mm-455
===
Subject: Re: order of operations
I think that this metaphor for Paul Tanner's position is
inaccurate.
As I understand his point, fluency is a matter of
understanding what
you are saying.
> I guess different people have different ideas about what
fluency is
in
> such a context. Many (most?) would say its a matter of
understanding,
> albeit it possibly informally, what you are saying/reading,
ie being able
to
> communicate to some effective degree without necessarily
knowing all the
> technical details, the formalities, etc, of the language.
> A reasonable person may indeed interpret Paul's idea as one
of implying
the
> formalties and technicalities are *neccessary* for fluency,
which goes
> against another popular school of thought on the matter.
I am using the term fluent in the same way as in She is fluent
in
Spanish and as in She knows Spanish well enough to get by, but
is
not fluent in it. To be fluent in Spanish, we have to have a
command
of some of the technical details of its grammar. Same in
algebra. And
I'm bringing up only a single technical detail, split into two
points:
The definitions of division and subtraction.
How could we say that a student is fluent in algebra (has a
command of
some of its grammar) but that the student does not see that
that a/b =
c/d is equivalent to a(1/b) = c(1/d) and that a-b = c-d is
equivalent
to a+(-b) = c+(-d)? How could we say that a student is fluent
in
algebra but that the student does not see that a/b*c is
equivalent to
a(1/b)c and that a-b+c is equivalent to a+(-b)+c? I'd like to
hear
from some algebra teachers who think that we could say such.
And Kevin, you are on target. Seeing the above equivalencies,
knowing
and being able to apply a couple of basic definitions, is
basic to
understanding what we say in algebra.
Paul
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Subject: Re: order of operations
> I am using the term fluent in the same way as in She is
fluent in
> Spanish and as in She knows Spanish well enough to get by,
but is
> not fluent in it.
OK, she's _both_ fluent and not fluent in Spanish, which of
course makes
perfect sense. Why didn't I see sooner what you were trying to
say?
Sorry...
> To be fluent in Spanish, we have to have a command
> of some of the technical details of its grammar. Same in
algebra. And
> I'm bringing up only a single technical detail, split into
two points:
> The definitions of division and subtraction.
> How could we say that a student is fluent in algebra (has a
command of
> some of its grammar) but that the student does not see that
that a/b =
> c/d is equivalent to a(1/b) = c(1/d) and that a-b = c-d is
equivalent
> to a+(-b) = c+(-d)?
I never said or implied that they shouldn't be able to justify
those
statements. All I said is that they don't *have* to look at it
quite that
way all the time.
> How could we say that a student is fluent in
> algebra but that the student does not see that a/b*c is
equivalent to
> a(1/b)c and that a-b+c is equivalent to a+(-b)+c? I'd like
to hear
> from some algebra teachers who think that we could say such.
Again, paul, as you so frequently do you are changing the
context of the
discussion significantly from what it was before. No one to my
knowlege,
and surely not myself, suggests that not seeing the
equivalency is OK and
they can still be fluent. No one said anything of the least.
What I
said
was, which was a direct counterpoint to what you initially
sad, was that
they don't *have* to *always* think of division in terms of
multiplication,
and subtraction in terms of addition. That's all. You seem to
think if
they don't always do so (more specifically if we as teachers
don't always
take the opportunity to point this out) that we are missing
out on an
opportunity. I am simply arguing that this opportunity you
speak of
simply does not *need* to always be mentoned everytime we
encounter a
division or subtraction problem. Somehow, and for some reason,
every time
someone posts an order of operations problem youi seem to go
out of
your
way, insisting that that there is some mathematical
requirement for the
agreement being what it is. There is not. Moreover, I think
it's a little
misleading to tell your students that there is.
> And Kevin, you are on target. Seeing the above
equivalencies, knowing
> and being able to apply a couple of basic definitions, is
basic to
> understanding what we say in algebra.
And I, Kevin, yourself, or anyone else could also state a few
things, such
as you can divide both sides by any nonzero number, etc. and
also that
would
apply to fluency however sould not be the subject of the
discussion,
really. You are listing, and apparently trying to argue under
the guide
that I have disagered somehow, with something that was *never*
in debate,
at
least not by me.
how such furthering is even possible) if we're going to keep
changing the
subject.
--
Darrell
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Subject: Re: order of operations
> OK, she's _both_ fluent and not fluent in Spanish, which of
course makes
> perfect sense. Why didn't I see sooner what you were trying
to say?
> Sorry...
The instances of she in the two given statements need not
apply to
the same person.
> Somehow, and for some reason, every time
> someone posts an order of operations problem youi seem to go
out of
your
> way, insisting that that there is some mathematical
requirement for the
> agreement being what it is. There is not. Moreover, I think
it's a
little
> misleading to tell your students that there is.
You seriously misread what I'm doing and what I'm saying.
I choose to do more than just give answers or solutions to
people
without exploring why they are having a problem getting the
answers or
solutions on their own. I don't see why anyone would object to
this.
There are proverbs to the effect that if people can't feed
themselves,
it's best to not merely feed them, but to explore why it is
they can't
feed themselves, so that that they can start feeding
themselves.
I'm convinced that students who have problems in the order of
operations area are having these problems because they are not
fluent
in applying the basic definitions of subtraction and division.
I have
yet to see a student who has problems in the order of
operations area
become fluent in applying these definitions and continue to
have a
problem in this area. I'm convinced that if these posters were
fluent
in such a way, then they just wouldn't be asking the questions
that
they are asking, asking for the help for which they ask. They
would
instead be able to answer their own questions - they would
instead be
able to feed themselves.
Paul
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Subject: Re: order of operations
As previously recommended, the subject should be dropped, at
least because
the
arguments have reached their best and are not further
productive.
If not fluent, the person must find a formal reason for each
step; if
fluent,
the person can find each step because the behavior for what to
do is built
in
to the person without needing to check which formal rule
justifies the step.
Algebra really is like natural language in this regard.
G C
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Subject: Re: order of operations
<... Algebra really is like natural language in this regard.
Exactly my point, but in natural language (eg a young child
communicating
with spoken English) its not that the formal rules are built
into the
person making unneccessary the need to check with some list of
rules or
something, its more like the formal rules are, well, just
simply *not
known*, at least not on any concious level. Yet young children
are often
sufficiently fluent in English.
It's quite possible, and in fact quite *common*, for beginning
algebra
students to perform much manipulation without really, really,
understanding
that well *what* they are really doing, and one school of
thought says
that's quite fine, because a certain level of fluency is
needed (long?)
before a deep understanding of the subject on a more formal
basis.
--
Darrell
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Subject: Simol math quesrion
HI
My name is rob and my wife and i were on a small road trio and
one of
things we were joking about was how many combinations of 5 can
we
make with the numbers one through forty nine ? can you help us
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Subject: possible answer
i'm not sure this is entirely correct, but i'll go thru all of
the
steps.
You can retrieve this answer by using the equation
C(n,r)=n!/(n-r)!r!
where n is the number of total numbers (49), and r is the
amount of
numbers in your combination (5).
C(n,r)=49!/(49-5)!5!
C(n,r)=49!/44!(120)
Getting the exact answer would take a lot of calculating...
but again,
i'm not exactly sure that this is correct. can anyone verify
my work?
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Subject: Re: possible answer
> i'm not sure this is entirely correct, but i'll go thru all
of the
> steps.
> You can retrieve this answer by using the equation
C(n,r)=n!/(n-r)!r!
> where n is the number of total numbers (49), and r is the
amount of
> numbers in your combination (5).
> C(n,r)=49!/(49-5)!5!
> C(n,r)=49!/44!(120)
> Getting the exact answer would take a lot of calculating...
but again,
> i'm not exactly sure that this is correct. can anyone verify
my work?
Slow down. 49! = 49 * 48 * 47 * 46 * 45 * 44 * 43 * 42 * ...
44! = 44 * 43 * 42 * ...
Does that help?
--Jeff
--
A man, a plan, a cat, a canal - Panama!
Ho, ho, ho, hee, hee, hee
and a couple of ha, ha, has;
That's how we pass the day away,
in the merry old land of Oz.
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Subject: math question
How many combinations of six are there between one to fifty
two??
and what is the formula so i can figure it out on my own
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Subject: Re: math question
> How many combinations of six are there between one to fifty
two??
> and what is the formula so i can figure it out on my own
Assuming you can only select each number once and the order
doesn't
matter, the total is 52 choose 6 = 52! / 6!(52-6!) = (52 * 51
* 50 * 49
* 48 * 47) / (6 * 5 * 4 * 3 * 2)
If the order does matter, you want 52 permute 6, which is
52!/(52-6)! =
52 * 51 * 50 * 49 * 48 * 47 (52 ways to pick the first number,
51 to
pick the second, and so on)
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Subject: Re: British subtraction vs north american method
Cheers,
Chris
> My wife is teaching my 7 yo son double digit subtraction.She
is from
> Scotland and they ( the scots) have a different method of
> borrowing.This method was new to me as well.It works just
fine
> though.The teacher at my sons school believes it will cause
him
> problems to learn this method , later on when he learns more
> complicated math.To explain this method I will use commas to
represent
> the digit carried. 50 - 49 .So 10 - 9 = 1 then 5 - 4, = 0 So
the
> answer is 1. another eg 83 - 58. So 13-8=5 8 - 5,= 2 answer
is 25.
> remember the comma represents the carried number. Any
opinions on
> possible pitfalls with this method would be apprieciated.
> Chris
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Subject: Re: Statistical Variance. Is it n or n-1 in the
denominator?
The sample mean generally differs slightly from the population
mean, because not all of the population is included in the
sample.
The difference is in the direction that reduces slightly the
computed variance of the sample, from what that variance would
be
if you used the population mean rather than the sample mean.
Because of that shift of the sample mean, in order to compute
the
population variance, you must multiply the sample variance by
the
factor n/(n-1) . That factor is derived in the paper-back book
The Elements Of Probability Theory and some of its
applications,
by Harold Cramer, published by Wiley, 1955. That is why the
variance formula denominator has the factor n-1 rather than n.
Dick Alvarez
alvarez@alumni.caltech.edu
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Subject: CSMP
I am a student who has greatly enjoyed this series of books
(the white
books as we call them). I think that the white books are very
informative indeed, and I have learned greatly from their
approach to
teaching mathematics. I was wondering if you knew who I should
contact
to obtain copies of the rest of the set, because our class
only went
up to Book 5 Functions.
Thank you much,
Andy
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Subject: order of ops
Man Abe! Give the guy a break.
What level do you teach? Even in high school the students mess
up the
order of operations because of PEMDAS. Mnemonics don't work for
everyone. Even after the teacher logically shows the students
the
right way to handle simplification of expressions involving
multiple
inline addition/substraction or multiplication/division they
still
make a very common mistake. They do what they want to do first
because they see addition and multiplication as being easier
than
subtraction and division.
It is a good idea to give another mnemonic device...how about
People
Enjoy McDonald's SAndwiches...? Notice the capitalization. Both
sequences should be presented and the differences should be
emphasized
to show the equal level of addition and subtraction. That
being said,
it is also important that the students get beyond the mnemonic
device.
Order of operations should be second nature whether you plan
on doing
exponential stuff or not.
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Subject: Re: order of ops
> Man Abe! Give the guy a break.
> What level do you teach? Even in high school the students
mess up the
> order of operations because of PEMDAS. Mnemonics don't work
for
> everyone. Even after the teacher logically shows the
students the
> right way to handle simplification of expressions involving
multiple
> inline addition/substraction or multiplication/division they
still
> make a very common mistake. They do what they want to do
first
> because they see addition and multiplication as being easier
than
> subtraction and division.
> It is a good idea to give another mnemonic device...how
about People
> Enjoy McDonald's SAndwiches...? Notice the capitalization.
Both
> sequences should be presented and the differences should be
emphasized
> to show the equal level of addition and subtraction. That
being said,
> it is also important that the students get beyond the
mnemonic device.
> Order of operations should be second nature whether you plan
on doing
> exponential stuff or not.
This emphasizes an important point: That PEMDAS, although
yielding
correct results, can confuse students about the relative
priority of
all of these operations. With PEMDAS, there is no qualitative
difference in the precedence that exponentiation has over
multiplication vs. the precedence that multiplication has over
division, or that addition has over subtraction for that
matter. I
agree that multiplication and division should be understood as
grouped
together, and addition and subtraction should be understood as
grouped
together. The acronym I gave earlier, PEMA, where
multiplication
subsumes division and addition subsumes subtraction via the
definitions of division and subtraction, gives students a
better view
of the relative priorities of all of these operations.
Unlike the priorities of exponentiation over multiplication
and of
multiplication over addition, the priorities of multiplication
over
division and of addition over subtraction in the left-to-right
context
are entirely rote in nature, devoid of providing any
understanding.
These priorities in this left-to-right context are just ways
to apply
division and subtraction without having to understand their
definitions.
This lack of understanding these definitions is a fundamental
problem
and more widespread than many might want to believe. I've seen
a
number of high school students who have come to me not
understanding
that an expression of the form (a/b)*c is equivalent to
(ac)/b. They
never learned the definitions of division and subtraction to
fluency
(or they never learned them at all). They just don't
understand what
they write. Instilling these definitions is the only fix.
Paul
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Subject: Re: order of ops
As earlier stated, this is a difficult discussion to carry.
Fluency here,
is
knowing what to do with an equation or an expression depending
on what is
wanted from the equation or expression; and without having to
formally
remember
and recite the relevant properties; and without needing to
look for the
relevant properties in a book. They just must be understood as
if they
were
acquired language.
Again; should I give an example?
G C
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Subject: Re: order of ops
What kind of example are you offering? A complex equation
proving the
necessity of 'knowing' order of ops fluently, or an example of
how to
instill that fluency in our students. If the later, please
provide an
example!
I am new to teaching and agree that the fluency goal is
important.
> As earlier stated, this is a difficult discussion to carry.
Fluency
here, is
> knowing what to do with an equation or an expression
depending on what is
> wanted from the equation or expression; and without having
to formally
remember
> and recite the relevant properties; and without needing to
look for the
> relevant properties in a book. They just must be understood
as if they
were
> acquired language.
> Again; should I give an example?
> G C
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Subject: Re: order of ops
My commentary on the importance of fluency in algebra:
> As earlier stated, this is a difficult discussion to carry.
Fluency
here,
>is
> knowing what to do with an equation or an expression
depending on what
is
> wanted from the equation or expression; and without having
to formally
>remember
> and recite the relevant properties; and without needing to
look for the
> relevant properties in a book. They just must be understood
as if they
>were
> acquired language.
 Again; should I give an example?
 G C
The concern and inquiry of ml.whall@rcn.com:
>What kind of example are you offering? A complex equation
proving the
>necessity of 'knowing' order of ops fluently, or an example
of how to
>instill that fluency in our students. If the later, please
provide an
>example!
My emphasis is on the fluency of algebra; the understanding of
order of
operations is built into the behavior.
Here is an example to show the importance of fluency.
Suitable for elementary algebra:
A rectangle has an area of 80 square units and perimeter of 42
units.
What is the length and width of the rectangle?
Translation from written natural language to mathematics:
L=length, w=width
L*w=80
42=2*L + 2*w
We want to find L and w.
Fluency essentially required in the solution to continue:
42-2*L=2*w
(42-2*L)/2=w
w = 21 - L
L*(21 - L) = 80
21*L - L*L - 80 = 0
L*L - 21*L + 80 = 0 Quadratic equation.
This should have been acquired or fairly natural to this
extent. Next,
check
to see if factorization is possible.
(L - 16)*(L-5)=0 Yes factorization was possible.
L=length = 16 units; then w=5 units
(or L is either 16 or 5, while w is either 5 or 16)
Between the translation and the factorization, one must be
fluent; not
needing
to look in the instructional book for various properties of
the operations
and
equations to determine what steps to take.
G C
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Subject: Re: order of ops
Disappointing if true that This lack of understanding these
definitions
is a
fundamental problem
and more widespread than many might want to believe;
Are these the same students who learned so well about dealing
with
fractions
and decimals, and then have great trouble in introductory
algebra?
The order of operations should become automatic to the
student, and
mnemonic
devices like PEMA, or PEMDA, or PEMDAS should be nothing more
than some
retarding crutch. Basically, at the level of this introductory
algebra,
after
4 or 6 weeks, the student moves from the properties of
equality,
inequality,
and the arithmetic operations, to just knowing what they see
when it is on
the
page (understanding the expressions upon looking at them each
for a few
seconds). If this discussion seems unclear, ask, and I could
give an
example.
G C
>This emphasizes an important point: That PEMDAS, although
yielding
>correct results, can confuse students about the relative
priority of
>all of these operations. With PEMDAS, there is no qualitative
>difference in the precedence that exponentiation has over
>multiplication vs. the precedence that multiplication has over
>division, or that addition has over subtraction for that
matter. I
>agree that multiplication and division should be understood
as grouped
>together, and addition and subtraction should be understood
as grouped
>together. The acronym I gave earlier, PEMA, where
multiplication
>subsumes division and addition subsumes subtraction via the
>definitions of division and subtraction, gives students a
better view
>of the relative priorities of all of these operations.
>Unlike the priorities of exponentiation over multiplication
and of
>multiplication over addition, the priorities of
multiplication over
>division and of addition over subtraction in the
left-to-right context
>are entirely rote in nature, devoid of providing any
understanding.
>These priorities in this left-to-right context are just ways
to apply
>division and subtraction without having to understand their
>definitions.
>This lack of understanding these definitions is a fundamental
problem
>and more widespread than many might want to believe. I've
seen a
>number of high school students who have come to me not
understanding
>that an expression of the form (a/b)*c is equivalent to
(ac)/b. They
>never learned the definitions of division and subtraction to
fluency
>(or they never learned them at all). They just don't
understand what
>they write. Instilling these definitions is the only fix.
>Paul
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Subject: Re: order of ops
> Disappointing if true that This lack of understanding these
definitions
is a
> fundamental problem
> and more widespread than many might want to believe;
> Are these the same students who learned so well about
dealing with
fractions
> and decimals, and then have great trouble in introductory
algebra?
> The order of operations should become automatic to the
student, and
mnemonic
> devices like PEMA, or PEMDA, or PEMDAS should be nothing
more than some
> retarding crutch. Basically, at the level of this
introductory algebra,
after
> 4 or 6 weeks, the student moves from the properties of
equality,
inequality,
> and the arithmetic operations, to just knowing what they see
when it is on
the
> page (understanding the expressions upon looking at them
each for a few
> seconds). If this discussion seems unclear, ask, and I could
give an
example.
> G C
This unconscious automaticity that you speak of should happen,
but it
many times doesn't. (These are for the most part students who
have had
problems with the arithmetic before the algebra s well.)
The problem is in uncovering why. I think it's because they
haven't
internalized, even if it is to this point of unconscious
automaticity,
the definitions I spoke of. They don't see to the point of
unconscious
automaticity the essential equivalency of addition/subtraction
or
multiplication/division. This is a hurdle that all must get
past to
become fluent, even if it is unconscious in the sense of an
acquired
language.
Paul
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Subject: Help...Never seen this before
I have been given this math problem and i have never seen this
symbol
before...it would be greatly appreciated if someone would lend
their
advice:
(superscript) 4
Find the value of: (uppercase pi symbol) (2k + 1)
k=0
sorry for the complicated layout, i know of no way to insert
the
symbols
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Subject: Re: Help...Never seen this before
<vaa12059iej4bdbdl1u724hs90ac0ev5q4@4ax.com>:
> I have been given this math problem and i have never seen
this symbol
> before...it would be greatly appreciated if someone would
lend their
> advice:
> (superscript) 4
> Find the value of: (uppercase pi symbol) (2k + 1)
> k=0
> sorry for the complicated layout, i know of no way to insert
the
> symbols
I believe you are referring to a product symbol (not sure of
the exact
terminology for it).
So, what you've described would mean to multiply all values of
the
expression (2k+1) where k takes on each of the values from 0
up to 4.
so:
[2(0)+1][2(1)+1][2(2)+1][2(3)+1][2(4)+1]
like that.
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Subject: Re: Help...Never seen this before
> I believe you are referring to a product symbol (not sure of
the exact
> terminology for it).
My Harper Collins DICTIONARY OF MATHEMATICS, by Borowski and
Borwein,
uses the phrase continued product while Mathworld says:
http://mathworld.wolfram.com/Product.html
(To all interested: For examples, look to number theory, where
continued products are common. The totient function or Euler's
phi
function [one of the most basic functions in number theory and
one of
my favorites] is a continued product.)
Paul
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Subject: Re: Help...Never seen this before
X-No-Archive: yes
><vaa12059iej4bdbdl1u724hs90ac0ev5q4@4ax.com>:
> I have been given this math problem and i have never seen
this symbol
> before...it would be greatly appreciated if someone would
lend their
> advice:
 (superscript) 4
> Find the value of: (uppercase pi symbol) (2k + 1)
> k=0
 sorry for the complicated layout, i know of no way to insert
the
> symbols
I believe you are referring to a product symbol (not sure of
the exact
>terminology for it).
>So, what you've described would mean to multiply all values
of the
>expression (2k+1) where k takes on each of the values from 0
up to 4.
>so:
>[2(0)+1][2(1)+1][2(2)+1][2(3)+1][2(4)+1]
>like that.
Sheila, I'm wondering if Potterwasp isn't referring to a
summation,
uppercase sigma. Either that, or educate me as to where this
sort of
operation, a progressive product might be used.
Please. :-)
--
charlie dick
The right to be left alone -- the most comprehensive
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Subject: Re: Help...Never seen this before
> Sheila, I'm wondering if Potterwasp isn't referring to a
summation,
> uppercase sigma. Either that, or educate me as to where this
sort of
> operation, a progressive product might be used.
> Please. :-)
> --
> charlie dick
> The right to be left alone -- the most comprehensive
> of rights, and the right most valued by a free people.
> - Justice Louis Brandeis, Olmstead v. U.S. (1928).
****************************************
A nice usage for the capital Pi, product notation is in
elementary number
theory.
For instance, let p_1, p_2, ... represent the infinite
sequence of prime
numbers.
Every positive integer m can then be written in the form m =
Pi(i=1 to
infinity) p_i^(e_i) where the e_i are nonnegative integer
exponents, only
finitely many of which are nonzero.
If n = Pi(i=1 to infinity) p_i^(f_i) is another positive
integer, then we
can write
gcd(m, n) = Pi(i=1 to infinity) p_i^(min{e_i, f_i})
and
lcm(m, n) = Pi(i=1 to infinity) p_i^(max{e_i, f_i}).
Please excuse the Pi(i=1 to infinity) expression; in
mathematical
typography this would be written entirely similar to
capital-sigma
notation, with the sigma replaced by capital Pi, of course.
--- Joe
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Subject: Re: Help...Never seen this before
X-No-Archive: yes
><vaa12059iej4bdbdl1u724hs90ac0ev5q4@4ax.com>:
>I have been given this math problem and i have never seen
this symbol
>before...it would be greatly appreciated if someone would
lend their
>advice:
 (superscript) 4
>Find the value of: (uppercase pi symbol) (2k + 1)
> k=0
> Sheila, I'm wondering if Potterwasp isn't referring to a
summation,
> uppercase sigma. Either that, or educate me as to where this
sort of
> operation, a progressive product might be used.
> Please. :-)
Hello Charlie,
Since she said pi and not sigma I doubt she is referring to
summation.
It really is hard to find an example of this on Google, since
I have no
idea what to call it. To my recollection, the best I can
remember is
product. lol. But of course that returns many other results as
well.
In any case I have found an example for you.
http://www.math.rutgers.edu/courses/573A/573-f00/S1/plnt0573.
pdf
See page five of the above document.
For one example, this is used in numerical methods for
interpolation of
polynomials. There are many uses for it in higher mathematics.
--
Sheila King
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Subject: Re: Help...Never seen this before
<vaa12059iej4bdbdl1u724hs90ac0ev5q4@4ax.com>:
I have been given this math problem and i have never seen this
symbol
> before...it would be greatly appreciated if someone would
lend their
> advice:
 (superscript) 4
> Find the value of: (uppercase pi symbol) (2k + 1)
> k=0
> Sheila, I'm wondering if Potterwasp isn't referring to a
summation,
> uppercase sigma. Either that, or educate me as to where this
sort of
> operation, a progressive product might be used.
 Please. :-)
 Hello Charlie,
> Since she said pi and not sigma I doubt she is referring to
summation.
> It really is hard to find an example of this on Google,
since I have no
> idea what to call it. To my recollection, the best I can
remember is
> product. lol. But of course that returns many other results
as well.
> In any case I have found an example for you.
>
http://www.math.rutgers.edu/courses/573A/573-f00/S1/plnt0573.
pdf
> See page five of the above document.
> For one example, this is used in numerical methods for
interpolation of
> polynomials. There are many uses for it in higher
mathematics.
In bioinformatics, we use the product symbol, an uppercase pi,
a lot.
We often have to take the joint probability of many events,
which (if
you assume independence) is the product of the probability of
the
individual events. I also get a lot of products when dealing
with
Dirichlet distributions, which are the most popular prior
distribution
for discrete alphabets in Bayesian statistics, since they form
a
conjugate prior (the posterior probability distribution is
from the
same family of functions as the prior probability
distribution, with a
particularly simple change of parameters).
See
http://www.soe.ucsc.edu/research/compbio/dirichlets/
dirichlet-papers.html
for a pointer to a paper on Dirichlet mixtures that has a lot
of
product symbols.
One also gets products a lot when doing combinatorics, though
many of
them can be re-written in terms of the factorial function and
the
binomial coefficients.
The upper-case pi is a completely standard and extremely useful
notational tool---it should be taught in pre-calculus classes
at
about the same time that ln(x) and exp(x) are taught.
--
Kevin Karplus karplus@soe.ucsc.edu
http://www.soe.ucsc.edu/~karplus
life member (LAB, Adventure Cycling, American Youth Hostels)
Effective Cycling Instructor #218-ck (lapsed)
Professor of Computer Engineering, University of California,
Santa Cruz
Undergraduate and Graduate Director, Bioinformatics
Affiliations for identification only.
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Subject: factoring quadratic equations
how do you factor an equation such as
15s^2 minus 16st plus 4t^2
I hope that that makes sense. Please if someone could just
list the
steps or somehow explain it, that would be very helpful
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Subject: Re: factoring quadratic equations
> how do you factor an equation such as
> 15s^2 minus 16st plus 4t^2
> I hope that that makes sense. Please if someone could just
list the
> steps or somehow explain it, that would be very helpful
Since 15*4 = 60
and
(-10)(-6) = 60
and
(-10) + (-6) = -16
Then re-write your expression as:
15 s^2 + (-10)st + (-6)st + 4 t^2
Now factor by grouping:
5s(3s - 2t) + -2t(3s - 2t)
(3s - 2t)(5s - 2t)
Check the above answer by multiplying it back out together to
see if it
gives the original expression.
Or, some people are just good at guessing/seeing the numbers
needed and can
go pretty much straight from the original expression you
provided to the
answer. I think to some extent, that comes with practice.
--
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Subject: Re: factoring quadratic equations
> how do you factor an equation such as
> 15s^2 minus 16st plus 4t^2
> I hope that that makes sense. Please if someone could just
list the
> steps or somehow explain it, that would be very helpful
that is just the product of two binomials in the form of:
(as + bt) x (cs - dt)
to factor this, you will have to create all the possible
combinations of
factors of 15 (for the s term) and 4 (for the t term) or in
another way,
the a and c will be factor pairs of 15 while b and d will be
factor pairs
of 4. It is basically trial and error until you get the right
combo to
multiply out to the one you seek. the work is left for you.
Steve
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Subject: Re: factoring quadratic equations
X-No-Archive: yes
>how do you factor an equation such as
>15s^2 minus 16st plus 4t^2
15s^2 = 15s * s or 5s * 3s
4t^2 = 4t * t or 2t * 2t
It follows that
(15s - 4t)(s - t) = 15s^2 - 19st + 4t^2 ... Nup
(15s - t)(s - 4t) = 15s^2 - 61st + 4t^2 ... Strike Two
(5s - 2t)(3s - 2t) = 15s^2 -16st + 4t^2 ... Eureka!
hth
--
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- Justice Louis Brandeis, Olmstead v. U.S. (1928).
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Subject: Middle school math
I am currently a University student majoring in elementary
education.
I am very interested in teaching math in the middle schools;
grades 6
through 8. If anyone happens to know about what topics of math
this
age group learns about, I would love to know. I have tutored a
few
students from various cities, and it seems as though the topics
covered differ greatly depending on what school district you
are in.
Any information on general math knowledge in these grades, as
well as
any good websites or lesson plans I could look at, would me
very much
appreciated.
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Subject: Re: Middle school math
> I am currently a University student majoring in elementary
education.
> I am very interested in teaching math in the middle schools;
grades 6
> through 8. If anyone happens to know about what topics of
math this
> age group learns about, I would love to know. I have tutored
a few
> students from various cities, and it seems as though the
topics
> covered differ greatly depending on what school district you
are in.
> Any information on general math knowledge in these grades,
as well as
> any good websites or lesson plans I could look at, would me
very much
> appreciated.
In the US, the NCTM (National Council of Teachers of
Mathematics)
provides a set of standards upon which most states (and most
textbook
publishers) base their own standards. They are subdivided by
grade
ranges as well as strand (Algebra, Geometry, Measurement,
etc.) You can
get an overview of these Principles & Standards for School
Mathematics
at this site: http://standards.nctm.org/document/index.htm.
I teach in Massachusetts, where our curriculum frameworks are
sliced in
several ways; for example, you can see them by grade range, or
by
strand, or by course (e.g., Algebra I). You can download the MA
frameworks from here:
http://www.doe.mass.edu/frameworks/current.html
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===
Subject: Re: Middle school math
Try a webinternet search for Mathematics Standards for your
State.
The topics should include Number Sense, Measurement,
Probablility and Data
(maybe measurement and data), Algebra, Geometry. The topics
will
probably
EXCLUDE Intermediate Algebra, Trigonometry, and Calculus.
G C
Trakakaka1@aol.com wants to know:
>I am currently a University student majoring in elementary
education.
>I am very interested in teaching math in the middle schools;
grades 6
>through 8. If anyone happens to know about what topics of
math this
>age group learns about, I would love to know. I have tutored
a few
>students from various cities, and it seems as though the
topics
>covered differ greatly depending on what school district you
are in.
>Any information on general math knowledge in these grades, as
well as
>any good websites or lesson plans I could look at, would me
very much
>appreciated.
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Subject: Re: Middle school math
Kaija <Trakakaka1@aol.com> asked,
> I am very interested in teaching math in the middle schools;
grades 6
> through 8. If anyone happens to know about what topics of
math this
> age group learns about, I would love to know. I have tutored
a few
> students from various cities, and it seems as though the
topics
> covered differ greatly depending on what school district you
are in.
Nothing can cure the variance from school district to school
district in
the
United States, but you can prepare now to teach middle school
mathematics
WELL, wherever you end up teaching.
A good source of reasonable standards for different age ranges
is
http://www.cde.ca.gov/cdepress/standards-pdfs/mathematics.pdf
which is based on reasonable international standards.
> Any information on general math knowledge in these grades,
as well as
> any good websites or lesson plans I could look at, would me
very much
> appreciated.
site
http://math.berkeley.edu/~wu
as soon as possible. That will lead you to more good reading.
Hope this helps!
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Subject: TouchMoney
I am a special education teacher in Boulder, CO. and am
interested in
getting my hands on a copy of Touch Money. Can you steer me in
the
right direction?
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Subject: times tables
there aren't any strategies per say just have them stand in
class and
say them other than rewarding them with a trinket it is just
something
they need to do. You do them a great disservice by not fully
impressing on them the joy in knowing these facts. Rote is not
a bad
word. Practice and repition are not evil concepts.
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Subject: Re: times tables
Practice and repition my not be evil concepts in your
classroom, but
as a high school math teacher of students who have no concept
of the
multiplication facts, they must be evil concepts in the
elementary
classrooms in my district. I do think there is a way to make
learning
the facts rewarding and not just a boring rote process. I my
room,
many rewards are used to encourage learning new concepts,
because most
of my students did not find the joy of knowing these facts.
> there aren't any strategies per say just have them stand in
class and
> say them other than rewarding them with a trinket it is just
something
> they need to do. You do them a great disservice by not fully
> impressing on them the joy in knowing these facts. Rote is
not a bad
> word. Practice and repition are not evil concepts.
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Subject: Prove that 7^203 ends in 07
hello,
this ia my math homework. been trying to prove it but failed.
basically, they told us that the last two digit of
7^2 => 49
7^3 => 43
7^4 => 01
7^5 => 07
now i have to prove that the last two digits of
7^201 is 07
help!
-thanks
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Subject: Re: Prove that 7^203 ends in 07
> hello,
> this ia my math homework. been trying to prove it but failed.
> basically, they told us that the last two digit of
> 7^2 => 49
> 7^3 => 43
> 7^4 => 01
> 7^5 => 07
> now i have to prove that the last two digits of
> 7^201 is 07
7^203
35884 10533920824 51103148256 68531670 853179816 727083901
45362898
0488240 48138226 5219523586132369 1391186 9250521380 15062
33857617
0367085788 347177 753570871369 75570713 80647449160343
7^201
732328680 39200500225 13229728 271769561 8734319851 191635
6196509805
882458803719 698357624 2067830436554 8352051456737 767824199218
442262405 78513831705527 9871378715742 462192840007
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Subject: Re: Prove that 7^203 ends in 07
> this ia my math homework. been trying to prove it but failed.
> basically, they told us that the last two digit of
> 7^2 => 49
> 7^3 => 43
> 7^4 => 01
> 7^5 => 07
> now i have to prove that the last two digits of
> 7^201 is 07
Here you say you have to prove 7^201 ends in 7 but in the
subject line
you say 7^203 ends in 7. FWIW, it is 7^201 that ends in 7.
Rich
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Subject: Re: Prove that 7^203 ends in 07
As Mr. Karplus says, do a few powers of seven, either by hand
(since
all you need to do is multiple each result by 7 to get the
subsequent
one, and it's good practice) or with your calculator.
Now, the degree of proof required will vary, based on what the
level of
your course is, but the important thing is to be able to
extract and
explain the generality, and use an earlier instance of it to
show that
it will repeat forever.
-Bill
> hello,
> this ia my math homework. been trying to prove it but failed.
