Wednesday, January 9, 2008

Wednesday Math, Vol. 10: But that trick never works!

Fractions are tricky when you are first learning them, and because a lot of people don't use them in their everyday lives, some people will fall back into bad habits when having to deal with them. Multiplying fractions is easy, but adding them is hard. Some students, seeing the easy formula for multiplication of fractions, will mistakenly believe that they can use a similar looking method to add, but it always, and that is ALWAYS, will give the wrong answer. I call this method baddition, and I try to impress on my students that it can't be used.

A simple example is 1/2+1/2, which everyone knows adds up to 1. With the baddition method, you would get 1/2+1/2 = (1+1)/(2+2) = 2/4, which reduces to 1/2. That is obviously the wrong answer and that is that.

Or is it?

Let's say I play in a game of softball, and I get 1 hit in 2 at-bats. My batting average is .500, which is the decimal way to represent 1/2 if we always round to the nearest thousandth. Let's continue this softball example, and say I hit 3 of 4 in the next game, which is .750. What is my overall batting average for the two games?

Use baddition, which I will represent as (+): 1/2 (+) 3/4 = (1+3)/(2+4) = 4/6, which rounds to .667. That is my batting average for the two games combined.

So it turns out that baddition is the right method for a completely different problem. Baddition will always give us an answer that is somewhere between the fractions we badd together. What baddition does is not add fractions but combine proportions. Another problem answered by 1/2 (+) 3/4 = 4/6 is that when you mix 2 quarts of 50% solution with 4 quarts of 75% solution, you get 6 quarts of 66.66...% solution, more than 50% but less than 75%.

Some formulas in statistics use baddition to combine proportions, which is how I 'splain it when we get to this part of the class.

And of course, we all know that the answer to "But that trick never works!" is "This time for sure." And this time, answering a different question, it does work for sure.


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3 comments:

Distributorcap said...

you take me back to 7th grade math and the hot Mrs DelVecchio...

we have this fight at work ALL time --- i wont tell you what the sides are or where i stand --- but maybe you can do a post on it

taking percents of percents....

jolie said...

yeah - percents of percents would be a great post. like when something goes from 5% to 7.5% - is that a 50% increase or 2.5%?

pollsters, take note!

dguzman said...

Oy vey, fractions are one of my nemeses! The other is exponents, especially fractional exponents! Curse them! Curse them all!

okay, I'm better now.