FairProspects Posted February 25, 2015 Share Posted February 25, 2015 Bear with me, math people. My math education was very procedural. When multiplying fractions, say: 2/3 x 3/4 Why can you reduce the factors from both the numerator and the denominator and not have it make a difference? Such as reducing to 1/1 x 1/2? The common factor of 2 is in the numerator of the first fraction and the denominator of the second. Why does this work? Don't the numerator and denominator mean different things? Is it because multiplication is commutative? It is not commutative across numerators and denominators though, right? Quote Link to comment Share on other sites More sharing options...
regentrude Posted February 25, 2015 Share Posted February 25, 2015 Bear with me, math people. My math education was very procedural. When multiplying fractions, say: 2/3 x 3/4 Why can you reduce the factors from both the numerator and the denominator and not have it make a difference? Such as reducing to 1/1 x 1/2? The common factor of 2 is in the numerator of the first fraction and the denominator of the second. Why does this work? Don't the numerator and denominator mean different things? Is it because multiplication is commutative? It is not commutative across numerators and denominators though, right? Yes, it is because multiplication is commutative. 2/3 x 3/4 = 2* 3/(3*4) you see the 3 is a common factor that occurs both in numerator and denominator so, you can write this is 3*2/ (3*4) or (3/3) * (2/4) and 3/3 is, of course 1. Which is what "canceling" really is: factoring out a common factor in numerator and denominator, which gives a factor of 1. Same with simplifying the 2/4. Quote Link to comment Share on other sites More sharing options...
FairProspects Posted February 25, 2015 Author Share Posted February 25, 2015 Yes, it is because multiplication is commutative. 2/3 x 3/4 = 2* 3/(3*4) you see the 3 is a common factor that occurs both in numerator and denominator so, you can write this is 3*2/ (3*4) or (3/3) * (2/4) and 3/3 is, of course 1. Which is what "canceling" really is: factoring out a common factor in numerator and denominator, which gives a factor of 1. Same with simplifying the 2/4. Why does your explanation make so much more sense than the HIG? The HIG just says it doesn't matter and doesn't really explain why. Quote Link to comment Share on other sites More sharing options...
regentrude Posted February 25, 2015 Share Posted February 25, 2015 Why does your explanation make so much more sense than the HIG? The HIG just says it doesn't matter and doesn't really explain why. They would have explained the concept way before multiplication of fractions, when simplifying fractions was first introduced Quote Link to comment Share on other sites More sharing options...
FairProspects Posted February 25, 2015 Author Share Posted February 25, 2015 Right, but some of us just never connect those concepts together that way without explicit instruction and/or reminders & review of relevant principles. I don't think I've ever applied that property to fractions with understanding. I think it is also somewhat spatial. I wouldn't think to flip fractions around in a way that makes the common factors easy to see without being told to do so (despite understanding that the communtative property makes it possible to do so) and the HIG assumes I would know to do that automatically. But it is also possible that I didn't read the HIG for teaching simplification of fractions, lol. Quote Link to comment Share on other sites More sharing options...
kiana Posted February 25, 2015 Share Posted February 25, 2015 When I teach multiplying fractions to our developmental students, I usually have them write out all the factors from the numerator and all the factors from the denominator before they cancel. For your example, I'd have them write it as (2)(3) / (3)(4), and then it is more obvious to see. With practice it becomes easier to see and shortcuts can be taken. Quote Link to comment Share on other sites More sharing options...
wapiti Posted February 25, 2015 Share Posted February 25, 2015 Yes - visually it's easier to see if it's "one giant fraction" rather than separate fractions. Quote Link to comment Share on other sites More sharing options...
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