Cross multiplication: the cross products of a proportion are equal
Cross multiplication has been a standard component in math courses for many years. So why should we cross out something? We are on a path to teach more math, instead of less, correct?
Picture my imaginary niece in 7th grade, a normal middle school girl, so a student with many other things on her teenage radar than mathematics. She happend to have zoned out a few times, but a cryptic message from the front of the class room did take root in her memory cells. It contained a word spoken with slightly more emphasis: ‘cross multiplication’ and it appeared to be a magical trick leading to an easy answer for an otherwise complicated equation.
She is surprised, what an interesting way to do math: “just multiply something form both sides of the equasion and something from above and below the division line… What about if you could even leave the awkward looking original equation behind and make your own new one: find some random numbers from both sides of the original equation and some from both sides of a division line and multiply them. Great, it looks easy now and it is fast!” Unfortunately for my imaginary niece this does not give her the right answer…
Istead of the introduction of a math ‘concept’ using a potentially confusing verbal sentence, students can (and should) try to figure it out themselves.
They are already familiar with the mantra that always holds true: to keep things equal, apply the same operation at both sides of the equation.
If they cannot figure it out on their own (and you would be surprised how many can, if you challenge them), walking them through the process should help understanding:
- a/b = c/d multiply both sides by b
- ab/b = bc/d b/b = 1, so cross out the b’s on the left side
- a = bc/d multiply both sides by d
- ad = bcd/d d/d = 1, so cross out the d’s on the right side
- ad = bc
Of course this boils down to the original cross multiplication that is super clear, even intuitive to Math teachers. For struggling students the veil of “magic” is lifted and they can understand what they are doing. For most of them the advantage of sticking to something they already know outweighs the small drawback of writing out more steps.