Let’s start with this problem. Which fraction is larger $$\frac{3}{8} \mbox{ or } \frac{4}{8}$$

Most students [and by default, I am considering those who are relatively new to fraction comparison] will say that \(\frac{4}{8}\) is larger because \(3 < 4\). And hopefully, some will also add in that the denominators are the same.

Now, suppose we want to compare $$\frac{3}{8} \mbox{ vs } \frac{2}{5}$$

Now things are a bit murky and students will give the gamut of responses. One common response, though incorrect, will be that \(\frac{3}{8}\) is larger because \(3 > 2\) *and* \(8 > 5\).

And finally, consider the same problem above, but with negative signs.

$$\frac{-3}{8} \mbox{ vs } \frac{-2}{5}$$

First, negative numbers tend to confuse students and throw off their sense of “greater than” because from a pure distance perspective \(-3\) is a greater distance from \(0\) than \(-2\) is. But in terms of numerical ordering \(-3 < -2\).

Second, by the logic used in the previous problem, \(8 > 5\), but [if they can reason this correctly] \(-3 < -2\). And now there is a perceived inconsistency. *Or* if they got their negative relationships incorrect, then they incorrectly, stumble onto the correct answer. And no good comes of this.

### Bad, But Common Way

Now, a typical method that students are taught in elementary school [and even basic skills classes in college] is to do the “cross multiplication” method for comparing fractions. The method works like this. We want to know what the relationship \(R\) is in \(\frac{3}{8} R \frac{2}{5}\), where \(R \in \{>, <, =\}\). To find \(R\) we “cross multiply” and compare \((3\times 5) \ R \ (2\times 8)\). Since \(15 < 16\), \(\frac{3}{8} < \frac{2}{5}\).

This is a common method that is taught and I think it is a terrible method. Here are a few reasons, in no particular order, why it’s bad and we should do away with this

- The method is a nice shortcut,
*if*one knows why they are doing it. Most students don’t know why because even if they are taught the reasons, the technique is all that’s remembered. Ask your students if they know why. Ask them a few weeks after they learned the method. - The method tends to have long-term confusing aspects for students because somehow \(\frac{3}{8}\) became \(15\) and \(\frac{2}{5}\) became \(16\) and the fractions magically disappeared. It also (opaquely) breaks Algebra rules (that they will learn later [phrases like “what you do to one side, you should do to the other”]. The technicality is that both sides of \(R\) are being multiplied by \(40\), but this detail is lost or never discussed.
- The method does not integrate well into Algebra where solving inequalities for rational expressions is an important topic. For example, find all \(x\) that satisfy \(\frac{1}{x} < 1 + x\). Clearing fractions is a no-no.
- With respect to the previous bullet-point, care has to be taken when multiplying across inequalities. In the arithmetic based examples, the denominators have been “nice-ified” to be positive, which is what allows for the method in question to work. In the Algebra problem, the denominator is \(x\) which may take on negative values and hence “clearing the fraction” by multiplying by \(x\) leads to trouble. A bad habit formed in arithmetic, persists through Algebra and another drop of blood is spilled in the death process of math education.

Don’t teach it this way. Instead, here are two better alternatives.

### Good Way #1 — Scale and Compare

If we want to compare \(\frac{3}{8}\) against \(\frac{2}{5}\), have students scale the fractions so that both fractions have a common denominator. This is essentially the “cross multiply” method, but the process is transparent and it uses a skill they have been acquiring for fraction arithmetic (adding and subtracting fractions). Thus, \(\frac{3}{8} = \frac{15}{40}\) and \(\frac{2}{5} = \frac{16}{40}\). From here it’s straightforward, \(\frac{2}{5} > \frac{3}{8}\).

### Good Way #2 — Subtract and Compare

Similar to Good Way #1, consider the difference \(\frac{3}{8} – \frac{2}{5}\). This will work out to \(\frac{-1}{40}\). Since the difference is negative, the subtrahend \(\Big(\frac{3}{8}\Big)\) is smaller than the minuend \(\Big(\frac{2}{5}\Big)\).

The huge bonus in both of these methods is that we are not adding to the cognitive load of “math rules”. In fact, we’re not creating a fake rule a la the Bad Way. When comparing fractions, via both Good Way #1 and #2, we are building on previous knowledge of fractions — addition, subtraction, finding common denominators, etc. (In the case where fraction comparison comes before basic fraction arithmetic, I would encourage that instructors rethink the sequence flow.)

There’s a side bonus, though small and nuanced. Comparators in `sort`

functions can use this type of pattern. For example, if I have `arr = [3,5,2,9,10,6]`

and I want to sort these from least to greatest, then I can write a simple comparator that works like this [in JavaScript] `arr.sort(function(x,y){return x-y;});`

If the contents of `arr`

were fraction objects endowed with some arithmetic operations, then we can sort similarly.

The two Good Ways I gave are consistent vertically through the math curriculum and at least have the capacity for further extension into coding. The Bad, But Common Way, needs to be dropped. I’d encourage that those who teach fraction comparison the common way, rethink this.

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