> basically, they told us that the last two digit of
> 7^2 => 49
> 7^3 => 43
> 7^4 => 01
> 7^5 => 07
> now i have to prove that the last two digits of
> 7^201 is 07
> help!
> -thanks
--
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http://www.billstevenson.org
Applied Cognitive Science Laboratory
Pennsylvania State University
http://acs.ist.psu.edu
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Subject: Re: Prove that 7^203 ends in 07
> hello,
> this ia my math homework. been trying to prove it but failed.
> basically, they told us that the last two digit of
> 7^2 => 49
> 7^3 => 43
> 7^4 => 01
> 7^5 => 07
> now i have to prove that the last two digits of
> 7^201 is 07
Hint: what does 7^6 end with? 7^7? Is there a pattern? Can you
prove
that the pattern continues?
--
Kevin Karplus karplus@soe.ucsc.edu
http://www.soe.ucsc.edu/~karplus
life member (LAB, Adventure Cycling, American Youth Hostels)
Effective Cycling Instructor #218-ck (lapsed)
Professor of Computer Engineering, University of California,
Santa Cruz
Undergraduate and Graduate Director, Bioinformatics
Affiliations for identification only.
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Subject: Fraction Strips
Try www.teachervision.com and click on printables
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Subject: Erdos Number Project update
The bi-annual update of the Erdos Number Project web site has
been
completed. The URL is
http://www.oakland.edu/~grossman/erdoshp.html
Briefly, this project studies the subject of collaboration in
mathematical research in general and the collaborations of
Paul Erdos
(1913-1996) in particular. On the web site are the list of the
509
people who have written joint papers with Erdos (they have
Erdos number
1) and the lists of their other coauthors (a total of 6984 of
them --
the people with Erdos number 2). The data are organized in
several
ways, and various statistics are summarized. We also include
an update
to the complete bibliography of Erdos (numbering more than
1500 papers),
preprints of papers about this subject, and dozens of links to
related
material, including the flurry of recent serious research
activity on
collaboration graphs.
Further information is contained in the README file, available
at the
site.
As always, we want to know about corrections, additions, and
related
information.
--
Jerrold W. Grossman, Professor VOICE: (248) 370-3443
Department of Mathematics and Statistics FAX: (248) 370-4184
Oakland University FLESH: 346 SEB
Rochester, MI 48309-4485 E-MAIL: grossman@oakland.edu
WEB: http://www.oakland.edu/~grossman/
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Subject: Integration - when it gets bad
Hi there,
I'm trying to do what looks like a simple integral but can't
find any
trigonometric relationships to do it. Does anyone know how to
do it?
I= (sin3x)/(1+cosx)
Any help would be much appreciated
SW
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Subject: an integral
> I = Integral((sin3x)/(1+cosx))dx
sin(3x) = sin(x)cos(2x) + sin(2x)cos(x) =
sin(x)*(2cos(x)*cos(x) - 1) + 2sin(x)cos(x)*cos(x) =
sin(x)*(4cos(x)*cos(x) - 1),
so
I = -Integral((4t^2 - 1)/(t+1))dt, where t = cos(x)
Alexander Bogomolny
http://www.cut-the-knot.org
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Subject: Re: Integration - when it gets bad
> I'm trying to do what looks like a simple integral but can't
find any
> trigonometric relationships to do it. Does anyone know how
to do it?
> I= (sin3x)/(1+cosx)
Are you sure you have the problem copied right?
The formula
sin(3x) / (1+ 2* cos(x))
simplifies much more easily.
--
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http://www.soe.ucsc.edu/~karplus
life member (LAB, Adventure Cycling, American Youth Hostels)
Effective Cycling Instructor #218-ck (lapsed)
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===
Subject: Integration - when it gets bad
Sarah Watt asked for suggestions on how to integrate
sin(3x)/[1+cos(x)]
Recall the addition formula for sin(a+b) and for cos(a+b). When
these are used, sin(3x)=sin(x)*[4cos^2(x)-1]. Then a
substitution
reduces this to the integral of a rational function.
Dick Askey
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Subject: Help with Daughter's Math Problem.
My daughter called me and asked for my help with a math
problem. I'm
usually pretty good with helping her but this problem threw me
for a
loop. The problem is:
What's In Store?
Store 88 is on the corner of 8th Street and 8th Avenue. But
the store
sells more than 8 things. In fact, it sells exactly 10 things.
Some items may cost the same amount as other items in the
store, but
only 8's appear on the price tags of everything in the store.
And, if
you bought one of each kind of thing, your total bill would be
$1,000!
What are the prices of the 10 things sold in Store 88?
I'm not seeing something here. Any suggestions?
Wade
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Subject: Re: Help with Daughter's Math Problem.
Not sure if my previous reply is stuck in cyberspace or what,
but the
prices are
1x $888.88
1x $88.88
2x $8.88
5x 88 cents
1x 8 cents
> My daughter called me and asked for my help with a math
problem. I'm
> usually pretty good with helping her but this problem threw
me for a
> loop. The problem is:
> What's In Store?
> Store 88 is on the corner of 8th Street and 8th Avenue. But
the store
> sells more than 8 things. In fact, it sells exactly 10
things.
> Some items may cost the same amount as other items in the
store, but
> only 8's appear on the price tags of everything in the
store. And, if
> you bought one of each kind of thing, your total bill would
be
> $1,000!
> What are the prices of the 10 things sold in Store 88?
> I'm not seeing something here. Any suggestions?
> Wade
--
Bill Stevenson
Editor in Chief
ACM Crossroads Magazine
http://www.acm.org/crossroads
http://www.billstevenson.org
Applied Cognitive Science Laboratory
Pennsylvania State University
http://acs.ist.psu.edu
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Subject: Re: Help with Daughter's Math Problem.
> Store 88 is on the corner of 8th Street and 8th Avenue. But
the store
> sells more than 8 things. In fact, it sells exactly 10
things.
> Some items may cost the same amount as other items in the
store, but
> only 8's appear on the price tags of everything in the
store. And, if
> you bought one of each kind of thing, your total bill would
be $1,000!
> What are the prices of the 10 things sold in Store 88?
Let's see.....there are exactly 10 different items, some of
which may
have the same price. Only 8s appear on the price tags (and all
the
items are less than $1000, obviously) so you can have the
following prices:
8 cents
88 cents
$8.88
$88.88
$888.88
the answer, but then I looked over it and realized it would be
a real
pain to understand, so I'll do it an easier way :-)
If all of the items cost $88.88, we would be short of $1000,
so there
must be at least one $888.88 item (and not more than one, for
obvious
reasons) So we have one item that costs $888.88.
Now we have $111.12 to spend. If all the remaining items cost
$8.88, we
would come to less than that, so we must have at least one
$88.88 item
(again, we can't have more than one, or we'd go over the
limit. So we
just have one)
Now we have two items (one cost $888.88 and one cost $88.88)
and $22.24
remaining. Again, we can't have all 88 cent items, so at least
one must
cost $8.88. In this case, we can squeeze in two items at that
cost, so
we'll do that, leaving us with $4.48 to spend.
Now we need 6 items to cost $4.48. All 8 cent items won't do
it, but we
can fit 5 88 cent items into it, leaving us with one item and
8 cents.
So the store sells:
1 8 cent item
5 88 cent iutems
2 $8.88 items
1 $88.88 item
1 $888.88 item
I really can't believe I typed out all that...
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Subject: Re: Help with Daughter's Math Problem.
888.88 + 88.88 + 8.88 + 8.88 + .88 + .88 + .88 + .88 + .88 +
.08 ('8
cents') = 1000.00
> My daughter called me and asked for my help with a math
problem. I'm
> usually pretty good with helping her but this problem threw
me for a
> loop. The problem is:
> What's In Store?
> Store 88 is on the corner of 8th Street and 8th Avenue. But
the store
> sells more than 8 things. In fact, it sells exactly 10
things.
> Some items may cost the same amount as other items in the
store, but
> only 8's appear on the price tags of everything in the
store. And, if
> you bought one of each kind of thing, your total bill would
be
> $1,000!
> What are the prices of the 10 things sold in Store 88?
> I'm not seeing something here. Any suggestions?
> Wade
--
Bill Stevenson
Editor in Chief
ACM Crossroads Magazine
http://www.acm.org/crossroads
http://www.billstevenson.org
Applied Cognitive Science Laboratory
Pennsylvania State University
http://acs.ist.psu.edu
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Subject: Re: Help with Daughter's Math Problem.
Had they had only 5 things, the prices might have been
$888, $88, $8, $8, $8.
Alexander Bogomolny
http://www.cut-the-knot.org
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Subject: Re: Help with Daughter's Math Problem.
> My daughter called me and asked for my help with a math
problem. I'm
> usually pretty good with helping her but this problem threw
me for a
> loop. The problem is:
> What's In Store?
> Store 88 is on the corner of 8th Street and 8th Avenue. But
the store
> sells more than 8 things. In fact, it sells exactly 10
things.
> Some items may cost the same amount as other items in the
store, but
> only 8's appear on the price tags of everything in the
store. And, if
> you bought one of each kind of thing, your total bill would
be $1,000!
> What are the prices of the 10 things sold in Store 88?
> I'm not seeing something here. Any suggestions?
> Wade
***********************************************
First a question: What grade is your daughter in?
Now, some hints:
Prices like $8,888.88 , $888.88, $88.88, $8.88, $.88 seem to
be valid, per
the description above.
Of course, $8,888.88 is way too big to be be part of a
$1,000.00 total
bill.
8 cents would seem to be invalid if it were written as $.08,
but valid if
the price tag were written 8 cents-symbol.
Try to reason it out:
Let A be the number of $888.88 items purchased
Let B be the number of $88.88 items purchased
Let C be the number of $8.88 items purchased
Let D be the number of $.88 items purchased.
For intance, A is at most 1.
If A = 1, then you are left with buying 9 items for $1000.00 -
$888.88 =
$111.12.
Continue, using $111.12 as your new starting point, and see
what
happens. It *may* turn out that A has to be zero.
It all worked out on the first try, but I had to use 8 cents
as a valid
price.
Please accept my sympathies. :) Have fun.
--- Joe
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Subject: Re: Limit that I can't find the answer
> Could anyone help with this one? My professor came out with a
> problem that I'm having trouble solving. It's not a homework
> assignment or a test question, it was just a challenge to the
> class. The problem is: find the limit as x -> 0 for
(1-cos(x))/x.
As another person responded, in Calculus L'Hopital's rule is
used to
find limits of this type. I am curious though. Did your
professor expect
you to evaluate the above expression for several values of x
that get
closer and closer to 0?
Dom Rosa
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Subject: 8th grade
Hi there,
I'm going to be teaching a lesson on slope for my methods
class next week.
The teacher tells me the kids are average, and doing ok in
their textbook.
My question is when is it appropriate to introduce Greek
letters?
Since I'm going to be talking about calculating slope, of
course the words
change in x and change in y are going to come up a lot.
Do you think they will understand if I tell them that DELTA x
means change
in x?
John
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Subject: Re: 8th grade
> Hi there,
> I'm going to be teaching a lesson on slope for my methods
class next
week.
> The teacher tells me the kids are average, and doing ok in
their
textbook.
> My question is when is it appropriate to introduce Greek
letters?
> Since I'm going to be talking about calculating slope, of
course the
words
> change in x and change in y are going to come up a lot.
> Do you think they will understand if I tell them that DELTA
x means
change
> in x?
The notation
f(x + h) - f(x) k
--------------- = -
(x + h) - x h
is _much_ better than
f(x + delta x) - f(x) delta y
---------------------- = --------
x + delta x - x delta x
But you should briefly explain the use of delta in case they
see it
elsewhere.
--
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Subject: Re: 8th grade
I wouldn't use delta. I use rise over run, y's over x's,
change in
verticle over change in horizontal.
> Hi there,
> I'm going to be teaching a lesson on slope for my methods
class next
week.
> The teacher tells me the kids are average, and doing ok in
their
textbook.
> My question is when is it appropriate to introduce Greek
letters?
> Since I'm going to be talking about calculating slope, of
course the
words
> change in x and change in y are going to come up a lot.
> Do you think they will understand if I tell them that DELTA
x means
change
> in x?
> John
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Subject: 8th grade
I think you should introduce the class to Greek letters. If you
explain it carefully, then they should have no problems. Just
tell
them it's a short cut to saying change in x.
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Subject: Re: 8th grade
I would say that the greek letter delta is appropriate to
teach to 7th
or 8th grade algebra students. As long as you are clear when
you
present it, explain exactly what it means and then continue to
use
correct terminology, everything should be fine.
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Subject: Re: 8th grade
Well I tried it.. and it bombed bad. The students had no clue
what I was
talking about....I tried to explain that is was just an
abbreviation in
this
case.. and it didn't work
teacher tells me they are low kids... so I don't know if it
was me or
not
I guess I learned something...
> I would say that the greek letter delta is appropriate to
teach to 7th
> or 8th grade algebra students. As long as you are clear when
you
> present it, explain exactly what it means and then continue
to use
> correct terminology, everything should be fine.
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Subject: Re: 8th grade
> Well I tried it.. and it bombed bad. The students had no
clue what I was
> talking about....I tried to explain that is was just an
abbreviation in
this
> case.. and it didn't work
> teacher tells me they are low kids... so I don't know if it
was me or
not
> I guess I learned something...
You've told us too different things here: that they are
average and now
that
they are low.
8th grade is certainly not too soon to introduce select Greek
letters,
which
is something they will see only more of in the next and coming
years. Even
low kids need to take algebra, surely. Better for them to
stumble a
little now that to fall flat on their faces when it really
matters; when
you
won't have any /choice/ but to show them the Greek. And if
their in 8th
grade now, that means next year probably, I guess?
Even apart from the Greek notation, the concept of 'delta' may
not be that
new to some of them. It is not all that uncommon to hear of
changes
referred to as deltas, e.g. Discovery...Houston, we have a
Delta for your
checklist on p.11... that's just one common ex. you can
probably think of a
few others.
It's just a symbol, FCOL, just like 'x' and 'y' and '+' and
'-'. Do you
shy
away from exposing them to pi just because its a Greek letter
too? I hope
not. I wouldn't necessarily go out of the way to introduce
this specific
letter, but if the situation presented itself you should not
avoid it (if
you do, you should be able to easily answer a self question of
why do so
many kieds have trouble w/ algebra?
--
Darrell
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Subject: Re: 8th grade
results from skyrookien@netscape.net:
>Well I tried it.. and it bombed bad. The students had no clue
what I was
>talking about....I tried to explain that is was just an
abbreviation in
this
>case.. and it didn't work
>teacher tells me they are low kids... so I don't know if it
was me or
not
>I guess I learned something...
> I would say that the greek letter delta is appropriate to
teach to 7th
> or 8th grade algebra students. As long as you are clear when
you
> present it, explain exactly what it means and then continue
to use
> correct terminology, everything should be fine.
Do not let the results fool you. Another group of possibly
similar
students
might understand these abbreviations. Often enough, one group
is not the
same
as another group. These groups of students vary from time to
time, and
place
to place.
G C
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Subject: Re: 8th grade
> Hi there,
> I'm going to be teaching a lesson on slope for my methods
class next
week.
> The teacher tells me the kids are average, and doing ok in
their
textbook.
> My question is when is it appropriate to introduce Greek
letters?
> Since I'm going to be talking about calculating slope, of
course the
words
> change in x and change in y are going to come up a lot.
> Do you think they will understand if I tell them that DELTA
x means
change
> in x?
> John
Personally, if I were teaching it, I'd say something like The
slope of
the
line is the change of y over the change in x, however, you
might see it
written like this... and show them the one with the deltas.
David Moran
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Subject: Re: 8th grade
> Hi there,
> I'm going to be teaching a lesson on slope for my methods
class next
week.
> The teacher tells me the kids are average, and doing ok in
their
textbook.
> My question is when is it appropriate to introduce Greek
letters?
> Since I'm going to be talking about calculating slope, of
course the
words
> change in x and change in y are going to come up a lot.
> Do you think they will understand if I tell them that DELTA
x means
change
> in x?
A year or two ago I was explaining slopes to 7th or 8th grade
students
(I forget which)
I asked if they knew was slope was, and they said rise over
run. I
explained how rise over run was the same as change in y over
change in
x, or the amount you go up divided by the amount you go
forward, and
that delta just means change. They all seemed to be ok with it.
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Subject: 9 digit number combination
I have a number combination that is minimum of 4 digits and
maximum is
8 digits and can be 0-9. The numbers can be used mutiple
times. My
question is what are the combinations you can use.
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Subject: Re: 9 digit number combination
<<
I have a number combination that is minimum of 4 digits and
maximum is
8 digits and can be 0-9. The numbers can be used mutiple
times. My
question is what are the combinations you can use.
>
[0-8 digits combinations] - [0-3 digits combinations]
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Subject: Re: 9 digit number combination
So this means you have can go from 0000 - 99999999
> I have a number combination that is minimum of 4 digits and
maximum is
> 8 digits and can be 0-9. The numbers can be used mutiple
times. My
> question is what are the combinations you can use.
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Subject: Re: math
>At 3:20 pm a jeweler set three antique clocks to the correct
time. The
>next afternoon at 3:20,she found that one clock was correct,
one clock
>was two minutes fast and the other was 2 minutes slow. At
those rates,
>how long will it take for all three clocks to show 3:20 again
>love Jayda
Let's see. Since one is going forward and one backwards at the
same
pace I think I can ignore one and still get the right answer.
Basically one clock has to race completely around the dial to
meet the
original 3:20. There are 720 minutes in a day. @ 2 minutes per
day it
would take 360 days. But we know the jeweler has already made
one
observation, so I'd say 359 days.
I'm I way off?
JD
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Subject: New jokes on math
New jokes on math added on www.bymath.com
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Subject: Defining percent in terms of ratio
Hi --
I'm proofreading translations of a math curriculum from
English to
another language. The abovementioned phrase appears in a list
of
skills the student should possess at a certain point in the
program,
and I want to make sure I have the meaning right.
Does this mean that given a percentage, the student can
restate it in
the form of a ratio -- e.g., turning 20% into 1:5? Or am I
totally
off the mark?
Thank you in advance for your help.
--
Steven
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Subject: Re: Defining percent in terms of ratio
> Hi --
> I'm proofreading translations of a math curriculum from
English to
> another language. The abovementioned phrase appears in a
list of
> skills the student should possess at a certain point in the
program,
> and I want to make sure I have the meaning right.
> Does this mean that given a percentage, the student can
restate it in
> the form of a ratio -- e.g., turning 20% into 1:5?
Yes. Specifically a% is the ratio a:100, although the ratio
may reduce, eg
20:100 = 1:5.
IIRC the English percent is derived from the Latin per centum,
meaning
per 100, or one part in 100, or some such.
--
Darrell
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Subject: Re: Defining percent in terms of ratio
>IIRC the English percent is derived from the Latin per centum,
meaning
>per 100, or one part in 100, or some such.
Sidebar, but not an etymology:
cent - cents in a dollar, years in a century, legs on a
centipede, Roman
numeral for 100, ...
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Subject: Re: Defining percent in terms of ratio
appropriate change and handed in the translation.
--
Steven
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Subject: Re: Defining percent in terms of ratio
Hi --
I'm proofreading translations of a math curriculum from
English to
another language. The abovementioned phrase appears in a list
of
skills the student should possess at a certain point in the
program,
and I want to make sure I have the meaning right.
Does this mean that given a percentage, the student can
restate it in
the form of a ratio -- e.g., turning 20% into 1:5?
> Yes. Specifically a% is the ratio a:100, although the ratio
may reduce,
eg
> 20:100 = 1:5.
> IIRC the English percent is derived from the Latin per
centum,
meaning
> per 100, or one part in 100, or some such.
> --
> Darrell
Good points. I've always taught percents as you mention,
explaining
what per and cent means, and that the equation a/100 = b/c is
involved, somehow someway in a percentage problem. I have them
write
this equation down in the process of any percentage problem.
Especially, so as to see what of the given information is to
replace
c, as to what corresponds to the 100 in the denominator. Then
they can
see through using algebra how and why of usually means
multiply:
Solving for instance for b yields, in words, b is a percent of
c.
It seems to me that the above provides a nice algebraic
framework and
approach to percentage problems, an approach that could be
used in a
pre-algebra framework quite easily.
Paul
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Subject: Re: Defining percent in terms of ratio
In English, your interpretation is correct. You should have no
trouble with
it
no matter what language you are translating.
>I'm proofreading translations of a math curriculum from
English to
>another language. The abovementioned phrase appears in a list
of
>skills the student should possess at a certain point in the
program,
>and I want to make sure I have the meaning right.
>Does this mean that given a percentage, the student can
restate it in
>the form of a ratio -- e.g., turning 20% into 1:5? Or am I
totally
>off the mark?
>Thank you in advance for your help.
>--
>Steven
>--
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Subject: Re: Limit that I can't find the answer
I am curious what might be a relevant homework problem if this
one is
a challenge?
Was there's a discussion about sin(x)/x as x->0?
Alexander Bogomolny
http://www.cut-the-knot.org
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Subject: Re: Limit that I can't find the answer
go to
http://lzkiss.netfirms.com/cgi-bin/igperl/igp.pl#
cut and paste this 'program' onto the empty textarea:
ZOOM(-pi,0,pi,1,1);
graph(
coord;
func=sin(x)/x;
);
and press submit.
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Subject: mechanics
A boy is tobogganing down a slope inclined at 25* to the
horizontal.
The ristances to his motion amounts to 15N. By modelling the
toboggan
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Subject: Re: mechanics
Remember that F = ma
However it's not as simple as 15N = m * 3.9 .
You need to consider the angle of the hill. Draw a picture and
label
everything! Don't forget the normal force and gravity.
Finally, you need to use a basic trig function to get your
answer.
Good Luck,
John
> A boy is tobogganing down a slope inclined at 25* to the
horizontal.
> The ristances to his motion amounts to 15N. By modelling the
toboggan
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Subject: Math tutors
My name is Demi, I majored in Math and just recently graduated
from
Central Washington University. I am currently residing in the
Seattle
Area, Kirkland Washington and would love to tutor again. I love
tutoring kids, and I seem to do very well at it!! I have
tutoring
experience and the children I have tutored always seem to
understand
the way I teach math. If you know of anyone who needs a math
tutor
please let me know, the websit that I was referred to has
apparently
moved, I do not know if you are still placing tutors but just
in case
Demi Marsh
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Subject: Re: Math tutors
>My name is Demi, I majored in Math and just recently
graduated from
>Central Washington University. I am currently residing in the
Seattle
>Area, Kirkland Washington and would love to tutor again. I
love
>tutoring kids, and I seem to do very well at it!! I have
tutoring
>experience and the children I have tutored always seem to
understand
>the way I teach math. If you know of anyone who needs a math
tutor
>please let me know, the websit that I was referred to has
apparently
>moved, I do not know if you are still placing tutors but just
in case
>Demi Marsh
You might want to go to the high schools and see if they will
let
you put up signs. There are a number of kids taking AP
Statistics
and AP Calculus at LW, and I understand that tutors for them
are hard
to find. The math teachers often get asked for recommendations
for
tutors, so talking to the department chairs might help, too.
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Subject: Re: Math tutors
I am currently residing in the Seattle Area, Kirkland
Washington and
would love to tutor
<snip You might want to go to the high schools and see if they
will let
> you put up signs.
Also talk to local colleges and universities. People ask them
for tutor
recommendations (even for lower grades).
If you want to advertise on Usenet, try local groups. perhaps
seattle.jobs.wanted, wa.ads.commercial, wa.jobs,
seattle.general, or
wa.general, if their respective charters/FAQs allow it.
msh210@math.wustl.edu Of a reply, then, if you have been
cheated,
http://math.wustl.edu/~msh210/ Likely your mail's by mistake
been deleted.
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Subject: TI-84
I wanted to get some idea about the evolution of graphing
calcs in high
school classrooms.
When I was in high school the TI-82 was the latest and
greatest....I have
one of the first ones with the Yellow printing on it. Lately
is has started
to turn it self off whenever it wants to (it's going on 10
years old now).
Not sure what the problem is, but it got me into looking at
new calcs. I
will begin teaching math in the Spring 2005 semester, and I
know all the
high school students are required these days to have a
graphing calc.
How long does it usually take for a new model to make its way
into
classrooms? I missed the whole TI-83 life cycle since I was
out of the math
loop for a while.
Is it safe to get the TI-84 when it comes out in the summer,
or will most
of
my students still be using the TI-83?
John
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Subject: Re: TI-84
> Is it safe to get the TI-84 when it comes out in the summer,
or will most
of
> my students still be using the TI-83?
The 84 will be keyboard and function identical to the 83. It
will run
the same programs and do the same stuff the same way. It
mainly just
adds a clock, a USB port (bye bye graphlink!), and the
overhead display
connection as standard features.
So, yeah, you'll be safe.
Matt T
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Subject: Re: TI-84
I think the TI-83 Plus is still the gold standard in a high
school
classroom. It's the calc for which there are the most
resources,
programs, etc.; and also the one that most textbooks use. I
still
encourage my students to get the TI-83, and I have one (albeit
the
Silver edition) because that's what I need to be most familiar
with.
Good luck -
Lisa
> I wanted to get some idea about the evolution of graphing
calcs in high
> school classrooms.
> When I was in high school the TI-82 was the latest and
greatest....I have
> one of the first ones with the Yellow printing on it. Lately
is has
started
> to turn it self off whenever it wants to (it's going on 10
years old
now).
> Not sure what the problem is, but it got me into looking at
new calcs. I
> will begin teaching math in the Spring 2005 semester, and I
know all the
> high school students are required these days to have a
graphing calc.
> How long does it usually take for a new model to make its
way into
> classrooms? I missed the whole TI-83 life cycle since I was
out of the
math
> loop for a while.
> Is it safe to get the TI-84 when it comes out in the summer,
or will most
of
> my students still be using the TI-83?
> John
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Subject: Graphing books/tools
My 13 year old daughter was given an assigment to create a
graph and the
teacher gave them the data and the x and y axis requirements -
but she
failed to teach them how to do it on the computer. She spent
quite some
time
on it and her friends all had the same issue - they could get
the data in
but the axis attributes were wrong. I figured it out this
morning but my
question is this - is there some book/web site that will help
kids with
charting. Where do you put the data in the spreadsheet - top
row, sides,
etc. What the different type of charts display best - in their
terms and
how
to format the x/y axis unit values. Pointers appreciated.
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Subject: Re: Graphing books/tools
> My 13 year old daughter was given an assigment to create a
graph and the
> teacher gave them the data and the x and y axis requirements
- but she
> failed to teach them how to do it on the computer. She spent
quite some
time
> on it and her friends all had the same issue - they could
get the data in
> but the axis attributes were wrong. I figured it out this
morning but my
> question is this - is there some book/web site that will
help kids with
> charting. Where do you put the data in the spreadsheet - top
row, sides,
> etc. What the different type of charts display best - in
their terms and
how
> to format the x/y axis unit values. Pointers appreciated.
I might be going out on a limb here, but maybe she wanted them
to do
something wacky like, I dunno, do it by hand?
Matt T
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Subject: Re: Graphing books/tools
nope - she said they should use the computer and it was for
science class.
The teacher did go over some basics the next day but any kid
who did not
even attempt the assignment had a rubber stamp on their
forehead (according
to my daughter) saying something about not trying. I take that
at face
value. Her teacher is a dynamo who at the beginning said that
earth science
is a very boring subject but she intented to get the kids
excited. Problem
is that several of her teachers this year are extremely good
and each think
that they are the only teacher of the student. The homework
burden this
year
My 13 year old daughter was given an assigment to create a
graph and
the
teacher gave them the data and the x and y axis requirements -
but she
failed to teach them how to do it on the computer. She spent
quite some
time
on it and her friends all had the same issue - they could get
the data
in
but the axis attributes were wrong. I figured it out this
morning but
my
question is this - is there some book/web site that will help
kids with
charting. Where do you put the data in the spreadsheet - top
row,
sides,
etc. What the different type of charts display best - in their
terms
and
how
to format the x/y axis unit values. Pointers appreciated.
> I might be going out on a limb here, but maybe she wanted
them to do
> something wacky like, I dunno, do it by hand?
> Matt T
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Subject: Re: Graphing books/tools
> nope - she said they should use the computer and it was for
science
class.
> The teacher did go over some basics the next day but any kid
who did not
> even attempt the assignment had a rubber stamp on their
forehead
(according
> to my daughter) saying something about not trying.
Huh. Well, that 's not cool.
> I take that at face
> value. Her teacher is a dynamo who at the beginning said
that earth
science
> is a very boring subject but she intented to get the kids
excited.
Hey, now, I teach earth science (among other things) and I
don't think
it's boting. Well, I get to teach the fun bits of what we put
in earth
science: astronomy and weather. Rocks? Yeah, hard to get
excited about
the rocks.
There's something to be said about teaching-through-discovery,
but you
still need to give them a nudge in the right direction.
Matt T
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Subject: Re: Graphing books/tools
> My 13 year old daughter was given an assigment to create a
graph and the
> teacher gave them the data and the x and y axis requirements
- but she
> failed to teach them how to do it on the computer. She spent
quite some
time
> on it and her friends all had the same issue - they could
get the data in
> but the axis attributes were wrong. I figured it out this
morning but my
> question is this - is there some book/web site that will
help kids with
> charting. Where do you put the data in the spreadsheet - top
row, sides,
> etc. What the different type of charts display best - in
their terms and
how
> to format the x/y axis unit values. Pointers appreciated.
There seems to be two issues here.
1. Did the teacher possibly intend for the students to do the
graph by
hand? What class is this for? What kind of graph? Is it a
business math
type of class and the students are supposed to draw bar charts
or some
such? Or is it an algebra class, and the students are supposed
to graph
an equation, say of a parabola?
2. Assuming that the teacher really did intend that the
students use the
computer, you have my sympathies. If the program to be used is
Microsoft
Excel, I would suggest that you go to a bookstore and look at
the scads of
books on using Excel. Since you already figured out how to do
the graph,
see if any of the books clearly explains the method that you
used.
Or... just give the teacher a call.
There are a number of algebraic/mathematical calculation and
graphing
programs. If it's not clear to you which program is to be
used, then give
the teacher a call.
Now for a rant about Excel:
I used Excel in analyzing some real-world data, and I decided
to graph the
results. There are so many settings and options available that
this
product richly deserves the appellation Bloatware. After a lot
of
effort using the help files and an old Excel book, I finally
was able to
accomplish just about what I wanted to do with the graphical
presentation.
same stuff. It was a three-page tutorial, that probably does
not apply
to anything other than that particular version of Excel.
End of rant about Excel.
Have fun :)
--- Joe
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Subject: Re: Graphing books/tools
analyzing some real-world data, and I decided to graph the
> results. There are so many settings and options available
that this
> product richly deserves the appellation Bloatware. After a
lot of
> effort using the help files and an old Excel book, I finally
was able to
> accomplish just about what I wanted to do with the graphical
presentation.
> same stuff. It was a three-page tutorial, that probably does
not
apply
> to anything other than that particular version of Excel.
My 7th graders use Excel to plot x-y data and even curve fit
using the
wizard. If you have three pages of instruction, something is
very wrong and
it has nothing to do with Excel!
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Subject: Re: Graphing books/tools
My 13 year old daughter was given an assigment to create a
graph and the
teacher gave them the data and the x and y axis requirements -
but she
failed to teach them how to do it on the computer. She spent
quite some
time
on it and her friends all had the same issue - they could get
the data in
but the axis attributes were wrong.
...
The teacher did go over some basics the next day but any kid
who did not
even attempt the assignment had a rubber stamp on their
forehead (according
to my daughter) saying something about not trying.
My 7th graders use Excel to plot x-y data and even curve fit
using the
wizard. If you have three pages of instruction, something is
very wrong and
it has nothing to do with Excel!
Well, Jackson, that sure convinces me! :)
Wouldn't it be ironic if it turns out that Jackson is the
teacher of
STEVEN STEIN's daughter?
--- Joe
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Subject: Re: Graphing books/tools
Usually if you create two columns in Excel and highlight both
columns and
click graph is graphs it for you. Then you can edit the graph.
> My 13 year old daughter was given an assigment to create a
graph and the
> teacher gave them the data and the x and y axis requirements
- but she
> failed to teach them how to do it on the computer. She spent
quite some
time
> on it and her friends all had the same issue - they could
get the data in
> but the axis attributes were wrong. I figured it out this
morning but my
> question is this - is there some book/web site that will
help kids with
> charting. Where do you put the data in the spreadsheet - top
row, sides,
> etc. What the different type of charts display best - in
their terms and
how
> to format the x/y axis unit values. Pointers appreciated.
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Subject: Need a Math Font Which Includes Congruent Symbol
Anyone know where to find a font which includes a congruent
symbol
(looks like the equals symbol with a wave on top)?
I'm typing up some worksheets and notes, and having to always
draw
this one in by hand isn't a big deal, but I know there must be
a font
which includes it.
I've searched the net with no luck... It's driving me BATTY!
SFS
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Subject: Re: Need a Math Font Which Includes Congruent Symbol
OK, after looking through all my fonts and searching the web
forever,
I posted here asking for help finding this symbol.
Within 12 hours I had 13 people respond (here and privately
through
email). In those replies I was shown no less than 7 different
ways I
could get that symbol from fonts currently installed on my
computer,
and several more places to find it on the net.
A windows illiterate walks away ashamed...
; )
SFS
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Subject: Re: Need a Math Font Which Includes Congruent Symbol
Math Type has everything you need.
> Anyone know where to find a font which includes a congruent
symbol
> (looks like the equals symbol with a wave on top)?
> I'm typing up some worksheets and notes, and having to
always draw
> this one in by hand isn't a big deal, but I know there must
be a font
> which includes it.
> I've searched the net with no luck... It's driving me BATTY!
> SFS
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Subject: Math 1 font for common symbols
If you are using MS Word just look under menu Insert/Symbol
and you
will see there a font Math 1 and in it the symbol for
character code
64 is the congruence symbol you wanted.
Go to the Tools/Autocorrect menu and insert into the list:
replace congr by [the char 64 symbol]
Then as you type along every time you type:
[space]congr[space] that
will get replaced by the congruence symbol -- easy and neat.
