"Simple" algebraic expression
Chris R Simmons
is asked to state algebraically the sentence: "What is the total cost for
all books if the bookstore charges $10 for the first 2 books and $6 for any
subsequent ones." The requirement is that she use only one variable ( X )
for the number of books. I am not able to put this into one expression.
Instead, I have to use a "decision" to state this as follows:
If X > 2 Then
{ 6 (X - 2) + 20 }
Else
{ 10 (X) }
This is a 7th grade accelerated math class she is in. I would think I
could figure this out (NOT). Any help anyone could provide us in putting
this into one algebraic expression, using one variable for the number of
books would be very much appreciated.
Dumb Dad
Rod
X Cost
0 0
1 10
2 20
3 26
maybe?
"Chris R Simmons" <csim...@antelecom.net> wrote in message
news:3e1d643d$1...@news.antelecom.net...
David W. Cantrell
> Cost=6X + 4*min( X , 2)
>
> X Cost
> 0 0
> 1 10
> 2 20
> 3 26
>
> maybe?
But technically "min" doesn't look algebraic either. It seems to require
that a decision be made also. See my comments below the original message.
> "Chris R Simmons" <csim...@antelecom.net> wrote in message
> news:3e1d643d$1...@news.antelecom.net...
> > She is asked to state algebraically the sentence: "What is the total
> > cost for all books if the bookstore charges $10 for the first 2 books
> > and $6 for any
> > subsequent ones." The requirement is that she use only one variable
> > ( X ) for the number of books. I am not able to put this into one
> > expression. Instead, I have to use a "decision" to state this as
> > follows:
> >
> > If X > 2 Then
> > { 6 (X - 2) + 20 }
> > Else
> > { 10 (X) }
> >
> > This is a 7th grade accelerated math class she is in. I would
> > think I could figure this out (NOT). Any help anyone could provide us
> > in putting this into one algebraic expression, using one variable for
> > the number of books would be very much appreciated.
> > Dumb Dad
No, you're not dumb! In fact, I'm almost certain that you've done exactly
the sort of thing that is desired here. But, just for the heck of it, if
you really want to see it done in just one clearly algebraic expression,
here it is:
Cost = 10*X - 2(X-2)(1 + (X-2.5)/Sqrt((X-2.5)^2))
Notes:
1. Instead of 2.5 in two spots, any number strictly between 2 and 3 could
have been used.
2. Don't ask how I got the expression! ;-)
Cheers,
David
QuickDraw
> Cost = 10*X - 2(X-2)(1 + (X-2.5)/Sqrt((X-2.5)^2))
>
> Notes:
> 1. Instead of 2.5 in two spots, any number strictly between 2 and 3 could
> have been used.
> 2. Don't ask how I got the expression! ;-)
so david, how _did_ you get this expression?
David W. Cantrell
OK. Well, if I were not worried about the expression _looking_ algebraic,
I would simply have used the Heaviside unit-step function
H(x) = 0 if x < 0
1 if x > 0
(BTW, at x = 0, definitions of this function vary, IIRC. But thankfully
that need not be of concern to us. Since we need our cost function to
work only for nonnegative _integers_ X, we can just put the discontinuity
of the step function strictly between 2 and 3.)
In terms of H, I had initially written
Cost = 10*X - 4(X-2)*H(X-c)
where c is any convenient number strictly between 2 and 3. [Readers should
not proceed until they see why this works.]
Of course H doesn't look algebraic, its definition obviously depending
on a decision. So what can we do? Well, H is closely related to the
signum (or sign) function
sgn(x) = -1 if x < 0
1 if x > 0
(Again, at x = 0, definitions vary.)
We have simply H(x) = (1 + sgn(x))/2, at least for all nonzero x.
Still, signum doesn't look algebraic, its definition obviously depending
on a decision also. But it is closely related to the absolute value
function
|x| = -x if x < 0
0 if x = 0
x if x > 0
by the identity sgn(x) = x/|x|, valid for nonzero x.
But doesn't |x| require a decision also? Well, yes, BUT we can now build
the decision-making process into a clearly algebraic expression by noting
that
|x| = Sqrt(x^2).
At this point, interested readers may now just backtrack to see how I got
the expression which QuickDraw asked about.
