I heard a similar story about when Paul Sally visited a grade school classroom. He asked the students what they were learning, and they said “Adding fractions. It’s really hard, you have to find the greatest common denominator....” Sally said “Forget about that, just multiply the numerator of each fraction by the denominator of the other and add them, and that’s your numerator.” The students loved this, and called it the Sally method.
That does not always produce a reduced fraction, of course. In order to do that, you need to go find a GCF just like before… but I agree, that should be presented as an *optimization* after teaching the basic idea.
I heard a similar story about when Paul Sally visited a grade school classroom. He asked the students what they were learning, and they said “Adding fractions. It’s really hard, you have to find the greatest common denominator....” Sally said “Forget about that, just multiply the numerator of each fraction by the denominator of the other and add them, and that’s your numerator.” The students loved this, and called it the Sally method.
That does not always produce a reduced fraction, of course. In order to do that, you need to go find a GCF just like before… but I agree, that should be presented as an *optimization* after teaching the basic idea.