You can
do the same for all the common symbols you might use. I have a
standard list of about 40 that I use and hand out to students
and
others -- that's more than enough for almost everyone.
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Subject: Re: Need a Math Font Which Includes Congruent Symbol
Using MS Word, I just Insert->Symbol. In the Font pulldown, I
have
Symbols font. I show congruence in the second row, 5th column.
If
it's
not there on yours, it's possible you just need to re-install
the fonts.
Mark
Mark Fowler
Grafton HS (Math)
403 Grafton Dr
Yorktown, VA 23692
(757) 898-0530
> Anyone know where to find a font which includes a congruent
symbol
> (looks like the equals symbol with a wave on top)?
> I'm typing up some worksheets and notes, and having to
always draw
> this one in by hand isn't a big deal, but I know there must
be a font
> which includes it.
> I've searched the net with no luck... It's driving me BATTY!
> SFS
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Subject: Re: Need a Math Font Which Includes Congruent Symbol
Lucinda Sans Unicode has the symbol.
SFS
>Anyone know where to find a font which includes a congruent
symbol
>(looks like the equals symbol with a wave on top)?
>I'm typing up some worksheets and notes, and having to always
draw
>this one in by hand isn't a big deal, but I know there must
be a font
>which includes it.
>I've searched the net with no luck... It's driving me BATTY!
>SFS
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Subject: Re: Need a Math Font Which Includes Congruent Symbol
> Anyone know where to find a font which includes a congruent
symbol
> (looks like the equals symbol with a wave on top)?
> I'm typing up some worksheets and notes, and having to
always draw
> this one in by hand isn't a big deal, but I know there must
be a font
> which includes it.
> I've searched the net with no luck... It's driving me BATTY!
> SFS
I've got two options for you, if you're using Microsoft Word.
The first is to use the Microsoft Equation - go Insert ->
Object to get
started; then near the top left corner you should see a button
that has
a few symbols (mine has a greater than or equal to, a not
equal to, and
a similarity symbol). Click on that and you'll get a bunch of
symbols
you can select.
If you don't have that, you can use the Euclid Symbol font -
go Insert
-> Symbol and select Euclid Symbol. You should see a congruent
symbol
near the top left corner. I don't know if you can download
that font if
you don't have it - it came with my version of Word, I
think....
-Lisa
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Subject: You'll like this method for all math symbols
Using MS Word, you have the option of inserting symbols into
your
document. However, constantly switching back and forth to the
symbol
list can be difficult. Here's what you do: Find the insert
symbol
list, choose a symbol that you use often, and hit the box to
create a
shortcut for it. That will allow you to assign the symbol to
any key
you wish. On my computer I've turned the F2 key into the
infinity
symbol, F3 into the null set, F4 into the square root, etc.
Label the
keys with small pieces of paper so you remember which is
which. The
change is permanent (until you change it back or your computer
crashes). Good luck.
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Subject: Re: Need a Math Font Which Includes Congruent Symbol
> Anyone know where to find a font which includes a congruent
symbol
> (looks like the equals symbol with a wave on top)?
> I'm typing up some worksheets and notes, and having to
always draw
> this one in by hand isn't a big deal, but I know there must
be a font
> which includes it.
> I've searched the net with no luck... It's driving me BATTY!
Symbol has the symbol you want. I believe that the Computer
Modern
and Lucida Bright Math families do also, but I don't have them
on my
laptop to double check.
--
Kevin Karplus karplus@soe.ucsc.edu
http://www.soe.ucsc.edu/~karplus
life member (LAB, Adventure Cycling, American Youth Hostels)
Effective Cycling Instructor #218-ck (lapsed)
Professor of Biomolecular Engineering, University of
California, Santa Cruz
Undergraduate and Graduate Director, Bioinformatics
Affiliations for identification only.
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Subject: Re: Ralston's review of California Dreaming
> The cited review can be read with an Acrobat reader at:
Some members of this NG may be interested in reading 4 letters
concerning Ralston's review at:
Scroll down to page 2 and 3.
Dom Rosa
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Subject: Calculus
im having difficulty with a problem in Calculus. I have to
prove the
following trigonometric identity:
sin 2A/1 + cos 2A = tan A
Please Help!!!
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Subject: Re: Calculus
> im having difficulty with a problem in Calculus. I have to
prove the
> following trigonometric identity:
> sin 2A/1 + cos 2A = tan A
Do you mean sin 2A/(1 + cos 2A) = tan A ?
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Subject: Re: Calculus
I had to look this stuff up.
Start by using the fact that: sin2x = 2(sinx)(cosx) and
1+cos2x =
2(cosx)^2
(2 sinx cosx)/(2(cosx)^2) = tanx
(2 sinx cosx)/(2(cosx)(cosx)) = tanx
The cosx's and the 2's cancel:
sinx/cosx = tanx QED.
Good Luck!
> im having difficulty with a problem in Calculus. I have to
prove the
> following trigonometric identity:
> sin 2A/1 + cos 2A = tan A
> Please Help!!!
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Subject: Re: Solving the stated problem not the problem you
think up
Re: Solving the stated problem not the problem you think up
My comment:
I wasn't educated in the US, and American skepticism of
testing has always
amused me.
IMHO, no testing instruments can possibly be perfect. You can
always find
fault with a test. But if I fail a test, should I blame that
on that
flaw
and ignore my own share of the blame? hmmm. (same with other
areas of
life...do we blame outcomes of athletic competitions or job
interviews on
the
scoring systems...academics is the one being singled out for
nitpicking in
that regard)
Besides, where I grew up, testing serves a very important
function:
It helps enforce a uniform curriculum so the rich and the poor
students are
more or less learning the same thing (more than can be said
for the American schools!) It gives poor and uneducated
parents a hope of
getting their kids a chance to go up the socioeconomic ladder.
To those
parents, the testing becomes the symbol of that hope. And my
parents were
those who believed in it. That's how my brother and I managed
to go to
college. We were taught not to blame anyone --not tests, not
teachers or
schools-- but ourselves for our academic results. If a student
can maintain
that attitude, no one can stop this child from excelling.
If I were taught in an American school, given my socioeconomic
status back
then, I'd be in an inner-city school, with limited access to
an advanced
curriculum. I wouldn't have learned as much.
Completely off the topic: given the American emphasis on
equality and
equity, I
find the discrepancy between the education a rich American kid
and a poor
American kid receives to be absolutely shocking.
>
katy (katyt88@yahoo.com <mailto:katyt88%40yahoo.com>)said:
In my mind, the reaction is more a political correct reaction
than an
educational correct action. The politicians do not want to
tell the parents
that their children may indeen error, so now the error is
caused by the test
writer. What stinks here is that the educators did not stand
up to the
occaction. (Or is it because they did not have a channel? Or
is it because
they
don't understand it in the first place? Or because they don't
care?)
-------------------------------------------------------
This mail sent through HKedCity http://www.hkedcity.net
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Subject: Re: Solving the stated problem not the problem you
think up
> Re: Solving the stated problem not the problem you think up
> My comment:
> I wasn't educated in the US, and American skepticism of
testing has always
> amused me.
Here's an example of why. Schools in Colorado are subject to
both
federal testing mandated by No Child Left Behind, and state
CSAP testing.
One school in the news recently recieved a perfect score on
the federal
tests. It may be closed soon due to inadaquete scores on the
CSAP.
(Actually, this is more an example of what happens when
someone other
than teachers designs the tests; the CSAP expects students to
know
things they won't be taught for another two years. Thus,
skeptism that
it serves any useful purpose)
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Subject: Re: Solving the stated problem not the problem you
think up
> Re: Solving the stated problem not the problem you think up
> My comment:
> I wasn't educated in the US, and American skepticism of
testing has always
> amused me.
> IMHO, no testing instruments can possibly be perfect. You
can always find
> fault with a test. But if I fail a test, should I blame that
on that
flaw
> and ignore my own share of the blame? hmmm. (same with other
areas of
> life...do we blame outcomes of athletic competitions or job
interviews on
the
> scoring systems...academics is the one being singled out for
nitpicking
in
> that regard)
If there are biases in the test questions that reflect
something other
than knowledge of the subject then the test isn't testing for
what it
purports to be testing - it is testing that other something. In
athletic competitions, the testing is of very simple
abilities. If
they added your shoe size into your high jump results then
jumping
alone wouldn't necessarily get you the gold medal - the test
would be
biased towards people with long feet. Sometimes people with
short
feet would do well in this new high jump, but they would have
to work
harder for reasons that had nothing to do with their jumping
ability.
The current academic tests can be like this new form of high
jump
scoring and many people feel it would make sense to make the
tests
more like the original form of high jump scoring.
> Besides, where I grew up, testing serves a very important
function:
> It helps enforce a uniform curriculum so the rich and the
poor students
are
> more or less learning the same thing (more than can be said
> for the American schools!)
Each state has an organization which sets the curriculum for
each
grade level and subject area, that should keep all schools in
each
state teaching all children the same things. How well the
subjects
are taught (and how well the students learn them) is dependent
on a
lot of other factors that may directly (fewer or inferior
resources)
or indirectly (psychological biases) reflect economic level.
A recent study, done in Arizona, showed that high stakes tests
do not
correlate with - in fact may be counterproductive to - future
success
in academia and the real world. Humans are not like widgets
and the
whole premise that you can quality control their education by
testing
them more appears to be faulty.
> It gives poor and uneducated parents a hope of
> getting their kids a chance to go up the socioeconomic
ladder.
The point is not to give people just a hope of this, but to
try to
make it so that there's NO correlation between where you start
off and
where you end up. That is supposed to be goal of the US
educational
system and that is a valid perspective from which to critique
standardized tests.
> To those
> parents, the testing becomes the symbol of that hope. And my
parents were
> those who believed in it. That's how my brother and I
managed to go to
> college. We were taught not to blame anyone --not tests, not
teachers or
> schools-- but ourselves for our academic results. If a
student can
maintain
> that attitude, no one can stop this child from excelling.
> If I were taught in an American school, given my
socioeconomic status back
> then, I'd be in an inner-city school, with limited access to
an advanced
> curriculum. I wouldn't have learned as much.
Testing is not a magic bullet for improving the schools. There
are
many issues involved and it is a thorny political problem.
But, that
said, there are many inner-city kids who do excel and who do
learn a
lot even here in the US.
> Completely off the topic: given the American emphasis on
equality and
equity, I
> find the discrepancy between the education a rich American
kid and a poor
> American kid receives to be absolutely shocking.
There are many ideals which attract more talk than action in
the US.
--Jeff
--
Ho, ho, ho, hee, hee, hee
and a couple of ha, ha, has;
That's how we pass the day away,
in the merry old land of Oz.
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Subject: Re: Math Education Sequence from 7th to 12th
Same with me. I had integrated math in China. For the first
three
years of secondary school (roughly 7-9th grade), I did algebra
and
geometry every year. At O-level (roughly 10th-11th), I did
trig,
pre-calculus and statistics all in one 2-year course. That's
the
liberal arts track. For science track, you'd have to do
applied Math
(which includes Calculus) in addition to regular O-level Math.
><<
>Thus geometry is integrated in with the algebra, not taught
separate
>from it.
This was and may be is now the method in Central Europe and
AFAIk in
>Germany. I was surprisaed that in the US the geometry is
separated from
>the algebra.
> This separation may not be universal; it probably varies
from school to
school,
> and maybe somewhat a changeable fashion. U S has in some
communities,
> something called Integrated Mathematics in high schools
which may be
good or
> bad depending on how the community manages their Integrated
Maths. The
> traditional separated courses method will suit many
students. When too
many
> different topics come in the same course, continuity of
learning is
destroyed
> for some students.
> G C
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Subject: 8th grade
The discussion of slope of a line in an 8th grade class seems
to me to
have veered off from the main point.
First, introducing a Greek letter (delta) is a useless
distraction.
Greek letters are never needed in elementary math - they just
distract
certain students, or cause a bit of confusion, or reinforce
the notion
some have that math is hard, abstract, and pointless. We
should be
spending our time on teaching about interesting math, not
seemingly
exotic notation. Ignore Greek letters entirely in K-14.
Second, the main point about 'slope' is that it is an INVARIANT
quantity, a constant, that is computable using ANY pair of
points on a
line L, given a Cartesian coordinate system X - Y in which L
is not
parallel to the Y axis. Such a fact would in physics be called
a
Conservation Law. Everthing about the connection between lines
and
linear equations falls out of this one fact, as do the
equations of
uniform linear motion (parametric representation of a line).
We should always make a big fuss whenever we come upon
invariants in
math -- it's an important and useful idea. It's the simplest
kind of
pattern to notice. For example, circumference/diameter for all
circles is an invariant called pi, (area of inscribed
circle)/(area of
square) is invariant pi/4, (volume of cone)/(volume of prism)
is
invariant 1/3 for a cone inscribed in a prism [same base of
any shape,
same height], a sequence is arithmetic iff its first
differences are
constant [invariant is amount of increase between successive
terms], a
sequence is geometric iff the ratios of its successive terms is
constant [growth factor], a sequence is shifted geometric iff
the
ratios of its successive first differnces is invariant, a
sequence is
quadratic iff its second differences are constant, a sequence
is cubic
iff its third differences are constant, first differences of
how far a
ball falls (or rolls down a slope, see Galileo) during
succesive time
periods (seconds, etc) is 2*(distance travelled in first
period), etc.
But if the kids don't know about similarity, how will I
convince them
that slope is indeed invariant? Well I might make a few
transparencies for an overhead projector, each one having a
Cartesian
grid and a single line L drawn on it, and on the line I would
have
marked maybe 6 points A,B,C,D,E,F and beside each one the pair
of its
coordinates -- which I should also be able to see from the
grid and
tic marks on the axes. There are then C(6,2)=15 pairs of
points that
can be used to calculate (change in Y)/(change in X) when
moving from
one point to the other, and these calculations could be
carried out by
students in small groups assigned to use a couple of pairs.
Everybody
gets the same number (we hope) for 1 line, so that is its
invariant
slope, and then we try it for some other lines, each yielding
its own
invariant slope calculation.
(Note that I only teach college students.)
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Subject: Re: 8th grade
I am surprised and disconcerted that my earlier message in
this thread
seems to have been misunderstood and ruffled the composure of
several
readers. No doubt in my haste I was unclear about my intent - I
thought I was being merely practical and not controversial.
Perhaps
you will allow me a few comments in reply to these agitated
responders.
=================================================
<<
First, introducing a Greek letter (delta) is a useless
distraction.
Greek letters are never needed in elementary math - they just
distract
certain students, or cause a bit of confusion, or reinforce
the notion
some have that math is hard, abstract, and pointless. We
should be
spending our time on teaching about interesting math, not
seemingly
exotic notation. Ignore Greek letters entirely in K-14.
We all hear stories these days that school math curricula are
filled
to overflowing with all kinds of miscellania, and unrelated
bits of
useless things that are to be fleetingly seen but not learned.
I was
merely pointing out that since there is no need for using Greek
letters in elementary math, that would be a good place to
simplify and
not contribute to the excessive clutter of things mentioned. I
was
mostly thinking of use of Greek letters for variables or
quantities
such as angles in geometry, and I assumed it would be
understood that
an exception would be made for such a commonly used irrational
constant like pi. (Though I see no reason why writing the Greek
letter named pi is any better that simply writing pi for school
math.) Likely only later in college might some of the students
see
additional examples of math constants or functions
traditionally
denoted by Greek letters (Euler's phi-function, the Gamma
function, or
perhaps the Euler-Mascheroni constant).
It is completely irrelevant to what is taught in school math
whether
some mathematicians and some scientists occasionally make use
of Greek
letters, or Hebrew letters, or German Fraktur letters, or
French names
for things, etc. Whether such use seems commonplace to us is
irrelevant, it may seem exotic and pointless to some students
-- just
one more unnecessary irritant associated with math.
Mr. Turner says Greek letters aren't scary and he knows kids
who
voluntarily memorize lots of weird names from games. True
enough, and
I look forward to hearing of his method of generating a
similar amount
of interest in kids for learnig Greek letters for use in math.
================================================
<<
Second, the main point about 'slope' is that it is an INVARIANT
quantity, a constant, that is computable using ANY pair of
points on a
line L, given a Cartesian coordinate system X - Y in which L
is not
parallel to the Y axis. Such a fact would in physics be called
a
Conservation Law. Everthing about the connection between lines
and
linear equations falls out of this one fact, as do the
equations of
uniform linear motion (parametric representation of a line).
I capitalized two words hoping thereby to emphasize them, to
make it
clear what I thought was critical to understanding precisely
what I
meant. I would be happy to hear suggestions on how I might
have done
this better without offending anyone by seeming to shout. But
I would
prefer to see some discussion about the content of this
paragraph.
================================================
I then gave examples of the idea of an invariant outcome of
many
similar computations that could be made on math object of
various
kinds -- generating a number associated with the object that
represents some intrinsic or characteristic property, like the
Euler
characteristic.
<<
We should always make a big fuss whenever we come upon
invariants in
math -- it's an important and useful idea. It's the simplest
kind of
pattern to notice. For example, circumference/diameter for all
circles is an invariant called pi, (area of inscribed
circle)/(area of
square) is invariant pi/4, (volume of cone)/(volume of prism)
is
invariant 1/3 for a cone inscribed in a prism [same base of
any shape,
same height], a sequence is arithmetic iff its first
differences are
constant [invariant is amount of increase between successive
terms], a
sequence is geometric iff the ratios of its successive terms is
constant [growth factor], a sequence is shifted geometric iff
the
ratios of its successive first differnces is invariant, a
sequence is
quadratic iff its second differences are constant, a sequence
is cubic
iff its third differences are constant, first differences of
how far a
ball falls (or rolls down a slope, see Galileo) during
succesive time
periods (seconds, etc) is 2*(distance travelled in first
period), etc.
The main point in these examples is not that we get out some
particular universal constant, but rather that for many
different
calculations that we can make on an object the resulting
derived value
is always the same. That is a theorem of course.
I don't think I understand the claim of slope not being an
invariant
associated with a line.
====================================================
Then I made a simple suggestion for a hands-on activity that
might
help students convince themselves of the invariance of the
slope
calculation. Without knowing about similarity, the only
alternative I
can think of is to simply *tell them frequently and
forcefully* that
this is a fact and maybe they will believe you, but will they
have
bought into it. Here is an opportunity for treating math as
experimental science.
<<
But if the kids don't know about similarity, how will I
convince them
that slope is indeed invariant? Well I might make a few
transparencies for an overhead projector, each one having a
Cartesian
grid and a single line L drawn on it, and on the line I would
have
marked maybe 6 points A,B,C,D,E,F and beside each one the pair
of its
coordinates -- which I should also be able to see from the
grid and
tic marks on the axes. There are then C(6,2)=15 pairs of
points that
can be used to calculate (change in Y)/(change in X) when
moving from
one point to the other, and these calculations could be
carried out by
students in small groups assigned to use a couple of pairs.
Everybody
gets the same number (we hope) for 1 line, so that is its
invariant
slope, and then we try it for some other lines, each yielding
its own
invariant slope calculation.
Mr. Turner says that calculating the slope of a single line
fifteen
times sounds dreadful. Perhaps, if any one child did that, but
I
suggested that the teacher would parcel out these calculations
to many
kids so each one would do just a few for any given line --
which
doesn't sound so onerous - especially if this is an important
calculation. As for just cutting out a right triangle and
sliding it
along the line, that fixes a specific horizontal and vertical
length
and so it represents the same computation of a ratio of
sidelenghts,
not quite the same thing as the notion of an invariant
calculated for
all possible pairs of points on the line. Why not make this
invariance the centerpiece of discussion of equations of lines?
Perhaps, instead of starting with a particular line in mind,
you'd
like to begin with any three points A,B,C and their
coordinates of
course, and ask how can we use calculation to determine
whether they
are collinear? (Now we are getting into Descartes' idea of
using
measurement and calculation to do geometry.) And as soon as we
know
the answer to this, we can write down an equation for the line
through A and B, that is, an equation that must be satisfied
by the
coordinates of every point on that line. We might rewrite the
equation of the line in the form p*x+q*y=r, and then we might
ask
exactly which equations we could get this way for the fixed
line using
various points on the line to derive such an equation. And the
answer
would be all equations of the form (t*p)*x+(t*q)*y=(t*r) for
all
nonzero real numbers t. At a higher level we might say that we
have
another invariant associated with the line, but this time it
represents a point in the real projective plane.
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Subject: Re: 8th grade
> I am surprised and disconcerted that my earlier message in
this thread
> seems to have been misunderstood and ruffled the composure
of several
> readers.
You're 'surprised' that your message may be misunderstood or
even 'ruffle'
some's composure? In Usenet? Really?
> No doubt in my haste I was unclear about my intent - I
> thought I was being merely practical and not controversial.
Perhaps
> you will allow me a few comments in reply to these agitated
> responders.
Well, not speaking for others, I am certainly not 'agitated!'
As for not
being clear about your intent, I must disagree. Statements
like Ignore
Greek letters entirely in K-14 don't leave much room for
misintepretation,
other than possibly a typo (maybe you meant K-12). Since Greek
letters are
indeed used in K-14 (and K-12), suggesting they not be is
certainly
controversial.
> =================================================
> <<
> First, introducing a Greek letter (delta) is a useless
distraction.
> Greek letters are never needed in elementary math - they
just distract
> certain students, or cause a bit of confusion, or reinforce
the notion
> some have that math is hard, abstract, and pointless. We
should be
> spending our time on teaching about interesting math, not
seemingly
> exotic notation. Ignore Greek letters entirely in K-14.
> We all hear stories these days that school math curricula
are filled
> to overflowing with all kinds of miscellania, and unrelated
bits of
> useless things that are to be fleetingly seen but not
learned.
Agreed, however I don't think Delta x used to denote a change
in x, is one
of them.
> I was
> merely pointing out that since there is no need for using
Greek
> letters in elementary math, that would be a good place to
simplify and
> not contribute to the excessive clutter of things mentioned.
...and neither is it necessary (in a mathematical sense) to
default to the
decimal system, so you suppose we should stop doing that too?
> I was
> mostly thinking of use of Greek letters for variables or
quantities
> such as angles in geometry, and I assumed it would be
understood that
> an exception would be made for such a commonly used
irrational
> constant like pi.
Again, statements such as Ignore Greek letters entirely in
K-14 don't
leave much wiggle room.
> (Though I see no reason why writing the Greek
> letter named pi is any better that simply writing pi for
school
> math.)
Irrlevent. What's relevent is, do you or do you not see any
reason why it
is any /worse/? That is, after all, the common way to do it in
typeset
notation, to use the actual letter and not a literal 'pi', so
unless you
have a compelling reason why this should be changed...
> Likely only later in college might some of the students see
> additional examples of math constants or functions
traditionally
> denoted by Greek letters (Euler's phi-function, the Gamma
function, or
> perhaps the Euler-Mascheroni constant).
For the sake of argument, let's stick to just what you
described earlier.
Using Greek letters for certain variables (angles, etc.) The
fact is, it
is
not only later in math that these are used. It can be (and
often is),
now.
> It is completely irrelevant to what is taught in school math
whether
> some mathematicians and some scientists occasionally make
use of Greek
> letters, or Hebrew letters, or German Fraktur letters, or
French names
> for things, etc.
/Some/ mathemeticians? /Some/ scientists? /Occasional/ use of
Greek,
Hebrew, etc.? You've got to be kidding. Since many (most)
indeed do it
(its hard to find even a high school math book that never
does) it most
certainly /is/ relevent to what is taught in school math.
> Whether such use seems commonplace to us is
> irrelevant, it may seem exotic and pointless to some
students -- just
> one more unnecessary irritant associated with math.
> Mr. Turner says Greek letters aren't scary and he knows kids
who
> voluntarily memorize lots of weird names from games. True
enough, and
> I look forward to hearing of his method of generating a
similar amount
> of interest in kids for learnig Greek letters for use in
math.
I suggest a good start would be not to go out of the way to
/avoid/ them.
That only enforces whatever fear someone may have of them.
Allow me to
give
an analogy, that has effected me personally. When I was young,
lightning
struck our house. The scariest moment of my childhood. Now,
don't ask
why,
but to this day (I'm 39) I get a little edgey during an
electical storm,
even when indoors. That fear has rubbed off on my son.
Apparently my own
fear is more pronounced than I originally thought. You see,
all his life
my
son has picked up on my reaction to lighting. To this day (and
he's 11) he
comes running to our bedroom, or wherever others are present,
during an
electical storm. When asked why he does this, he did not
hesitate to tell
me that it is because I am also 'scared.'
Keep /treating/ Greek letters like they are something they
should be afraid
of, and they most certainly /will/ be afraid of them.
> ================================================
> <<
> Second, the main point about 'slope' is that it is an
INVARIANT
> quantity, a constant, that is computable using ANY pair of
points on a
> line L, given a Cartesian coordinate system X - Y in which L
is not
> parallel to the Y axis. Such a fact would in physics be
called a
> Conservation Law. Everthing about the connection between
lines and
> linear equations falls out of this one fact, as do the
equations of
> uniform linear motion (parametric representation of a line).
> I capitalized two words hoping thereby to emphasize them, to
make it
> clear what I thought was critical to understanding precisely
what I
> meant. I would be happy to hear suggestions on how I might
have done
> this better without offending anyone by seeming to shout.
I think enough discussion (at least on my part) about your
actual point in
unneccessary. As for the netiquette, know that it is
longstanding
netiquette that use of caps should generally not be done
because it looks
like you are SHOUTING. Hence, when the need to place emphasis
arises in an
ascii only forum, most use some other, less visually
intimidating, means.
For ex. you could do _emphasis_ (usually taken to be equiv. to
underscoring), or *emphasis*, or maybe even as I recently
began doing,
/emphasis/.
Just not EMPHASIS. This is the norm.
Yes, I know a fixed line has constant slope. I'll agree not to
further
split hairs over your choice of invariant in this context. It
is,
after
all, far removed from any real relevence to whether or not
Greek letters
should be introduced in 8th grade.
<... Mr. Turner says that calculating the slope of a single line
fifteen
> times sounds dreadful. Perhaps, if any one child did that,
but I
> suggested that the teacher would parcel out these
calculations to many
> kids<...>
Perhaps, but to me at least it still sounds somewhat dreadful.
People,
even
aside from a mathematical context, have an intuitive notion of
slope.
The
slope of a hill, for ex. Just point to a line and say see, the
slope
never
changes and there....you have just convinced them that the
slope never
changes 9the lijne is obviously 'straight' after all.) Once
they
understand
that on an intuitive level (which they already did, you only
reminded them
of it) then they have no problem also believing, given that
two distinct
points can be used to calculat the slope, that /any/ two
distinct points
can
be used to calculate that slope.
<...>
--
Darrell
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Subject: Re: 8th grade
It is completely irrelevant to what is taught in school math
whether
some mathematicians and some scientists occasionally make use
of Greek
letters, or Hebrew letters, or German Fraktur letters, or
French names
for things, etc.
> /Some/ mathemeticians? /Some/ scientists? /Occasional/ use
of Greek,
> Hebrew, etc.? You've got to be kidding. Since many (most)
indeed do it
> (its hard to find even a high school math book that never
does) it most
> certainly /is/ relevent to what is taught in school math.
I believe that what Ladnor was getting at here is that variable
notation is irrelevant to what it denotes, in that no
particular
variable is required.
Whether such use seems commonplace to us is
irrelevant, it may seem exotic and pointless to some students
-- just
one more unnecessary irritant associated with math.
 Keep /treating/ Greek letters like they are something they
should be
afraid
> of, and they most certainly /will/ be afraid of them.
I've seen a distinct difference in student response when
presented
with the same idea with or without Greek letter variables (not
constant pi). Some of them have volunteered that they like it
better
without the Greek letter variables, that they feel more
comfortable
without them. (I didn't teach them to be afraid of them. They
just
don't like them. I could have said, It's like eating your
broccoli.
It's good for you and you have to have it, so I'm forcing it
on you
for your own good. But I chose not to.) Besides, even if using
Greek
letter variables in certain situations is traditional, why
should I
make my students bow down and kiss the feet of Tradition if
they don't
want to? Evidently, what these students are communicating is
that not
using the Greek letter variables makes the symbolism more
transparent
to them, that they better see the math behind the symbolism, a
good
thing. I think that this is what Ladnor was getting at.
Mr. Turner says that calculating the slope of a single line
fifteen
times sounds dreadful. Perhaps, if any one child did that, but
I
suggested that the teacher would parcel out these calculations
to many
kids<... Perhaps, but to me at least it still sounds somewhat
dreadful. People,
even
> aside from a mathematical context, have an intuitive notion
of slope.
The
> slope of a hill, for ex. Just point to a line and say see,
the slope
never
> changes and there....you have just convinced them that the
slope never
> changes 9the lijne is obviously 'straight' after all.) Once
they
understand
> that on an intuitive level (which they already did, you only
reminded
them
> of it) then they have no problem also believing, given that
two distinct
> points can be used to calculat the slope, that /any/ two
distinct points
can
> be used to calculate that slope.
Making connections under a unifying theme such as similarity
is a very
good way to teach math. It is one of the best recommendations
made by
the NCTM.
In addition, making connections between the abstract symbolism
and any
of its visual, geometrically intuitive representations is a
problem
area. You'd be surprised as to how common it is that middle
school and
early high school students do not see that for a given
non-vertical
line, m is invariant. One tells them that the slope of a given
hill is
invariant, and no problem, they see that. But one can design
questions
to ask along the lines of If I change this, then will m
change? to
uncover the fact that many of them are not making the
connection
between the abstract symbolism and the visual, geometrically
intuitive
notion.
Ladnor is pointing out that we should make a fuss about the
why of it
to forge the connections that so many of the students miss.
The why of
it is the concept of similarity, under which mention of other
invariants can be made along the line of This value of slope is
invariant just as these other values are invariant and all for
the
same reason, this reason being similarity.
And to those who wonder about whether similarity is something
appropriate for beginning algebra students: Topics in geometry
are
covered in middle school and even late elementary school math
curricula.
Paul
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Subject: Re: 8th grade
 It is completely irrelevant to what is taught in school math
whether
> some mathematicians and some scientists occasionally make
use of
Greek
> letters, or Hebrew letters, or German Fraktur letters, or
French
names
> for things, etc.
/Some/ mathemeticians? /Some/ scientists? /Occasional/ use of
Greek,
Hebrew, etc.? You've got to be kidding. Since many (most)
indeed do
it
(its hard to find even a high school math book that never
does) it most
certainly /is/ relevent to what is taught in school math.
> I believe that what Ladnor was getting at here is that
variable
> notation is irrelevant to what it denotes, in that no
particular
> variable is required.
Then why Ladnor's fuss about using Greek letters, if it really
doesn't
matter? That's the point. To her/him, it clearly *matters*.
The argument
is being presented under the guise that only a few, not most,
mathemticians
use Greek letters which is totally silly.
It is totally silly to acknowlege it to be OK to use Greek
letters to
denote
certain constants, but never a variable. That simply does not
compute. If
they're not afraid of seeing pi they should equally not be
afraid to see
theta for an angle.
> I've seen a distinct difference in student response when
presented
> with the same idea with or without Greek letter variables
(not
> constant pi). Some of them have volunteered that they like
it better
> without the Greek letter variables, that they feel more
comfortable
> without them. (I didn't teach them to be afraid of them.
They just
> don't like them. I could have said, It's like eating your
broccoli.
> It's good for you and you have to have it, so I'm forcing it
on you
> for your own good. But I chose not to.) Besides, even if
using Greek
> letter variables in certain situations is traditional, why
should I
> make my students bow down and kiss the feet of Tradition if
they don't
> want to? Evidently, what these students are communicating is
that not
> using the Greek letter variables makes the symbolism more
transparent
> to them, that they better see the math behind the symbolism,
a good
> thing. I think that this is what Ladnor was getting at.
We might as well be arguing over whether or not we should use
x for a
variable. ..and don't even try to tell me that when first
introduced to
even this _English_ letter to denote a variable, some are not
confused for
a
while.
--
Darrell
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Subject: O the fearsome variable!
I thought this was a curious and telling comment.
<<<
We might as well be arguing over whether or not we should use
x for
a variable. ..and don't even try to tell me that when first
introduced to even this _English_ letter to denote a variable,
some
are not confused for a while.
I have a good friend who teaches first and second grade, and
she
assures me that introducing letter names for numbers in simple
equations doesn't faze her students.
So I wonder what sort of spiny shell a teacher might be
creating in
the minds of students around the introduction of a letter as a
variable in middle school that could cause confusion, and other
problems reported by some teachers. Students are perfectly at
home
naming cats, dogs, stuffed animals, game characters, cars,
trucks,
trains, and various other machines -- so one would not expect
them to
even think it odd to name (even temporarily) some numbers, or
formulas, or functions when the discussion turns to math. These
students learned before going to school how pronouns
(substitute names
that may be without specific object reference) work and how
they can
be used in sentences, and variables are just the pronouns of
math.
How is the teacher whose students react negatively hedging the
natural
naming propensity about with fearsome extra verbiage, and why?
(Sorry, I wouldn't normally write anything related to
education that
used the word fear or any variant of it - I must have been
infected by
the frequent exposure to it in one responder's missives.)
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Subject: Re: O the fearsome variable!
Seems to me that when early grades use problems such as open
box or star
or circle or any other symbol + 3 = 5 and expect the youngster
to write
2 in the symbol, that's introducing the concept of a variable.
Don
> I thought this was a curious and telling comment.
> <<<
> We might as well be arguing over whether or not we should
use x for
> a variable. ..and don't even try to tell me that when first
> introduced to even this _English_ letter to denote a
variable, some
> are not confused for a while.
> I have a good friend who teaches first and second grade, and
she
> assures me that introducing letter names for numbers in
simple
> equations doesn't faze her students.
> So I wonder what sort of spiny shell a teacher might be
creating in
> the minds of students around the introduction of a letter as
a
> variable in middle school that could cause confusion, and
other
> problems reported by some teachers. Students are perfectly
at home
> naming cats, dogs, stuffed animals, game characters, cars,
trucks,
> trains, and various other machines -- so one would not
expect them to
> even think it odd to name (even temporarily) some numbers, or
> formulas, or functions when the discussion turns to math.
These
> students learned before going to school how pronouns
(substitute names
> that may be without specific object reference) work and how
they can
> be used in sentences, and variables are just the pronouns of
math.