David
Rod
Rod
"David W. Cantrell" <DWCan...@sigmaxi.org> wrote in message
news:20030109103004.918$B...@newsreader.com...
David W. Cantrell
> Is not taking only the +ve root of sqrt also a decision?
Indeed it was. (Note past tense.) That decision _was_ made. There is no
decision to make now. That decision was built into standard algebra in
the sense that Sqrt, which I use here in lieu of the common radical
symbol, is taken to denote the principal (i.e., nonnegative) square root.
Regards,
David
Greg
If this is an elementary course I think the answer required is for x>=2. It
could be expressed as a piecewise function. If y is the cost and x belongs
to N.
y: { x --> 10, x<3 }
{ x --> 6(x-2) + 10, x>2 }
I have read the question differently from you. I read it that one or two
books cost ten dollars. This is not logical as no-one would buy four books.
It would be cheaper to buy two lots of two books. If the question means "...
$10 each for the two first books... ", then your logic is correct.
--
Greg
(remove BALL from address to reply)
Greg
Hi David,
In a spreadsheet the decisions could be built into a single expression (x
belongs to N or W)
cost = (x<3)*(10*x) + (x>2)*(6*(x-2)+20).
Assuming the question means $10 'each' for the first two books. Is this
algebraic?
Chris R Simmons
I returned from work this afternoon and checked for any responses to the
post I made this morning about my daughter's algebra problem. We (my
daughter and I) want to thank all who responded. We are going to print
these out and show them to her teacher. The problem is obviously above her
grade level or the question in the book was not edited out originally. At
any rate, both my daughter and myself have learned from this.
If it is appropriate to this newsgroup, I will post the teacher's
response/answer to the question when my daughter turns in the assignment.
The assignment is due tomorrow (Friday).
Thanks to all,
Chris
"Greg" <greg...@contact.net.nz> wrote in message
news:3e1e...@clear.net.nz...
David W. Cantrell
> "David W. Cantrell" <DWCan...@sigmaxi.org> wrote in message
> news:20030109111654.518$r...@newsreader.com...
> > "Rod" <RodRod...@hotmail.com> wrote:
> > > Is not taking only the +ve root of sqrt also a decision?
> >
> > Indeed it was. (Note past tense.) That decision _was_ made. There is no
> > decision to make now. That decision was built into standard algebra in
> > the sense that Sqrt, which I use here in lieu of the common radical
> > symbol, is taken to denote the principal (i.e., nonnegative) square
> > root.
>
> belongs to N or W)
>
> cost = (x<3)*(10*x) + (x>2)*(6*(x-2)+20).
>
> Assuming the question means $10 'each' for the first two books. Is this
> algebraic?
I believe that it is. OTOH, short of giving an argument like I did
earlier in this thread, it certainly wouldn't seem obvious that the
two "decision functions" which you used are algebraic.
David
flip
> news:20030109111654.518$r...@newsreader.com...
> > "Rod" <RodRod...@hotmail.com> wrote:
> > > Is not taking only the +ve root of sqrt also a decision?
>
>
No, the term (x < 3) is the same as If x < 3, which evaluates to true.
Note, this can be dangerous, as true could be a nonzero integer in some
cases and when (x < 3) evaluates to true, it could set that value to 3
instead of the desired 1.
Looks like excel is setting these to 1 when true, so no problem.
Same comment goes for (x > 2).
In both cases, it is safer to use If (x < alpha) then contructs.
Flip
flip
This one is certainly algebraic is and is the best answer.
I would only change the y to be
f (x ) = { 10*x, 0 <= x <= 2}
{ 6(x-2) + 10x, x >= 2}
This way the limit is continuous throughout.
Darrell
What you have is essentially OK. It is "algebraic," so to speak. Yes, it
is a "decision" as well, so to speak, but that's no problem. It can be
considered a single expression. Actually, a single "function" would be a
more appropriate choice of words. Just because a function is defined
differently over different parts of the domain does not stop it from being a
'single' function. Using standard functional notation (which she is
probably not even aware of yet), you could write (C, cost, in dollars):
C(x) = 10x , x an integer such that 0 =< x =< 2,
= 6x+8 , x an integer such that x > 2
Note I multiplied out 6(x-2)+20, not that it need be done though. It just
takes less keystrokes that way, i.e. it just looks a little "neater."
