[Numeracy 115] Re: introductions
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Sun Feb 7 22:49:49 EST 2010
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Hi, George,
Prime factorization is great for /reducing/ fractions ("...how do I know
if I'm in lowest terms?"), but looking for the least common denominator
is a good way to /raise/ fractions to higher terms in order to add,
subtract, or compare them. Another (only slightly different) way is to
go through the times tables of the largest denominator, asking along the
way if the other denominator(s) can be factors of the products of the
largest denominator (times 2, times 3, times 4, etc.) For example, if
you want to add 1/20, 1/2, and 1/8: start with the 20 (times 2) equals
40. Since both 2 and 8 are factors of 40 (of course, 2 was also a
factor of 20...but the 8 wasn't), then 40 will be your least common
denominator.
I don't know if I really understood it at the time, but when our
teachers said (back there in elementary school) that you can't add
apples and oranges, /this/ is what they were talking about...that the
denominators had to be the same to add, subtract, or compare fractions.
Claire
On 2/7/2010 1:53 PM, George Demetrion wrote:
> Thanks Barbara,
>
> From what I'm gathering is based on what you are laying out here is
> that the least common multiple is the least common denominator, but is
> a different way than prime factorization to get there.
>
> In terms of instructional usefulness (and I'll know more once we're
> actually on the topic), I'm assuming prime factoring for adding
> or subtracting fractions with different numbers is important in
> finding the Lowest Common Denominator. Where that becomes difficult,
> following the methodology you describe below provides an alternative.
> Fortunately, my AM students get prime factorization so should be able
> to solve for the LCD.
>
> In terms of my own mastery it took me a fair amount of time to get
> prime factorization and calculate accurately; to get LCD and and
> Greatest Common Factor (a method for reducing fractions) and then to
> truly get what their purposes were.
>
> Thus far, when explaining and practicing with prime factorization the
> students readily get it. Whether that is due to (1) the care I've
> taken in seeking both to grasp the purposes of these procedures as
> well as to master them through a great deal of practice; (2) to the
> innate learning strengths of the students themselves, or (3) some
> combination thereof would make an interesting teacher research
> project. What I do sense is that straight forward and accurate
> explanation of procedure and purpose of any given function or
> operation is useful combined with a lot of practice, and working with
> students through their strengths and at any current level of mastery.
> Longer term, building increasing math awareness as a metacognitive
> capacity (in my case as a newbie within myself as well as within the
> students) is also an important pedagogical goal that I'll be working
> on through the course.
>
> George Demetrion
>
>
> ------------------------------------------------------------------------
> From: bamurr at metrocast.net
> To: gdemetrion at msn.com; numeracy at nifl.gov
> Subject: Re: [Numeracy 109] introductions
> Date: Sat, 6 Feb 2010 18:22:16 -0500
>
> The least common multiple is used to find the lowest common denominator.
> Example: Add 1/2 + 1/3 + 1/4 + 1/6
> The multiples:
> of 2 are 2 4 6 8 10 *12* 14 16 18 20 22 24. . .
> of 3 are 3 6 9 *12* 15 18 21 24 . . .
> of 4 are 4 8 *12* 16 20 24. . .
> of 6 are 6 *12* 18 24 . . .
> The least common multiple is 12 and this is also the lowest common
> denominator. Notice that 24 is also a common multiple, but it is not
> the least common multiplie. It is also not the lowest common
> denominator, although it would be A common denominator.
> 1/2 = 6/12
> 1/3 = 4/12
> 1/4 = 3/12
> 1/6 = 2/12
> The sum is 15/12 = 1 3/12 = 1 1/4
> ----- Original Message -----
>
> *From:* George Demetrion <mailto:gdemetrion at msn.com>
> *To:* numeracy at nifl.gov <mailto:numeracy at nifl.gov>
> *Sent:* Saturday, February 06, 2010 2:56 PM
> *Subject:* [Numeracy 109] introductions
>
>
> Good afternoon all.
>
> While I am an experienced adult educator I am a newbie math
> teacher, but I'm plugging away in my first transitions to college
> basic math course.
>
> We've had two three hour sessions thus far in a 15 week course and
> things are moving along okay.
>
> To be sure I've put a lot of time practicing my math through basic
> algebra and concentrating on the assignments in our weekly sessions.
>
> I'm learning and I'm also getting a good experiential dose of math
> phobia, which in turn, in the process of transforming in the
> process of learning and then drawing on my overall teaching
> skills, especially incorporating basic explanation, a lot of
> practice and collaborative scaffolding instructional processes.
>
> One technical question:
>
> What is the difference between the Lowest (or least) Common
> Denominator and the Least Common Multiple and what different
> functions do they accomplish?
>
> Keep it simple and straightforward, please.
>
>
> George Demetrion
> East Hartford, CT
> ------------------------------------------------------------------------
>
>
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