> How is the teacher whose students react negatively hedging
the natural
> naming propensity about with fearsome extra verbiage, and
why?
> (Sorry, I wouldn't normally write anything related to
education that
> used the word fear or any variant of it - I must have been
infected by
> the frequent exposure to it in one responder's missives.)
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Subject: Re: O the fearsome variable!
> (Sorry, I wouldn't normally write anything related to
education that
> used the word fear or any variant of it - I must have been
infected by
> the frequent exposure to it in one responder's missives.)
Quite OK. Although not p.c., everything pretty much is related
to fear in
some way or another, at least in the U.S. The media controls
the masses
via
fear. The government also controls by fear. We're literally
scared into
consumption, in general. Look at Y2K and all the stuff
purchased in
preparation. Fear is a very powerful thing, and the ability to
instill
fear
makes for the power to control others. As a society, we are
essentially
scared into compliance for just about everything, including
education.
Don't pass, and look, _here's_ what happens. Seen _Bowling for
Columbine_?
So although not p.c. in many circles, there is much truth and
relevance in
discussing fear in an educational context.
You do raise some interesting questions, though. I submit the
possibility
that those who are scared of algebra at first, specifically
variables
like
x and y, (if you'll forgive my continued references to fear,)
are just
having problems because what used to be straightforward
concepts (eg 5,
231,
1/2) are now being presented in a somewhat abstract fashion
(x, y, etc.)
Once the fear of the abstract is removed, then usually the
fear of whatever
particular 'name' is chosen for an unidentified number, is
also removed.
I applaud the person you spoke of that had the fortitude to
introduce
letters to name numbers, and was successful. Chances are,
those 2nd
graders
will not have any problem at all if, when in 8th grade, their
math teacher
shows tells them to denote a change in x by a certain Greek
letter for they
learned long before there was nothing really to be 'scared' of.
--
Darrell
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Subject: Re: O the fearsome variable!
Learning to use variables was not so bad. Introductory
Algebra, naturally
relying on variables, rectified many basic difficulties about
Math and
arithmetic.
Mathematics: many definitions but less vocabulary;
Natural Language: Much more vocabulary, and not too much
relies on precise
quantities.
Many people have difficulty focusing; vocabulary takes care of
this focus.
As
soon as more precise clauses and phrases come to issue, many
people are
uncomfortable because the topic is too technical.
In Algebra, one first studies some important number and numer
sentence
properties in a formal and somewhat rigorous way.
G C
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Subject: X education
I get a certain amount of mileage at the beginning of each year
telling my classes of eighth grade Algebra students that they
are in a
class all about x education. Even the writers of the
McDougal-Littel text support this by so frequently using 6x in
a
problem. I think they followed the Disney animators lead by
slipping
in in these sorts of in jokes (slow down the Lion King at the
point
where Simba is so downhearted about not having his Dad
around-he flops
to the ground, kicks up some dust which spells out sex).
I tell my young people that I know they think about x a lot,
but that
other things are important too. Nevertheless, x is a part of
their
lives and eighth grade is a good time to learn as much as
possible
about it.
At least for a couple of weeks, students come to my class
curious
about what I'm going to say.
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Subject: Re: 8th grade
> Then why Ladnor's fuss about using Greek letters, if it
really doesn't
> matter? That's the point. To her/him, it clearly *matters*.
The
argument
> is being presented under the guise that only a few, not most,
mathemticians
> use Greek letters which is totally silly.
> It is totally silly to acknowlege it to be OK to use Greek
letters to
denote
> certain constants, but never a variable. That simply does
not compute.
If
> they're not afraid of seeing pi they should equally not be
afraid to see
> theta for an angle.
(snip)
> We might as well be arguing over whether or not we should
use x for a
> variable. ..and don't even try to tell me that when first
introduced to
> even this _English_ letter to denote a variable, some are
not confused for
a
> while.
In terms of how students respond to things, I note that you
keep
saying should. One can say should all one wants in this regard,
but then there's reality. All I did was report on what is, on
how some
students actually respond, and how I responded to what is.
Some (as in
many) students actually do respond differently to using an
alphabet
with which they are familiar vs. using one with which they are
not.
Some teachers are more sensitive to what they actually observe
in some
students. This is a good thing. As I see it, that's all Ladnor
was
doing (and by extension, what some other math teachers do) on
using a
foreign language alphabet - being more sensitive to what is,
and for
the sake of these students, responding accordingly, a good
thing.
What's your actual argument here? That teachers who are more
sensitive
to what they actually observe in some students should not be?
Yes, it
is a judgement call as to how far to go in this regard. But I
don't
understand the ferocity in the response to what he said, which
I think
all quite reasonable.
Paul
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Subject: Re: 8th grade
> In terms of how students respond to things, I note that you
keep
> saying should. One can say should all one wants in this
regard,
> but then there's reality.
Sure. the reality is, using seemingly strange things like x,
y, f(x) and
what not, can be just as if not more confusing to some, than
seeing a Greek
letter. the simple fact of the matter is, if a student is
comfortable with
the concepts aside from whatever notation is being used to
convey them,
then
they, well, understand the concepts _aside_ from the notation
being used to
convey them.
Steering clear of introducing a commom notation for a common
concept not
only in h.s. math but beyond (namely Delta x) for fear of some
hindrance to
understanding, is more suggestive of, well, a lack of
understanding of the
basic concept than an implication that the notation is getting
in the way.
Those that understand the concepts are not the ones who need
steer clear of
the Greek notation.
> All I did was report on what is, on how some
> students actually respond, and how I responded to what is.
Some (as in
> many) students actually do respond differently to using an
alphabet
> with which they are familiar vs. using one with which they
are not.
Then that settles it. Remove pi from the texts. Call it
something else.
Why should this be any different?
> Some teachers are more sensitive to what they actually
observe in some
> students. This is a good thing. As I see it, that's all
Ladnor was
> doing (and by extension, what some other math teachers do)
on using a
> foreign language alphabet - being more sensitive to what is,
and for
> the sake of these students, responding accordingly, a good
thing.
Brushing it under a rug only makes it worse. One day, like it
or not, on
some standardized test or some (yes even some _high school_)
course and/or
text, they are going to be exposed to it. What you are
suggesting is that
you just blow the issue off until then. Hoipefuly it will be
someone else
(I guess is the logic) that introduces them to it, and must
deal with all
the I've never seen that before...
> What's your actual argument here?
Are you really that confused? My argument is clear. It's silly
to suggest
all Greek letters be removed from all k-12 (actually, I think
the claim was
k-14) mathematics. Even if pi is excluded, which BTW there was
never any
logic revealed why this should be an exception, there are
other Greek
letters that are used in h.s. mathematics. uppercase Sigma
comes to mind,
as do a few others (alpha, beta, gamma, theta...)
That teachers who are more sensitive
> to what they actually observe in some students should not be?
I never argued that. Instead of putting words in my mouth, why
don't you
respond to what I actually am writing instead of what you
think I am
writing. of course they should be sensitive to such
observations, but that
does not come close to implying a complete removal of all
Greek letters
from
the literature and chalkboard. you seem to be making the
argument that, if
one should be sensitive, then that implies one should avoid
Greek letters.
Yes, it
> is a judgement call as to how far to go in this regard. But
I don't
> understand the ferocity in the response to what he said,
which I think
> all quite reasonable.
Completely avoid Greek letters entirely in K-14? You find that
reasonable?
Yoiu are so deep into your defense of your comrad, that you
don't even see
the obvious harm that would come from that. I can see it
now.....make sure
we adapt only texts that never use any Greek, until then
scratch out all
occurances of Greek letters in rthe current texts, redo any
exams,
handouts,
and other materials that conmtain them, etc. etc. all for
naught since the
very next class they take, they probably *will* see a Greek
letter and oh
no, now whatever fear they had is even worse, since you, the
teacher,
reinforced their fear from the year before instead of trying
to show that
there is nothing to be afraid _of_.
Suggesting to do away with all Greek in k-14 mathematics, is
akin
suggesting
to do away with all latin in pre-med. Its simply silly to
suggest such a
thing, as it would be silly for me to contiue after tonight
even having
this
debate with those that are obviously so closed minded to even
consider that
I may have a point. and we wonder sometimes why we are so far
behind.
let's just say it's getting clearer to me by the day. Hint--
it's *not*
because our students are less intelligent, or somehow less
capable of
learning the same things kids in other countries do.
--
Darrell
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Subject: Delta
As someone noted, x is a problem right from the first for many
students.
After I teach about the coordinate plane, plotting points, x-
and y-
value tables (input/output tables, t- tables, whatever),
y=mx+b and
all that, then in the midst of explaing the slope formula and
the rise
over the run, then I appeal to the inner superior eighth
grader by
mentioning that change-in-y over change-in-x can be written
delta y
over delta x, but only really advanced Algebra 1 students will
do
that. Catches about 10% each year.
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Subject: Re: 8th grade
>The discussion of slope of a line in an 8th grade class seems
to me to
>have veered off from the main point.
Well, I do think you make a useful point here, that the main
issue
should be how to introduce the idea of slope, not what we call
it. If
slope is new to the students, or if they are slow, some effort
may
be needed to do that -- a good use of class time. But the
nature of
slope is not affected by its name. To (mis)quote a famous
American
economist, you can call it a banana if you want, but that does
not
affect the meaning.
>First, introducing a Greek letter (delta) is a useless
distraction.
So you have banished pi from the curriculum? (Or you are one
of those
who think the word is pie?) And I assume you have also wiped
out the
biol and chem classes? Both capital and lower case delta are
needed in
basic chem. I did not choose the terminology.
This is a bit of over-reaction!
bob
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Subject: Re: 8th grade
>The discussion of slope of a line in an 8th grade class seems
to me to
>have veered off from the main point.
> Well, I do think you make a useful point here, that the main
issue
> should be how to introduce the idea of slope, not what we
call it.
<...>
...so long as we don't call it Delta x, that is. Seems to me
one of her
main
points was that we shouldn't call it that at this level, so at
least to
some, the name indeed seems to matter. ...which I find a
little odd, since
every elementary algebra text at my current disposal makes a
good amount of
use out of it.
Why shun something we know they will, like it or not, _have_
to know (and
relatively soon if not sooner, I might add)?
--
Darrell
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Subject: Re: 8th grade
> The discussion of slope of a line in an 8th grade class
seems to me to
> have veered off from the main point.
> First, introducing a Greek letter (delta) is a useless
distraction.
> Greek letters are never needed in elementary math - they
just distract
> certain students, or cause a bit of confusion, or reinforce
the notion
> some have that math is hard, abstract, and pointless. We
should be
> spending our time on teaching about interesting math, not
seemingly
> exotic notation. Ignore Greek letters entirely in K-14.
> Second, the main point about 'slope' is that it is an
INVARIANT
> quantity, a constant, that is computable using ANY pair of
points on a
> line L, given a Cartesian coordinate system X - Y in which L
is not
> parallel to the Y axis. Such a fact would in physics be
called a
> Conservation Law. Everthing about the connection between
lines and
> linear equations falls out of this one fact, as do the
equations of
> uniform linear motion (parametric representation of a line).
> We should always make a big fuss whenever we come upon
invariants in
> math -- it's an important and useful idea. It's the simplest
kind of
> pattern to notice. For example, circumference/diameter for
all
> circles is an invariant called pi, (area of inscribed
circle)/(area of
> square) is invariant pi/4, (volume of cone)/(volume of
prism) is
> invariant 1/3 for a cone inscribed in a prism [same base of
any shape,
> same height], a sequence is arithmetic iff its first
differences are
> constant [invariant is amount of increase between successive
terms], a
> sequence is geometric iff the ratios of its successive terms
is
> constant [growth factor], a sequence is shifted geometric
iff the
> ratios of its successive first differnces is invariant, a
sequence is
> quadratic iff its second differences are constant, a
sequence is cubic
> iff its third differences are constant, first differences of
how far a
> ball falls (or rolls down a slope, see Galileo) during
succesive time
> periods (seconds, etc) is 2*(distance travelled in first
period), etc.
> But if the kids don't know about similarity, how will I
convince them
> that slope is indeed invariant? Well I might make a few
> transparencies for an overhead projector, each one having a
Cartesian
> grid and a single line L drawn on it, and on the line I
would have
> marked maybe 6 points A,B,C,D,E,F and beside each one the
pair of its
> coordinates -- which I should also be able to see from the
grid and
> tic marks on the axes. There are then C(6,2)=15 pairs of
points that
> can be used to calculate (change in Y)/(change in X) when
moving from
> one point to the other, and these calculations could be
carried out by
> students in small groups assigned to use a couple of pairs.
Everybody
> gets the same number (we hope) for 1 line, so that is its
invariant
> slope, and then we try it for some other lines, each
yielding its own
> invariant slope calculation.
> (Note that I only teach college students.)
To Ladnor Geissinger:
Let me express my appreciation for the lucidity and on-point
qualities of
your writing. I try to write as clearly, and hope that others
also
appreciate such clarity.
Here are a couple of points related to this slope stuff that I
would bet
you have given thought to, but did not express in the above
post.
First, the shorter point: I agree with the idea of *not* using
the
delta x, change in x terminology, and much rather would stay
with
using *pairs of points* as the central idea/terminology of
computing
slopes.
One of the points that could also be mentioned in connection
with *not*
using the delta notation is that it looks exactly like we are
*multiplying* delta times x. To apply a new kind of
notation/connotation, that of juxtaposing delta and x. *without
multiplying* is needlessly confusing. After all, it is a
geometric
concept that we are trying to get across, in an algebraic
setting of
course.
Second, a bit longer point, at least content-wise:
In talking about slope, it seems natural to remark that the
ratio, however
you notation it, is *invariant* because of that property of
similar
triangles.
I have never taught eighth or ninth grade math, so I can only
wonder about
the following: Isn't the concept of similar triangles
typically taught in
tenth grade geometry, whereas the concept of slope is taught
in ninth
grade algebra? If so, how is the *invariance* property of
slope explained
to those ninth (or eighth) grade students?
A small aside for those who like to split hairs over notation:
Rather
than using *new notation/terminology* to define and explain a
new topic,
we can reverse the emphasis: Explain well the new concept
using a minimum
of new notation/terminology, and then when the new concept is
well-understood, use the new concept to introduce the new
notation.
This is just my way of saying, that yeah, sooner or later,
*some* of the
students may have to become familiar with delta notation, but
using delta
notation at the very beginning is hardly the best way to
explain slope of
a straight line.
--- Joe (And I didn't use the word context once!) Sroka
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Subject: Re: 8th grade
> The discussion of slope of a line in an 8th grade class
seems to me to
> have veered off from the main point.
> First, introducing a Greek letter (delta) is a useless
distraction.
> Greek letters are never needed in elementary math - they
just distract
> certain students, or cause a bit of confusion, or reinforce
the notion
> some have that math is hard, abstract, and pointless. We
should be
> spending our time on teaching about interesting math, not
seemingly
> exotic notation. Ignore Greek letters entirely in K-14.
How can you call something (use of Greek alphabet) that is
mainstay in
mathematics exotic notation?
Are you implying pi should not be introduced until at least
yr. 15??? Or
do
you instead replace the Greek letter with something else, and
call it
something else?
> Second, the main point about 'slope' is that it is an
INVARIANT
> quantity, a constant, that is computable using ANY pair of
points on a
> line L, given a Cartesian coordinate system X - Y in which L
is not
> parallel to the Y axis.
Stop shouting (I thought this was a moderated group...I
shouldn't have to
be
the one to say this)
Slope is not necessarily constant. For a determined line,
perhaps, but why
must everything in k-12 math be a determined line in the
context of
slope.
> Such a fact would in physics be called a
> Conservation Law. Everthing about the connection between
lines and
> linear equations falls out of this one fact, as do the
equations of
> uniform linear motion (parametric representation of a line).
> We should always make a big fuss whenever we come upon
invariants in
> math -- it's an important and useful idea. It's the simplest
kind of
> pattern to notice. For example, circumference/diameter for
all
> circles is an invariant called pi,
You shouldn't be allowed, according to your own statements, to
use any
Greek
here. Seems its OK for you to do it, though, just not anyone
else.
Or did you forget pi is a Greek letter?
Pi, is what you call 'invariant' because it's a _constant_.
Why not just
call it a constant, since that's what it is. In this context,
its not any
different from any other constants (0, 1, etc.)
_Slope_ is not necessarily constant. Even in the context of
lines, it
doesn't have to be. Consider a general line of the form y=mx+b
where m can
be any real. There. The slope is _not_ constant.
(area of inscribed circle)/(area of
> square) is invariant pi/4, (volume of cone)/(volume of
prism) is
> invariant 1/3 for a cone inscribed in a prism [same base of
any shape,
> same height], a sequence is arithmetic iff its first
differences are
> constant [invariant is amount of increase between successive
terms], a
> sequence is geometric iff the ratios of its successive terms
is
> constant [growth factor], a sequence is shifted geometric
iff the
> ratios of its successive first differnces is invariant, a
sequence is
> quadratic iff its second differences are constant, a
sequence is cubic
> iff its third differences are constant, first differences of
how far a
> ball falls (or rolls down a slope, see Galileo) during
succesive time
> periods (seconds, etc) is 2*(distance travelled in first
period), etc.
> But if the kids don't know about similarity, how will I
convince them
> that slope is indeed invariant?
Because maybe its just _not_ so invariant as you think it is.
You seem to
be using invariant differently than most. Normally, in
mathematics
when
we say something is an invariant, we mean that it doesn't
really change
after going through some sort of transformation. You seem to
be using it
as
a synonym for constant. They are really not the same concept.
Invariants
can be variable; constants are just, well, constants.
Well I might make a few
> transparencies for an overhead projector, each one having a
Cartesian
> grid and a single line L drawn on it, and on the line I
would have
> marked maybe 6 points A,B,C,D,E,F and beside each one the
pair of its
> coordinates -- which I should also be able to see from the
grid and
> tic marks on the axes.
Again, you have a determined line. Why do you think any
discussion of a
line at all must involve such a determined line? Can't we (and
in fact,
_don't_ we) speak of general cases, where the slope has not
been specified?
There are then C(6,2)=15 pairs of points that
> can be used to calculate (change in Y)/(change in X) when
moving from
> one point to the other, and these calculations could be
carried out by
> students in small groups assigned to use a couple of pairs.
Everybody
> gets the same number (we hope) for 1 line, so that is its
invariant
> slope, and then we try it for some other lines, each
yielding its own
> invariant slope calculation.
> (Note that I only teach college students.)
Are they juniors yet (so that you can show them Greek
letters?) Just
curious. You said Greek letters should not be introduced at
all in k-14.
Let me get this straight. You teach _college_ students how to
find the
slope of a given line by giving them a pair of points on the
line? so when
_do_ you get around showing them any Greek? Grad school?
--
Darrell
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Subject: Re: 8th grade
> Pi, is what you call 'invariant' because it's a _constant_.
Why not just
> call it a constant, since that's what it is. In this
context, its not
any
> different from any other constants (0, 1, etc.)
...
> Because maybe its just _not_ so invariant as you think it
is. You seem
to
> be using invariant differently than most. Normally, in
mathematics
when
> we say something is an invariant, we mean that it doesn't
really change
> after going through some sort of transformation. You seem to
be using it
as
> a synonym for constant. They are really not the same concept.
Invariants
> can be variable; constants are just, well, constants.
A constant is a member of some set (the reals, the complex
numbers,
the integers, ...)
An invariant is a predicate that evaluates to true over all
settings
of its free variables.
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Subject: Re: 8th grade
This from dr6583@bellsouthsnip.net, responding to
ladnor@email.unc.edu:
>Let me get this straight. You teach _college_ students how to
find the
>slope of a given line by giving them a pair of points on the
line? so
when
>_do_ you get around showing them any Greek? Grad school?
Recheck the subject line: 8th grade.
First year Algebra students in high school (OR in 8th grade)
may study
linear
equations, and the slope for any particular line is constant.
It may be
unknown and treated as a variable until it is known. That is
the general
idea
about a line; some of the specifics may vary depending on the
exact topic
being
instructed for.
About greek letters, why all the fuss? Do beginning algebra
students use
them?
The greek symbols come to common use in Geometry and
Trigonometry.
G C
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Subject: Re: 8th grade
> This from dr6583@bellsouthsnip.net, responding to
ladnor@email.unc.edu:
>Let me get this straight. You teach _college_ students how to
find the
>slope of a given line by giving them a pair of points on the
line? so
when
>_do_ you get around showing them any Greek? Grad school?
> Recheck the subject line: 8th grade.
<...>
algebra students study with regards to lines and slopes. And,
I'm aware
that this person stated (s)he teaches only college, not h.s.
Obviously, (s)he was giving an example of how (s)he might
teach the concept
to an 8th grader, if (s) taught 8th grade. Add an appropriate
emoticon to
the above passage of mine you quoted, since it was written
with tongue in
cheek. I thought it clear that I really don't think this
person teaches
college level mathematics students about lines by making them
find the
slopes using several pairs of points, just to see that it
matters not which
two they use.
...but for that matter, there really /are/ students in college
receiving
such remedial training in elementary algebra, but that's
another story.
One of the points (the original point, actually) was whether
or not to
introduce the notation Delta x to denote a change in x to an
8th grade
class, and some of the arguments against it are for reasons of
little more
than no, its Greek, so don't do it.
The fact of the matter is the Delta notation is commonly used,
even by a
first year algebra student.
> First year Algebra students in high school (OR in 8th grade)
may study
linear
> equations, and the slope for any particular line is
constant. It may be
> unknown and treated as a variable until it is known.
Exactly. That's what I was saying, actually. It may indeed
(and is)
treated like a variable, in certain senses.
<... About greek letters, why all the fuss? Do beginning algebra
students use
them?
Yes, unless beginning algebra has changed significantly since
the last
time I took it ;-)
> The greek symbols come to common use in Geometry and
Trigonometry.
Indeed, but certain ones also come to common use earlier. Pi
(lowercase)
and yes Delta (uppercase) are two.
--
Darrell
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Subject: Re: 8th grade
<nop830thf51cr6kmej4f63629vduqdvgoj@4ax.com>:
> One of the points (the original point, actually) was whether
or not to
> introduce the notation Delta x to denote a change in x to an
8th grade
> class, and some of the arguments against it are for reasons
of little
more
> than no, its Greek, so don't do it.
> The fact of the matter is the Delta notation is commonly
used, even by a
> first year algebra student.
I did not read Ladnor's post that way at all. However, it may
be due in
part to my own background. I have extensive experience as a
high school
teacher, including teaching many years of 9th grade algebra
(which is not,
I assume, unsimilar to teaching 8th grade algebra).
My interpretation of Ladnor's post, was that the introduction
of Delta was
not necessary to the topic of instruction for that lesson, and
so omit it.
I didn't really think he meant to omit any mention of greek
letters in
beginning algebra for 8th graders. Indeed, the greek letter Pi
is commonly
used in first year algebra, and sometimes even earlier than
that, depending
on the middle school curriculum.
However, in regards to a first lesson on slope for either 8th
or 9th grade
students, here is my own experience and recommendations:
First of all, it is important to be aware that for WHATEVER
reason, there
are students who are uncomfortable with algebra, or who have
difficulty
with it. It is important for the teacher, when planning a new
presentation/lesson, to identify what the main objectives are
of the lesson
and to keep his/her audience in mind. Anything that does not
contributed
directly to the main objectives can be considered optional, or
extra
baggage, and possibly eliminated from the presentation.
A first lesson on slope should emphasize (imo) the meaning of
that
measurement. What does it MEAN for a line to have slope of 2?
1? 1/2? -1/2?
Students should be able to associate this value both with the
graph and
with the equation of the line. Given two points they should be
able to
calculate it both symbolically and also by plotting the points
in a plane
and counting out the vertical vs. horizontal change. Also, the
relationship
to similar triangles and constancy of the slope are also
extremely
important ideas.
Amazingly, though the concept is simple, students often have
difficulty in
determining slope due to overlooking details and being
imprecise/careless.
I have usually found that in teaching computing symbolically
from two given
points, one must emphasize that the Y coordinates must always
be on the top
of the fraction and the X's on the bottom. Also, in the formula
(y1-y2)/(x1-x2)
Students are often careless and choose x1 from one of the
coordinate pairs
and y1 from the other.
In counting out on the graph, I try to emphasize to them
choosing a
beginning point and then an ending point, and traveling along
the legs
of
a right triangle, paying attention to the directed distance (a
term
that
I do not use with beginning algebra students, but I'm certain
that the
readers here will understand my meaning).
All of the above usually belong to a first lesson on slope as
I have
usually taught it.
Students who continue on in math will have plenty of
opportunity to see the
letter Delta later, at a time when they are already
comfortable with the
idea and meaning of what slope is and represents.
Contrary to what others have stated in this discussion, I have
seldom seen
the symbol Delta used in textbooks for first or second year
algebra. It is
not uncommon for it to first be introduced in a first year
calculus class,
although it is sometimes introduced in the class prior to
that. And I
honestly see no benefit to introducing it earlier, as to what
advantage the
student will gain by the earlier introduction. It is merely a
symbol and
adds no benefit in conceptual understanding.
However, for some students, the strange symbol is
intimidating, confusing.
I do not know why, but it is simply a fact that teachers can
observe. I
know for my own part, that I personally do find new notation
and symbols
off-putting and sometimes intimidating. Obviously, I just
brush that aside,
work through and get on with the task at hand. But for a 13
year old kid
who already doesn't like algebra, the introduction of notation
that is not
necessary for the lesson can make the difference between
whether the kid
comes out of the classroom feeling like he understood what was
presented
that day or not, which can also affect whether or not he feels
like he
knows what to do for homework that night or not.
I also have taught a number of remedial algebra courses at the
university
level, and we do not introduce delta in those courses either
when the topic
of the lesson is the first introduction to the topic of slope.
FWIW, I have
seen students start in the earliest of the remedial math
courses and work
their way up to courses where plenty of Greek letters are
used. But there
is really no need to introduce them in beginning algebra.
Except for Pi.
That one is so commonly used, and the students are quite
comfortable with
it, because it is a constant. Other than that, I can see no
reason or
benefit for throwing Greek letters in before at least
Trigonometry (usually
a third year of math for most high school students?)
--
Sheila King
http://www.thinkspot.net/sheila/
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Subject: Re: 8th grade
>A first lesson on slope should emphasize (imo) the meaning of
that
>measurement. What does it MEAN for a line to have slope of 2?
1? 1/2?
-1/2?
>Students should be able to associate this value both with the
graph and
>with the equation of the line.
A small interjection:
The first lessons on slope should happen long before these
abstractions.
Using T-tables and Cartesian graphs to models scenarios such
as total
earnings from a job that pays $8 an hour, or the charge for
ordering
silk-screened shirts with a $25 setup fee and a rate of $12
per shirt, is
both utilitarian and helpful for the beginner in his quest to
grasp the
notion of slope. I'd also have at least a discussion, or
better a
demonstration of how to use a framing square (pitch of a
roof), and
talking about how water drains from athletic fields that have
been
excavated with a 1 to 2 percent grade. I might even toss in
some mention
of monter - Fr. to climb or to mount, (apologies to Karen D.
;-) ) and
certainly I'd mention that most of the rest of the classrooms
around the
world use the word gradient, rather than slope.
--
charlie dick
The right to be left alone -- the most comprehensive
of rights, and the right most valued by a free people.
- Justice Louis Brandeis, Olmstead v. U.S. (1928).
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Subject: Re: 8th grade
>A first lesson on slope should emphasize (imo) the meaning of
that
>measurement. What does it MEAN for a line to have slope of 2?
1? 1/2?
-1/2?
>Students should be able to associate this value both with the
graph and
>with the equation of the line.
> A small interjection:
> The first lessons on slope should happen long before these
abstractions.
> Using T-tables and Cartesian graphs to models scenarios such
as total
> earnings from a job that pays $8 an hour, or the charge for
ordering
> silk-screened shirts with a $25 setup fee and a rate of $12
per shirt, is
> both utilitarian and helpful for the beginner in his quest
to grasp the
> notion of slope. I'd also have at least a discussion, or
better a
> demonstration of how to use a framing square (pitch of a
roof), and
> talking about how water drains from athletic fields that
have been
> excavated with a 1 to 2 percent grade. I might even toss in
some mention
> of monter - Fr. to climb or to mount, (apologies to Karen D.
;-) ) and
> certainly I'd mention that most of the rest of the
classrooms around the
> world use the word gradient, rather than slope.
> --
> charlie dick
Oh, come now, Charlie. You know that such discussions should
best be saved
for the Applications lesson that comes at the end of each
chapter in
the
algebra text. ;)
--
Sheila King
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Subject: Re: 8th grade
Well it seems that I have a way of putting my foot in my mouth
- but
since that appears to generate some interesting discussion
(rants?) on
this forum, I'll not be hesitant to try again.
The question of just what geometric facts are commonly, or may
in very
good programs be, discussed in elementary and middle schools
came up
in a couple of the earlier notes to this thread. Since I'm
curious
about this and have no idea about it, I ask for someone to
make a list
for us so we can consider it seriously.
When (in K-12) is the notion of area introduced, and when are
area
formulas developed and for what kinds of regions?
What early versions or elements of similarity are taught, and
at what
grade level?
Regardless of whether it is taught specifically, what
geometric facts
(invariance of slope of a line, for example) can we pretty
much depend
on most kids knowing, and by what age?
I have some provisional ideas about the simplest appealing,
memorable
and workable forms of some geometric facts which perhaps we can
discuss later (the road theorem, the sidewalk theorem, and the
boardwalk [or grill] theorem). Can you recommend some papers
or books
or web pages that are directly relevant to these kinds of
considerations?
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Subject: Re: 8th grade
> The question of just what geometric facts are commonly, or
may in very
> good programs be, discussed in elementary and middle schools
came up
> in a couple of the earlier notes to this thread. Since I'm
curious
> about this and have no idea about it, I ask for someone to
make a list
> for us so we can consider it seriously.
 When (in K-12) is the notion of area introduced, and when
are area
> formulas developed and for what kinds of regions?
> What early versions or elements of similarity are taught,
and at what
> grade level?
> Regardless of whether it is taught specifically, what
geometric facts
> (invariance of slope of a line, for example) can we pretty
much depend
> on most kids knowing, and by what age?
> I have some provisional ideas about the simplest appealing,
memorable
> and workable forms of some geometric facts which perhaps we
can
> discuss later (the road theorem, the sidewalk theorem, and
the
> boardwalk [or grill] theorem).
To follow up what Sheila said about a google search, I offer
the
following.
Here are the NCTM Principles & Standards for School
Mathematics:
http://standards.nctm.org/document/index.htm
Here are the Mathematics Content Standards for California
Public
Schools:
http://www.cde.ca.gov/standards/math/
In both sets of standards, check out the algebra and geometry
expectations for middle school, and check out what is expected
for
late elementary. The NCTM standards have been used as a model
for the
state standards of most states outside of CA.
I very much would like to see these provisional ideas posted
here at
k12.ed.math. I think that I speak for many when I say that I've
thoroughly enjoyed the math content that you've posted here and
elsewhere.
Paul
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Subject: Re: 8th grade
<okbd30tdi6s44offq6utga4t37s376d8aq@4ax.com>:
> Well it seems that I have a way of putting my foot in my
mouth - but
> since that appears to generate some interesting discussion
(rants?) on
> this forum, I'll not be hesitant to try again.
;)
> The question of just what geometric facts are commonly, or
may in very
> good programs be, discussed in elementary and middle schools
came up
> in a couple of the earlier notes to this thread. Since I'm
curious
> about this and have no idea about it, I ask for someone to
make a list
> for us so we can consider it seriously.
I would think that searching on Math standards at Google.com
might produce
some such lists.
> When (in K-12) is the notion of area introduced, and when
are area
> formulas developed and for what kinds of regions?
I've only taught at grade 9 and above, but I have observed
that my own
children did study area for at least certain shapes in
elementary school.
> What early versions or elements of similarity are taught,
and at what
> grade level?
I seem to recall working with my daughter on some worksheets
about similar
triangles when she was in middle school.
<snip>
--
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Subject: Re: 8th grade
> <nop830thf51cr6kmej4f63629vduqdvgoj@4ax.com>:
One of the points (the original point, actually) was whether
or not to
introduce the notation Delta x to denote a change in x to an
8th grade
class, and some of the arguments against it are for reasons of
little
more
than no, its Greek, so don't do it.
The fact of the matter is the Delta notation is commonly used,
even by
a
first year algebra student.
> I did not read Ladnor's post that way at all.
> My interpretation of Ladnor's post, was that the
introduction of Delta
was
> not necessary to the topic of instruction for that lesson,
and so omit
it.
> I didn't really think he meant to omit any mention of greek
letters in
> beginning algebra for 8th graders.
Really. How else do you interpret
Ignore Greek letters entirely in K-14. ?
That's a quite clear statement, with a quite clear meaning,
even clarified
by ladnor's supporting content (w/ the exception of pi, that
is).
> Indeed, the greek letter Pi is commonly
> used in first year algebra, and sometimes even earlier than
that,
depending
> on the middle school curriculum.
It's utterly ridiculous (not just simply misspeaking, but
utterly
_ridiculous_) to suggest to ignore Greek letters completely in
K-14, even
if
you exclude pi. If exotic notation is to be feared for the
simple fact
that it's exotic, therefore may be a hindrance at least for
the moment,
then
even pi would never be introduced. Nor would x. heck, nort
would
algebra
alltogether.
> However, in regards to a first lesson on slope for either
8th or 9th
grade
> students, here is my own experience and recommendations:
<...>
That's all quite fine, but avoiding Delta initially when first
discussing
the concept of slope, is very different from suggesting it (or
any other
Greek letter) be avoided in *all* K-14 math. That's what was
suggested.
You can interpret enirely as with the exception of pi if you
want
to,
but that doesn't make much sense. Entirely is, well, entirely,
ie
without
exception. Even clarifying later that pi should be excepted,
does not in
any way discredit the use of any other Greek letter.
Specifically, Ladnor
made it clear (beyond even your doubt, surely) that Greek
should not be
used
for angles, other variables, etc. That's just silly. The
letter x can
also
be intimidating on day 1 of algebra I too, but we don't go
around
suggesting
it be avoided in *all* of K-12 math. On day 1 of alg. 1,
perhaps, but
certainly for not much longer. They _will_ encounter it, as
they will
encounter some Greek letters as well, uless you want to
rewrite all the
textbooks that contain them.
