[Numeracy 266] Re: Is an absolute value positive?
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Wed Mar 31 16:30:49 EDT 2010
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Aaron and All,
It's a kind of philosophical question. I can count marbles, for example.
If I have 3 marbles and take away 2, I'm left with 1. If I have 3
marbles, and I take away 4... Wait a minute; everyone knows I can't take
away 4 marbles from 3. When it comes to counting things you can touch
and see, you can't have a number less than zero. We only use negative
numbers to count or measure conceptualized quantities. For example, we
have negative numbers on the temperature scale because we have chosen by
convention to start the scale at 0 where water freezes. If we start at
absolute zero, we can only have positive numbers. (Temperature measures
the average motion of molecules in a substance. At absolute zero, all
molecular motion stops. You can't have less motion than none.)
Similarly, sea level is an arbitrary spot for measuring heights and
depths. If we started instead from the center of the earth, there would
be no negative heights or depths. Negative numbers are a mathematical
abstraction that allow us to find answers to certain kinds of problems,
primarily subtraction problems where the number being subtracted is
greater than the one being subtracted from.
If I could take 4 marbles from 3, then I would have an absence to
contend with, and the mind recoils from the presence of an absence. I
would have to add a marble to this absence in order to have nothing. In
this sense, Michael is right; negative numbers have no existence. I was
merely being glib when I made the crack about my bank account.
On 3/31/2010 12:06 PM, Kohring, Aaron M wrote:
>
> Michael,
>
> I’m not sure I understand your statements.
>
> If amounts to the left of zero on a number line are ‘negative’ and
> distances below sea level are considered ‘negative’ and temperatures
> below zero are considered ‘negative’. Then doesn’t that distance to
> the left or below or downward serve as a definition for ‘negative’?
>
> Aaron
>
> Aaron Kohring
>
> Research Associate
>
> UT Center for Literacy Studies
>
> 600 Henley St, Ste 312
>
> Knoxville, TN 37996-4135
>
> Ph: 865-974-4258
>
> Main: 865-974-4109
>
> Fax: 865-974-3857
>
> akohring at utk.edu <mailto:akohring at utk.edu>
>
> *From:* numeracy-bounces at nifl.gov [mailto:numeracy-bounces at nifl.gov]
> *On Behalf Of *Michael Gyori
> *Sent:* Wednesday, March 31, 2010 12:24 PM
> *To:* The Math and Numeracy Discussion List
> *Subject:* [Numeracy 261] Re: Is an absolute value positive?
>
> Greetings Chip and all,
>
> Unfortunately, what you owe is a positive amount (the distance from
> zero to the left or downward). This might serve as a good example of
> why absolute values are positive.
>
> The negative has no existence other than a notation on a balance sheet.
>
> Michael
>
> Michael A. Gyori
>
> Maui International Language School
>
> www.mauilanguage.com <http://www.mauilanguage.com/>
>
> ------------------------------------------------------------------------
>
> *From:* Chip Burkitt <chip.burkitt at orderingchaos.com>
> *To:* numeracy at nifl.gov
> *Sent:* Wed, March 31, 2010 3:23:50 AM
> *Subject:* [Numeracy 259] Re: Is an absolute value positive?
>
> Wow! I wish I could get my bank to agree that negative numbers have no
> existence. I'd never be overdrawn again.
>
> On 3/31/2010 1:38 AM, Michael Gyori wrote:
>
> Greetings Carol and all,
>
> I suppose the question is what is meant by "positive." If we view
> numbers to represent quantities (of whatever), then they are
> intrinsically positive. Negative quantities have no existence, and
> perhaps we can regard the notion of absolute value to reflect just that.
>
> As for teaching absolute values to "struggling" learners, my sense is
> we shouldn't underestimate their ability to make sense of things.
> Quite on the contrary, the challenge lies in our (educators') ability
> to facilitate meaning.
>
> Michael
>
> Michael A. Gyori
>
> Maui International Language School
>
> www.mauilanguage.com <http://www.mauilanguage.com/>
>
> ------------------------------------------------------------------------
>
> *From:* Carol King <cking at lyon.k12.nv.us> <mailto:cking at lyon.k12.nv.us>
> *To:* The Math and Numeracy Discussion List <numeracy at nifl.gov>
> <mailto:numeracy at nifl.gov>
> *Sent:* Tue, March 30, 2010 9:12:47 AM
> *Subject:* [Numeracy 250] Re: what is the difference between....
>
> I would point out that technically an absolute value is not a positive
> number. It represents the distance from 0 either negatively or
> positively on the number line. It operates mathematically like a
> positive number, but it is not the same as the positive number. I
> don’t know if I would share that with struggling students.
>
> Carol King
>
> cking at lyon.k12.nv.us <mailto:cking at lyon.k12.nv.us>
>
> Fernley Adult Education.
