To cross multiply or not cross multiply

When I was in Grade 7 our homeroom/math teacher told the class in a hushed voice, “Don’t cross multiply.” At the time I didn’t understand what he meant, nor did I likely understand what cross multiplication was. Ever since I have occasionally wondered why he would admonish students from using such a useful tool. Cross-multiplication can tell you whether two proportions are equivalent and get you out of some tricky algebraic situations involving fractions.

Showing that two proportions are equivalent

img01_equiprop

Solving an algebraic equation with a variable in the numerator

img02_alg1var

Solving an algebraic equation with a variable in the numerator and in the denominator

img03_alg2var
So, cross-multiplication is a powerful yet easy-to-use shortcut. What’s the harm in it? None really, but, if I was to guess at the thinking behind my teacher’s warning, it would be that he was worried that algebra neophytes like myself would use cross-multiplication without knowing how it really worked, i.e. the underlying steps that make cross-multiplication a valid method.

Both Wikipedia and Math is Fun have good explanations how cross-multiplication works, and I recommend readers to check them out. This is the way I think about it:

Take the second above example img04_inlineex . You want to get rid of the ‘5’ denominator on the lefthand side. What do you do? Well, if you multiply both sides by 5, you would get the following (Remember: whatever operation you do one side of an equation you must do on the other for the sides to remain equal to each other).

img05_exhalf01

Both sides in the last line of the equation are presented in reduced form. Always remember to reduce your fractions!*

We can then repeat the procedure on the righthand side of the equation to get rid of the ‘4’ denominator. From there on, the last steps of the solution to the problem should be straightforward.

img06_exhalf02

As can be seen, cross-multiplication is a lot faster than the several operations I used above. Its relative speed makes it a favourite of students—anything to finish homework faster :-).

That said, I don’t think it’s a waste of time to see how a useful thing like cross-multiplication is not just some random, isolated skill students have to accept blindly. By the time students learn about cross-multiplication, they should have enough math behind them to understand how it works. Sometimes it’s possible to show students what’s under the roof of a formula or a theorem, such as in this case or in learning that the formula for the distance between two Cartesian points is essentially an application of Pythagorean Theorem. Sometimes it’s not possible: understanding why the formula for the volume of a sphere is img07_fn2volsphererequires either a close familiarity with Archimedes’ “On the Sphere and Cylinder” or some facility with integral calculus, both of which most high school students don’t have (nor do I).

I think it’s worthwhile for math teachers and tutors to show how a formula or math technique works when appropriate. When this is done, students use the logic faculties of their minds rather than rote memory. Logic is the glue that connects everything in mathematics together and makes possible, along with knowledge and imagination, the ability to solve complex problems.

*The question of whether to reduce fractions comes up quite a lot from students, and my usual answer is that you should always reduce. The only exception I can think to this is when calculations are made easier by keeping fractions in non-reduced form (e.g. adding up probabilities of events). Typically, however, it’s fraction reduction that makes calculations less unwieldy.

†I have to admit that I have sometimes not shown students how cross-multiplication works. I justify this by persuading myself that they are already overburdened with so much to learn in math class. Understanding is gravy, which maybe can be had later.

‡ It is tempting to poo-poo rote memorization when compared to “understanding”, but burning things into memory is actually extremely useful and effective in many areas of study. It’s sad to see high school students use their calculators to figure out what “4 x 6” is because they were never taught to learn their times tables by heart in elementary school. To not have to look everything up on Google, such as the capital of Iceland (Reykjavik) is admirable and classy, the latter in the sense that our educated forebears valued knowing and retaining things. Also, after having memorized a fact or technique, I suspect that it’s easier to understand it than vice-versa. At least for myself, I understand through example and often need to see many applications of a particular formula or theorem to even begin to hope of understanding how it works or why it’s true.

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