Cross Multiplication: A Slippery Slope

As students progress towards their last few years of elementary school, they are often tasked with learning to compare fractions. A very common approach to teaching students to compare the value of two fractions is to use cross multiplication, which is GREAT if all you care about is getting the "right answer." IS all we care about getting the right answer? Absolutely not. In fact, cross multiplication only works as expected in a very narrow range of problems and is not a good foundation for understanding proportional relationships at their core. The most insidious issue about the use and overuse of this technique is that it neither requires nor supports a conceptual understanding of patterns and relationships in numbers. 

Cross multiplication was originally designed to determine the proportionality of two ratios or to solve for a missing value in a proportion problem. Historically, there has been little to no instruction for students explaining HOW or WHY cross multiplication works. So, ask yourself: do YOU know how or why it works? If not, how would you go about explaining the approach to a student without it coming off as a "math magic trick?" The perception on behalf of students that math is full of magic is dangerous, to say the least. At some point, the student comes to view the teacher or parent as the magician and gives up trying to understand how the rabbit magically appeared in the hat. In math terms, that's the moment a student hits the proverbial wall and decides that he or she "isn't good at math."

So, HOW does cross multiplication work, and WHY? I'll try not to get too "mathy" here. Cross multiplication is a basic shortcut for finding the lowest common denominator, then comparing numerators. There are two common approaches to cross multiplication:

A) Multiply both sides of the equation by a fractional equivalent of 1 to yield a common denominator OR
B) Multiply both sides of the equation by the product of the denominators.

Remember that if you have two equal quantities and multiply them 
by the same amount, the products will again be equal. So if we 
multiply the fractions a/b and c/d by b, the results are equal:

     a         c
    --- * b = --- * b
     b         d

which can be written as

         bc
    a = ----
          d

Now we can multiply both fractions by d:

          bc
    ad = ---- * d
           d

which, of course, means

    ad = bc

There are a number of pitfalls of using cross multiplication. One is that it detracts from paying 

attention to the relationship between the two values. Are we comparing unit price to unit 

amount? 

Cross multiplication gained ground and its popularity swelled in a time when rote methods wereapplied without thought or context. The fundamental drawbacks of cross multiplication may not be as obvious in elementary school as they will surely become in middle and high school, when functions, graphs, linear equations, and other key algebraic ideas depend upon a more             dynamic understanding of the relationships between and among numbers. It is our responsibility as educators to ensure that we are setting up our students for success and that we are keeping in mind the need for vertical articulation from start to finish. After all, it takes a village.