I suspect that a specific notation may not be the primary focus of the
exercise, but rather that what she writes is logically correct given the
context of whatever notation she uses. IOW, I for one feel what you already
have is fine. Ironically, although the latter form is a more common
mathematics 'lingo' and is usually the notation she will most encounter in
standard mathematics courses yet to come, she will probably be much more
likely to encounter the prior form in any real life setting.
Although it is often desirable to make statements as concise and elegant as
possible by using a single 'rule' to define a function over its entire
domain, sometimes it just can't be done that way. The only thing I see that
could use improvement (in the former notation) is you have not indicated
anything that requires x to be a nonnegative integer. I'm not sure what
sets of numbers she is aware of at 7th grade level, so this restriction may
or may not be so important, i.e. it may be OK just to let this restriction
be assumed. Otherwise, what you have is perfectly fine already (at least it
is if I were grading it.)
--
Darrell
Russ
(in message <avlkq5$grdkd$1...@ID-117088.news.dfncis.de>):
> The only thing I see that
> could use improvement (in the former notation) is you have not indicated
> anything that requires x to be a nonnegative integer. I'm not sure what
> sets of numbers she is aware of at 7th grade level, so this restriction may
> or may not be so important, i.e. it may be OK just to let this restriction
> be assumed. Otherwise, what you have is perfectly fine already (at least it
> is if I were grading it.)
Given the statement of the problem(dealing with books) the domain is the set
of natural numbers so it probably can be assumed
Darrell
news:0001HW.BA43D4F1...@news.verizon.net...
I must be hearing echoes. I thought I just said it may be OK to assume as
much. Not necessarily so for a machine you wish to program with that
if-then statement in a real life setting. My point was, the context
determines if it can be safely assumed or not. In this case, one may wish
to require the input to be a nonnegative integer before the program
continues.
--
Darrell
Doug Magnoli
what you want. You want:
f(x) = 10x (0 <=x <=2)
6x+8 (2 <= x)
i.e., where you have 6(x-2) + 10x, you meant to say 6(x-2) + 20.
-Doug Magnoli
[Delete the two and the three for email.]
Doug Magnoli
All Participants To This Thread:
I returned from work this afternoon and checked for any responses to the
post I made this morning about my daughter's algebra problem. We (my
daughter and I) want to thank all who responded. We are going to print
these out and show them to her teacher. The problem is obviously above her
grade level
My own take on this is that your answer is fine, although rather than writing it in computer-language fashion, I'd write it in math fashion:
C = cost
X = number of books (positive integer)
C = 10x (0
<= x <= 2)
6(x-2) + 20 (x >= 2)
-Doug Magnoli
[Delete the two and the three for email.]
or the question in the book was not edited out originally. At
David W. Cantrell
> Chris R Simmons wrote:
>
> > All Participants To This Thread:
> > I returned from work this afternoon and checked for any responses
> > to the post I made this morning about my daughter's algebra problem.
> > We (my daughter and I) want to thank all who responded. We are going
> > to print these out and show them to her teacher. The problem is
> > obviously above her grade level
>
> When I read this much of the sentence, I understood it as 'the problem is
> obviously above the teacher's grade level,' which may be true. My guess
> is that the teacher will be hard pressed to understand the answer given
> by David Cantrell.
Hmm. Was my explanation not clear? ;-)
Seriously, I do hope that people will not forget that the first thing I
said in this thread was "No, you're not dumb! In fact, I'm almost certain
that you've done exactly the sort of thing that is desired here.", thus
concurring with Doug's
> My own take on this is that your answer is fine,
I went on to say "But, just for the heck of it, if you really want to see
it done in just one clearly algebraic expression,..." merely as a curious
exercise to demonstrate that such _could_ be done, not implying (hopefully)
that doing so was preferable.
David
Darrell
> All Participants To This Thread:
> I returned from work this afternoon and checked for any responses to
the
> post I made this morning about my daughter's algebra problem. We (my
> daughter and I) want to thank all who responded. We are going to print
> these out and show them to her teacher. The problem is obviously above
her
> grade level or the question in the book was not edited out originally.
<...>
I think you have misinterpreted some responders. I doubt that the problem
is above her grade level. I feel the suggstions of some others are, though,
and surely they will concur. I'm sure if you decide to show those responses
to the 7th grade teacher, he/she will confirm that type of complexity for an
answer was never intended.