How do you express a sum in Sigma notation, or is this another
exception to
entirely? Do you just completely avoid it? Call it something
else?
Of
course not. When the time comes you know as well as I do that
you use the
standard, Greek notation, *and* you also know full well that
this time may
first come in K-12 somewhere.
> First of all, it is important to be aware that for WHATEVER
reason, there
> are students who are uncomfortable with algebra, or who have
difficulty
> with it.
Why is everyone shouting? Has everyone forgot common
netiquette?
> It is important for the teacher, when planning a new
> presentation/lesson, to identify what the main objectives
are of the
lesson
> and to keep his/her audience in mind. Anything that does not
contributed
> directly to the main objectives can be considered optional,
or extra
> baggage, and possibly eliminated from the presentation.
Fine, but that's a far cry from suggesting that *all* extra
baggage
*always*
be eliminated, at *all* times in K-14, as statements such as
Ignore Greek
letters entirely in K-14 imply without question.
Snip long explanation of slope introduction; its fine but
doesn't really
apply to the issue at hand, which is whether or not Greek
should be
completely avoided at *all* times in K-14, not just on the day
they are
first introduced to slope of a line. _That's_ the issue.
Suggesting it
isn't, or never was intended to be the issue by Ladnor, is
simply refusing
to see Ladnor's post for what it is; a clear suggestion that
Greek be
completely avoided, with the exception of pi, in all of K-14
mathematics.
The reasons given for the suggestion, if taken to have merit,
also imply
that we should give equal merit to refraining from introducing
even English
letters as variables in all of K-14. After all, there is
little argument
that they (symbols like x and what not) have indeed proven to
be a
hindrance
to some, at least initially, to the overall concepts being
addressed. In
short, some seem to be discriminating here against the Greek
notation for
no
more reason that it not being English. If Greek was something
they could
not expect to see again, I may support the idea more, but the
reality is
they will see it again, assuming they stay in math, and many
times this is
even before they get out of h.s.
--
Darrell
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Subject: Re: 8th grade
<i76d305g6thuqtqoi4737ti8sinfiblroj@4ax.com>:
<nop830thf51cr6kmej4f63629vduqdvgoj@4ax.com>:
> One of the points (the original point, actually) was whether
or not
to
> introduce the notation Delta x to denote a change in x to an
8th
grade
> class, and some of the arguments against it are for reasons
of little
> more
> than no, its Greek, so don't do it.
 The fact of the matter is the Delta notation is commonly
used, even by
a
> first year algebra student.
I did not read Ladnor's post that way at all.
My interpretation of Ladnor's post, was that the introduction
of Delta
was
not necessary to the topic of instruction for that lesson, and
so omit
it.
I didn't really think he meant to omit any mention of greek
letters in
beginning algebra for 8th graders.
> Really. How else do you interpret
> Ignore Greek letters entirely in K-14. ?
Well, I thought that either he was exaggerating, or (as he
noted) that he
was not familiar with the K-12 curriculum, so I just didn't
really focus on
that one, short sentence out of his entire, rather
well-written and lengthy
meant literally.
Indeed, the greek letter Pi is commonly
used in first year algebra, and sometimes even earlier than
that,
> depending
on the middle school curriculum.
> It's utterly ridiculous (not just simply misspeaking, but
utterly
> _ridiculous_) to suggest to ignore Greek letters completely
in K-14, even
if
> you exclude pi. If exotic notation is to be feared for the
simple
fact
> that it's exotic, therefore may be a hindrance at least for
the moment,
then
> even pi would never be introduced. Nor would x. heck, nort
would
algebra
> alltogether.
To be precise, he called the notation seemingly exotic. As I
read that,
he is considering the student's point of view, and that
because the student
has not seen these symbols used in mathematical statements
previously, the
notation may seem exotic to the student.
I guess you've decided to focus on this one brief statement of
his and I
really don't have anything more to say to that particular
statement. He has
also, in other posts, indicated that he was rushed and may
have made
statements that were not 100% accurate as to how he feels on
the matter.
However, in regards to a first lesson on slope for either 8th
or 9th
grade
students, here is my own experience and recommendations:
> <... That's all quite fine, but avoiding Delta initially when
first discussing
> the concept of slope, is very different from suggesting it
(or any other
> Greek letter) be avoided in *all* K-14 math. That's what was
suggested.
See above.
> The letter x can also
> be intimidating on day 1 of algebra I too, but we don't go
around
suggesting
> it be avoided in *all* of K-12 math. On day 1 of alg. 1,
perhaps, but
> certainly for not much longer. They _will_ encounter it, as
they will
> encounter some Greek letters as well, uless you want to
rewrite all the
> textbooks that contain them.
Actually, it is entirely possible to complete the required 3
years of math
required for college admission and not encounter any greek
letters other
than Pi. I suppose if they actually enroll in college and take
the required
GE course, which will probably include at least statistics if
it does not
require some sort of practical calculus course, then they will
eventually
need to deal with some additional greek letters. But at that
stage of the
game, I have rarely seen adult students have problems
accepting the
notation used. It's quite different teaching a 12 year old vs.
an adult who
has had quite a bit more exposure to algebra.
--
Sheila King
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http://www.k12groups.org/
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Subject: Re: 8th grade
> The discussion of slope of a line in an 8th grade class
seems to me to
> have veered off from the main point.
> First, introducing a Greek letter (delta) is a useless
distraction.
> Greek letters are never needed in elementary math - they
just distract
> certain students, or cause a bit of confusion, or reinforce
the notion
> some have that math is hard, abstract, and pointless. We
should be
> spending our time on teaching about interesting math, not
seemingly
> exotic notation. Ignore Greek letters entirely in K-14.
Greek letters aren't scary. If you imply that using greek
letters is
grown up (_We_ call the change in x delta x) it might even make
them want to learn it. Then USE deltas all over the place. My
nephews memorized the names of lots of Yu-Gi-Oh! monsters at
the ages
of six and seven, and Pokemon before that, and so did all their
friends. If they can comprehend the concept of change in x, I
don't
think that the name delta x is going to be a problem.
> Second, the main point about 'slope' is that it is an
INVARIANT
> quantity, a constant, that is computable using ANY pair of
points on a
> line L, given a Cartesian coordinate system X - Y in which L
is not
> parallel to the Y axis. Such a fact would in physics be
called a
> Conservation Law. Everthing about the connection between
lines and
> linear equations falls out of this one fact, as do the
equations of
> uniform linear motion (parametric representation of a line).
> We should always make a big fuss whenever we come upon
invariants in
> math -- it's an important and useful idea. It's the simplest
kind of
> pattern to notice. For example, circumference/diameter for
all
> circles is an invariant called pi, (area of inscribed
circle)/(area of
> square) is invariant pi/4, (volume of cone)/(volume of
prism) is
> invariant 1/3 for a cone inscribed in a prism [same base of
any shape,
> same height], a sequence is arithmetic iff its first
differences are
> constant [invariant is amount of increase between successive
terms], a
> sequence is geometric iff the ratios of its successive terms
is
> constant [growth factor], a sequence is shifted geometric
iff the
> ratios of its successive first differnces is invariant, a
sequence is
> quadratic iff its second differences are constant, a
sequence is cubic
> iff its third differences are constant, first differences of
how far a
> ball falls (or rolls down a slope, see Galileo) during
succesive time
> periods (seconds, etc) is 2*(distance travelled in first
period), etc.
> But if the kids don't know about similarity, how will I
convince them
> that slope is indeed invariant? Well I might make a few
> transparencies for an overhead projector, each one having a
Cartesian
> grid and a single line L drawn on it, and on the line I
would have
> marked maybe 6 points A,B,C,D,E,F and beside each one the
pair of its
> coordinates -- which I should also be able to see from the
grid and
> tic marks on the axes. There are then C(6,2)=15 pairs of
points that
> can be used to calculate (change in Y)/(change in X) when
moving from
> one point to the other, and these calculations could be
carried out by
> students in small groups assigned to use a couple of pairs.
Everybody
> gets the same number (we hope) for 1 line, so that is its
invariant
> slope, and then we try it for some other lines, each
yielding its own
> invariant slope calculation.
> (Note that I only teach college students.)
Math should be fun. Calculating the slope of a single line
fifteen
times sounds dreadful. If you really must show that the slope
of a
line doesn't change, why not just cut out a right triangle and
slide
it along the line? Not a terribly exciting demonstration but
it gets
it over with more quickly so you can discuss more interesting
topics.
--Jeff
--
Ho, ho, ho, hee, hee, hee
and a couple of ha, ha, has;
That's how we pass the day away,
in the merry old land of Oz.
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Subject: learning disabilities
I am trying to find information on the Everyday Math
curriculum and if
there is a parallel curriculum for students with Learning
Disabilities, as the students I have are being left behind by
the
regular Everyday Math curriculum. Thank you!
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Subject: Score data
Hi there,
I'm doing a project for my stats class...we need to find a set
of data and
analyze it.
I thought about doing ACT/SAT data and how it effects college
performance..or lack of effect..
Problem is I don't know where to find such data...
If this data isn't available, does anyone have any other ideas
where I can
find a set of data that interesting to analyze?
John
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Subject: Re: Score data
Go to the US Census web site. You'll find data up the wazoo.
http://www.census.gov/
Matt T
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Subject: Re: Score data
> Hi there,
> I'm doing a project for my stats class...we need to find a
set of data
and
> analyze it.
> I thought about doing ACT/SAT data and how it effects college
> performance..or lack of effect..
> Problem is I don't know where to find such data...
> If this data isn't available, does anyone have any other
ideas where I
can
> find a set of data that interesting to analyze?
The testing companies do publish some data about their tests,
as they
try marketing them to college admissions offices. You may not
be able
to get your hands on any raw data from them though, as they
don't
really want competing analyses to come out. I have recently
seen some
tables showing the correlation between GRE scores and
first-year grad
GPAs, but there were a whole bunch of caveats that should have
been
with the table and weren't. (It was a secondary source and
only an
extract from that, so the original publication may have had the
necessary information about the sample sizes and probable
sample
biases.)
--
Kevin Karplus karplus@soe.ucsc.edu
http://www.soe.ucsc.edu/~karplus
life member (LAB, Adventure Cycling, American Youth Hostels)
Effective Cycling Instructor #218-ck (lapsed)
Professor of Biomolecular Engineering, University of
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Undergraduate and Graduate Director, Bioinformatics
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Subject: Available data
Approach any school Principal and ask if you can see data from
the
testing done by students over the last few years.
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Subject: Re: Available data
> Approach any school Principal and ask if you can see data
from the
> testing done by students over the last few years.
The data needed for tracking students (how well does their
performance
on one test predict their performance in future courses or on
future
tests) needs careful processing to remove student identities,
or
you'll be violating federal law about the confidentiality of
student
records. Thus it may be (and should be) difficult to get
permission
from a school to use such data, unless you are employed by the
school
district.
You may be able to get some anonymized data from the
admissions office
of a local university---many do keep records allowing them to
determine such things as the correlation between various test
scores
and first-year GPAs or successful completion of bachelor's
degrees.
In California, the continual budget cuts and staff cuts to the
universities have made it more difficult to gather such data,
so
finding it may be harder.
--
Kevin Karplus karplus@soe.ucsc.edu
http://www.soe.ucsc.edu/~karplus
life member (LAB, Adventure Cycling, American Youth Hostels)
Effective Cycling Instructor #218-ck (lapsed)
Professor of Biomolecular Engineering, University of
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Subject: Re: Available data
Approach any school Principal and ask if you can see data from
the
testing done by students over the last few years.
> The data needed for tracking students (how well does their
performance
> on one test predict their performance in future courses or
on future
> tests) needs careful processing to remove student
identities, or
> you'll be violating federal law about the confidentiality of
student
> records. Thus it may be (and should be) difficult to get
permission
> from a school to use such data, unless you are employed by
the school
> district.
> You may be able to get some anonymized data from the
admissions office
> of a local university---many do keep records allowing them to
> determine such things as the correlation between various
test scores
> and first-year GPAs or successful completion of bachelor's
degrees.
> In California, the continual budget cuts and staff cuts to
the
> universities have made it more difficult to gather such
data, so
> finding it may be harder.
Check the National Center for Education Statistics
http://nces.ed.gov/.
NCES collects data on a myriad of topics and provides free
access to their
data files. They tend to be pretty complex data files so
you'll need to
read the documentation carefully.
---
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Subject: Eighth grade Algebra 1 in a 42 minute period
I am interested in any and all suggestions regarding this
situation,
no matter how small nor radically sweeping.
Here's the problem:
I teach Algebra 1 to eighth graders, the same Algebra 1 (as
based on
the 25 California standards) as is taught in the high schools.
I have
my students for five periods per week, four of which have 42
minutes
and one has 32 minutes. My 200 minutes should be compared to
the time
high school teachers have, which is somewhere around 250
minutes. I
teach the same thing to younger students in four-fifths of the
time.
I am expected to provide the same Algebra education as a high
school
teacher. Both of us will send qualified, passing students to
the next
class in the chain: Geometry.
How do you do this? How has anyone done this? How do you save
time?
How do you present content? Do you limit homework? How much
homework
do you check and answer questions about? What materials work
best for
you? What stories, comparisons, costumns, group work, etc.,
etc.
works for you? Do you short-shrift some of the standards, and
if so,
which? How often do you quiz, test, work one-on-one with
students?
Thank you, in advance, for any contribution.
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Subject: Re: Eighth grade Algebra 1 in a 42 minute period
> I am interested in any and all suggestions regarding this
situation,
> no matter how small nor radically sweeping.
> Here's the problem:
 I teach Algebra 1 to eighth graders, the same Algebra 1 (as
based on
> the 25 California standards) as is taught in the high
schools. I have
> my students for five periods per week, four of which have 42
minutes
> and one has 32 minutes. My 200 minutes should be compared to
the time
> high school teachers have, which is somewhere around 250
minutes. I
> teach the same thing to younger students in four-fifths of
the time.
> I am expected to provide the same Algebra education as a
high school
> teacher. Both of us will send qualified, passing students to
the next
> class in the chain: Geometry.
> How do you do this? How has anyone done this? How do you
save time?
> How do you present content? Do you limit homework? How much
homework
> do you check and answer questions about? What materials work
best for
> you? What stories, comparisons, costumns, group work, etc.,
etc.
> works for you? Do you short-shrift some of the standards,
and if so,
> which? How often do you quiz, test, work one-on-one with
students?
> Thank you, in advance, for any contribution.
I agree with prior responses that the time you're provided is
too short,
but it sounds like that's what you're faced with so I'll put
in my two
cents.
First, if you're teaching Algebra I to eighth graders, they
must be (or
have been judged to be) pretty advanced for eighth graders.
The rest of
my response is based on this assumption.
Given that, I'd suggest you tell them that they're honors
students and
that you'll treat them as such. This means that (for example)
when you
go over an assignment or exam, you shouldn't need to analyze
every
problem and that THEY need to be responsible for asking
questions; I'd
lean toward having homework answers on the board or overhead
at the
beginning of class and let them check their own work; you're
going to
need to use class time to TEACH as much as possible.
I'd try NOT to short-shrift standards but would probably
minimize
one-on-one time, just out of necessity - you can give that in
extra help
sessions, to those who ask for it! I'd give frequent but small
assessments, so the students know how they're doing before
they fall too
far behind.
Finally, if I'm correct in my assumption that these are
advanced eighth
graders, I'd remember that if they need to drop back to
pre-algebra (or
whatever your school's normal 8th-grade course is called),
they WILL
NOT be risking graduating on time. And if they don't do that
and end up
failing Algebra I, they can repeat it as 9th-graders - in all
likelihood, still not a risk to graduation.
Good luck!
Lisa
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Subject: Re: Eighth grade Algebra 1 in a 42 minute period
> First, if you're teaching Algebra I to eighth graders, they
must be (or
> have been judged to be) pretty advanced for eighth graders.
The rest
of
> my response is based on this assumption.
That may not be a good assumption in a lot of places. In CA
where I
teach, the standard for 8th grade is algebra, which means that
every 8th
grader is expected to be in algebra. When we have an eighth
grader who
is not in algebra we get an automatic fail for that student on
the state
standardized tests for purposes of our school API. Since that
is the
measure of our schools quality and it is used to identify
underperforming
schools (and underperforming administrators whose jobs are on
the line)
we have our administrators forcing a lot of kids into algebra
classes
that they are completely unprepared for. Perversely, if the
kid goes to
9th grade ands still doesn't take algebra the high school's
API is not
impacted because their API is more based on the rate of the
students'
success on the state High School Exit Exam.
And, FWIW, we have the same problem the original poster
mentioned, we are
expected to teach the same curriculum in 180 45-minute periods
that the
high school covers in 180 55-minute periods.
Ahhh, public education, the land of cognitive dissonance.
Rich
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Subject: Re: Eighth grade Algebra 1 in a 42 minute period
>First, if you're teaching Algebra I to eighth graders, they
must be (or
>have been judged to be) pretty advanced for eighth graders.
The rest
> of
>my response is based on this assumption.
> That may not be a good assumption in a lot of places. In CA
where I
> teach, the standard for 8th grade is algebra, which means
that every 8th
> grader is expected to be in algebra. When we have an eighth
grader who
> is not in algebra we get an automatic fail for that student
on the state
> standardized tests for purposes of our school API. Since
that is the
> measure of our schools quality and it is used to identify
underperforming
> schools (and underperforming administrators whose jobs are
on the line)
> we have our administrators forcing a lot of kids into
algebra classes
> that they are completely unprepared for. Perversely, if the
kid goes to
> 9th grade ands still doesn't take algebra the high school's
API is
not
> impacted because their API is more based on the rate of the
students'
> success on the state High School Exit Exam.
> And, FWIW, we have the same problem the original poster
mentioned, we are
> expected to teach the same curriculum in 180 45-minute
periods that the
> high school covers in 180 55-minute periods.
> Ahhh, public education, the land of cognitive dissonance.
> Rich
are writing the performance standards can't see the nonsense
in their
requirements? Maybe they should go back to ninth - er, eighth
- grade
algebra! Go figure.....
Good luck -
Lisa
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Subject: Re: Eighth grade Algebra 1 in a 42 minute period
l.belec_nospam@comcast.net comments about reply:
>are writing the performance standards can't see the nonsense
in their
>requirements? Maybe they should go back to ninth - er, eighth
- grade
>algebra! Go figure.....
>Good luck -
>Lisa
standards HAVE met those standards during their time as
students. How do
we
know for sure?
No matter! 9th grade in high school is not too late for many
students to
learn
elementary algebra IF they obtain the proper remediation
before studying
algebra. For most students, doing well in elementary algebra
is just a
matter
of having the right remediation and having good academic
discipline for the
subject. In regard to the time per day in class and number of
days in the
school year, 45 minutes should be the absolute minimum. Nine
months needed
G C
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Subject: Re: Eighth grade Algebra 1 in a 42 minute period
>I teach Algebra 1 to eighth graders, the same Algebra 1 (as
based on
>the 25 California standards) as is taught in the high
schools. I have
>my students for five periods per week, four of which have 42
minutes
>and one has 32 minutes. My 200 minutes should be compared to
the time
>high school teachers have, which is somewhere around 250
minutes. I
>teach the same thing to younger students in four-fifths of
the time.
In my earlier days, we studied algebra in class for 50 minutes
per day, 5
days
per week, for about 8.5 to 9.0 months (the typical 'year'
term). Anyway,
this
was 250 minutes per week. The algebra teacher usually gave
various
discussions
both on and off topic because he was social and skilled at
having a group
conversation about school events and school and community
situations and
life
experiences. This cut down about 6 to 8 minutes per class
meeting. Your
42
minutes per class meeting might be adequate, but do NOT goof
around with
the
time; get to business right away. No funny speeches, just do
the work.
Really, the 250 minutes would be a much better situation. You
also need to
be
very organized.
G C
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Subject: Re: Eighth grade Algebra 1 in a 42 minute period
> I teach Algebra 1 to eighth graders, the same Algebra 1 (as
based on
> the 25 California standards) as is taught in the high
schools. I have
> my students for five periods per week, four of which have 42
minutes
> and one has 32 minutes. My 200 minutes should be compared to
the time
> high school teachers have, which is somewhere around 250
minutes. I
> teach the same thing to younger students in four-fifths of
the time.
> I am expected to provide the same Algebra education as a
high school
> teacher. Both of us will send qualified, passing students to
the next
> class in the chain: Geometry.
> How do you do this? How has anyone done this? How do you
save time?
> How do you present content? Do you limit homework? How much
homework
> do you check and answer questions about? What materials work
best for
> you? What stories, comparisons, costumns, group work, etc.,
etc.
> works for you? Do you short-shrift some of the standards,
and if so,
> which? How often do you quiz, test, work one-on-one with
students?
I have no experience with this age group or this problem.
The courses I teach at UCSC are all 210 minutes a week (either
3
70-minute classes or two 135-minute ones), and we are expected
to
cover material at a much faster pace (the year of high school
algebra
is a 10-week remedial course here).
The way most of us end up doing it is to spend most of class
time on
direct instruction and assign lots of homework. (Our students
take 3
classes at a time, and the typical load is 12-15 hours a week
per
class, only 3.5 hours of which is class time.) You should
probably
aim for 5 hours a week, 3.3 of which are in class--so about 1-2
hours of homework each week.
All homework should be graded and discussed in class after
grading, so
that common errors can be rectified.
Leave out a lot of the froo-froo (costumes??? stories???) and
concentrate on the essential material---being able to take a
statement
of a problem in English, convert it to algebraic equations,
and solve for
the unknown variable(s). If your school has a good science
teacher
who teaches the same students, connect up with him or her to
come up
with some exercises that work for both of you (collecting the
data in
the science class, fitting a straight line in the algebra
class, for
example). That way you can increase the effective instruction
time
without needing more of the student's day.
--
Kevin Karplus karplus@soe.ucsc.edu
http://www.soe.ucsc.edu/~karplus
life member (LAB, Adventure Cycling, American Youth Hostels)
Effective Cycling Instructor #218-ck (lapsed)
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Subject: re: Solving the stated problem not the problem you
think up
Mister, I've read a lot of anti-high-stake-testing posts in
this forum. I
understand why you guys are against it.
I merely ask that people think beyond the American perspective
for a moment,
and consider all the possible social functions it could serve
in other
societies. OTOH, I am not saying that testing would solve THE
American
problem. Not at all. (If you don't believe in something, it
will never
work
in the first place.) I merely ask that you do not consider
testing as
*EVIL*
for everyone in the world, simply because it doesn't suit the
American
social
and cultural taste.
For simple-minded people like my parents back home, their
children's success
in
a high-stake test is their glimmer of hope to get out of the
bottom rung of
the
social ladder. Are you telling me that this is bad for them?
This was the
only
hope they had.
(of course, this is something I have trouble getting Americans
to
understand,
cause the cultural environment is so different)
Jeffrey Turner (jturner@localnet.com) said:
The point is not to give people just a hope of this, but to
try to
make it so that there's NO correlation between where you start
off and
where you end up. That is supposed to be goal of the US
educational
system and that is a valid perspective from which to critique
standardized tests.
A recent study, done in Arizona, showed that high stakes tests
do not
correlate with - in fact may be counterproductive to - future
success
in academia and the real world. Humans are not like widgets
and the
whole premise that you can quality control their education by
testing
them more appears to be faulty.
-------------------------------------------------------
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Subject: Re: Solving the stated problem not the problem you
think up
> Recently, several questions in this group made me thinking
of our
> math education in general.
> Seems to me we put too much emphasis on using daily language
to
> describe math problems and in the process lost the beauty of
> precise of math. As a result, the students sometimes are
trying
> to solve a problem they think are asked.
> bashing the examine designers about item 8 of this test. see
> item 8.
> http://www.mathforum.com/epigone/mathed-news/clarcorplee
This problem was discussed by Fernando Q. Gouvea in page 6 of
the Dec.
America.
Students had to calculate the perimeter, to the nearest
centimeter, of
a pictured regular octagon with the following given data:
1. The radius is 4.6 cm (distance from the center to a vertex).
2. The apothem is 4cm (perpendicular distance from the center
to a
side).
The choices were 41cm, 36cm, 27cm, and 18cm.
As Gouvea pointed out, no octagon with the given measurements
exists:
If the radius is 4.6, then the apothem would be 4.6(cos22.5) =
4.2.
Students who used the Pythagorean theorem would obtain the
length of
half a side to be 2.2715633, yielding a perimeter that would
round to
36 cm.
Students who used trigonometry, would obtain the length of
half a side
to be 4(tan22.5) = 1.6568542, yielding a perimeter that would
round to
27 cm.
How did the test preparers come up with the numbers 4.6 and 4?
Gouvea
sees the following two possibilities:
1. For a Hexagon, a radius of 4.6 yields an apothem of
4.6(cos30) =
3.9837169, which rounds to 4.0 cm. So the authors did the
calculations for a hexagon and then changed to an octagon
without
changing the numbers.
2. A more intriguing one is this: If you compute 4.6(cos22.5)
with
the calculator set to radians you get -4.01. Ignore that minus
sign,
and tadah: ...
It just occurred to me that there may be a third possibility:
The test
preparers may have calculated an apothem of 4 cm by carelessly
using a
central angle of 60 degrees for an Octagon.
posted at:
http://mathforum.org/epigone/math-teach/plemzitai
The first from The Hartford Courant deals with a scoring
debacle that
has occurred with the Connecticut Mastery Tests. The second
from the
Seattle Times is a grim expose of how scoring mills operate.
Dom Rosa
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Subject: Re: Solving the stated problem not the problem you
think up
> Recently, several questions in this group made me thinking
of our
> math education in general.
> Seems to me we put too much emphasis on using daily language
to
> describe math problems and in the process lost the beauty of
> precise of math. As a result, the students sometimes are
trying
> to solve a problem they think are asked.
> bashing the examine designers about item 8 of this test. see
> item 8.
> http://www.mathforum.com/epigone/mathed-news/clarcorplee
This problem was discussed by Fernando Q. Gouvea in page 6 of
the Dec.
America.
Students had to calculate the perimeter, to the nearest
centimeter, of
a pictured regular octagon with the following given data:
1. The radius is 4.6 cm (distance from the center to a vertex).
2. The apothem is 4cm (perpendicular distance from the center
to a
side).
The choices were 41cm, 36cm, 27cm, and 18cm.
As Gouvea pointed out, no octagon with the given measurements
exists:
If the radius is 4.6, then the apothem would be 4.6(cos22.5) =
4.2.
Students who used the Pythagorean theorem would obtain the
length of
half a side to be 2.2715633, yielding a perimeter that would
round to
36 cm.
Students who used trigonometry, would obtain the length of
half a side
to be 4(tan22.5) = 1.6568542, yielding a perimeter that would
round to
27 cm.
How did the test preparers come up with the numbers 4.6 and 4?
Gouvea
sees the following two possibilities:
1. For a Hexagon, a radius of 4.6 yields an apothem of
4.6(cos30) =
3.9837169, which rounds to 4.0 cm. So the authors did the
calculations for a hexagon and then changed to an octagon
without
changing the numbers.
2. A more intriguing one is this: If you compute 4.6(cos22.5)
with
the calculator set to radians you get -4.01. Ignore that minus
sign,
and tadah: ...
It just occurred to me that there may be a third possibility:
The test
preparers may have calculated an apothem of 4 cm by carelessly
using a
central angle of 60 degrees for an Octagon.
posted at:
http://mathforum.org/epigone/math-teach/plemzitai
The first from The Hartford Courant deals with a scoring
debacle that
has occurred with the Connecticut Mastery Tests. The second
from the
Seattle Times is a grim expose of how scoring mills operate.
Dom Rosa
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Subject: Re: Solving the stated problem not the problem you
think up
In my opinion, the only reason that we have an outside test to
determine
graduation is that the public can't understand how some people
graduate from high school and can barely read, write, or do
basic
computation. Until we stop social promotions and demand that
kids show
real accomplishment to move on to higher grades, diplomas
cannot be
trusted.
Although we have implemented a high stakes test in
Massachusetts, there
is still no provision to record on the pupil's transcript the
grade or
the date it was taken. Further, the test is given in Mar- May
and the
scores are not returned until December. Those who failed have
been most
likely promoted into the next year.
> Mister, I've read a lot of anti-high-stake-testing posts in
this forum. I
> understand why you guys are against it.
> I merely ask that people think beyond the American
perspective for a
moment,
> and consider all the possible social functions it could
serve in other
> societies. OTOH, I am not saying that testing would solve
THE American
> problem. Not at all. (If you don't believe in something, it
will never
work
> in the first place.) I merely ask that you do not consider
testing as
*EVIL*
> for everyone in the world, simply because it doesn't suit
the American
social
> and cultural taste.
> For simple-minded people like my parents back home, their
children's
success in
> a high-stake test is their glimmer of hope to get out of the
bottom rung
of the
> social ladder. Are you telling me that this is bad for them?
This was the
only
> hope they had.
> (of course, this is something I have trouble getting
Americans to
understand,
> cause the cultural environment is so different)
> Jeffrey Turner (jturner@localnet.com) said:
> The point is not to give people just a hope of this, but to
try to
> make it so that there's NO correlation between where you
start off and
> where you end up. That is supposed to be goal of the US
educational
> system and that is a valid perspective from which to critique
> standardized tests.
> A recent study, done in Arizona, showed that high stakes
tests do not
> correlate with - in fact may be counterproductive to -
future success
> in academia and the real world. Humans are not like widgets
and the
> whole premise that you can quality control their education
by testing
> them more appears to be faulty.
> -------------------------------------------------------
> This mail sent through HKedCity http://www.hkedcity.net
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Subject: In-Depth, Self-Guided Consumer Math Curriculum for
Teens
True Life Activities include lessons on creating a personal
budget,
the real cost of credit cards, planning for savings and
retirement,
calculating a loan payment, estimating taxes and insurance,
researching job salaries, discovering the costs of higher
education,
Best of all, the lessons are integrated into a process: True
Life
guides the user to envision their lifestyle, figure out what
that
lifestyle will truly cost, find out what jobs will provide
enough
income, and then commit to getting enough education to get the
right
jobs.
It's more than individual lessons on money, finances, and
self-management  it's a method of aligning a teen's
expectations with
the reality of their capabilities and motivation.
True Life Activities include lessons on the working world,
creating a
personal budget, finding and furnishing an apartment,
calculating loan
payments, shopping for a car, computing transportation costs,
discovering the real cost of credit, planning savings and
retirement,
estimating taxes and insurance, finding the right job or
career,
locating a college or university, setting goals for the
future, making
commitments to change, and much more. Most importantly, all
these
lessons are wrapped a process of aligning teens' expectations
with the
realities of their capabilities.
Visit http://truelifeinteractive.com/consumer-math.htm
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Subject: Math made easy
I am working at an alternative school and have bought the math
made
easy on ebay. I find them a way to individualize the
instruction.
Ron Crisp Math teacher for 31 years.
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===
Subject: Why integrated math
What's so great about integrated math. I haven't been able to
figure
this out yet. If someone can please tell me a GOOD reason for
it,
i'll listen. I am trying to get it kicked out of my district.
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Subject: Re: Why integrated math
<130d30lq3p2fhkv2gthbifevsa47vuspqa@4ax.com>:
> What's so great about integrated math. I haven't been able
to figure
> this out yet. If someone can please tell me a GOOD reason
for it,
> i'll listen. I am trying to get it kicked out of my district.
I think it is important to distinguish between the notion of
an integrated
curriculum, vs. the available published curriculums that use
integrated
math.
I think the concept of teaching geometry and algebra in
parallel so that
the ideas from one branch can be used in the other and thereby
a connection
and emphasis between the two branches can be established, is
an excellent
goal.
However, the last time I looked (about 5 years ago...) I did
not see any
integrated math series from the usual textbook publishing
companies that I
would have felt worthwhile.
--
Sheila King
http://www.thinkspot.net/sheila/
http://www.k12groups.org/
http://www.mathxy.com
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Subject: Re: Why integrated math
How about Math books from SingaporeMath.com? The New Elementary
Mathematics series (7-10th grade) is integrated.
(I'm not advertising for them or anything).
> I think the concept of teaching geometry and algebra in
parallel so that
> the ideas from one branch can be used in the other and
thereby a
connection
> and emphasis between the two branches can be established, is
an excellent
> goal.
> However, the last time I looked (about 5 years ago...) I did
not see any
> integrated math series from the usual textbook publishing
companies that
I
> would have felt worthwhile.
> --
> Sheila King
> http://www.thinkspot.net/sheila/
> http://www.k12groups.org/
> http://www.mathxy.com
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Subject: Re: Why integrated math
> What's so great about integrated math. I haven't been able
to figure
> this out yet. If someone can please tell me a GOOD reason
for it,
> i'll listen. I am trying to get it kicked out of my district.
I know that the primary reason that integrated math was
instituted in my
district is this:
- the state-mandated tests for my state are first given in
10th grade
and include both algebra and geometry
- a student who takes algebra in 9th grade (as is typical) but
fails
that year generally repeats algebra, and so has to take the
state tests
without ever having seen geometry; and thus is doomed on that
test.
- our poorer math students take integrated math in order to be
exposed
to both algebra and geometry concepts during both 9th and 10th
grades,
in the hope that they will retain SOME of each and have a
better chance
to attain the minimum passing requirements.
Probably 20-25% of our students are in the integrated math
track (oh,
that's right, we don't call it tracking). Since we have a 93%
pass
rate on the first try (and 99% have passed by their fourth
try), it
actually seems to be achieving its goal pretty well. Prior to
our
integrated math courses, the pass rate was measureably lower.
Just my two cents....
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===
Subject: Why integrated math
===
>Subject: Why integrated math
>Author: Laura Wilkinson <l_wilky@hotmail.comWhat's so great about integrated math. I haven't been able to
figure
>this out yet. If someone can please tell me a GOOD reason for
it,
>i'll listen. I am trying to get it kicked out of my district.
Laura,
There are certain ideas that seem on their faces to be so
compelling and people are so utterly infatuated with them that
people
will not give up on them despite many years of failed efforts.
Three
examples that come immediately to mind are the IQ test,
learning
styles, and marxism. Integrated mathematics is another.
