## Tuesday, January 21, 2014

### 2: Improperly Canceling Fractions

Simple fraction reduction ...  \$\dfrac{26}{65} = \dfrac{2}{5}\$ ... Wait a minute. Reduce by canceling the 6s?

Is this a fluke or is there something more sinister at work here?
Does this trick work in any other fractions?

If the Latex isn't working, this problem is 26/65 = 2/5.

1. It seems to be a fluke to me and this is one of the only times this worked. If I observed a student of mine cancelling like that to get the new fraction I'd be inclined to think that there are misconceptions in their understanding of place value and division because in reality, the 6 in the numerator specifies 6 ones and the 6 in the denominator specifies 6 tens (or 60).

2. So ... what other fractions does this work for? The ones that cancel a factor of 11 all work, e.g., 11/99 = 1/9. There are four others.

1. But in 11/99 you are cancelling a 1 with a 9, whereas in the given example 26/65 you cancelled a 6 with a 6, so I don't see how they are comparable.

2. I was thinking in terms of somewhat random canceling that just happened to work. You're right, it not the same exactly.

3. I find this exercise to be invaluable in that, having found that four other exceptions, they will never do this again ... and it leads us to a discussion of why (x^2-9)/(x+3) doesn't reduce by canceling an x and then a 3.

But that might be giving up a hint as to a future post ...

4. 19/95
49/98
26/65
16/64

5. "Does this trick work in any other fractions? " - If so, can you find a pattern (i.e. can you find a way to generate these fractions besides guess-and-check)?