>
> ------------------------------------------------------------------------
>
> *From:* numeracy-bounces at nifl.gov <mailto:numeracy-bounces at nifl.gov>
> [mailto:numeracy-bounces at nifl.gov] *On Behalf Of *Michael Gyori
> *Sent:* Monday, March 29, 2010 4:55 PM
> *To:* The Math and Numeracy Discussion List
> *Subject:* [Numeracy 241] Re: what is the difference between....
>
> Hi again George and all,
>
> If /-6/ means the absolute value of negative 6, then I stand
> corrected. I didn't realize the forward slashes might have been bars.
>
> In that case, absolute values are positive. So, the negative of the
> absolute value of -6, which is +6, = -6. On the other hand, -(-6)
> [i.e, in parentheses] would be positive 6.
>
> Also, I always teach part numbers (fractions, decimals, and percents)
> before I teach integers. I find it interesting that you delve into
> integers first. Do you have a reason for doing so?
>
> Michael
>
>
> Michael A. Gyori
>
> Maui International Language School
>
> www.mauilanguage.com <http://www.mauilanguage.com/>
>
> ------------------------------------------------------------------------
>
> *From:* Michael Gyori <tesolmichael at yahoo.com>
> <mailto:tesolmichael at yahoo.com>
> *To:* The Math and Numeracy Discussion List <numeracy at nifl.gov>
> <mailto:numeracy at nifl.gov>
> *Sent:* Mon, March 29, 2010 1:30:03 PM
> *Subject:* [Numeracy 237] Re: what is the difference between....
>
> Hello George and all,
>
> See my attempt at making sense of your message. I will embed my
> thoughts in green into your post below:
>
> Michael
>
> Michael A. Gyori
>
> Maui International Language School
>
> www.mauilanguage.com <http://www.mauilanguage.com/>
>
> ------------------------------------------------------------------------
>
> *From:* George Demetrion < gdemetrion at msn.com
> <mailto:gdemetrion at msn.com> >
> *To:* Numeracy List <numeracy at nifl.gov> <mailto:numeracy at nifl.gov>
> *Sent:* Mon, March 29, 2010 10:18:55 AM
> *Subject:* [Numeracy 235] what is the difference between....
>
> Good afternoon colleagues.
>
> In my newly articulated and highly pleasant role as a Transition to
> College math teacher, I've come ac ross the following
>
> *-2^2 *^
>
> ^ ^When a number is positive, we don't sign it. For example, 2+2=4
> really means +2 (+) + 2 = + 4.
>
> ^ In that vein, we really have negative *times* _positive 2_ squared
> (or the negative *of* positive 2 squared) equals -4.
>
> *(-2)^2 *
>
> ^In this case, the 2 is signed as a negative, or you can see it as the
> negative of positive 2 = negative two, times itself, = +4.
>
> ^I teach my students rather early on that unsigned numbers are
> actually signed by an invisible "+" before them, just like whole
> number have an invisible point (.) to their right, which is the border
> separating whole numbers from decimals (part numbers). 345 is the
> same as 345 *^. *^
>
> ^Something "of something" is always multiplication. In the case of -
> +, we are saying the negative* of* positive, which leads me to
> negative *times* positive. That leads me to teach (also quite early
> on) the *golden rules*, namely, positive x positive and negative x
> negative = positive, while negative times positive or positive times
> negative = negative.
>
> According to my book the answer to the first problem is -4 while the
> answer to the second is 4. The examples are easy enough to follow,
> but a little light on the explanation. In the first problem the
> notation in the book states that 2 is the base; thus (2.2)=4 and, I
> assume, we keep the negative sign, so that the answer becomes -4. The
> second problem is easy enough. I get (-2.-2)=4.
>
> What's missing as far as I'm concerned is a clear and simple
> explanation of the reasoning behind the first problem - 2^2 d
>
> I deduced that the second problem is based on an order of operations
> problem solving menthodology and I'm thinking the same thing for the
> first problem in which the negative sign indicated a -1. Thus, on
> this hypothesis, I am carrying out an order of operation (exponent
> first, including the implied paranthesis (2.2) multiplied by -1 in
> which this later stage is last on the order of operations process.
>
> Questions:
>
> 1. Is my hypothesis for problem #1 correct?
>
> 2. If not, what would be the correct explanation?
>
> 3. Whether or not the hypothesis is correct what woould be the
> simplist accurate explanation to provide my students with?
>
> One more question -/-6/= -6, which I translate to mean is that the
> opposite of the absolute number -6 has an absolute value of 6;
> therefore its opposite is -6.
>
> *-/-6/ = -6 strikes me as being incorrect. If I'm wrong, PLEASE
> CORRECT ME: In language, the equations reads to me, negative times
> negative 6 = POSITIVE 6, because a negative times a negative = a
> positive.*
>
> a) is this correct
>
> b) If so, is there an easier way to state it?
>
> c) If it is correct what is the best way to teach it to TCC students
> with limited mathematical experience
>
> d) If it is incorrect, what would be the correct answer?
>
> Okay, we're just about through with integers. Onto fractions.
>
> Best,
>
> George DemetrionImage removed by sender.
>
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