And, for just the same reasons as with IQ tests, learning
styles,
and marxism, people are convinced---truly and profoundly
**convinced**, mind you---that the next time they are going to
get it
right. That is why, while I wish you all successes, I predict
you
will fail to kick integrated math out of your district.
When you do fail, do not blame yourself. You are right but
the race is not to the swift,
nor the battle to the strong,
neither yet bread to the wise,
nor yet riches to men of understanding,
nor yet favor to men of skill;
but time and chance happeneth to them all.
Haim
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Subject: Re: Why integrated math
>What's so great about integrated math. I haven't been able to
figure
>this out yet. If someone can please tell me a GOOD reason for
it,
>i'll listen. I am trying to get it kicked out of my district.
Nothing is so good about integrated math. You are well
justified to want
it
gone from where you work. Realize that not all integrated
mathematics
course
arrangments are equivalent. In any case, have you tried to
convince your
colleagues or administrators to reorganize your integrated
mathematics?
Mathematics should work this way:
You learn Mathematics from dedicated subject matter courses
focusing on one
major topic of Mathematics; then you apply this knowledge in
an integrated
manner. This is much more lifelike. A statistical situation
may use some
relatively simple algebra. A mix-blend situation may rely on
some fairly
simple geometry and a bit of algebra. A technical data
collection
situation
may rely on algebra (including some linear algebra) to produce
a graph and
find
an equation which comes close to defining the graph.
G C
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Subject: Re: Why integrated math
> What's so great about integrated math. I haven't been able to
> figure this out yet. If someone can please tell me a GOOD
reason
> for it, i'll listen. I am trying to get it kicked out of my
> district.
Mathematics education has become a veritable Tower of Babel --
integrated math, connected math, math connections, sequential
math, core plus, etc. There is nothing great about these
programs,
other than the fact that the National Science Foundation has
awarded
multimillion-dollar grants to produce and promote them. Our
standard
textbooks--Algebra I, Geometry, Algebra II, Precalculus--have
become
so abominable that these programs are being adopted by
misguided
school districts.
Dom Rosa
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===
Subject: 1st order Diff Eqns
Hi there,
I've been studying some kinetics and have come up with this
differential eqn from my experiment.
dy/dx = 2y-xy^2
I can't for the life of me see how you can solve it. As far as
I see
it using the substitution y=vx doesn't help. Can anyone lend
me a
hand?
Thank you for your help, it is really really appreciated.
Sarah
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Subject: Re: 1st order Diff Eqns
> Hi there,
> I've been studying some kinetics and have come up with this
> differential eqn from my experiment.
> dy/dx = 2y-xy^2
> I can't for the life of me see how you can solve it. As far
as I see
> it using the substitution y=vx doesn't help. Can anyone lend
me a
> hand?
> Thank you for your help, it is really really appreciated.
> Sarah
Don't you look at the replies you get?
This is what I gave you on sci.physics 20 minutes after you
asked:
| > take
| > z = 1/y
| > so
| > dz/dx = -1/y^2 dy/dx
| > Then
| > -1/y^2 dy/dx = -1/y^2 (2y - xy^2)
| > = -2/y + x
| > so
| > dz/dx = -2z + x
| >
| > hth
|
| By the way, my taking z = 1/y was a bit of a lucky guess,
induced
| by your complaint about having y^2 screwing it up :-)
| But in fact your equation is a simple example of the
Bernoulli
equation.
| See for instance
|
http://www.sosmath.com/diffeq/first/bernouilli/bernouilli.html
| In this case
| p(x) = -2
| q(x) = -x
| n = 2
|
| Dirk Vdm
Dirk Vdm
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===
Subject: Re: 1st order Diff Eqns
===
Subject: 1st order Diff Eqns
>I've been studying some kinetics and have come up with this
>differential eqn from my experiment.
>dy/dx = 2y-xy^2
Doesn't look dimensionally balanced.
>I can't for the life of me see how you can solve it.
>using the substitution y=vx doesn't help.
1/y dy/dx = 2 - xy
u = log y
du/dx = 2 - xe^u
u = 2x - v
2 - dv/dx = 2 - xe^2x e^-v
e^v dv/dx = xe^2x
w = e^v
dw/dx = xe^2x
w = (1/4)(2x - 1)e^2x + c
v = log w
u = 2x - log w
y = e^2x / w
= 4 / (2x - 1 + 4ce^-2x)
= 4 / (2x - 1 + ce^-2x)
let c = 0
y = 4/(2x-1)
y' = -8/(2x - 1)^2
2y - xy^2 = 8/(2x-1) - 16x/(2x-1)^2
c /= 0 asymptotically approaches c = 0.
----
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===
Subject: Re: 1st order Diff Eqns
> Hi there,
> I've been studying some kinetics and have come up with this
> differential eqn from my experiment.
> dy/dx = 2y-xy^2
> I can't for the life of me see how you can solve it. As far
as I see
> it using the substitution y=vx doesn't help. Can anyone lend
me a
> hand?
> Thank you for your help, it is really really appreciated.
> Sarah
Mathematica gives
sol=DSolve[y'[x]==2*y[x]-x*y[x]^2,y[x],x]; Print[sol];
{{y[x] -> (4*E^(2*x))/(^(2*x) + 2*E^(2*x)*x + 4*C[1])}}
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Subject: Re: 1st order Diff Eqns
>I've been studying some kinetics and have come up with this
>differential eqn from my experiment.
>dy/dx = 2y-xy^2
>I can't for the life of me see how you can solve it. As far
as I see
>it using the substitution y=vx doesn't help. Can anyone lend
me a
>hand?
On both sides, subtract 2y and divide by y^2 to get
1 dy 1
--- -- - 2 - = -x [1]
y^2 dx y
Using u = 1/y and negating both sides gives
du
-- + 2u = x [2]
dx
This is of the form (D+2)u = x, where D = d/dx. Equations of
this form
can be handled using an integrating factor.
We want to find a function m, the integrating factor, which
satisfies
(D+2)m = 0, because then,
(D+2)(mz) = z(D+2)m + mDz = mDz [3]
Substituting u/m for z, [3] becomes
(D+2)u = mD(u/m) [4]
Noting that (D+2)e^{-2x) = 0, combine [2] and [4] to get
-2x 2x
x = (D+2)u = e D(e u) [5]
Multiplying both sides of [5] by e^{2x} yields
2x 2x
e x = D(e u) [6]
Equation [6] can be solved simply by integration; don't forget
the
constant of integration.
To verify that [6] really is the same as [2], use the product
rule:
2x 2x 2x
e x = 2 e u + e Du [7]
divide [7] by e^{2x}:
x = 2u + Du [8]
and [8] is the same as [2].
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===
Subject: Re: 1st order Diff Eqns
>Hi there,
>I've been studying some kinetics and have come up with this
>differential eqn from my experiment.
>dy/dx = 2y-xy^2
>I can't for the life of me see how you can solve it. As far
as I see
>it using the substitution y=vx doesn't help. Can anyone lend
me a
>hand?
>Thank you for your help, it is really really appreciated.
>Sarah
This is a Bernoulli equation. Multiply by -y^(-2):
-y^(-2)y' + 2y^(-1) = x and let u = y^(-1)
Then u' = -y^(-2)y' giving:
u' + 2u = x. This has complementary solution Ce^(-2x) and
particular
solution (1/2)x - 1/4 so its general solution is
u = Ce^(-2x) + (1/2)x - 1/4
and y = 1/u.
Note this method precludes the y == 0 solution.
--Lynn
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===
Subject: Re: 1st order Diff Eqns
Well, I've come up with the solution 1/y = x/2 - 1/4 +
C*e^(-2x), for any
constant C.
-Michael.
> Hi there,
> I've been studying some kinetics and have come up with this
> differential eqn from my experiment.
> dy/dx = 2y-xy^2
> I can't for the life of me see how you can solve it. As far
as I see
> it using the substitution y=vx doesn't help. Can anyone lend
me a
> hand?
> Thank you for your help, it is really really appreciated.
> Sarah
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===
Subject: First (?) follow-up
In no particular order:
In a public intermediate school, even the best teacher will
lose
from four to eight minutes at the beginning of each period
while
students take out needed materials, present notes to the
teacher
(absence slips, grade tracking forms from the
counselors/assistant
principals, etc.), finish (or try to start!)
conversations...The
quality of an intermediate school teacher can be judged by how
many
and how long students are actively involved in the class. I
recall a
movement a few years back where teachers were encouraged to
teach to
the bell, an idea that swept through a conference for
administrators
somewhere. I know many of my colleagues give in for the last
ten
minutes, particularly after the end of the first semester.
I teach as many minutes as I can. It requires a tremendous
amount of
energy. I don't know many jobs that require that kind of manic
behavior.
Last year I taught two classes of Algebra 1 and four of
Transitional
Algebra. Transitional Algebra is a class for students who did
poorly
(or at least not well) in 7th grade pre-Algebra. The idea was
to give
instruction in the basics of Algebra while remediating
problems from
seventh grade. Last year there were six Algebra 1 classes on
campus
and nine of Transitional Algebra.
This year I teach six classes of Algebra 1. There are ten
Algebra 1
classes and four Transitional Algebra. The decision was made
to give
those kids,many of whom scored in the below basic category on
those
tests we all cringe about, a big shove into Algebra 1. It's
not clear
to me why this was done, though the post about losing API
points makes
a scary amount of sense now. Guess who has most of these young
people...
I do not teach honors students.
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Subject: Geometry later in education
What happens with Geometry later in a Mathematics education,
after (long
after?) the study of the remedial lower division kind taken
just before
standard trigonometry? Does any coursework reexamine it for
purposes of
review
plus further development? I have the feeling that most
mathematical
activities
rely on analytic geometry and trigonometry if any sort of
geometry is used.
What would be the title of a course which examines Geometry at
and above
that
lower division remedial course?
G C
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Subject: Re: Geometry later in education
When I was in college I had a class called A Survey of
Geometry which
did
some brief review of plane geometry and then very quickly went
into
nonuclidian topics such as geometry on different solids like
spheres and
saddles (please forgive, but I can't for the life of me
remember the
correct mathematical term). If I remember right it was a
junior level
course and had several prerequisites that seemed to have
nothing to do with
geometry.
Amy
> What happens with Geometry later in a Mathematics education,
after (long
> after?) the study of the remedial lower division kind taken
just
before
> standard trigonometry? Does any coursework reexamine it for
purposes of
review
> plus further development? I have the feeling that most
mathematical
activities
> rely on analytic geometry and trigonometry if any sort of
geometry is
used.
> What would be the title of a course which examines Geometry
at and above
that
> lower division remedial course?
> G C
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Subject: Re: Geometry later in education
> What happens with Geometry later in a Mathematics education,
after (long
> after?) the study of the remedial lower division kind taken
just
before
> standard trigonometry? Does any coursework reexamine it for
purposes of
review
> plus further development? I have the feeling that most
mathematical
activities
> rely on analytic geometry and trigonometry if any sort of
geometry is
used.
> What would be the title of a course which examines Geometry
at and above
that
> lower division remedial course?
Most mathematicians never see axiomatic Euclidean geometry
again after
high school. Analytic geometry and vector spaces are
ubiquitous. The
axiomatic *approach* to math they'll see again, generally in
the most
detail in group theory, number theory, and set theory classes.
--
Kevin Karplus karplus@soe.ucsc.edu
http://www.soe.ucsc.edu/~karplus
life member (LAB, Adventure Cycling, American Youth Hostels)
Effective Cycling Instructor #218-ck (lapsed)
Professor of Biomolecular Engineering, University of
California, Santa Cruz
Undergraduate and Graduate Director, Bioinformatics
Affiliations for identification only.
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Subject: 3 number combinations 2.
I need to compose a list of all possible three number sets
using the
numbers 1 through 4. can anyone help me compile a list? All
help will
be greatly appriciated(spelling)
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Subject: Re: 3 number combinations 2.
> I need to compose a list of all possible three number sets
using the
> numbers 1 through 4. can anyone help me compile a list? All
help will
> be greatly appriciated(spelling)
Can you use numbers more than once?
If you can, start with:
1111
1112
1113
1114
1121
1122
..
..
4444
(ie, increment the last digit as many times as you can, then
let it roll
over to the next digit, just like a clock or odometer)
If you can only use each number once, simply eliminate
everything from
the above list with duplicate digits. So you'd have (again,
smallest to
largest)
1234
1243
1324
etc
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Subject: Re: 3 number combinations 2.
>I need to compose a list of all possible three number sets
using the
>numbers 1 through 4. can anyone help me compile a list? All
help will
>be greatly appriciated(spelling)
> Can you use numbers more than once?
> If you can, start with:
> 1111
> 1112
> 1113
> 1114
> 1121
> 1122
> ..
> ..
> 4444
> (ie, increment the last digit as many times as you can, then
let it roll
> over to the next digit, just like a clock or odometer)
> If you can only use each number once, simply eliminate
everything from
> the above list with duplicate digits. So you'd have (again,
smallest to
> largest)
> 1234
> 1243
> 1324
> etc
Uh, those are _four_ number sets, but they do give the basic
idea.
--Jeff
Next trick, counting to 1023 on your fingers.
--
Ho, ho, ho, hee, hee, hee
and a couple of ha, ha, has;
That's how we pass the day away,
in the merry old land of Oz.
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Subject: equations
what are the steps to finding out for example y=2b+5 what are
you
suppose to do? Please help thankyou
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Subject: Re: equations
> what are the steps to finding out for example y=2b+5 what
are you
> suppose to do? Please help thankyou
Finding what out? I can see quite a few things you could be
doing here:
1) Solving for y
2) Solving for b
3) Finding the slope
4) Graphing the line
etc
When you ask a math question, the first thing to do is to
precisely
define the problem. Once you know what it's asking, THEN you
can look
for the answer.
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Subject: Re: equations
> what are the steps to finding out for example y=2b+5 what
are you
> suppose to do? Please help thankyou
Finding what out? I can see quite a few things you could be
doing here:
1) Solving for y
2) Solving for b
3) Finding the slope
4) Graphing the line
etc
When you ask a math question, the first thing to do is to
precisely
define the problem. Once you know what it's asking, THEN you
can look
for the answer.
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Subject: HELP~!!!!!!!!!!!!!!!!!!!!~
My math teacher keeps confusing me. Ok, we are on chapter 10
section
5(finding the area, apothem, and perimeter of regular polygons
and
circle). I know how to find the area of regular polygons, but
I don't
know how to find the apothem of them. For example, if you have
a
triangle and the line to the dot in the middle of triangle is
8in.,
how do I find the apothem of that triangle?
Signed,
Confused
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Subject: Re: HELP~!!!!!!!!!!!!!!!!!!!!~
> My math teacher keeps confusing me. Ok, we are on chapter 10
section
> 5(finding the area, apothem, and perimeter of regular
polygons and
> circle). I know how to find the area of regular polygons,
but I don't
> know how to find the apothem of them. For example, if you
have a
> triangle and the line to the dot in the middle of triangle
is 8in.,
> how do I find the apothem of that triangle?
> Signed,
> Confused
A polygon of n sides has 180(n-2) degrees. If you want to work
with a
triangle, you know there is 180 degrees. If you draw the
medians, you get a
60 degree angle. When the apothem is drawn, it will bisect the
angle so
basically, you have a 30-60-90 triangle. In this case, the
apothem is 8 in.
David Moran
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Subject: Re: HELP~!!!!!!!!!!!!!!!!!!!!~
> ... For example, if you have a triangle and the line to the
dot
> in the middle of triangle is 8in., how do I find the apothem
of
> that triangle?
If the line to the dot in the middle of triangle is the line
from a
vertex to the center of the equilateral triangle, you should
be able
to find the apothem by using the properties of a 30-60-90
degree
triangle.
Dom Rosa
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Subject: Re: HELP~!!!!!!!!!!!!!!!!!!!!~
> My math teacher keeps confusing me. Ok, we are on chapter 10
section
> 5(finding the area, apothem, and perimeter of regular
polygons and
> circle). I know how to find the area of regular polygons,
but I don't
> know how to find the apothem of them. For example, if you
have a
> triangle and the line to the dot in the middle of triangle
is 8in.,
> how do I find the apothem of that triangle?
> Signed,
> Confused
A polygon of n sides has 180(n-2) degrees. If you want to work
with a
triangle, you know there is 180 degrees. If you draw the
medians, you get a
60 degree angle. When the apothem is drawn, it will bisect the
angle so
basically, you have a 30-60-90 triangle. In this case, the
apothem is 8 in.
David Moran
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Subject: What is the equation to answer the problem
In this particular problem, I am given only the answer the
product),
and I am to figure out how they arrived at product. What is the
equation to figure this out???
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Subject: Re: What is the equation to answer the problem
> In this particular problem, I am given only the answer the
product),
> and I am to figure out how they arrived at product. What is
the
> equation to figure this out???
er...not knowing anything about the problem, that's kind of
hard to
answer :-)
However, product means multiply.
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Subject: Sylvan Learning center to catch up in math/reading?
Our fourth grade son was recently assessed at a Sylvan Learning
Center. We had him assessed because he is falling behind at
school.
The special education services he has received have allowed
him to
fall drastically behind in the area of math. The total math
score was
a grade level of 3.2, and our son is currently in his 6th
month of
fourth grade. We also suspected a possible comprehension
problem with
reading. The reading assessment showed only 70% comprehension,
all
other reading scores were average. We are considering spending
a
great deal of time and money to hopefully have our son catch
up to
grade level by the time he begins fifth grade in September. We
would
like to hear about good/bad experiences regarding Sylvan from
people
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Subject: Re: Sylvan Learning center to catch up in
math/reading?
Find a Kumon and have them do that for the Math. At the end of
2nd grade
our son was the worst math student (Thank you new math which
stinks) by the
end of 3rd he was the best. I just had his IQ tested 1/2
through 4th and
he is testing in the genius level for math, had his IQ tested
in
kindergarten and it was 100 in math. The same psychologist did
both tests
and they have no idea about him doing kumon. This was the best
thing we
ever did. $90 a month 10-15 minutes a day. His self esteem is
1000x
better.
I doubt Sylvan or a tutor can do this. We didn't do the Kumon
English
since
he has always done well in it. I expect them to be as good in
English
though.
Good luck,
Paul
> Our fourth grade son was recently assessed at a Sylvan
Learning
> Center. We had him assessed because he is falling behind at
school.
> The special education services he has received have allowed
him to
> fall drastically behind in the area of math. The total math
score was
> a grade level of 3.2, and our son is currently in his 6th
month of
> fourth grade. We also suspected a possible comprehension
problem with
> reading. The reading assessment showed only 70%
comprehension, all
> other reading scores were average. We are considering
spending a
> great deal of time and money to hopefully have our son catch
up to
> grade level by the time he begins fifth grade in September.
We would
> like to hear about good/bad experiences regarding Sylvan
from people
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Subject: Re: Sylvan Learning center to catch up in
math/reading?
> Find a Kumon and have them do that for the Math.
I'll second the Kuman learning center. Much cheaper and
better. :o)
--
Sue (mom to three girls)
I'm Just a Raggedy Ann in a Barbie Doll World...
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Subject: Re: Sylvan Learning center to catch up in
math/reading?
> Our fourth grade son was recently assessed at a Sylvan
Learning
> Center. We had him assessed because he is falling behind at
school.
> The special education services he has received have allowed
him to
> fall drastically behind in the area of math. The total math
score was
> a grade level of 3.2, and our son is currently in his 6th
month of
> fourth grade. We also suspected a possible comprehension
problem with
> reading. The reading assessment showed only 70%
comprehension, all
> other reading scores were average. We are considering
spending a
> great deal of time and money to hopefully have our son catch
up to
> grade level by the time he begins fifth grade in September.
We would
> like to hear about good/bad experiences regarding Sylvan
from people
You should call a new IEP meeting to disucss this.Take his
tests and graph
them for impact. Then insist that the school provide tutoring
1:1 or have
them pay for Sylvan.
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Subject: Creative Publications... Middle School Math with
Pizzazz
I need to order the math books called Middle School Math With
Pizzazz. I use them for tutoring my students and they love it.
I am
unable to find information. Can you please let me know where
to find
these books? Thank you in advance, Jaime Gall
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Subject: Re: Creative Publications... Middle School Math with
Pizzazz
> I need to order the math books called Middle School Math With
> Pizzazz. I use them for tutoring my students and they love
it. I am
> unable to find information. Can you please let me know where
to find
> these books? Thank you in advance, Jaime Gall
I did a quick Google search, first with
middle school math with pizzazz
then with
middle school math with pizzazz creative publications
Some of the websites you'll want to check are
the authors' web site: http://www.marcymathworks.com/
the publisher's web site: http://www.wrightgroup.com/
You can order direct from the publisher:
Middle School Math With Pizzazz! All 5 Binders
Author: Steve Marcy and Janis Marcy
Series: Pizzazz!
ISBN: 156107098X
Price: $155.95
Learn to use google--it can save you the embarassment of asking
questions that can be answered in a minute or two with a quick
search.
--
Kevin Karplus karplus@soe.ucsc.edu
http://www.soe.ucsc.edu/~karplus
life member (LAB, Adventure Cycling, American Youth Hostels)
Effective Cycling Instructor #218-ck (lapsed)
Professor of Biomolecular Engineering, University of
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Undergraduate and Graduate Director, Bioinformatics
Affiliations for identification only.
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Subject: Slovenian Mathematics
I just posted this message on a Slovenian Culture newsgroup,
and I
wanted to include a copy here in case anybody could provide me
with
any information or thoughts. Even information or thoughts on
where I
can go next for information about Slovenian Mathematical
Curriculum
~Joseph
My name is Joseph Sustar, and I am a high school Mathematics
teacher
in New York City, USA. I am also currently pursuing my Masters
degree
in Secondary Mathematics at the City College of New York. As
part of
my studies, I am doing a comparative study of how Mathematics
is
taught in various countries.
I am interested in anybody's insights as former or current
students,
teachers, citizens of Slovenia, or from people who know the
culture
well.
My grandparents are originally from Slovenia, and given the
recent
political shifts in Slovenia, I am particularly interested in
how
Mathematics is taught there.
Any information that anybody can provide would be most
appreciated.
Also, if you would be able to forward this message to any
Mathematics
teachers (or any teachers) who would be interested in
responding, I
would be very grateful.
I am looking for answers to the following questions:
1) What is the basic outline of the Mathematics curriculum from
grades K-12 (from the start of schooling up through high
school)?
2) When do you (they) teach what Mathematics subject? For
instance,
when is Algebra taught? When is Geometry taught? When is
Calculus
taught?
3) How is Mathematics taught in Slovenia?
4) Do all students follow the same academic path? Do all
students go
into academic studies? Or are there various tracks for
students? How
do the various tracks affect how Mathematics is taught?
5) When Slovenia regained their independence from Yugoslavia
at the
end of the Cold War, how was the educational system, and
Mathematics
in particular, affected? Are there remnants of the old system?
or
was there an effort to change things?
Thank you very much for your time and for any answers that you
are
able to provide.
Sincerely,
Joseph C Sustar
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Subject: Re: Slovenian Mathematics
I am Hungarian, that country belonged the same cultural cradle
as
Slovenia, and the curriculum was prescribed under the same
Ratio
Educationis introduced first in 1756 and renewed occasionally
Here are the answers, how was the education till the end of the
communist era (interestingly communist did not changed the
former
curriculum, math and science were intact from the ideology)
<<
1) What is the basic outline of the Mathematics curriculum from
grades K-12 (from the start of schooling up through high
school)?
2) When do you (they) teach what Mathematics subject? For
instance,
when is Algebra taught? When is Geometry taught? When is
Calculus
taught?
3) How is Mathematics taught in Slovenia?
>
No math curriculum for Kindergarten, most children did not
attended
Kindergarten at all.
Grades 1-2 : counting, adding subtracting 1-20
Grade 3 : numbers 1-100, multiplication
Grade 4 : division, rations, numbers to infinity
Grades 5-6 : repetition of the previous years, intro to
geometry.
Grade 7 : pre-algebra
Grade 8 : Algebra
Grade 9 : repetition of the previous years
Grade 10 : Introduction to trigonometry and analytic geometry
Grade 11 : Pre-calculus
Grade 12 : Calculus, projective geometry.
any difference in the given school.
<
4) Do all students follow the same academic path? Do all
students go
into academic studies? Or are there various tracks for
students? How
do the various tracks affect how Mathematics is taught?
>
No.
About half of the student after the 8-th grade went to 2 years
trade
schools, which did not provided new topics in Math.
About 25% went to technical schools which were 4 year schools
with
emphasis on some topic, this schools did not covered the
pre-calculus
and calculus.
For the rest 25%, the gymnasium, the curriculum was the same,
it was the
college preparatory curriculum.
<<
5) When Slovenia regained their independence from Yugoslavia
at the
end of the Cold War, how was the educational system, and
Mathematics
in particular, affected? Are there remnants of the old system?
or
was there an effort to change things?
>
Again I don't know Slovenia just Hungary. The above described
200+ years
old curriculum was abandoned, the schools decide themselves
what to
teach, a lot of electives instead of the obligatory
requirement. The
process started in the eighties. In 1992 Hungary was still in
the first
5 in the international test, by 1998 it fall down to the
15-16-th place.
About 10% of the schools keeps the original curriculum, in the
rest
the problem is the lack of discipline, which makes impossible
the
teaching. (This later part are know only from my frien
laszlo
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Subject: 3 digit combinations
the answers are 45 for 2 combinations and 120 for 3
combinations.
use the formula C(n,r)=n!/(n-r)!r!
where n is your total number of objects (10 in this case) and
r is the
number that you have in your combination (1,2,3 in this case)
so just
plug the numbers in
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Subject: I need help with this word problem please
An inheritance of $30,000 is divided into two investments
earning 8.5%
and 10% simple interest respectively. Your goal is to have a
total
annual interest income of $2700. What is the smallest amount
you can
invest at 10% in order to meet your objective?
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Subject: Re: I need help with this word problem please
> An inheritance of $30,000 is divided into two investments
earning 8.5%
> and 10% simple interest respectively. Your goal is to have a
total
> annual interest income of $2700. What is the smallest amount
you can
> invest at 10% in order to meet your objective?
Let A be the amount of money you invest at 8.5% and B be the
amount you
invest at 10%. You now have two equations in two unknowns:
A + B = 30,000
..085A + .1B = 2700
Now solve for B. As this is the amount you'll need in B to
earn exactly
$2700, it is also your minimum.
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Subject: Re: I need help with this word problem please
There are no choices
8.5*x/100 + 10*y/100 = 2700
x + y = 30000
<<
An inheritance of $30,000 is divided into two investments
earning 8.5%
and 10% simple interest respectively. Your goal is to have a
total
annual interest income of $2700. What is the smallest amount
you can
invest at 10% in order to meet your objective?
>
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Subject: Re: I need help with this word problem please
> An inheritance of $30,000 is divided into two investments
earning 8.5%
> and 10% simple interest respectively. Your goal is to have a
total
> annual interest income of $2700. What is the smallest amount
you can
> invest at 10% in order to meet your objective?
If you put all the money into the 10% account, how much
interest would
you get? All in the 8.5% account? See what happens as you
change
some of the money from the 8.5% account to the 10%. Maybe
$1,000 at a
time. Good luck.
--Jeff
--
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Ho, ho, ho, hee, hee, hee
and a couple of ha, ha, has;
That's how we pass the day away,
in the merry old land of Oz.
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Subject: Re: I need help with this word problem please
An inheritance of $30,000 is divided into two investments
earning 8.5%
and 10% simple interest respectively. Your goal is to have a
total
annual interest income of $2700. What is the smallest amount
you can
invest at 10% in order to meet your objective?
> If you put all the money into the 10% account, how much
interest would
> you get? All in the 8.5% account? See what happens as you
change
> some of the money from the 8.5% account to the 10%. Maybe
$1,000 at a
> time. Good luck.
Certainly such experimentation has its place, but a
mathematical method
(read...not a guess) may be in order.
Let x be the amount invested at 8.5%, and 30000-x the amount
invested at
10%.
.085x + .1(30000-x) = 2700
x=?
Reverse it. Let x be the amount invested at 10%, 30000-x the
amount
invested at 8.5%
.085(30000-x) + .1x = 2700
x=?
Which x is smaller?
--
Darrell
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Subject: Re: I need help with this word problem please
> Let x be the amount invested at 8.5%, and 30000-x the amount
invested at
> 10%.
> .085x + .1(30000-x) = 2700
> x=?
> Reverse it. Let x be the amount invested at 10%, 30000-x the
amount
> invested at 8.5%
> .085(30000-x) + .1x = 2700
> x=?
> Which x is smaller?
Of course, no sooner than I hit send (never can it be before)
I see my
blunder. In the former, x is the amount at 8.5%, and in the
latter it is
the amount at 10%. In either case, the amount invested at 10%
is the same.
Obviously, only one of the equations was necessary <doh>.
--
Darrell
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Subject: i need another help here
here is another problem i also need help on:
The sum of the squares of two consecutive odd positive
integers is
130. so i need possible answers for this.
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Subject: Re: i need another help here
>here is another problem i also need help on:
>The sum of the squares of two consecutive odd positive
integers is
>130. so i need possible answers for this.
x and x+2 are the consecutive odd integers
x^2 + (x+2)^2 = 130
x^2 + x^2 + 4x + 4 = 130
2x^2 + 4x - 126 = 0
x^2 + 2x - 63 = 0
(x+9)(x-7)=0
x=-9,7
So there are two sets of solutions: -9 and -7; 7 and 9
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Subject: Re: i need another help here
>here is another problem i also need help on:
>The sum of the squares of two consecutive odd positive
integers is
>130. so i need possible answers for this.
> x and x+2 are the consecutive odd integers
> x^2 + (x+2)^2 = 130
> x^2 + x^2 + 4x + 4 = 130
> 2x^2 + 4x - 126 = 0
> x^2 + 2x - 63 = 0
> (x+9)(x-7)=0
> x=-9,7
> So there are two sets of solutions: -9 and -7; 7 and 9
REREAD the problem---it calls for POSITIVE integers.
--
Kevin Karplus karplus@soe.ucsc.edu
http://www.soe.ucsc.edu/~karplus
life member (LAB, Adventure Cycling, American Youth Hostels)
Effective Cycling Instructor #218-ck (lapsed)
Professor of Biomolecular Engineering, University of
California, Santa Cruz
Undergraduate and Graduate Director, Bioinformatics
Affiliations for identification only.
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Subject: Re: i need another help here
here is another problem i also need help on:
>The sum of the squares of two consecutive odd positive
integers is
>130. so i need possible answers for this.
 x and x+2 are the consecutive odd integers
 x^2 + (x+2)^2 = 130
> x^2 + x^2 + 4x + 4 = 130
> 2x^2 + 4x - 126 = 0
> x^2 + 2x - 63 = 0
> (x+9)(x-7)=0
> x=-9,7
 So there are two sets of solutions: -9 and -7; 7 and 9
>REREAD the problem---it calls for POSITIVE integers.
No need to SHOUT perfesser. I stand as a corrected and ever so
humble
Martyr before you.
Shoes for Industry, Comrade. :-)
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Subject: Re: i need another help here
> here is another problem i also need help on:
> The sum of the squares of two consecutive odd positive
integers is
> 130. so i need possible answers for this.
You mean, what two numbers satisfy that condition? Again,
translate. How
can you express two consecutive odd integers? Assuming x is
odd (we'll
find
out for sure later) how about:
x
x+2 (since adding 2 to any odd integer x gives the next odd
integer)
What are their squares?
What is the sum of these squares?
What does this equal?
Again, this is a quadratic equation (two solutions) so check
both solutions
to see if they works (assuming you're looking for more than
one pair of
numbers that satisfy the condition). Note that x and x+2 are
*not* the
only
was to describe these numbers. How about x-2 and x? What
happens when you
set up your equation using x-2 and x? Do you get the same
solutions as
before? Also don't forget that your two solutions to that
equation are
*not* the two numbers asked for. Rather, they give you two
possibilities,
x, for the first number. The other number is therefore x-2.
Next time please try one yourself, and if you still need help
you can
always
ask for it while showing us what you have tried. So far, you
seem to be
just posing your problems to us without showing what you have
tried.
--
Darrell
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Subject: Re: i need another help here
> here is another problem i also need help on:
> The sum of the squares of two consecutive odd positive
integers is
> 130. so i need possible answers for this.
(2n-1)^2 + (2n+1)^2 = 130
General method:
Rearrange to look like a normal quadratic equation in n,
solve for n, write out 2n-1 and 2n+1, if the quadratic quation
has an
integer solution.
Specific short-cut for this example:
130 is a very small number---one of the squares needs to be
bigger
than half of it, so just enumerate the odd squares between 65
and 130:
81
121
Only one of these can work, 49+81=130.
--
Kevin Karplus karplus@soe.ucsc.edu
http://www.soe.ucsc.edu/~karplus
life member (LAB, Adventure Cycling, American Youth Hostels)
Effective Cycling Instructor #218-ck (lapsed)
Professor of Biomolecular Engineering, University of
California, Santa Cruz
Undergraduate and Graduate Director, Bioinformatics
Affiliations for identification only.
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Subject: Re: i need another help here
> here is another problem i also need help on:
> The sum of the squares of two consecutive odd positive
integers is
> 130. so i need possible answers for this.
130 = 7^2 + 9^2
How many possible answers do you think there can be?
--
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Subject: Re: i need another help here
> here is another problem i also need help on:
> The sum of the squares of two consecutive odd positive
integers is
> 130. so i need possible answers for this.
Two consecutive odd positive integers: x, x+2
Their squares: x^2, (x+2)^2
The sum of their squares: x^2 + (x+2)^2 = 130
--Jeff
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Subject: Re: i need another help here
> here is another problem i also need help on:
 The sum of the squares of two consecutive odd positive
integers is
> 130. so i need possible answers for this.
>Two consecutive odd positive integers: x, x+2
>Their squares: x^2, (x+2)^2
>The sum of their squares: x^2 + (x+2)^2 = 130
More elegant, perhaps:
Two consecutive odd positive integers: x-1, x+1
Their squares: x^2 - 2x + 1
x^2 + 2x + 1
Mean of their squares: m = x^2 + 1
we have m = half of 130 = 65, so its pretty obvious now.
- -
Ken, __O
-<,_
(_)/ (_)
Virtuale Saluton.
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Subject: need help on this problem.
If the sides of a square are lengthened by 7 in., the area
becomes 121
in.(exponent 2). Find the length of a side of the orginal
square.