I feel that some others are answering _your_ question quite literally
instead of answering your daughter's question in the context of her 7th
grade math class, with the assumption that this is perfectly clear to you.
After all, you did clearly state that the only requirement (from your
daughter's perspective) is that a single variable, x, be used to represent
the number of books. The part about forming a "single" expression seems to
be an additional curiosity of yours, and is not necessary (in fact, too
complex and beyond her scope) expected as a requirement of the teacher.
--
Darrell
Chris R Simmons
From all the responses this question has generated I am truly excited
and thankful. It appears, after presenting the list of replies from this
newsgroup to her teacher, the teacher sort of shrugged it all off. She
indicated to my daughter that she was reading too much into the question.
The solution she (the teacher) had for the problem was (20 + (X - 2) * 6).
A classmate of my daughter's then asked the teacher what about the
likelihood of someone just purchasing one or two books. The teacher replied
with a joke and continued to indicate too much thought has gone into this
question. Her teacher, to my regret, didn't take the 3 page list of
responses from this newsgroup from my daughter either.
I don't want to sound too critical of the teacher though. She is a hard
working, well liked, competent instructor. But, the question did cause my
daughter, other classmates of her's, and myself to ponder a single algebraic
expression that would cover all scenarios. Thanks again to David W.
Cantrell, for showing us that this is indeed possible. My daughter's
teacher could have learned something also from this if she would have
accepted those pages of responses from this newsgroup when my daughter
offered them to her.
Maybe my excitement is a little dampened by the teachers' response but
it has been a fun crusade. Again, many thanks to everyone in this newsgroup
for such an outpouring of responses.
Chris
flip
> Dear Participants,
> Maybe my excitement is a little dampened by the teachers' response but
> it has been a fun crusade. Again, many thanks to everyone in this
newsgroup
> for such an outpouring of responses.
> Chris
Actually, it bothers me greatly that a teacher doesn't acknowledge that
mathematics isn't being a monkey, but asking questions.
Wanting to understand how and why? How did we get there? Why do we do it
this way? Can I improve upon the method? Is there another solution? Is
there a better solution? Are there multiple solutions? This is what math
is about and the students should be "PRAISED" for questioning like this, not
ignored!
This excusing bad behavior for what may indeed be a good teacher, but she is
totally missing the boat here!
My 2 cents, Flip
Albert Lai
inherit the restrictions given several centuries ago. The scope of
"number" or "counting number" certainly changed. The ancient Greeks
did not include "one" in their counting numbers; they counted from
two. (This convention is still reminiscent in for example English: "I
have a number of wives" is thought to imply "I have at least two
wives", as if 0 and 1 are not legal numbers.) When Euclid attempted
something that we now recognize as induction proofs, he wrote one base
case for 0, one base case for 1, and one base case for 2, even though
all three cases read exactly the same. We now know better. If
"counting numbers" can change to make way for progress, "algebraic
expression" certainly can too.
The operators min, max, if-then-else, and even |x| (absolute value)
are not considered "algebraic" by our ancestors, but that should
constitute no resistance. In the present context, the ultimate goal
and meaning of "algebra" is:
1. to symbolically model a numerical situation we encounter
2. to calculate with and reason about the model by symbolic manipulations,
so that it helps us solve problems and answer questions about the
original situation
It is a fact of experience that the four operators above are
invaluable for #1. The present rejection of these operators by some
people is, candidly, due to the apparent absence of relevant
manipulation methods (algebraic laws) for the sake of #2. But this we
can fix.
Here are some algebraic laws for min and max.
min(x,y) = min(y,x)
min(min(x,y),z) = min(x,min(y,z))
max(x,y) = max(y,x)
max(max(x,y),z) = max(x,max(y,z))
Thus they are commutative and associative. Because of this, and
because they are useful binary operators, they deserve infix notations
like addition and multiplication do. I write x/\y for max(x,y), and
x\/y for min(x,y); this is only due to the ASCII restriction here, and
I would prefer using upward and downward arrows instead. Here are
the above laws again, plus more laws, using the new notation:
x\/y = y\/x
(x\/y)\/z = x\/(y\/z)
x/\y = y/\x
(x/\y)/\z = x/\(y/\z)
x\/x = x
x/\x = x
x\/(x/\y) = x
x/\(x\/y) = x
x\/(y/\z) = (x\/y)/\(x\/z)
x/\(y\/z) = (x/\y)\/(x/\z)
x<=y iff x\/y=x
x<=y iff x/\y=y
(x\/y)+z = (x+z)\/(y+z)
(x/\y)+z = (x+z)/\(y+z)
-(x\/y) = (-x)/\(-y)
-(x/\y) = (-x)\/(-y)
The if-then-else operator can be said to come from computer
programming, but there is nothing wrong in incorporating it into
algebra and mathematics. Physics, surveying, statistics, and
economics have all inspired additions to mathematics; so can computer
programming.
