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.
As someone who teaches computers to do low-level math efficiently, I am rather surprised by this article. All of the “good ways” listed in this article are demonstrably more complicated and slower algorithms than the cross-multiplication. (That is, 2 multiplications and then an integer comparison.)
For example, finding the least common multiple for the denominator of either of the “good” ways requires either:
1.) factorization, then finding the maximum of the power of each factor, then multiplying all of these together. The difficulty of factorization is an unresolved problem in algorithmic complexity theory, but even if you just do it by trial division, lots of trial divisions is much more complicated than 2 multiplications.
2.) calculating the greatest common divisor (by what, Euler’s method by hand? Do people do this?) and then a multiply and divide. both of which have significantly higher computational complexity than two multiplications. In other words, LCM[a,b] = a * b / GCD[a,b], then several more multiply and divide operations.
Both of these are demonstrably more complicated than 2 multiplies.
I would suggest implementing these in a low-level programming language (or as a step-by-step list of procedures that the student will follow) and you’ll see the relative complexities of each operation. If you can write an algorithm that is simpler or faster than cross-multiplication, it would be of great interest to the computing community.
While I understand the rationale for students being able to turn two fractions into equivalent fractions that share a common denominator, (such as for addition or subtraction,) we have to be honest that following one of the “good” methods to simply compare two fractions is a significantly more complex algorithm by any analysis, and needs to be very strongly justified. You almost certainly wouldn’t do this more complex algorithm in software. (I certainly don’t; I cross-multiply.) Why would we make students perform an operation that’s significantly, provably more complex, and requires many more multiplications, divisions, remainder operations, etc? That’s why we need a better justification for doing so than, I believe, this article demonstrates.
The article’s focus was on math pedagogy. Verbatim, here are the points from the article why the bad way is bad.
The argument in favor for “cross multiply” about excess computations for humans when compared against how a computer would do it, is bad math pedagogy. A computer doesn’t have to know why it is doing anything. A human, at least until the AI revolution fulfills its most fanciful vision, still has to think about why.
As a mild reconciliation with your point, the first bullet point above does concede that the cross multiply method is a nice shortcut. However, the preference is for knowing why. And even if you were to make a computation argument, the programmer would have to ensure that the fractions are nice-ified to having a positive denominator. There are enough cases where by the result of some other computation can result in a negative denominator. And if you were to blindly apply the algorithm in a computer program, your answers would be incorrect because the math wasn’t handled correctly. And that is the why.
As a final aside, the article never mentions use of LCM or GCD. The article states “find common denominators” … not necessarily “least”. In fact, Good Way #1 explicitly says that it is the crossmultiply method, but more transparent.
That’s what I was thinking here. It’s the core concept that makes all of this work, and it doesn’t really get better than that. Don’t let it get lost.
In practice I find other methods can be more convenient. Humans are really good at identifying known cases and estimation. You can confirm 1/6 < 5/18 with ease without consciously going through some method. If you do need some help, it's easy to spot that 18 = 3*6. Spotting that at a glance is faster than going through cross-multiplication, at least for me. But when all of that fails, I'd definitely go with cross-multiplication.
Sometimes, it’s easier to do common numerator.
It comes down to understanding what fractions are…
A) denominator = number of pieces in a whole. Therefore, bigger denominator means smaller pieces
B) numerator = number of pieces we have.
Given the comparison 3/8 to 2/5 then, we have 3 smaller pieces compared to 2 bigger pieces…
Lest make common numerators:
3/8 = 6/16. … 2/5 = 6/15
6 smaller pieces compared to 6 bigger pieces. 2/5 wins.
Sometimes numerator are easier, sometimes denominators are.
Sometimes it’s benchmarks, sometimes it’s decimals.
Let the context define the method.
As always, just Something To Think About!
I like to encourage numerator and denominator comparison, then comparison to a landmark. Unfortunately, my preservice teachers love the methods they’ve been taught. A few but not many cross multipliers, a few common denominator, and many divide and compare by decimal. Hard to encourage fraction thinking and understanding.
Yeah, the cross multiplying I don’t particularly care for. Common denominator I find to be pretty good and sound (Good Way #1 — unless you meant something else?). I don’t care for converting to decimal and comparing. Only works up to the numerical precision one is willing to compare to, though at least the method doesn’t introduce new rules.
For numerator, denominator comparison, what do you mean? Something like \(\frac{5}{3} < \frac{5}{2}\) because the numerator is the same but the small denominator wins out? or something else?