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Subject: Re: need help on this problem.
>If the sides of a square are lengthened by 7 in., the area
becomes 121
>in.(exponent 2). Find the length of a side of the orginal
square.
x=original length
(x+7)^2 = 121
x+7 = 11 Note: -11 is an extraneous root, as lengths preclude
negative
values in this context, or something like that
x = 4
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Subject: Re: need help on this problem.
> If the sides of a square are lengthened by 7 in., the area
becomes 121
> in.(exponent 2). Find the length of a side of the orginal
square.
What are you looking for? Answers? processes? Hopefully the
latter.
Translate into algebra. If you let x be the side length of the
original
square, then lengthened by 7 means x+7, so that the area of
this new,
bigger square is side^2 ==> (x+7)^2 and you are told what this
equals.
Solve the equation. Note that it is quadratic (two solutions),
but one
solution to that equation will not make sense.
--
Darrell
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Subject: Re: need help on this problem.
> If the sides of a square are lengthened by 7 in., the area
becomes 121
> in.(exponent 2). Find the length of a side of the orginal
square.
(x + 7)^2 = 121
x + 7 = sqrt(121)
= 11
x = 4
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Subject: Re: need help on this problem.
> If the sides of a square are lengthened by 7 in., the area
becomes 121
> in.(exponent 2). Find the length of a side of the orginal
square.
We want to know the length of a side of the original square: x
The new square has sides that are 7in longer than x: x + 7
The area of the new square is: (x + 7)^2 = 121
--Jeff
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Subject: How to solve your problem
First, ask yourself what is the problem asking you to solve?
It's
asking you to find the length of the original side of the
square.
So let x = length of original side of the square.
It's length is increased by 7. This means, the sides of the
square are
now the original length + 7, or x + 7.
The area of this bigger square is given as 121.
If you square the side of the square (i.e., raise it to the 2nd
power), you will get the new area of 121, or:
(x+7)^2 = 121
You can solve this problem in one of two ways: the easy way,
or the
more involved way.
Here's the easy way. We know 121 is 11^2 so the equation is now
(x + 7)^2 = 11 ^2
Take the square root of both sides to obtain x + 7 = 11, or x
=4.
THe second method is to expand the binomial, collect terms on
one side
of the equation and use the quadratic formula:
(x + 7)^2 = x^2 + 14x + 49
So now the equation is x^2 + 14x + 49 = 121
Subtract 121 from both sides so the equation is in proper form
and use
the quadratic formula to solve it. You'll get the same answer
(discared the negative answer).
BG
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Subject: word problems
i can do math once its in an equation form, but im garbage at
word
problems. my question is The base of the temple is a rectangle
with
a perimeter of 300m. The area of the base is 4400msquared.
What are
the dimensions of the base? any help is appreciated,
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Subject: Re: word problems
> i can do math once its in an equation form, but im garbage
at word
> problems. my question is The base of the temple is a
rectangle with
> a perimeter of 300m. The area of the base is 4400msquared.
What are
> the dimensions of the base?
For a rectangle, perimeter is twice the length plus twice the
height, since
opposite sides are equal. You're told what this equals.
Area is the product of length and height. You're told what
this equals
too.
Looks like two equations in two unknowns (the unknowns being
length and
height). Write the equations and solve the system.
--
Darrell
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Subject: Re: word problems
> i can do math once its in an equation form, but im garbage
at word
> problems. my question is The base of the temple is a
rectangle with
> a perimeter of 300m. The area of the base is 4400msquared.
What are
> the dimensions of the base? any help is appreciated,
This is a good problem to attack by drawing a picture. So
start by
drawing a rectangle.
Then look at what the problem is asking you to find. It wants
you to
find the dimensions of the rectangle. Once you've identified
what you
are being asked to find it is generally a good idea to assign
variable
name(s) to what you are trying to find. In this case you are
asked to
find the dimensions of the rectangle, i.e., the length and
width of the
rectangle. So assign the variable name l to the length of the
rectangle
and the variable name w to the width of the rectangle.
Now go back and reread the problem to see what information you
are given
that relates to l and w. The first thing you are told is that
the
perimeter is 300. What is the formula for the perimeter of the
rectangle
in terms of l and w? That's right, P = 2l + 2w, so since you
are told
the perimeter is 300 we have our first equation:
300 = 2l + 2w
Now we are also told that the area is 4400. What is the
formula for the
area of the rectangle in terms of l and w? That's right, A=
(l)(w).
Since we are told that the area is 4400 we now have our second
equation:
4400 = (l)(w)
Now just solve the two equations.
Rich
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Subject: Re: word problems
h_robyn@hotmail.com wants to understand this from words to
symbolism:
>i can do math once its in an equation form, but im garbage at
word
>problems. my question is The base of the temple is a
rectangle with
>a perimeter of 300m. The area of the base is 4400msquared.
What are
>the dimensions of the base? any help is appreciated,
That sounds very straightforward, and uncomplicated.
>The base of the temple is a rectangle
The base forms or covers a rectangular area.
> with
>a perimeter of 300m
The distance around this base consists of the sides; when you
find their
sum,
this sum is 300 m.
>The area of the base is 4400msquared.
When you calculate area, which will be length * width, the
value will be
4400
square meters.
Perimeter: length + length + width + width
Area: length * width
Now you can organize the symbolic information better, and then
solve the
problem.
G C
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Subject: Re: word problems
> i can do math once its in an equation form, but im garbage
at word
> problems. my question is The base of the temple is a
rectangle with
> a perimeter of 300m. The area of the base is 4400msquared.
What are
> the dimensions of the base? any help is appreciated,
What you need to do is read the problem and see what can be
put into
equation form. For example:
>The base of the temple is a rectangle with a perimeter of 300m
You know (one would hope) that the perimeter of a figure is
the sum of
the sides, and since it's a rectangle the opposing sides are
the same
length, so you can rewrite the above sentence as:
2a + 2b = 300
>The area of the base is 4400msquared
Again, rewrite as a formula, using the fact that the area of a
rectangle
is the base times height (or length times width, whichever you
prefer)
ab=4400
Now you have two equations that you can combine. Rewriting the
first
one, you get 2a=300-2b => a=150-b. Plugging this in to the
second
equation gives:
(150-b)b=4400
150b-b^2=4400
Rewritin again, we have:
b^2 - 150b + 4400 = 0, which you can solve with the quadratic
equation.
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Subject: Formulae trouble
Currently this is my biggest problem. I just don't know what
order to
do the operations in.
1/a = 1/b - 1/c ; find b
1/a + 1/c = 1/b
1/a + 1/c = b
_________
1
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Subject: Re: Formulae trouble
>1/a = 1/b - 1/c ; find b
1/a + 1/c = 1/b
c/ac + a/ac = 1/b
(c+a)/ac = 1/b
ac/(c+a) = b/1 = b
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Subject: Re: Formulae trouble
> 1/a = 1/b - 1/c ; find b
> 1/a + 1/c = 1/b
okay
> 1/a + 1/c = b
> _________
> 1
whoops!! while the reciprocal of 1/b is b, the left side is
incorrect
1
--------- = b
1/a + 1/c
this still is messy, let's try to simply the left hand side
What is 1/a + 1/c ?
1 1 c a c+a
- + - = -- + -- = ---
a c ac ac ac
then
1 ac
--------- = ---
1/a + 1/c a+c
thus
ac
--- = b
a+c
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Subject: Re: Formulae trouble
> Currently this is my biggest problem. I just don't know what
order to
> do the operations in.
> 1/a = 1/b - 1/c ; find b
> 1/a + 1/c = 1/b
> 1/a + 1/c = b
> _________
> 1
Wrong way up:
b = 1/(1/a + 1/c)
--
G.C.
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Subject: Re: Formulae trouble
> Currently this is my biggest problem. I just don't know what
order to
> do the operations in.
> 1/a = 1/b - 1/c ; find b
> 1/a + 1/c = 1/b
> 1/a + 1/c = b
> _________
> 1
Close. The inverse of 1/b is 1/(1/b).
--Jeff
--
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Ho, ho, ho, hee, hee, hee
and a couple of ha, ha, has;
That's how we pass the day away,
in the merry old land of Oz.
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Subject: Re: Formulae trouble
> Currently this is my biggest problem. I just don't know what
order to
> do the operations in.
> 1/a = 1/b - 1/c ; find b
The order of the steps you take to solve an equation is
largely a matter of
personal taste, and experience will largely determine the best
choice to
make. For instance, you can first clear fractions by
multiplying
everything
by abc, or you can first add 1/c to both sides. Either way is
a valid
'first step'.
> 1/a + 1/c = 1/b
Fine, you added 1/c to both sides.
> 1/a + 1/c = b
> _________
> 1
You appear to be trying to take reciprocols of both sides. On
the right,
the reciprocol of 1/b is indeed b, but on the left, that is
not the
reciprocol of 1/a+1/c. The reciprocol would be 1 over
(1/a+1/c), not
[1/a+1/c] over 1.
1/(1/a + 1/c) = b
That's a 'complex' fraction, and most instructors at this
level will
probably make you reduce it. So, multiply everything on the
left by the
LCD
of all the 'little' fractions, in this case ac.
1(ac) / [(1/a)ac + 1/c(ac)]
ac / (c+a)
In either event, it appears some simplification is needed in
order to
express the solution in reduced form. You could also cross
multiply (a
shortcut form of clearing fractions by multiplying thru by the
LCD), but
again this will involve, sooner or later, some simplification.
...which
raises the question, would it be any more straightforward to
multiply
everything by abc as a 'first' step, as was initially
suggested?
1/a = 1/b - 1/c
(1/a)(abc) = (1/b)(abc) - (1/c)(abc)
...each of the involved terms is fairly straightforward to
reduce as
opposed
to one larger expression which looks a little 'more' involved
to reduce.
Choose your poison.
bc = ac - ab
...and solve for b from there.
--
Darrell
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Subject: heron's formula
evryone please help! i need a derivation of heron's
formula..thakns a
lot
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Subject: Re: heron's formula
part:
> evryone please help! i need a derivation of heron's formula.
Don't you mean Someone please help!?
Drop an altitude to the longest side and use the Pythagorean
Theorem on
the resulting two smaller triangles to express the altitude's
length in
terms of the three sides' respective lengths. Then you use the
usual
formula for area in terms of altitude and base to express the
area in
terms of the three sides. After some simplification, you get
Heron's
formula.
msh210@math.wustl.edu Of a reply, then, if you have been
cheated,
http://math.wustl.edu/~msh210/ Likely your mail's by mistake
been deleted.
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Subject: Re: heron's formula
Recentyly Rhea <tsinelas71@yahoo.com> queried:
> i need a derivation of heron's formula.
Michael Hamm (mhamm@artsci.wustl.edu) replied:
: Drop an altitude to the longest side and use the Pythagorean
Theorem on
: the resulting two smaller triangles to express the
altitude's length in
: terms of the three sides' respective lengths. Then you use
the usual
: formula for area in terms of altitude and base to express
the area in
: terms of the three sides. After some simplification, you get
Heron's
: formula.
Nice! All elementary. Although, if you don't already know the
formula,
the factoring at the end is a little fierce...
I've always derived Heron's formula using the incircle.
Let 'r' be the radius of the incircle and let 'x', 'y', and 'z'
be the lengths of the three tangents.
So the three sides of the triangle are: a=x+y, b=y+z, and
c=x+z.
[Or, inverting that system: x=(a-b+c)/2, y=(a+b-c)/2,
z=(-a+b+c)/2]
Also, the semiperimeter becomes: s=x+y+z [or s=(a+b+c)/2].
It is easily seen that the area of the triangle is just: A=rs.
Looking at the central angles we see that:
InverseTan(x/r) + InverseTan(y/r) = Pi - InverseTan(z/r)
Take the Tangent of both sides and use the Tangent sum formula
to obtain: r^2 = xyz/s or r = Sqrt(xyz/s).
Substituting into the above area formula makes: A = Sqrt(sxyz)
which is Heron's formula.
The tangent sum formula part is not so elementary.
Can anyone see a nice geometric arguement that Ar=xyz?
Robert, who is:
|)|/| || Burnaby South Secondary School
|| |orewood@olc.ubc.ca || Beautiful British Columbia
Mathematics & Computer Science || (Canada)
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Subject: Re: heron's formula
> evryone please help! i need a derivation of heron's
formula..thakns a
> lot
A proof from the law of cosines is here:
http://mathworld.wolfram.com/HeronsFormula.html
--
G.C.
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Subject: Homework assignments
Are we really suppose to be helping people do their homework
(or take
home tests)? ;)
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Subject: Re: Homework assignments
> Are we really suppose to be helping people do their homework
(or take
> home tests)? ;)
According the FAQ, the link to which is included by the mod at
the bottom
of
every post, yes, we certainly can ;).
--
Darrell
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Subject: Re: Homework assignments
> Are we really suppose to be helping people do their homework
(or take
> home tests)? ;)
Why not. I just don't believe in doing all the work for them.
--Jeff
--
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and a couple of ha, ha, has;
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Subject: i need help
solving it.
Let U be an open set in R^n.
Consider
f in C^{infty}(U, R)
such that
exists lim_{x -> x_0} f(x) / ||x-x_0||^{k-1} = 0 .
Then f belongs to I^k_{x_0}(U, R) .
--------------------------------------------------------------
--------------
--
I^k_{x_0}(U, R) denotes the product of the ideal I_{x_0}(U, R)
k-times with itself and I_{x_0}(U, R) denotes the ideal in
C^{infty} (U,R)
of function vaniscing at p.
In other words, f belongs to I^k_{x_0}(U, R) if and only if f
is of the form:
f = h_0 g_01 * g_02 * .. * g_0{k-1}*g_0k +
+...+
+ h_m g_m1 * g_02 * .. * g_m{k-1}*g_0k
where m is a natural number , say m=1 or m >1 ,
h_i belongs to C^{infty}(U, R)
and
g_ij belongs to I_{x_0}(U,R) <==> g_ij(x_0)= 0 in R.
Note that for m=0 , the proposition above is false,
consider for example k=2 and f(x,y)=x^2+y^2.
Then, exists lim_{(x,y) -> (00)} (x^2+y^2) / ||(x,y)||^{2-1}
= lim_{(x,y) ->(00)} (x^2+y^2)^{1/2}=0
,on the other hand, it's impossible to write f as
f = h_0 g_01 * g_02 with g_01 and g_02 in C^{infty}(U, R) and
vanishing
at p.
My ask is:
Is the proposition above true with m=1 or m>1 ?
Tern
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Subject: Re: i need help
: Let U be an open set in R^n.
: Consider f in C^{infty}(U, R)
: such that lim_{x -> x_0} f(x) / ||x-x_0||^{k-1} = 0
:
: Then f is of the form:
: f = h_0 g_01 * g_02 * .. * g_0{k-1}*g_0k +
: +...+
: + h_m g_m1 * g_m2 * .. * g_m{k-1}*g_mk
:
: where m is a natural number,
: h_i and g_ij belong to C^{infty}(U, R), and g_ij(x_0)=0
A fairly straight-forward induction on 'k'.
: Note that for m=0 , the proposition above is false,
:
: Consider for example k=2 and f(x,y)=x^2+y^2.
: Then, lim_{(x,y) -> (00)} (x^2+y^2) / ||(x,y)||^{2-1}
: = lim_{(x,y) ->(00)} (x^2+y^2)^{1/2}=0
:
: On the other hand, it's impossible to write f as f = h_0
g_01 * g_02
: with g_01 and g_02 in C^{infty}(U, R) and vanishing at x_0.
True. x^2+y^2 does not factor over R.
: My question is:
: Is the proposition above true with m=1 or m>1 ?
Yes. h_0=h_1=1, g_01(x,y)=g_02(x.y)=x, g_11(x,y)=g_12(x,y)=y.
In general 'm', should be one less than the number of monomials
of degree 'k' in 'n' variables, or [(k+n-1) choose (n-1)] - 1.
Actually, you might be able to do better. For example, with
n=2 and k=1, you can always have m=0. For n=2 and k=2, by
completing the square we can get m=1.
Robert, who is:
|)|/| || Burnaby South Secondary School
|| |orewood@olc.ubc.ca || Beautiful British Columbia
Mathematics & Computer Science || (Canada)
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Subject: Don't understand question - A group of 20 people...
I have a question:
A group of 20 people, adults and children, pay $64 to go to a
matinee.
The entrance fee for adults is $6 and children $2. How many
adults are
in the party?
I have no IDEA !!!
I think it is a quadratic equation? Where do I start?
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Subject: Re: Don't understand question - A group of 20
people...
> I have a question:
> A group of 20 people, adults and children, pay $64 to go to
a matinee.
> The entrance fee for adults is $6 and children $2. How many
adults are
> in the party?
> I have no IDEA !!!
You have gotten a number of responses showing you how to solve
it with
simultaneous equations so I'd like to give a different method.
Say all of the attendees were kids. Then the total fees would
be $40 (20
x $2). Every time we replace a child with an adult the total
receipts go
up by $4 (the $6 adult fee minus the $2 child fee). To go from
$40 to
$64 you have to go up $24 dollars. This represents 6
increments of $4,
so 6 children would have to have been replaced by 6 adults.
Therefore
there are 6 adults and 14 children.
This approach will work with most problems of this type, e.g.,
a farmer
has a total of 30 cows and chickens and there are a total of
76 legs.
How many cows and how many chickens are there? If they were
all chickens
there would be 60 legs. When a cow replaces a chicken the
number of legs
goes up by 2. To go from 60 to 76 we have to go up 2 8 times.
So there
are 8 cows and 22 chickens.
Rich
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Subject: Re: Don't understand question - A group of 20
people...
two equations:
a= adults
c= children
a + c = 20 (first equation, number of people attending.)
6a + 2c = 64 (second equation, cost of matinee, $6 per adult,
$2 per child)
a + c = 20
6a + 2c = 64
using the elimination method you can multiply the first
equation by -2 and
that
will get rid of the c and 2c. You are left with 4a = 24. You
can then
figure
out a = 6. If there are 6 adults, there are 14 children.
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Subject: Re: Don't understand question - A group of 20
people...
jasonpush@hotmai.com who has no idea asks:
>A group of 20 people, adults and children, pay $64 to go to a
matinee.
>The entrance fee for adults is $6 and children $2. How many
adults are
>in the party?
>I think it is a quadratic equation? Where do I start?
Two simple linear equations; certainly not quadratic.
G C
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Subject: Re: Don't understand question - A group of 20
people...
This problem needs a set of linear equations.
a = # of adults
c = # of children
a + c = 20
6a + 2c = 64
Solve the system.
John
> I have a question:
> A group of 20 people, adults and children, pay $64 to go to
a matinee.
> The entrance fee for adults is $6 and children $2. How many
adults are
> in the party?
> I have no IDEA !!!
> I think it is a quadratic equation? Where do I start?
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Subject: Re: Don't understand question - A group of 20
people...
> I have a question:
> A group of 20 people, adults and children, pay $64 to go to
a matinee.
> The entrance fee for adults is $6 and children $2. How many
adults are
> in the party?
> I have no IDEA !!!
> I think it is a quadratic equation? Where do I start?
This type of problem is a linear system problem. This is what
to be on
the lookout for, and what to do, and why:
In general, whenever you have a set of objects (people, for
instance)
that is partitioned into nonempty subsets (adults and
children, for
instance), then you have at least one linear equation of at
least 2
variables that you can write. Here, the set has 20 people,
with two
nonempty subsets, adults and children. We don't know yet the
number of
adults and children. So using ax + by = c we can write x for
the
number of adults and y for the number of children:
(1)x + (1)y = 20
where here the coefficient a = 1 and the coefficient b = 1 and
the sum
c = 20.
In word problems like this, be on the lookout for more
information
that you can use to write another linear equation of the same
number
of variables, using the same variables. Here, we have that
information: We have a set of dollars, numbering $64, that is
also
partitioned into two nonempty subsets, where each subset of
dollar
amounts is a function of our subsets of people. This is
crucial, in
that it allows us to use the same variables x and y to write
another
linear equation. Here are the functions I just mentioned: For
each
adult, we have 6 dollars and so we can write 6x. For each
child we
have 2 dollars and so we can write 2y. The sum is 64 dollars.
So using
ax + by = c, we can write
6x + 2y = 64
where here the coefficient a = 6 and the coefficient b = 2 and
the sum
c = 64.
So here is our linear system that we can also write as an
augmented
matrix:
1x + 1y = 20
6x + 2y = 64
or
1 1 : 20
6 2 : 64
The hard part I think for word problems is setting them up,
changing
the words into pure mathematics. Once this is done, then
everything
should be more of a routine. See if you can't solve this
system or
matrix!!
Check your answer by substituting what you find for x and y
into both
equations: It's important that you do this for both equations,
to make
sure.
Paul
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Subject: Re: Don't understand question - A group of 20
people...
There are x at $6 and (20 - x) at $2.
> I have a question:
> A group of 20 people, adults and children, pay $64 to go to
a matinee.
> The entrance fee for adults is $6 and children $2. How many
adults are
> in the party?
> I have no IDEA !!!
> I think it is a quadratic equation? Where do I start?
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Subject: Re: Don't understand question - A group of 20
people...
> I have a question:
> A group of 20 people, adults and children, pay $64 to go to
a matinee.
> The entrance fee for adults is $6 and children $2. How many
adults are
> in the party?
> I have no IDEA !!!
> I think it is a quadratic equation? Where do I start?
Start as always, how else. Determine what you are asked to
identify and
assign it a variable (why not a for #of adults). Let #of
children be
c.
Immediately, you can eliminate one of these variables (why not
eliminate c
since it's a you're after) since you know their sum is 20:
a + c = 20
...and solve that for c.
Now you are working with a single variable, a, so hereafter
where you need
to refer to #of children, instead of writing c you write
whatever the
solutuion for c you got above. Next, write a single equation
utilizing the
other given information. Alternatively, you can keep c in
there and
write
a system of two equations to be solved simultanously (I even
told you what
one of these equations is). In either case, you won't see any
quadratic
equation here.
--
Darrell
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Subject: Re: Don't understand question - A group of 20
people...
> I have a question:
> A group of 20 people, adults and children, pay $64 to go to
a matinee.
> The entrance fee for adults is $6 and children $2. How many
adults are
> in the party?
> I have no IDEA !!!
> I think it is a quadratic equation? Where do I start?
A + C = 20
6A + 2C = 64
etc.
--
G.C.
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Subject: Re: Don't understand question - A group of 20
people...
>I have a question:
>A group of 20 people, adults and children, pay $64 to go to a
matinee.
>The entrance fee for adults is $6 and children $2. How many
adults are
>in the party?
>I have no IDEA !!!
>I think it is a quadratic equation? Where do I start?
Not quadratic, it's a linear system... with co-varying inputs
and
constant outputs.
x=# of adults
y=# of children
x+y=20
6x+2y=64
There's more than a start for you ...
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Subject: Re: Don't understand question - A group of 20
people...
> I have a question:
> A group of 20 people, adults and children, pay $64 to go to
a matinee.
> The entrance fee for adults is $6 and children $2. How many
adults are
> in the party?
> I have no IDEA !!!
> I think it is a quadratic equation? Where do I start?
The question asks for the number of adults, let's call that x.
The total number of people is 20, so if the number of children
is y
then x + y = 20.
We know that the amount of money the adults pay is $6 * x.
And the amount the children pay is $2 * y.
The total amount is $64 = ($6 * x) + ($2 * y).
So we know:
x + y = 20
6x + 2y = 64
Can you solve that or do you need more help?
--Jeff
--
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Subject: Re: Don't understand question - A group of 20
people...
<<
I have a question:
A group of 20 people, adults and children, pay $64 to go to a
matinee.
The entrance fee for adults is $6 and children $2. How many
adults are
in the party?
I have no IDEA !!!
I think it is a quadratic equation? Where do I start?
>
Without equation:
10 adult and 2 children would pay $64
If we substitute 1 adult for 3 children, the payment will be
unchanged, so
10 - 2
9 - 5
8 - 8
7 - 11
6 - 14
5 - 17
4 - 20
3 - 23
2 - 26
1 - 29
0 - 32
etc are satisfiying the payment
You can simplify by observing that each sustitution will
increase the
number of the group memebers by 2, and for the original we
need the
increase of 8 persons, or stating from the minimum of adults,
de
decrease of 12 persons.
With equation
x + y = 20
6*x + 2*y = 64
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Subject: Re: Don't understand question - A group of 20
people...
> I have a question:
> A group of 20 people, adults and children, pay $64 to go to
a matinee.
> The entrance fee for adults is $6 and children $2. How many
adults are
> in the party?
> I have no IDEA !!!
> I think it is a quadratic equation? Where do I start?
Set up a system of equations
Let a=adults and c=children.
a+c=20
6a+2c=64
a=20-c
6(20-c)+2c=64
120-4c=64
-4c=-56
c=14
a=20-14
a=6
David Moran
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Subject: Solution
Actually, it's not a quadratic equation. It's actually a
system of
equations. You can write two equations to solve this problem.
First let, the number of adults be a, and the number of
children be b.
Now you can write:
6a+2c=64
a+c=20
The first equation means $6xNumber of Adults + $2xNumber of
Children =
$64 Total
The second means the Number of Adults + Number of Children =
20.
You you can solve this buy either substitution of subtracting
the
equations.
You can divide the first equation by a 2 to simplify.
3a+c=32
a+c=20
Subtract the second from the first you you'll get:
2a=12
a=6
Hence, there are 6 adults in the party.
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Subject: OK ! - Thank you all, I have another
FIND M;
K = CUBE ROOT (L^2 + 2M^2 - N) ; This means cube root of ()?
______________
4
K^3 = L^2 + 2M^2 - N
______________
4
4K^3 = L^2 + 2M^2 - N
4K^3 + N = L^2 + 2M^2
(4K^3 + N)^2 - L = 2M^2
SQRT (4K^3 + N)^2 - L
______________________ = M
2
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Subject: Re: OK ! - Thank you all, I have another
> FIND M;
> K = CUBE ROOT (L^2 + 2M^2 - N) ; This means cube root of ()?
> ______________
> 4
> K^3 = L^2 + 2M^2 - N
> ______________
> 4
> 4K^3 = L^2 + 2M^2 - N
> 4K^3 + N = L^2 + 2M^2
So far, so good. But the next one goes off the tracks. Just
subtract
L^2 from both sides.
4K^3 + N - L^2 = 2M^2
Should be smooth sailing from there.
--Jeff
> (4K^3 + N)^2 - L = 2M^2
> SQRT (4K^3 + N)^2 - L
> ______________________ = M
> 2
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===
Subject: Incorrect process
As per my example, at the level where:
4K^3 = L^2 + 2M^2 - N
I should have just transposed L^2 accross instead I took the
sqrt from
L then took L. I Can just move (L^2) as a whole unit. This
would
result in:
4K^3 + N - L^2
I should not reduce it to L and ^2. If this is correct I think
this is
the solution to my problem with formulea.
Another point. I am still a little confused as to what order I
complete this process in. -N then L^2 or the other way?
Thank you!!!
Jason
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Subject: Re: Incorrect process
> As per my example, at the level where:
> 4K^3 = L^2 + 2M^2 - N
> I should have just transposed L^2 accross instead I took the
sqrt from
> L then took L. I Can just move (L^2) as a whole unit. This
would
> result in:
> 4K^3 + N - L^2
> I should not reduce it to L and ^2. If this is correct I
think this is
> the solution to my problem with formulea.
> Another point. I am still a little confused as to what order
I
> complete this process in. -N then L^2 or the other way?
> Thank you!!!
That's it. It doesn't matter whether you deal with N or L
first, the
result will be the same either way since there is only
addition and
subtraction involved.
--Jeff
--
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and a couple of ha, ha, has;
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===
Subject: Quadratic equations
I have a question:
3X^2 - 12X -13 = 0
my answer is:
4.89 or - 0.89
The question asks: comment on your solution
My response is:
b^2 - 4ac not = 0
b^2 - 4ac = (-12)^2 - 4(3)(-13)
= 144 - (-156)
= 300
b^2 - 4ac > 0;
(Well I don't know what else to say. But I know there is more
to my
explenation?)
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Subject: Re: Quadratic equations
>I have a question:
>3X^2 - 12X -13 = 0
>my answer is:
>4.89 or - 0.89
>The question asks: comment on your solution
My suspicion is that the examiner hopes that,
inspired by the closeness of your answers to 5 and -1,
and wondering why the difference is 0.11 in each case,
you will notice that the equation can be written:
Given f(x) == (x-5)(x+1) = x^2 - 4x - 5
3 f(x) = 3x^2 - 12 x - 15
Solve:
f(x) = -2/3
First solve for f(x) = 0, which gives the two solutions x=5 or
x=-1
Next use a linear approximation, using slope:
Adjustment in x = Error in f(x) / slope
= 2 / 3 * slope
Now we have slope = f'(x) == 2x - 4
so f'(5) = 6
and f'(-1) = -6
So the adjustments in x are 2 / ( 3 x 6 )
= 1 / 9
= 0.1111 (approx) as discovered above.
- -
Ken, __O
-<,_
(_)/ (_)
Virtuale Saluton.
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Subject: Re: Quadratic equations
> I have a question:
> 3X^2 - 12X -13 = 0
> my answer is:
> 4.89 or - 0.89
> The question asks: comment on your solution
The first 'comment' that comes to my mind is these are only
approximate,
not
exact, solutions (why).
> My response is:
> b^2 - 4ac not = 0
> b^2 - 4ac = (-12)^2 - 4(3)(-13)
> = 144 - (-156)
> = 300
Since you have gone this far, why not finish the process to
find the exact
solutions? 300=3*100 and 100 is a perfect square, thus
sqrt(b^2-4ac)=sqrt(300)=?
..thus [-b +- sqrt(b^2-4ac)] = what, in simplest form?
> b^2 - 4ac > 0;
> (Well I don't know what else to say. But I know there is
more to my
> explenation?)
With respect to the discriminant being >0, you could say the
solutions are
real and distinct before going any further (why). You may also
say
something else (particularly what kind of reals, rational or
irrational)
depending on whether or not the entire discriminate b^2-4ac is
a perfect
square or not. In this case I think you'll find, even in
simplest form,
there is a radical you can't get rid of (because 300 is not a
perfect
square) thus the solutions are not only real but __________.
If b^2-4ac turned out to be =0, what can you say about the
solutions then?
What if its less than 0?
Since you have posted several questions of late, if you wish
you can post
to
alt.algebra.help in the future (that's not to say its
inappropriate to ask
here; just giving you another forum you may find useful). That
groups also
seems to be more populated than this one (read...you'll
probably get more
useful replies to your questions.)
--
Darrell
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===
Subject: Re: Quadratic equations
> I have a question:
> 3X^2 - 12X -13 = 0
> my answer is:
> 4.89 or - 0.89
> The question asks: comment on your solution
> My response is:
> b^2 - 4ac not = 0
> b^2 - 4ac = (-12)^2 - 4(3)(-13)
> = 144 - (-156)
> = 300
> b^2 - 4ac > 0;
> (Well I don't know what else to say. But I know there is
more to my
> explenation?)
Well, you could say b^2 - 4ac > 0 implies two real solutions.
But
since a, b and c don't appear in the original problem that may
read a
bit oddly, so try 3X^2 - 12X -13 = 0 has positive discriminant
so it
has two real solutions.
--
G.C.
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Subject: Re: Quadratic equations
 I have a question:
 3X^2 - 12X -13 = 0
 my answer is:
 4.89 or - 0.89
 The question asks: comment on your solution
 My response is:
 b^2 - 4ac not = 0
 b^2 - 4ac = (-12)^2 - 4(3)(-13)
 = 144 - (-156)
 = 300
 b^2 - 4ac > 0;
 (Well I don't know what else to say. But I know there is
more to my
> explenation?)
> Well, you could say b^2 - 4ac > 0 implies two real
solutions. But
> since a, b and c don't appear in the original problem that
may read a
> bit oddly, so try 3X^2 - 12X -13 = 0 has positive
discriminant so it
> has two real solutions.
If I asked someone to comment on your solution, and they just
provided me with a series of equations, without any connecting
text, I would consider the question unanswered. I strongly
recommend
reading How to write mathematics, an essay by P.R. Halmos. The
essay is published in How to Write Mathematics edited
by N. E. Steenrod. Knuth's short book called Mathematical
Writing
is also worth reading.
A pile of equations is not a commentary.
--
Kevin Karplus karplus@soe.ucsc.edu
http://www.soe.ucsc.edu/~karplus
life member (LAB, Adventure Cycling, American Youth Hostels)
Effective Cycling Instructor #218-ck (lapsed)
Professor of Biomolecular Engineering, University of
California, Santa Cruz
Undergraduate and Graduate Director, Bioinformatics
Affiliations for identification only.
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===
Subject: Re: Quadratic equations
>I have a question:
>3X^2 - 12X -13 = 0
>my answer is:
>4.89 or - 0.89
>The question asks: comment on your solution
>My response is:
>b^2 - 4ac not = 0
>b^2 - 4ac = (-12)^2 - 4(3)(-13)
>= 144 - (-156)
>= 300
>b^2 - 4ac > 0;
>(Well I don't know what else to say. But I know there is more
to my
>explenation?)
It's worth mentioning that the roots you have found are
irrational, as the
value of the discriminant indicates. Along that line of
thinking it would
be appropriate to give the roots in their radical notattion
forms, as well
as rounded decimals.
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Subject: Different technique for solving quadratic equation
when A > 1
Down and dirty way to factor quadratics (assuming they factor)
when
the first coefficient A is greater than one.
Ex: 8m^2 - 10m + 3
Multiply AC or (8)(3) = 24
Since the second sign is positive, thinking addition, find
other factors
of 24 that will add to 10. In this case 6 and 4. If the second
sign were
negative, look for the difference of the factors.
Here's the trick: Open parentheses and put the A value in BOTH
parentheses: (8m - )(8m - ). Note first degree of m.
If signs were different, the placement of the second value is
not
optional, and you need to put the higher absolute value with
the proper
sign. When signs are the same, not a problem.
(8m - 6)(8m - 4)
In all cases where this works, one of the parenthesis MUST a
common
reduceable factor and both MIGHT.
In this case, divide the first factor by 2 and the second by 4:
(4m - 3)(2m - 1)
No trial and error necessary for the middle term.
