taught that way), but these hacks will do far more harm than good.

### Tactile Alternatives for Scaffolding

The pencil and paper route to mathematics is not without its problems. Oftentimes with mathematics, we need some visual or tactile interaction with the objects of study. In this case, we are studying fractions and inequalities (more formally, rational inequalities).

Rational inequalities are, in essence, an exercise in obtaining common denominators. The question we should try to get students to answer is, why? Why must we obtain common denominators?

However, the tricky part is that with visual methods like drawing a unit bar and dividing it into ten pieces and shading in 6 to represent \(\frac{6}{10}\) and comparing it to a similarly drawn one for \(\frac{7}{8}\) makes it immediately obvious that \(\frac{7}{8}\) is larger. And then students don’t actually learn the mechanics of solving rational inequalities.

Rather, I prefer to do this.

- Have students draw three rectangles of equal dimensions (and draw them large enough).
- Have students divide rectangle #1 and #2 into 10 equal parts and the third rectangle into 8 equal parts (bring out the ruler!)
- For rectange #1 have them color in, from left to right, 6 boxes. This represents \(\frac{6}{10}\)
- Now, the task! Subdivide rectangle #2 and rectangle #3 so that when we shade in six-tenths of both rectangles, we will have shaded in the
*same number of pieces*in both - What should we find? Well, one solution is that each of the mini-rectangles in rectangle #2 had to be subdivided further into 4 pieces and rectangle #3 had subdivisions of 5. And thus, both rectangles now are subdivided into 40 pieces and it should be plain to see that we color in 24 pieces in both

This should drive home the point of what common denominators do for us. They allow us to compare two different ratios on the same *scale*.

Beyond this, there are other things we can do, if students are unable to grasp what’s going on. Feel free to reach out and I’m happy to help. If your school has the budget, I’m glad to do a workshop with your team!

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