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Re: Concrete to abstract in dividing fractions?
Posted:
Aug 23, 2014 4:45 PM



Another thing I've been wondering about is how formally do we care to introduce the concept of "fraction" in relation to "rational number"?
In some contexts we tend to equate these two, but the 1/pi has meaning, and is what I'd call "a fraction" in the sense that it has a numerator (1) and a denominator (pi), it's just that they're not both rational.
Even if we don't hand compute "continued fractions" in elementary grades, I do think we should more than hint that these exist. If you have a computer handy, then it's a great workout with parentheses and such to practice writing one. Let me try:
>>> 1 + 1/(1 + 1/(1 + 1/(1 + 1/(1 + 1/(1 + 1/1))))) 1.6153846153846154
I see "+ 1/(1" as a repeating pattern that needs an ending and beginning. I might write a program that returns an expression such as the above as a string type thing, a quoted expression, that I then evaluate for real:
>>> def cont(n): ... return "1 " + "+ 1/(1" * n + "+ 1/1" + ")" * n ... >>> cont(10) '1 + 1/(1+ 1/(1+ 1/(1+ 1/(1+ 1/(1+ 1/(1+ 1/(1+ 1/(1+ 1/(1+ 1/(1+ 1/1))))))))))'
>>> eval(cont(10)) 1.6180555555555556
I'll be having stack problems pretty soon maybe. This is *not* the most efficient way to write a continued fractions calculator, believe you me.
Some of you may have recognized that the number that's emerging or being converged to is possibly the familiar constant Phi, right up there with Pi in name recognition surveys, though more people can say 3.141 than 1.618 from recall memory (an empirical fact, not evidence that either is a morally superior number).
[Sorry: I don't have any citations handy and if you know of a survey where people are asked about "famous numbers" (e, pi, phi, 911) I'd be interested in a link. Thanks in advance. ]
Confining "fractions" to mean "rational numbers" is a worthwhile step in at least some lesson plans, as then we're getting back into the fact that numbers come in types, as in "types of number".
The schema we follow is usually: N < W < Z < Q < R < C where W is big during New Math days as "whole numbers" i.e. the set of counting numbers (N) plus zero. Those were but a subset of integers (Z) which are rational numbers (Q) embedded in reals (R) which are all themselves the real part of complex (C) i.e. the reals are a subset of the complex numbers.
I'd say Whole Numbers on the whole get less attention since the Reagan Era, and we just go straight from N to Z and don't make that much buzz about zero being added to N to give W. I could be wrong about that and it would vary by curriculum and school. The complex plane had its hay day with the Mandelbrot Tshirts. I suggest keeping all that and continuing the reap the benefits. However I digress from the topic at hand.
We should distinguish between "fractional notation" which uses the division operator, from "fractions as number p/q where p, q are members of Z such that p/q is a member of Q". The expression 1/pi has plenty of meaning even if pi is a member of R but not of Q.
Kirby
On Sat, Aug 23, 2014 at 10:45 AM, Robert Hansen <bob@rsccore.com> wrote:
> > On Aug 23, 2014, at 1:55 AM, Jay Wiegmann < > jay.wiegmann@palmbeachschools.org> wrote: > > > My questions is how to help students make the connection between the > concrete and the abstract when dividing fractions. > > > > I've seen and taught how to divide fractions concretely with fraction > bars. And I've seen and taught how to do it abstractly via "keep change > flip"/KCF. But where is the lesson that helps students understand how to go > FROM fraction bars TO keep change flip? > > > > Ex: > > Model 2 / 1/4 with fraction bars. > > Lay down 2 wholes. Lay down 8 1/4 beneath. Therefore, 1/4 goes into 2 8 > times. Little Johnny nods. > > > > Ok well it's a lot easier to just do it like this in the future. > > 2 / 1/4 = ? > > 2 x 4 = 8 > > Keep change flip, got it? Doesn't that make sense given what we just > modeled? Little Johnny is clearly puzzled. > > > > What instructional sequence do you recommend to help students make the > connection b/t concrete and abstract? > > > > First off, when you teach ?Keep, Change, Flip? it sounds like your > students don?t even know what a fraction is or what its structure is. It > has a numerator and a denominator, call them that. And the proper way to > say flip is ?take the reciprocal?. Childish mnemonics are part of the issue. > > When you divide a number by another number (not equal to 0), that is the > same as multiplying by the reciprocal of that number. > > Why? > > Because when you multiply the numerator and denominator of a fraction by > the same value then you are multiplying by 1, which doesn?t change the > value of the fraction because any number times 1 is the same number. Thus, > given the fraction 2 / 1/4, if we multiply the numerator and denominator by > 4/1 we are left with 8 / 1. In other words, (2 / 1/4) * (4/1 / 4/1) = 2 * > 4/1 = 8. > > Since you are at the point of teaching fraction division, the students > must understand fraction equivalence by now and how 1/2, 4/8 and 50/100 are > the same. > > You can also show why 2 / 4/1 is the same as 2 * 1/4. > > > 1. There are some basic things in mathematics that you just see. For > example a(b + c) = ab + ac is one of those things. Sure, we have a name for > it, it is called the distributive property, and you would have to recall > the name. But the phenomenon itself, that a(b + c) = ab + ac is intuitive > and axiomatic. > > 2. There are some basic things in mathematics that are not axiomatic nor > intuitive, but are encountered so often that they should be committed to > long term memory and would be committed to to long term memory in the > normal course of doing math. The rule for dividing by a fraction is one of > those things. Another example is a^b * a^c = a^b+c. > > 3. Then there are things that we teach, like the quadratic formula, while > important enough to teach, don?t fit the criteria for long term memory. > > > Using bars is part of the process (of teaching), but that is just a visual > confirmation that 2 / 1/4 is 8. It isn?t going to provide insight into the > rule of multiplying by the reciprocal. You will have to settle for doing > that with arithmetic, as shown above, and later revisit it with algebra > (which only shows it more formally). > > The only thing that will commit all of this to long term memory and make > it familiar and automatic to a student, as it is to us, is using it > continuously. And when I say familiar and automatic, I mean the rule > itself, not the proof or insight into why it works. > > Bob Hansen >