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Subject: Re: Different technique for solving quadratic
equation when A > 1
> Down and dirty way to factor quadratics (assuming they
factor) when
> the first coefficient A is greater than one.
> Ex: 8m^2 - 10m + 3
> Multiply AC or (8)(3) = 24
> Since the second sign is positive, thinking addition, find
other factors
> of 24 that will add to 10. In this case 6 and 4. If the
second sign were
> negative, look for the difference of the factors.
> Here's the trick: Open parentheses and put the A value in
BOTH
> parentheses: (8m - )(8m - ). Note first degree of m.
Neat trick, but I would advise against it for the simple
reason it looks an
awful lot like you are rewriting the original expression as
(8m-something)(8m-something) which of course is not exactly
clear since
8m*8m is 64m^2, not 8m^2 as is the original. I suggest instead
to rewrite
at each step in terms of an equivelent expression only. It
keeps things
logical, easy to follow, and no confusing 'tricks' are needed.
The method that you are dancing very closely 'round is not an
uncommon one.
Stated appropriately, it is a quite common method of
factoring. I know it
by the name 'master product' but there may be other names for
it.
Let's start from the beginning, a trinomial in standard,
decreasing order
of
powers:
8m^2 - 10m + 3
8*3=24. What factors of 24 sum to -10? -6 and -4.
Using these numbers, rewrite the middle term, -10m, as -6m-4m
8m^2 - 6m - 4m + 3
Now group them:
(8m^2 - 6m) - (4m - 3)
..and factor:
2m(4m - 3) - (4m - 3)
(4m - 3)(2m - 1)
--
Darrell
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Subject: calculus question
I'm a 36 year old adult trying to learn calculus (I remember
little of
it from taking and passing business Calculus I and II in
college in
the 80's). I am using Calculus Made Easy. I have a question
about
multiplication of independent variables.
Let's use a simple example: y=(x^2+1)(x^2+1)
You can either multiply it out to get y=x^4+2X^2+1, and
differentiate
to:
dy/dx=4x^3+4x
OR, you can use the product rule. Silvanus Thompson's book
explains
it roughly so:
Go back to 'first principles', and say that y=u*v, where u and
v are
both functions of x. Then you add some minute amount dx to x,
giving
you the equation:
y+dy=(u+du)(v+dv)
Multiply that out and subtract the original equation, and you
get:
dy=(u*dv)+(v*du)+(dv*du)
You throw out the last bit (dv*du) because it is so minute, and
differentiate, and you get:
dy/dx=(u*dv/dx)+(v*du/dx)
Plug the original formulas back in, and you get:
dy/dx=((x^2+1)2x)+(2x(x^2+1))=4x^3+4x
This is the exact result arrived at originally, so it would
seem that
all is fine. BUT, my problem is that it IS exactly the same.
Even if
the product of dv*du is insignificant, it should be some amount
greater than zero, shouldn't it?
Actually, after writing all this it occurs to me that perhaps
the way
that it was explained was just simplified, and that perhaps dx
and dy
are merely fictions to help the student wrap his mind around
the
concepts, and that they are therefore actually equal to zero.
This
would obviously make their product also equal zero.
On second thought, that would make the dy/dx notation absurd.
Oh
well. Any help would be appreciated.
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Subject: Re: calculus question
> I'm a 36 year old adult trying to learn calculus (I remember
little of
> it from taking and passing business Calculus I and II in
college in
> the 80's). I am using Calculus Made Easy. I have a question
about
> multiplication of independent variables.
> Let's use a simple example: y=(x^2+1)(x^2+1)
> You can either multiply it out to get y=x^4+2X^2+1, and
differentiate
> to:
> dy/dx=4x^3+4x
> OR, you can use the product rule. Silvanus Thompson's book
explains
> it roughly so:
> Go back to 'first principles', and say that y=u*v, where u
and v are
> both functions of x. Then you add some minute amount dx to
x, giving
> you the equation:
> y+dy=(u+du)(v+dv)
> Multiply that out and subtract the original equation, and
you get:
> dy=(u*dv)+(v*du)+(dv*du)
> You throw out the last bit (dv*du) because it is so minute,
and
> differentiate, and you get:
> dy/dx=(u*dv/dx)+(v*du/dx)
<... Plug the original formulas back in, and you get:
> dy/dx=((x^2+1)2x)+(2x(x^2+1))=4x^3+4x
> This is the exact result arrived at originally, so it would
seem that
> all is fine. BUT, my problem is that it IS exactly the same.
Even if
> the product of dv*du is insignificant, it should be some
amount
> greater than zero, shouldn't it?
Depends. There are ways of looking at it such that dv and du
really are
'minutely' small yet not zero, therefore a derivative (e.g.
dy/dx) can be
thought of as a ratio of nonzero, yet infinitesimally small,
quantities.
If
I recall my calculus history correctly, that's roughly how it
was thought
of
in the early days.
Standard mathematics (ie mathematical operations on real
numbers and what
not) can get a little fuzzy when dealing with infinitesimal
quantities. In
more standard treatments (e.g. a standard cal I course) the
calculus is
made
algebraically rigorous by means of limiting processes,
although there are
some courses you can take that do use the infinitesimal
approach. I do not
want to purport to be familiar enough with the infinitesimal
approaches to
be of any real assistance, so I will not attempt to offer any
specific
help,
just general comments.
> Actually, after writing all this it occurs to me that
perhaps the way
> that it was explained was just simplified,
In a book, ore series, or what not, entitled _Calculus Made
Easy_ I'd say
that's a pretty safe assumption.
> and that perhaps dx and dy are merely fictions to help the
student wrap
his mind around the
> concepts, and that they are therefore actually equal to
zero. This
> would obviously make their product also equal zero.
That's one of the fuzzy things about such an approach. On one
hand, they
are treated as nonzero and on the other, they are treated as
zero. IIRC,
this (the inconsistent treatment of infinitesimals) was one of
the main
reasons people like Cauchy and Weierstrass came along and
placed calculus
on
a more algebraically rigorous footing (i.e. the 'standard'
approach used in
so many classrooms today.) Such standard approaches uses
limiting
processes
which make sense (i.e. is rigorous) in terms of standard
mathematics on
real
numbers, not infinitesimals which are not real numbers. To
give you
another
oversimplification: in such limiting processes you never
really *get* to
zero, so to speak.
> On second thought, that would make the dy/dx notation
absurd. Oh
> well. Any help would be appreciated.
That's because the dy/dx notation to some extent *is* absurd!
The dy/dx
notation to denote a derivative, IMHO, is never intended to be
fully
understood by even those taking a standard cal I course, much
less by
someone reading a made easy book. It appears at face value to
be a
quotient
of two mysterious very small things that aren't 0, but the
actual
(standard)
definition of the notation dy/dx is in terms of a particular
limiting
process. The actual dy and dx, as *individual* entities, are
never really
rigorously defined at this level, yet many of the theorems
involving
derivatives seem to 'work out' when translated into the
differential
notation like this, as if these are actual nonzero quantities
that obey
usual algebraic properties. E.g. a common way to remember the
chain rule
goes...
dy/dx = (dy/du)(du/dx)
...such that it appears to be valid by simple way of the du's
cancelling.
In
standard calculus treatment, however, that's not the actual
proof. The
actual proof is a little deeper than that.
The notation is attributed to Leibniz, one of the inventors of
calculus,
and
IIRC he actually did use an infinitesimal approach, so he
really did think
of dy and dx as infinitesimals. So why use the same notation
in standard
approaches that don't use infinitesimals. Suffice to say the
notation has
proved to be convenient in some cases for precisely this
reason (the
algebra
works out as if they really are fractions) so the notation has
stuck
around.
In short, there are many students right now going around
cancelling du's
and
what not without really understanding what they are doing,
since the above
formula 'works out' and it looks like the du's just *have* to
cancel. In
an
infinitesimal approach to calculus, I think it can indeed be
rigorously
shown that they do, but in standard treatments, they do not.
It just looks
like they do.
Something similar is probably the case with this treatment of
the product
rule you are referring to. The author is probably
oversimplifying the
concept, by your standards. IMO, it appears by your very good
questions
about such things and desire for rigor (i.e. you want things
to make
algebraic 'sense' and be algebraically justifiable) you may
benefit from a
standard-type review in terms of limits. This is, after all,
the way its
usually done in a first course in calculus, to include
probably the courses
you already took some time ago. That's not to say an
alternative
treatment,
if even informal, can't be useful too. It just means when
using an
informal
learning aid, you can't expect everything to be rigorously
justified.
--
Darrell
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Subject: Re: calculus question
The thing that makes both methods reach the same result is
that we are
using
limits. The derivative of f(x), for example, is the limit as
dx goes to
zero
of [f(x+dx).-f(x)]/dx. Your concern that there should be
something due to
du*dv is unnecessary. If either du or dv approaches zero, then
their
product
will too. Before you take limits, they will be different, but
the limiting
process will indeed give you the same answer for either method
of
approaching the problem.
> I'm a 36 year old adult trying to learn calculus (I remember
little of
> it from taking and passing business Calculus I and II in
college in
> the 80's). I am using Calculus Made Easy. I have a question
about
> multiplication of independent variables.
> Let's use a simple example: y=(x^2+1)(x^2+1)
> You can either multiply it out to get y=x^4+2X^2+1, and
differentiate
> to:
> dy/dx=4x^3+4x
> OR, you can use the product rule. Silvanus Thompson's book
explains
> it roughly so:
> Go back to 'first principles', and say that y=u*v, where u
and v are
> both functions of x. Then you add some minute amount dx to
x, giving
> you the equation:
> y+dy=(u+du)(v+dv)
> Multiply that out and subtract the original equation, and
you get:
> dy=(u*dv)+(v*du)+(dv*du)
> You throw out the last bit (dv*du) because it is so minute,
and
> differentiate, and you get:
> dy/dx=(u*dv/dx)+(v*du/dx)
> Plug the original formulas back in, and you get:
> dy/dx=((x^2+1)2x)+(2x(x^2+1))=4x^3+4x
> This is the exact result arrived at originally, so it would
seem that
> all is fine. BUT, my problem is that it IS exactly the same.
Even if
> the product of dv*du is insignificant, it should be some
amount
> greater than zero, shouldn't it?
> Actually, after writing all this it occurs to me that
perhaps the way
> that it was explained was just simplified, and that perhaps
dx and dy
> are merely fictions to help the student wrap his mind around
the
> concepts, and that they are therefore actually equal to
zero. This
> would obviously make their product also equal zero.
> On second thought, that would make the dy/dx notation
absurd. Oh
> well. Any help would be appreciated.
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Subject: Great Book for Math Teachers
There is a great new book called The Great Math Experience.
You can find it
at
www.thegreatmathexperience.com !
Enjoy!
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Subject: Thank you all very much for your help !
Thank you
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Subject: Identity Crisis!!!
OMG! It's been my 5 days of my school life that I can't solve
this
problem... Anyways, I'm a 15 yr. old kid who can't simply find
an
answer w/ the everyone I know here in the Philippines... Pls.
by the
name of 3rd yr. HS Autobiography Diary of my life solve this!!!
cos30'cot30'=csc30'-sin30'
note thatyou could use the identities which it could be
understandable
by lamens? terms
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Subject: Re: Identity Crisis!!!
>cos30'cot30'=csc30'-sin30'
cos(pi/6) = sqrt(3)/2
cot(pi/6) = cos(pi/6)/sin(pi/6) = (sqrt(3)/2) / (1/2) =
sqrt(3)/2 * 2 =
sqrt(3)
So the left member of the identity is
sqrt(3)/2 * sqrt(3) = 3/2
csc(pi/6) = 1/sin(pi/6) = 1/(1/2) = 2
sin(pi/6) = 1/2
2 - 1/2 = 3/2
Quidnelda and Plano, Consternation turns to Lucidation!!
Shoes for Industry, Comrade.
--
charlie dick
The right to be left alone -- the most comprehensive
of rights, and the right most valued by a free people.
- Justice Louis Brandeis, Olmstead v. U.S. (1928).
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Subject: Re: Identity Crisis!!!
> OMG! It's been my 5 days of my school life that I can't
solve this
> problem... Anyways, I'm a 15 yr. old kid who can't simply
find an
> answer w/ the everyone I know here in the Philippines...
Pls. by the
> name of 3rd yr. HS Autobiography Diary of my life solve
this!!!
> cos30'cot30'=csc30'-sin30'
> note thatyou could use the identities which it could be
understandable
> by lamens? terms
cosxcotx=cosx(cosx/sinx)=(cos^2x)/sinx=(1-sin^2x)/sinx=
cscx-sinx
David Moran
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Subject: Re: Identity Crisis!!!
> OMG! It's been my 5 days of my school life that I can't
solve this
> problem... Anyways, I'm a 15 yr. old kid who can't simply
find an
> answer w/ the everyone I know here in the Philippines...
Pls. by the
> name of 3rd yr. HS Autobiography Diary of my life solve
this!!!
> cos30'cot30'=csc30'-sin30'
> note thatyou could use the identities which it could be
understandable
> by lamens? terms
cot x = cos x / sin x
csc x = 1/sin x
rewriting it we get:
cos x * cos x / sin x = (1/sin x) - sin x
multiply both sides by sin x and it becomes crystal clear.
--Jeff
--
A man, a plan, a cat, a canal - Panama!
Ho, ho, ho, hee, hee, hee
and a couple of ha, ha, has;
That's how we pass the day away,
in the merry old land of Oz.
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Subject: Re: Identity Crisis!!!
> OMG! It's been my 5 days of my school life that I can't
solve this
> problem... Anyways, I'm a 15 yr. old kid who can't simply
find an
> answer w/ the everyone I know here in the Philippines...
Pls. by the
> name of 3rd yr. HS Autobiography Diary of my life solve
this!!!
> cos30'cot30'=csc30'-sin30'
What are the 's? Is that your sign for multiplication?
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Subject: Seeking Math Teachers
My name is Dr. Jill Carr and I am VP of Education for Learning
Today,
Inc. located in Plantation, Fl. We are actively seeking to
employ a
full-time elementary ed. math teacher to work at our corporate
headquarters and/or full-time or part-time virtual math
teachers.
Here is some information about our full time position:
Minimum requirements: Teaching Certificate (valid)
2 years elem. ed. teaching exper.
Powerpoint & email skills
Salary: $31,200/year + health benefits
For Virtual teaching positions, please contact me for more
information.
Dr. Jill Carr
Learning Today, Inc.
954-584-5250
jill@learningtoday.com
www.learningtoday.com
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Subject: integrated math
If any of you are trying, like I am to get rid of Integrated
math,
Ivestigative math, and all those others, Try
TeachUtahkids.com. It's
a website situated in my area for alpine school district. They
have
an e petition, a regular petition to pass out, flyers to
download,
bumper stickers to buy, ect. They're going to have tee-shirts
too.
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Subject: Re: calculus question
Jim on queries about the product rule, wondering why that
tiny dv*du term makes not a small difference, but NO
difference:
> Go back to 'first principles', and say that y=u*v, where u
and v
> are both functions of x. Then you add some minute amount dx
to x,
> giving you the equation:
> y+dy=(u+du)(v+dv)
> Multiply that out and subtract the original equation, and
you get:
> dy=(u*dv)+(v*du)+(dv*du)
At this point, it might be best to say Divide by dx and then
take the Limiting Value as dx approaches zero
dy/dx=(u*dv/dx)+(v*du/dx)+dv*(du/dx)
Now dy/dx and dv/dx and du/dx all have finite limits.
(Assuming that those functions HAVE derivatives, then these
limits ARE those derivatives.)
But in that last term, the dv does not have a dx under it.
As dx goes to zero, so does dv. (Assuming v is continuous,
which it better be since it is supposed to be differentiable.)
So we have a finite (du/dx) multiplied by (dv) which is
approaching
zero. The Limiting Value of that product is ZERO. Not just
insignificant, but exactly zero.
Robert, who is:
|)|/| || Burnaby South Secondary School
|| |orewood@olc.ubc.ca || Beautiful British Columbia
Mathematics & Computer Science || (Canada)
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Subject: Octagon-all sides are 4'
Pretty sad when I can't even figure out the area of an
octagon, even
when you give me the formula...
The octagonal room has 4' sides.
I'll mail aynone a $5 gift certificate to Dunkin Donuts for
the answer
:-D
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Subject: Re: Octagon-all sides are 4'
> Pretty sad when I can't even figure out the area of an
octagon, even
> when you give me the formula...
> The octagonal room has 4' sides.
> I'll mail aynone a $5 gift certificate to Dunkin Donuts for
the answer
> :-D
The formula for finding the area of a regular polygon is
A=(1/2)Pa
where P is the perimeter and a is the apothem. The apothem is
the
segment that comes from the center of the polygon andis
perpendicular
to a side...basically the height of each of the congruent
triangles
formed.
In your problem the perimeter is 32 ft. To find the apothem
you will
need to use trigonometry. Each interior angle of the octagon
is 135
degrees. The formula for that is 180(number of sides - 2). You
then
divide that angle by two and get 67.5 for the base angle of
one of the
triangles formed. The apothem will 2 X tan(67.5). 2 is half the
length of one of your sides. That will give you 4.828. Your
perimeter is 32. Use the formula and you get 77.25 square feet.
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Subject: Re: Octagon-all sides are 4'
> Pretty sad when I can't even figure out the area of an
octagon, even
> when you give me the formula...
> The octagonal room has 4' sides.
> I'll mail aynone a $5 gift certificate to Dunkin Donuts for
the answer
>:-D
To figure out the area of any right polygon, you need to break
it into
triangles. in this case you divide it into eight equal
triangles. next you
need to get the area of one of these triangles.
in this case each central angle is 360/8 degrees or 45. the
height
of
the triangle is determined using trigonometry (tangent) to be
tan(22.5) =
2/x and x comes out to be 4.828 ft. using the base of 4 and a
height of
4.828 you get an area for one of the triangles of 0.5 * 4 *
4.828 = 9.656.
add up eight of these and you get 77.25 ft^2 as the total area.
I'm pretty sure that is correct way to do it.
Steve
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Subject: Re: Octagon-all sides are 4' (an even better way)
<snip> previous quote
Naturally as soon as I posted I found a way to get the exact
answer
using nothing more than the Pythagorean theorem. Inscribe the
octagon
in a square in the obvious way. The area of the octagon is the
area of
the square less the four triangles. The hypotenuse of a little
triangle is a side of the octagon so is 4, and by the
Pythagorean
theorem each leg of the little triangles is 4/sqrt(2).
Therefore the
total area of the four little triangles is:
T = 4 * (1/2)*(4/sqrt(2))^2 = 16
The side of the square is one side of the octagon plus two
triangle
sides so s = 4 + 8/sqrt(2) and the area of the square is
A = (4 + 8/sqrt(2))^2 = 16 + 64/sqrt(2) + 32 = 48 + 64/sqrt(2)
and the area of the octagon is therefore:
A - T = 32 + 64/sqrt(2) = 32 + (64*sqrt(2))/2
= 32 + 32*sqrt(2).
It is an interesting exercise to show this is equal to the
exact
answer given in my previous post.
--Lynn
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Subject: Re: Octagon-all sides are 4'
> Pretty sad when I can't even figure out the area of an
octagon, even
> when you give me the formula...
 The octagonal room has 4' sides.
 I'll mail aynone a $5 gift certificate to Dunkin Donuts for
the answer
>:-D
To figure out the area of any right polygon, you need to
break it into
>triangles. in this case you divide it into eight equal
triangles. next you
>need to get the area of one of these triangles.
>in this case each central angle is 360/8 degrees or 45. the
height
of
>the triangle is determined using trigonometry (tangent) to be
tan(22.5) =
>2/x and x comes out to be 4.828 ft. using the base of 4 and a
height of
>4.828 you get an area for one of the triangles of 0.5 * 4 *
4.828 = 9.656.
>add up eight of these and you get 77.25 ft^2 as the total
area.
>I'm pretty sure that is correct way to do it.
>Steve
Of course, you might want an exact answer. Since h =
2*cot(22.5) you
can use the half angle formula:
cot(a/2) = sqrt( (1+cos(2a))/(1-cos(2a)) )
with a = 22.5 so 2a = 45 degrees and cos(45) = 1/sqrt(2)
This gives, after simplifying, h = 2 * sqrt(
(2+sqrt(2))/(2-sqrt(2)) )
so Area = 8 * (1/2)*4*h = 32*sqrt( (2+sqrt(2))/(2-sqrt(2)) )
which is exact and which is approximately 77.25483395.
--Lynn
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Subject: Re: Octagon-all sides are 4'
>To figure out the area of any right polygon, you need to
break it into
>triangles. in this case you divide it into eight equal
triangles. next you
>need to get the area of one of these triangles.
This is one perfectly correct method, but it is not the best
one in
this case, firstly because it does not use the fact that we
have a
regular polygon.
The better formula works for any polygon that can be drawn
touching a
circle, a set that includes ALL triangles, cyclic
quadrilaterals, and
all regular polygons. The second reason it is better is that it
leads convincingly to the formula for the area of a circle.
It is: Area = half of Perimeter x Radius
The Radius, in this case, is conveniently found (as you
explained)
using trigonometry. It is the common altitude of all the
triangles.
**********************
Historical note: You don't actually need trigonometry for
a triangle or a square, or for any polygon that can
be derived from them by doubling the number of sides.
All you need is Pythagoras' theorem. This is what Archimedes
did
when he calculated an approximate value for pi by using a
(6 x 2^n)-gon instead of a true circle.
- -
Ken, __O
-<,_
(_)/ (_)
Virtuale Saluton.
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Subject: I'm confused and need help please!
I'm really having trouble with a few types of problems and any
help
will be great!
here are some examples:
1) Given csc(x)=-3 and tan(x)>0, find cos(x)
2) Factor and simplify cos^2(x)-sin^2(x)cos^2(x)
3) 1-cos^2(x)
----------
tan^2(x)
4) find the exact value of sin 165 degrees
5) rewrite as sum: 9sin(3x)cos(7x)
Please help me! Thank you
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Subject: Re: I'm confused and need help please!
> I'm really having trouble with a few types of problems and
any help
> will be great!
> here are some examples:
> 1) Given csc(x)=-3 and tan(x)>0, find cos(x)
Which quadrant is tangent positive and cosecant negative? You
should know
that pretty much immediately, right, it's QIII. Sketch a
reference
triangle
in QIII with reference angle a, such that csc(a)=3, i.e.
hyp/opp=3, i.e.
hyp
is 3, opp is 1. Don't worry about signs (+/-) at this
point--the sign
assignmentn need not be done until the end.
^
|
|
|
|
|
|
|
-------------------------->
| a *
| *
1 | * 3
| *
|*
and solve for the remaining side b
b^2 + 1^2 = 3^2
b= ?
...and now you know all three sides, thus can quickly cite any
trig value
of
x, not just cosine. Finally, for the sign (+/-), you know that
cosine is
negative in QIII.
<... 4) find the exact value of sin 165 degrees
QII angle. The reference angle is 15 deg. This is half of 30
deg, which
is
a special angle you can find the exact trig values of. Do
that, then you
can use a half-angle identity to finish.
> 5) rewrite as sum: 9sin(3x)cos(7x)
Use an appropriate product-to-sum identity.
--
Darrell
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Subject: Re: I'm confused and need help please!
> I'm really having trouble with a few types of problems and
any help
> will be great!
sin^2(x) + cos^2(x) = 1
sin^2(x) = 1 - cos^2(x)
cos^2(x) = 1 - sin^2(x)
csc(x) = 1/sin(x)
sec(x) = 1/cos(x)
tan(x) = sin(x)/cos(x) = sin(x)sec(x)
cot(x) = cos(x)/sin(x) = cos(x)csc(x)
> here are some examples:
> 1) Given csc(x)=-3 and tan(x)>0, find cos(x)
csc(x) = 1/sin(x), so sin(x) = ?, sin^2(x) = ?
cos^2(x) + sin^2(x) = 1, so cos^2(x) = ?
tan(x) = sin(x)/cos(x), so which of the two square roots of
cos^2(x)
fits?
> 2) Factor and simplify cos^2(x)-sin^2(x)cos^2(x)
Hrm... a^2 - b^2 * a^2 = a^2 * (1 - b^2)...
> 3) 1-cos^2(x)
> ----------
> tan^2(x)
tan(x) = sin(x)/cos(x), tan^2(x) = sin^2(x)/cos^2(x)
> 4) find the exact value of sin 165 degrees
165 = 180 - 15, 15 = 30/2, sin (x/2) = +/- square
root((1-cos(x))/2)
sin(x) = -sin(180 - x)
> 5) rewrite as sum: 9sin(3x)cos(7x)
> Please help me! Thank you
--
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and a couple of ha, ha, has;
That's how we pass the day away,
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Subject: Re: I'm confused and need help please!
> I'm really having trouble with a few types of problems and
any help
> will be great!
I'll start you off with some hints on the first two problems.
Lotsa luck!
> here are some examples:
> 1) Given csc(x)=-3 and tan(x)>0, find cos(x)
You know that csc = 1 / sin, and that tan = sin / cos. Since
tan(x)>0,
sin and cos are either both positive or both negative. Since
sin is
negative (as 1/sin is), that means cos is negative. So find
the values
of x such that csc(x)=-3 and choose the one for which cos(x)
is negative.
> 2) Factor and simplify cos^2(x)-sin^2(x)cos^2(x)
You have a cos^2(x) in both sides, so start by factoring that
out; that
should give you something you'll recognize.
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Subject: Obtaining a job with a PhD?
Hi there,
I'm about to finish my undergrad degree in math with a
secondary
certification. I want to go straight into grad school before I
get a job,
since I know it will be hard for me to go back once I start
teaching full
time. While looking into different programs at different
schools I came
across a program in which one can obtain a masters in math
education and
then seamlessly go into their PhD program in mathematics
education. The
program sounds like a great fit for me, but I'm afraid if I do
go that far
I'm going to have a lot of trouble finding a job in a public
high school
because I'll be to expensive. Of course I guess I could go as
far as to get
a PhD with ABD status, then finish it while I'm working as a
teacher. The
program says that gradates from it usually end up in the
private sector or
teach college. I could teach college in the interim but I
think I really
want to teach in high school. Some grads from the program went
to work for
school districts in curriculum development also, but I want to
be in the
classroom. So far I've only seen one school district's website
that has a
lane for PhDs on their salary schedule. I wouldn't mind being
a department
head if necessary that way I could still teach.
How many PhDs out there are regular high school teachers?
What other questions should I be asking myself and of the
program?
Are their any high school math department heads here? How
would look at a
candidate like me?
John
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Subject: Re: Obtaining a job with a PhD?
> Hi there,
> I'm about to finish my undergrad degree in math with a
secondary
> certification. I want to go straight into grad school before
I get a job,
> since I know it will be hard for me to go back once I start
teaching full
> time. While looking into different programs at different
schools I came
> across a program in which one can obtain a masters in math
education and
> then seamlessly go into their PhD program in mathematics
education. The
> program sounds like a great fit for me, but I'm afraid if I
do go that
far
> I'm going to have a lot of trouble finding a job in a public
high school
> because I'll be to expensive. Of course I guess I could go
as far as to
get
> a PhD with ABD status, then finish it while I'm working as a
teacher. The
> program says that gradates from it usually end up in the
private sector
or
> teach college. I could teach college in the interim but I
think I really
> want to teach in high school. Some grads from the program
went to work
for
> school districts in curriculum development also, but I want
to be in the
> classroom. So far I've only seen one school district's
website that has a
> lane for PhDs on their salary schedule. I wouldn't mind
being a
department
> head if necessary that way I could still teach.
> How many PhDs out there are regular high school teachers?
> What other questions should I be asking myself and of the
program?
> Are their any high school math department heads here? How
would look at a
> candidate like me?
> John
John,
I have a PhD in engineering, and worked in industry for about
10 years
before deciding to teach high school math. After I got my
first level
of certification, I didn't have too many problems getting
interviews and
offers to teach high school math - I probably had 20+ first
interviews
and follow-up interest on about half of those. While many
districts
(allegedly) have budget issues, that didn't seem to stop them
from
having interest in someone who's in the right-hand column of
the pay scale.
At least in my state (MA), staying certified requires you to
frequently
take in-service and/or graduate-level courses. Once you're
hired - at
whatever level you're hired - as long as you're doing a good
job, get
tenure (or professional status, or whatever it's called where
you
live) and aren't laid off, you have to take courses. If those
credits
result in a lane change (say, from the Master's to the
Master's + 30
column), you'll get the corresponding raise. If you get enough
credits
to move all the way over to the PhD column, you'll get that
money....
In my district's contract, the highest pay scale isn't labeled
PhD but
rather CAGS - which includes CAGS (Certificate of Advanced
Graduate
Study), PhD, EDD, and multiple Master's degrees. I've seen
CAGS in a
number of districts' contracts....
As for the department-head thing, I have two thoughts: first,
I don't
think a PhD would rule out being a teacher (as opposed to a
department
head); and second, I don't think it'd really be sensible to be
a
department head without having been a pure teacher for awhile.
Until
you've been responsible for your own classroom (and five or
six classes
full of kids, three or four different subjects to prepare for,
etc.),
there's no way you can be responsible for a group of teachers
who have
to deal with that! If you want to be a teacher, then be a
teacher -
find a district that'll hire you to do that, and if,
eventually, you
want to manage a department, make that switch...
Good luck!
Lisa
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Subject: Re: Obtaining a job with a PhD?
When I was in a teacher certification program for secondary
education, a
few
of my professors were PhDs and still teaching, even in junior
high.
However, for most of them, they went for their PhD while they
were teaching
high school or junior high, and the district they worked for
foot the bill.
I, on the other hand, have quit that program and decided to
stick with
teaching community college and pursure my PhD over the summer.
While a district may not hire you outright with a PhD, they
will surely
hire
you while you work on your MA and then move into your PhD. You
may not
want
to mention the PhD during the interview. Once hired, I would
think it
difficult for them to relieve you of your duties.
John
> Hi there,
> I'm about to finish my undergrad degree in math with a
secondary
> certification. I want to go straight into grad school before
I get a job,
> since I know it will be hard for me to go back once I start
teaching full
> time. While looking into different programs at different
schools I came
> across a program in which one can obtain a masters in math
education and
> then seamlessly go into their PhD program in mathematics
education. The
> program sounds like a great fit for me, but I'm afraid if I
do go that
far
> I'm going to have a lot of trouble finding a job in a public
high school
> because I'll be to expensive. Of course I guess I could go
as far as to
get
> a PhD with ABD status, then finish it while I'm working as a
teacher. The
> program says that gradates from it usually end up in the
private sector
or
> teach college. I could teach college in the interim but I
think I really
> want to teach in high school. Some grads from the program
went to work
for
> school districts in curriculum development also, but I want
to be in the
> classroom. So far I've only seen one school district's
website that has a
> lane for PhDs on their salary schedule. I wouldn't mind
being a
department
> head if necessary that way I could still teach.
> How many PhDs out there are regular high school teachers?
> What other questions should I be asking myself and of the
program?
> Are their any high school math department heads here? How
would look at a
> candidate like me?
> John
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Subject: state tests for third grade math
Does anyone have good practice tests for third grade math?
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Subject: Geometry
How to get the centdoids of parabola? i need it for my reporst
yhis
affternnon!
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Subject: Re: Geometry
> How to get the centdoids of parabola? i need it for my
reporst
> yhis affternnon!
Do you mean the Focus or Directrix?
DR
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Subject: Standard Deviation Question
Hello!
We learned two methods of calculating Standard Deviation - one
involving the mean and the other not. The one with mean is:
[E(x - u)^2] / N
(where E is epsilon (sum of..), u is the mean, and N is the
number of
items in the list.)
The one not involving mean is:
[E(x)^2 - (Ex)^2) / N] / N
However, we've been asked to equate these two formulae, and
I'm a bit
flustered. Can anyone give me a hand?
Mike
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Subject: Re: Standard Deviation Question
> Hello!
> We learned two methods of calculating Standard Deviation -
one
> involving the mean and the other not. The one with mean is:
> [E(x - u)^2] / N
> (where E is epsilon (sum of..), u is the mean, and N is the
number of
> items in the list.)
> The one not involving mean is:
> [E(x)^2 - (Ex)^2) / N] / N
> However, we've been asked to equate these two formulae, and
I'm a bit
> flustered. Can anyone give me a hand?
Replace u by Ex, expand the square, then simplify. You may
need to
use some simple properties of the expectation operator.
--
Kevin Karplus karplus@soe.ucsc.edu
http://www.soe.ucsc.edu/~karplus
life member (LAB, Adventure Cycling, American Youth Hostels)
Effective Cycling Instructor #218-ck (lapsed)
Professor of Biomolecular Engineering, University of
California, Santa Cruz
Undergraduate and Graduate Director, Bioinformatics
Affiliations for identification only.
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Subject: SAT after school Prep class
I am looking for teachers who teach, or have taught, an after
school SAT
Prep class. Our after school SAT Prep class is not working as
well as it
should. I am looking for someone to exchange ideas with on the
topic.
Hopefully, I will see ways of making it better.
Terry W Antoine
Fossil Ridge High School
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Subject: question
15) What is the solution to the system of equations x
(exponent 2) -x
+ 6 = y and 2x + 26 = 36?
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Subject: Re: question
> 15) What is the solution to the system of equations x
(exponent 2) -x
> + 6 = y and 2x + 26 = 36?
Solve the second equation for x and plug that into the other.
You should
get
(5,26).
David Moran
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Subject: Re: question
the correct answers are in fact (5, 26) not (5,14) ... a good
example of
why
it's important to check your answers! (or in my case the
original
equations). Thank you David for the correction.
Colin
15) What is the solution to the system of equations x
(exponent 2) -x
+ 6 = y and 2x + 26 = 36?
> Solve the second equation for x and plug that into the
other. You should
get
> (5,26).
> David Moran
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Subject: Re: question
> 15) What is the solution to the system of equations x
(exponent 2) -x
> + 6 = y and 2x + 26 = 36?
> --
One way to solve this type of problem is to first solve the
second
equation:
2x+26=36. Isolating the x on one side you should see that x=5.
Now you
can
substitute x=5 into the second equation. By the way the second
equation
can
be written as x^2 - x + 6=y. If you do this you'll find that
y=14.
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Subject: Geometry
Can anyone recommend a good hands-on resource for teaching
solids?
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