# [Numeracy 502] Re: Equality, unary signs, and operations...

Share:

## Archived Content Disclaimer

This page contains archived content from a LINCS email discussion list that closed in 2012. This content is not updated as part of LINCS’ ongoing website maintenance, and hyperlinks may be broken.

Michael Gyori michael_gyori at yahoo.com
Wed Aug 18 19:14:45 EDT 2010

Chip and all,

My personal challenge with temperature scales is that they are entirely
aribitrary, meaning, as you say, that  we could also create a scale that begins
with absolute zero.   On the other hand, when we deal with debt, we can
illuminate the abstractness of negative numbers (i.e. their non-existence in
physical reality)  by showing that the money that we owe is a positive amount,
while in balancing our checkbooks, it shows as a negative amount.

In terms of performing operations, using a temperature scale works well in
illustrating how, as the temperature falls (using °F or °C) we can arrive at
negative values and how that occurs. In accounting terms, however, a negative
value can be associated with what we don't have or owe.  This serves as a better
bridge, IMO, in linking math with its practical daily applications, especially
in ABE.

Michael

Michael A. Gyori
Maui International Language School
www.mauilanguage.com

________________________________
From: Chip Burkitt <chip.burkitt at orderingchaos.com>
To: Michael Gyori <michael_gyori at yahoo.com>
Cc: The Math and Numeracy Discussion List <numeracy at nifl.gov>; Ladnor Geissinger
Sent: Wed, August 18, 2010 3:57:15 AM
Subject: Re: Equality, unary signs, and operations...

Michael,

Yes. Temperature scales we use every day are also arbitrary. The Celsius scale
makes the most sense since it takes its 0 and 100 from the freezing and boiling
points of water. The Fahrenheit scale takes its 0 from a brine solution and its
96 (originally) from normal human body temperature. It's easy to avoid negative
temperature values, however, by using a scale that starts at absolute zero, such
as the Kelvin scale. Nevertheless, most students are at least familiar with the
notation of below-zero temperatures expressed with a unary minus sign. They also
quickly grasp the idea that adding positive degrees to a negative value causes
it to go up the scale, while subtracting causes it to go down. So it provides a
fairly natural way to introduce addition and subtraction involving negative
numbers.

Net worth is also a useful concept for dealing with negative numbers, but it
requires introducing assets and liabilities as well. Still, it's not hard to
describe assets as what you own and liabilities as what you owe. When you owe
more than you own, you have a negative net worth.

Chip Burkitt

On 8/18/2010 1:32 AM, Michael Gyori wrote:
Chip,

>

>My only reservation is that a unit of measurement of temperature is entirely

>abstract, while being in the red is not.  When we *owe* money, our balance is

>negative, even if the money we owe is positive...

>

>Michael

>

>Michael A. Gyori

>Maui International Language School

>www.mauilanguage.com

>

>

>

>

>

>

>

________________________________
From: Chip Burkitt <chip.burkitt at orderingchaos.com>

>To: Michael Gyori <michael_gyori at yahoo.com>

>Cc: The Math and Numeracy Discussion List <numeracy at nifl.gov>; Ladnor Geissinger

>Sent: Tue, August 17, 2010 7:06:27 PM

>Subject: Re: Equality, unary signs, and operations...

>

>Michael,

>

>I agree. A balance scale makes an excellent metaphor for equality.

>

>I remember a bit about the unary minus sign and your contention that it has no

>physical representation. Of course, the same could be said for the concept of

>number. Counting is actually a fairly sophisticated exercise. It requires

>knowledge of a number system, understanding of one-to-one correspondence, and

>confidence that the quantity does not change if items are counted in a different

>order. Even tallying, which does not require a number system, still requires

>fairly powerful abstract reasoning. For example, if I say, "I have fourteen

>chickens," the number fourteen is an abstraction. It cannot be derived from the

>chickens; it has to be brought to the chickens. So fourteen (or any number, for

>that matter) also has no physical representation. It could be argued that

>fourteen is what all groups of fourteen items have in common, but that reasoning

>is circular and neglects other properties that groups of physical items have in

>common (e. g., occupies space, color, mass, duration).

>

>Nevertheless, numbers certainly have more common applications in the lives of

>students than unary minus signs do. When I introduce negative numbers, I give

>examples likely to be common to everyone: below zero temperatures (especially

>here in Minnesota), overdrawn bank account, and below sea level elevations. It

>is especially easy to understand operations on temperatures (e. g., Most

>students know that warming up 10° from −17° is −7°).

>

>Chip Burkitt

>

>On 8/17/2010 5:42 PM, Michael Gyori wrote:

>Hello Chip,

>>

>>I wonder whether Ladnor might be willing to weigh in further at this poin with

>>respect to the nature of equality.   I don't know what a "real" mathemetician

>>is, but if there is such a person, I would consider him to be one... :)

>>

>>As for equality, the reason I like the metaphor of a scale in discussing

>>equality is that it connects numbers with physical physical phenomenon and

>>shared background knowledge (by and large).   Scales don't compare what's on

>>either side, they just tell you that weights are the same when they're balanced.

