[Numeracy 269] Re: Is an absolute value positive?
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Wed Mar 31 17:36:34 EDT 2010
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To help with such questions and occasional inaccuracies in our material
I bought a mathematics dictionary (looks like I will be modifying the
text again.).
According to the Collins Dictionary of Mathematics: 1. Absolute value is
the positive real number equal to a given real but disregarding its
sign; Where r is positive, absolute value of r = r = absolute value of
-r.
Less helpful is definition 2. it is a modulus. A Modulus is defined as
a positive real number that is a measure of the magnitude of a complex
number, and its equal to the square root of the sum of the squares of
the real and imaginary parts of the given number. Thus the absolute
value of x+iy is the absolute value of x+iy = the square root of x
squared + y squared.
For me this creates the question of if I have the absolute value of -6
plus 10 is it saying I have 16 of something? If so why didn't the
question just occur as 6+10? As I understand it absolute value is used
in problems that are actually expressing range type ideas. What is the
ultimate purpose of this concept; into what context can a student put an
absolute value question?
________________________________
From: numeracy-bounces at nifl.gov [mailto:numeracy-bounces at nifl.gov] On
Behalf Of Chip Burkitt
Sent: Wednesday, March 31, 2010 1:31 PM
To: numeracy at nifl.gov
Subject: [Numeracy 266] Re: Is an absolute value positive?
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
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>
<mailto: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
Fernley Adult Education.
________________________________
From: 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 >
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
-22
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 - 22 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 Demetrion
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