When Wrong Becomes Right!
Too often, students compute by blindly moving symbols around, not paying any attention to their meaning. And, they sometimes happen to get correct answers, which falsely reinforces their poor habits.
As an example, consider the addition of fractions. Too often, students transfer their process for multiplication to the addition situation, amd ignore the need for a common denominator.
a/b + c/d = (a+c)/(b+d)
And examples abound where this "misalgorithm" works. Using the above "rule," try these two problems:
49/35 + 9/15 = ?
9/6 + 1/2 = ?
You should find that both produce the correct answer by the wrong method. In fact, there exist an infinite number of examples that work.
You are perhaps crying FOWL! because of the speciallyconstructed examples, using negative numerators and nonreduced fractions (i.e. 3/2 + 1/2 = ? does not work!).
Your Task: Determine the special conditions on numbers a, b, c, and d such that the "algorithm" a/b + c/d = (a+c)/(b+d) works.
Source: L. Carmony's "Adding Fractions Incorrectly?" Mathematics Teacher, Dec. 1978, pp. 737738
Hint: Start with a/b + c/d = (a+c)/(b+d), and add the lefthand side correctly...then...
Solution Commentary: As Lowell Carmony proved so nicely, you should determine that c = (ad^{2})/(b^{2}).
But, you still need to check to see if there are any constraints on a, b, c, d....?
