reducing fractions

[Greg Scott]

Factors? We don't need no steenking Factors!
Pardon me while I indulge my propensity for a rant:
1. Explain the concept. Then have them write a computer program to do it. If
they can do it, they'll learn something. If not, they might also.
2. Don't bother. The average man on the street barely understands prime
numbers and factoring, or doesn't really understand it at all. So just
concentrate on survival skills. Teach them to use a calculator and use
decimals. Forget fractions.
3. Get concrete. Get tactile. Get musical. Etc. Teach kids to read musical
notation, not for the notes, but for the time. Teach them to measure stuff,
to read an analogue clock, to use a slide rule, to use a ruler and tape
measure. Have them cook and map and sew. Play games.
4. Hit the concept. Do the math. If they don't get it, don't sweat it. Save
it for later. Let them enjoy success later, rather than wallow in failure
now. Piaget said if they're not ready, they're not ready. This is one of the
concepts some people never really get. Especially prime factoring. Just hit
it again. Get out the cuisnaire rods, and run them thru the Cuisnart.  Just
don't beat them to death with it. I've been doing database design for years,
and I still don't grok all of the laws of database normalization. Sometimes
I do, sometimes I don't.
5. Don't torture the kids. Eschew the following:
Little Jack Horner sat in the corner
Extracting cube roots to infinity
An assignment for boys that will minimize noise
And produce a more peaceful vicinity
            -- Space Child's Mother Goose
6. All of the above. Above all, have fun.
7. Everybody's different. If you just know you can beat it into them, then
maybe you're right. Just don't leave any bruises! Sometimes "drill, drill,
drill" results in something other than a hole in your head.
8. I home schooled my kids. They did O.K. I found that you don't need
success on your schedule if your kids can find it on their own schedule.
9. Teach the kids to do spreadsheets.
Write a spreadsheet to reduce fractions.
Write a spreadsheet to employ a brute-force algorithm.
Give credit for each new and better algorithm developed during the year.
Give credit for the most "stupid" algorithm that works.
Who can write the slowest program to do the job without delay loops?



"Kirby Urner" <urner at alumni.princeton.edu> wrote in message
news:lalgpsou36vvrrbhnj2hr0qapk955p3fdc at 4ax.com...
> "Janet Johnson" <johnsonje at adelphia.net> wrote:
>
> >I teach 6th grade and every year my students seem to have a lot of
> >difficulty with fractions, specifically, reducing or recognizing that a
> >fraction isn't reduced.  I have given them many ideas on how to tell,
even
> >to the point of writing out the factors for both the numerator and
> >denominator.  Does anyone have any suggestions on how to get this concept
> >across to the students?  Any suggestions would be greatly appreciated.
> >Janet Johnson
>
> You could unsimplify some fractions e.g. 2/3 -> 10/15
> i.e. show the inverse of what it means to "simplify".
>
> If 'prime number' is already a concept, you could try
> 'relative prime' meaning no factors in common (i.e.
> "a fraction is in lowest terms when the numerator and
> denominator are both integers and are relative primes").
>
> Along these lines, I really like your idea of writing
> out the prime factors, crossing out those in common e.g.:
>
>    150/210 -> (3 x 5 x 2 x 5) / (7 x 3 x 2 x 5) -> 5/7
>
> That ties back to "intersection of sets" i.e. the
> intersection of {3,5,2,5} and {7,3,2,5} is {3,5,2}
> (multiple appearances of the same factor constitute
> unique members of each set).
>
> As the other poster mentioned, you're looking for the
> greatest common divisor or "biggest gzinta (goes into)",
> in the above example 3 x 2 x 5 = 30.
>
> Pictorially, you could do a pie divided into 3 wedges
> (thirds) and then slice each third in half (sixths).
> Getting 1/3 of the pie is the same as getting 2 x 1/6s
> or 2/6s.  Lots of pictorials along these lines -- including
> in 3D (i.e. volumes in space).[1]
>
> Historically speaking, finding the greatest common
> divisor (gcd) is one of the oldest algorithms on record,
> and is credited to Euclid.
>
[quoted algorithms and code snipped by moderator]
>
> Long division problem (good to show because calculators and
> even floating point processors choke on this many digits,
> but our paper and pencil algorithms do not):
>
>       12089163724642057
>       ------------------
>     6|72534982347852342
>
> Doing a long divisions such as the above on paper needn't
> be too tedious if the whole class works together a few times.
> Good review of the times table for 6.
>
> I teach putting the remainder as a tiny digit to the upper
> left of the next, so we don't have this huge long series of
> subtractions trailing down like the tendrils of a jelly fish.
>
> E.g. 6|7... leaves 1, so the 1 goes to the upper left of
> the 2, making 12, and so on (I'm sure all the K-12 math
> teachers know what I mean).
>
> I think it's important to clue kids early about these limitations
> of floating point and even long integer arithmetic on computers,
> plus most calculators won't even touch 17-digit numbers (or
> greater).  As I wrote re the Math Summit in Oregon (1997):
>
>   "(as the official note taker, I managed to interject twice
>    -- once about the wierdnesses in floating point math as
>    implemented in computers, reason enough to learn the
>    algorithms on paper as well)"[4]
>
> I.e. this is how to respond to students who ask why we're still
> learning long division on paper when we have calculators and
> computers:  because calculators wimp out, and with computers
> you may need to check its math, especially if any floating
> point operations were involved.
>
> For example in Python, using floating point:
>
>  >>> 72534982347852342.0/6.0
>  12089163724642056.0
>
> Which is off by a digit (because of the limited precision of
> floating point numbers).
>
> DrScheme gives the same answer, but warns the result is
> imprecise using the # symbol.  From the manual:
>
>   * Print inexact numbers with #i -- Prints inexact numbers
>   with a leading #i to emphasize that they represent
>   imprecise results (or even effectively incorrect results,
>   depending on the intended calculation).
>
>  > 72534982347852342/6.0
>  #i12089163724642056.0
>
> If the above were really the right answer, then 72534982347852342
> should be divisible by 36, since 12089163724642056 is again
> divisible by 6.  Python might lead us to believe this is true,
> because the / operator is designed to give only integer answers
> no matter what -- unless one of the arguments is floating point
> (i.e. we'll get the wrong answer whether we use / as an integer
> or floating point operator in this case).
>
> So:
>
>  >>> 72534982347852342L/36L
>  2014860620773676L
>
> -- but we can't really trust this result.  The way to check is
> to ask for a remainder is to use the % operator:
>
>   >>> 72534982347852342L%36L
>   6L
>
> .... which shows 36 isn't really a 'gzinto'.
>
> DrScheme doesn't implement the / operator in the same way, and
> 72534982347852342/36 gives a correct answer, a fraction:
>
>  > 72534982347852342/36
>  12089163724642057/6
>
>
> Kirby
[quoted footers trimmed by moderator]

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