Consider the following "easy" way to add fractions:
a/b + c/d = (a+c) / (b+d).
An elementary school teacher's nightmare! With this sort of reasoning, 1/2 + 1/2 = 2/4 = 1/2. Bad news.
But we can find a situation where this actually works, as my professor pointed out the other day: mixed drinks. If the fractions represent the concentrations of alcohol in a mixed drink -- the numerator being the amount of vodka, and the denominator being the total amount of screwdriver (vodka + orange juice), the math works out perfectly!
For instance, if you have one drink that has 1 oz of vodka in a total 8 oz beverage, and another with 2 oz vodka in a 4 oz beverage, you pour them together and you get 3 oz in a 12 oz beverage:
1/8 + 2/4 = (1+2) / (8+4) = 3/12.
Do not show this trick to your ten-year-old child!
The resulting fraction is always between the two that you added together, which makes sense in the context of drinks -- if you mix together two drinks, the resulting drink can't be weaker than both of the original ones, nor stronger than both of the original ones; it has to be in between.
I am wondering whether I should show this to my students this summer. Last summer, we did many problems of the form "two gallons of a 20% concentration are mixed with five gallons of a 10% concentration. What is the resulting concentration of the mixture?" Even after the initial confusion of "what the heck is a gallon?" (most of the students were from abroad) these kinds of problems were beyond many of the students. They liked to add the percentages, average the percentages... anything but "total amount of concentrate over total volume of the mixture!" as I would always repeat, tailoring it to the situation. ("Total amount of cheerios over total amount of cereal! Total number of cloudy days over total number of days!)
Perhaps if I twisted it so that they would be "allowed" to do this "crazy" fraction addition after they set it up properly, they would think that was fun and learn to set it up properly. (Although some of them came in not knowing how to add fractions, so that could be dangerous! "What did your crazy American teacher tell you about adding fractions?")
18 hours ago