What happens when you add both the numerator and denominator
In an article (Rowland et al) is mentioned a kid who conjectures that the fraction you get by adding the numerators and denominators will be between the two original fractions. Example: you get a fraction between 1/2 and 3/4 by calculating (1+3)/(2+4) = 4/6 = 2/3.
This works, of course. One explanation I like is the following: Let's say that 2 persons together own 1 dollar and 4 other persons together own 3 dollars. If they decide to pool their resources, 6 persons will own 4 dollars. It's common sense that not all of them are now richer and not all of them are now poorer.
Another explanation concerns vectors. (Generally it's a terribly bad idea to illustrate fractions with vectors, but in this case it actually works.) If you add the vectors [2,1] and [4,3], you get the vector [2+4,1+3] = [6,4]. Obviously, the slope of the vector [6,4] (which is 2/3) is somewhere between the slopes of the two addend vectors (which are 1/2 and 3/4).
For some reason, some of my colleagues and I started to look at problems associated with this, such as: When is (a+c)/(b+d) exactly halfway between a/b and c/d? (Answer: only when b=d).
In a way, this whole thing illustrates why teachers need skills that other professions don't. While many professions need mathematics to get the right answer, mathematics teachers need to be able to see whether the solution a student proposes is actually correct. That may be with methods within the students' reach or with other methods.
In addition, the teacher of course needs to decide whether he/she should follow up on this - that is, whether the student will learn anything from going deeper into the problem. That is another (but related) kind of skill.
All of this relates to an article by Deborah Ball that I will blog about later...
This works, of course. One explanation I like is the following: Let's say that 2 persons together own 1 dollar and 4 other persons together own 3 dollars. If they decide to pool their resources, 6 persons will own 4 dollars. It's common sense that not all of them are now richer and not all of them are now poorer.
Another explanation concerns vectors. (Generally it's a terribly bad idea to illustrate fractions with vectors, but in this case it actually works.) If you add the vectors [2,1] and [4,3], you get the vector [2+4,1+3] = [6,4]. Obviously, the slope of the vector [6,4] (which is 2/3) is somewhere between the slopes of the two addend vectors (which are 1/2 and 3/4).
For some reason, some of my colleagues and I started to look at problems associated with this, such as: When is (a+c)/(b+d) exactly halfway between a/b and c/d? (Answer: only when b=d).
In a way, this whole thing illustrates why teachers need skills that other professions don't. While many professions need mathematics to get the right answer, mathematics teachers need to be able to see whether the solution a student proposes is actually correct. That may be with methods within the students' reach or with other methods.
In addition, the teacher of course needs to decide whether he/she should follow up on this - that is, whether the student will learn anything from going deeper into the problem. That is another (but related) kind of skill.
All of this relates to an article by Deborah Ball that I will blog about later...
Labels: fractions, mathematics, vectors
posted by Bjørn at 12:46 PM
6 Comments:
Interesting.
That's nice, I had never thought about this idea of adding simultaneously on the top and below. There are always new things one learns about fractions. Here is another interesting exercise (no.2)about fractions.
Really Good..
Strangely interesting stuff :) I'd love to know if this has been used anywhere (perhaps in some mathematical proofs?)...
Interesting method especially example which have given in the blog in interesting and easy to understand.This blog makes easy for teachers to teach the usage of denominators and nominators.
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