2.
>"Why does 1/2 + 1/3 = 5/6?
>Explain using a step by step process giving a reason for each step."These
>questions are difficult to answer using a calculator.
I would like to put my own spin on these two postings. This from someone who
believes that community college students should use calculators with
fraction keys from day one - if they want to.
In the first case, I would present the algorithm to students. In the second,
I would not (beyond demonstration in class).
In the first case, casting out nines is a terrific tool to get some students
interested in pure mathematics. It's easier to use than the written
explaination makes it seem (and that isn't too bad anyway). It can provide a
great check on addition, which some on this list believe should be done by
hand, at least by students. Frankly I've only seen it used for
multiplication, but it seems that it should work for addition as well.
Just like divisibility by 9, the math is easy. For example, step 1 for 83275
would be 8+3 == 2 mod 9; 2 + 2 + 7 == 2 mod 9; 2 + 5 == 7 mod 9, so 83275 ==
7 mod 9. But why one can fold the calculations like this needs some algebra.
To understand why casting out 9s works, one uses the fact that 10^n == 1 mod
9 for any whole number n. To show this one may need to show that (x - 1) is
a divisor of x^n - 1 for any whole number n. Then one uses the fact that if
a == c mod 9, b == d mod 9, then a + b == c + d mod 9. It seems to me that
this would be a wonderful investigation for a college algebra student.
In the case of 2: Why does a step by step process explain why 1/2 + 1/3 =
5/6? It would show that the addition algorithm produces 5/6, but to me that
does not explain why. To do that one draws the "usual" box cut up into 6
parts, marks off 1/2 of them (3 of them), 1/3 of them (2 of them) and notes
that 5 out of the 6 boxes are marked off. That tells me why. And I would
argue that's worth knowing. And it shows why a common denominator is
necessary to add fractions. But to me the usual algorithm does not shed any
light on why the sum is what it is - it just shows me how to get it. And my
calculator will do that too.
I might not have replied to these postings individually, but when I found
myself on the opposite sides of both, and given that they both deal with the
ongoing how-much-to-use-a-calculator controversy, I couldn't resist.
And I haven't thought much about casting out nines, so if I'm wrong about
the theory I hope someone will correct me. But my point is that the
curiosity of some students could lead them to study more math, through
casting out nines.
Phil Mahler
Middlesex CC
Bedford, MA
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