There is existing mathematical notation related to if-then-else.
We can define piecewise functions by writing something like
f(x) = { blah if x is prime
{ bleh otherwise
But the if-then-else operator is more general. It does not require
defining a function just so as to use it once. I can write directly
"if x>y then x+y else x-5y"; I don't have to write indirectly "f(x,y)
where f(x,y) = ..."
Only two things distinguish if-then-else from other algebraic
operators: It is a tenary operator, and its first operand is a boolean
rather than a number. But it is useful enough that we should include
and master it, rather than hide our heads in sand.
Here are some laws for if-then-else.
if true then x else y = x
if false then x else y = y
f(if b then x else y) = if b then f(x) else f(y)
e.g. z+(if b then x else y) = if b then z+x else z+y
if b then x else x = x
The absolute value operator is familiar enough, so I won't repeat
its laws here.
A note on "making decisions": we make decisions all the time, even
with "algebra" as we knew it two centuries ago. Whenever we see
x*y/y, we say to ourselves, "if y is 0, x*y/y is undefined; otherwise,
x*y/y is x." The division operator requires a decision all the time.
There is nothing wrong with an algebraic operator prescribing a
decision.
A note on the construction sqrt(x^2) to mimic |x|, min/max, and
if-then-else: it is technically correct, it is clever, and it even
fits the bill of "algebraic expression" in its narrowest, most ancient
sense. But it is obsfucation! A good vocabulary is one that allows
you to say directly what you mean. If I say "13 + (x-2) * (2.5 + 1.5
* (x-2)/sqrt((x-2)^2)), and by the way pretend 0/0=1", do you know
what I am trying to describe? Don't you appreciate it if I just
plainly say "if x<2 then x+11 else 4x+5"? The use of clever tricks to
communicate something supposedly straightforward does not underline
the cleverness of the writer; rather, it underlines the flaws of the
language and vocabulary used. A good set of operators and notations
does not glorify the writer by way of intimidating the reader.
Let us extend the scope of algebra for our grandchildren to inherit.
Let us turn heritage into a blessing, not a curse.
Darrell
> Dear Participants,
> From all the responses this question has generated I am truly excited
> and thankful. It appears, after presenting the list of replies from this
> newsgroup to her teacher, the teacher sort of shrugged it all off. She
> indicated to my daughter that she was reading too much into the question.
> The solution she (the teacher) had for the problem was (20 + (X - 2) * 6).
> A classmate of my daughter's then asked the teacher what about the
> likelihood of someone just purchasing one or two books. The teacher
replied
> with a joke and continued to indicate too much thought has gone into this
> question. Her teacher, to my regret, didn't take the 3 page list of
> responses from this newsgroup from my daughter either.
That's just plain inexcusable. Apparently the teacher has not given enough
thought herself to the problem. These students are being short changed by
being ridiculed for asking a very intelligient question instead of being
given a useful answer. The teacher should encourage, not discourage by
ridiculing, the question of purchasing one or two (or even 0) books.
The model given simply is incorrect. If 3 books are purchased, you end up
correctly paying:
(20 + (3 - 2) * 6) = $26
If two books are purchased you correctly pay:
(20 + (2 - 2) * 6) = $20
But if one book is purchased, according to the same model you pay:
(20 + (1 - 2) * 6) = $14, which is incorrect because the problem clearly
states the first two books purchased are 10 bucks a piece.
To make matters worse, if 0 books are purchased you end up paying $20 bucks.
> I don't want to sound too critical of the teacher though. She is a
hard
> working, well liked, competent instructor. But, the question did cause my
> daughter, other classmates of her's, and myself to ponder a single
algebraic
> expression that would cover all scenarios. Thanks again to David W.