>>

>>On another note, we had some very interesting discussions about unary signs a

>>while ago.  I contended and still contend that they had no physical

>>existence.   I'd be really interested in further discussion about unary signs

>>vs. operations.

>>

>>Michael

>>

>>

>>Michael A. Gyori

>>Maui International Language School

>>www.mauilanguage.com

>>

>>

>>

>>

>>

>>

>>

________________________________
From: Chip Burkitt <chip.burkitt at orderingchaos.com>

>>To: numeracy at nifl.gov

>>Sent: Tue, August 17, 2010 7:28:47 AM

>>Subject: [Numeracy 496] Re: What does equality mean?

>>

>>Michael,

>>

>>You are probably right. There is undoubtedly a consistent definition we can

>>apply. I was not accurate in saying that certain transformations change the

>>equality. What may change, however, is the amount of information present in the

>>expressions. For example, squaring an expression loses information about whether

>>it is less than or greater than 0. Conversely, taking the square root of an

>>expression introduces ambiguity about the sign (unless there are contextual

>>clues to eliminate one of the possibilities). Likewise, in the second example,

>>transforming the equation from y/(x − 1) = 3 to y = 3x − 3, loses the

>>information that x ≠ 1 unless we specifically preserve it. The nature of the

>>equality is unchanged.

>>

>>The point I intended to make was that the balance metaphor serves as an

>>excellent touchstone for students learning about what the equal sign means. For

>>basic operations (+, −, ×, ÷), it is flawless as far as I can tell. If anyone

>>has an alternative or wishes to critique the balance metaphor, please reply.

>>

>>

>>Equality is reflexive, symmetric, and transitive. That is, for any real numbers

>>a, b, c:

>>  a = a

>>  if a = b, then b = a

>>  if a = b and b = c, then a = c

>>I think that these three properties are sufficient to define equality for real

>>expressions, but I am not sure. Perhaps a real mathematician could weigh in on

>>this.

>>

>>Chip Burkitt

>>

>>On 8/17/2010 12:34 AM, Michael Gyori wrote:

>>Chip,

>>>

>>>I fail to understand why the definition of equality is subject to variation -

>>>say in relation to the level of numeracy of our students.  I have

>>>always understood equality in math to signify equality in value. If you perform

>>>the same operation on identical values with different expressions, of

>>>course doing so might result in inequality, as you appear to state in your post

>>>below.  My question is, why can we not apply a consistent definition to equality

>>>when teaching math at whatever level we may be doing so?

>>>

>>>Michael

>>>

>>>Michael A. Gyori

>>>Maui International Language School

>>>www.mauilanguage.com

>>>

>>>

>>>

>>>

>>>

>>>

>>>

________________________________
From: Chip Burkitt <chip.burkitt at orderingchaos.com>

>>>To: numeracy at nifl.gov

>>>Sent: Sat, August 14, 2010 6:59:24 AM

>>>Subject: [Numeracy 493] Re: What does equality mean?

>>>

>>>I think the difficulty is that mathematics requires rigorous definitions and

>>>logic, especially as one advances in it. However, for ABE or GED students, it is

>>>usually enough to know that the equal sign is like a balance scale. In order for

>>>the sides to be in balance, the expressions on both sides must have the same

>>>value. If you add something to one side, you must add it to the other side as

>>>well to maintain the balance. If you subtract from one side, you must subtract

>>>it from the other side as well. When students get into algebra, they need to

>>>know that some transformations of an expression can change the character of the

>>>equality. For example (−a)^2 = a^2, but it does NOT follow by taking the square

>>>root of both sides that −a = a. Likewise, y/(x − 1) = 3 needs to be qualified by

>>>x ≠ 1, even though the equation can be transformed to y = 3x − 3, which has a

>>>solution for x = 1 at y = 0. For most purposes in ABE or GED classes, the

>>>balance analogy works well without getting into abstract discussions about

>>>various kinds of equivalence relations and the transformations that change the

>>>relation or leave it unchanged. If anyone has a better explanation of the equal

>>>sign for ABE and GED students, I would like to hear it.

>>>

>>>Chip Burkitt

>>>

>>>On 8/14/2010 1:04 AM, Michael Gyori wrote:

>>>Greetings all,

>>>>

>>>>After all this discussion about what the equal sign (or equality) means, I find

>>>>myself somewhat in a maze.  A discussion of equality takes us into a potentially

>>>>esoteric realm from the perspective of our students.

>>>>

>>>>Might it be time to attempt to more clearly (and simply!) define terms among

>>>>those who teach math?

>>>>

>>>>

>>>>Michael

>>>>

>>>>Michael A. Gyori

>>>>Maui International Language School

>>>>www.mauilanguage.com

>>>

>>> ---------------------------------------------------- National Institute for

>>>Literacy Math & Numeracy discussion list Numeracy at nifl.gov To unsubscribe or

>>>http://www.nifl.gov/mailman/listinfo/numeracy Email delivered to

>>>chip.burkitt at orderingchaos.com

>>>

>>

>

-------------- next part --------------
An HTML attachment was scrubbed...
URL: http://lincs.ed.gov/pipermail/numeracy/attachments/20100818/ca82d24f/attachment.html