> Cantrell, for showing us that this is indeed possible. My daughter's
> teacher could have learned something also from this if she would have
> accepted those pages of responses from this newsgroup when my daughter
> offered them to her.
Apparently this teacher is one that already has her mind made up on the
issue. Being closed-minded is not a trait of a good mathematician, or
teacher in general. Even if the teacher has the preconceived notion that
she is right and everyone else is wrong, she should welcome such questions
as an opportunity to TEACH someone WHY she is right. Either this teacher is
plain stupid and doesn't know why her answer is insufficient (possibly
explaining why she is avoiding the issue) or she is too arrogant to consider
the possibility that the student may actually have a POINT. Either way,
your child is being short changed by this teacher. You give this teacher
too much credit. If she isn't willing to acknowledge that the equation
given is only valid for x>=2 (meaning there must be some OTHER equation for
the other possible cases and can explain what this other equation is) then
that just isn't right, period. I feel for the math teacher that *I* catch
telling *my* son something like that when asked a perfectly intelligient and
legitimate question. I'm sorry for going off on a rant, but it's just
inexcusable. It's because of teachers like this that so many of our young
people do not learn the math that they should (and deserve to) learn.
Charming, perhaps. Well-liked, probably. Good teacher, no...
> Maybe my excitement is a little dampened by the teachers' response but
> it has been a fun crusade.
You make an excellent and very positive point. Although it is
dissapointing, all is not lost because you have taken enough interest in the
matter to be able to explain to your daughter why the teacher's response is
a crock. I admire your attitude, because if *I* was there and it was *my*
son I would probably take the chalk from her hand and answer the question
myself. these kids *deserve* to know the answer to such an intelligent
question.
--
Darrell
Doug Magnoli
> Dear Participants,
> From all the responses this question has generated I am truly excited
> and thankful. It appears, after presenting the list of replies from this
> newsgroup to her teacher, the teacher sort of shrugged it all off. She
> indicated to my daughter that she was reading too much into the question.
> The solution she (the teacher) had for the problem was (20 + (X - 2) * 6).
> A classmate of my daughter's then asked the teacher what about the
> likelihood of someone just purchasing one or two books. The teacher replied
> with a joke and continued to indicate too much thought has gone into this
> question.
Bad response. The question:
"What is the total cost for all books if the bookstore charges $10 for the
first 2 books and $6 for any subsequent ones."
seems to be asking for an equation describing the cost of a pile of books. If
that pile is only one book (or zero books) big, the question still seems to be
asking about it. To shrug off the thoughtful question of a student about
something like this strikes me as irresponsible teachering.
Just my two-cents' worth.
Hans van Duijnhoven
I am simply astonished by this teacher's behavior!
In my almost 40 years of experience as a teacher in mathematics in
Holland (on different levels) I never experienced something like that.
For many people mathematics must be some kind of hokus-pokus with that
kind of teachers and I begin to understand, why it is not a shame, if
you say you know nothing about mathematics or admitting "not being
good at it".
I think you're much too polite calling her a "hard working, well
liked, competent instructor". May be she'd nice but a bad
math-teacher.
Regards, Hans
(remove the anti-spam x for e-mail)
Russ
(in message <3E1FABD7...@attbi.com>):
> Chris R Simmons wrote:
>
>> Dear Participants,
>> From all the responses this question has generated I am truly excited
>> and thankful. It appears, after presenting the list of replies from this
>> newsgroup to her teacher, the teacher sort of shrugged it all off.
> "What is the total cost for all books if the bookstore charges $10 for the
> first 2 books and $6 for any subsequent ones."
>
> seems to be asking for an equation describing the cost of a pile of books.
> If
> that pile is only one book (or zero books) big, the question still seems to
> be
> asking about it.
The publishers of the textbook should revise the question for the next
edition along the lines of "the bookstore is running a special promotion,$10
for the first 2 books and $6 for any subsequent ones, if the customer buys 2
or more books in a single purchase. What is the equation if a customer takes
part in this deal?"
David W. Cantrell
It seems to me that the author of the question _intended_ the students to
consider the possibilities that just one book or no book was purchased.
That intent seems perfectly reasonable. It need not be altered, unless a
simpler question is more appropriate for a certain group of students.
As to the wording, IMO, the necessary change is the insertion of "each of"
at two spots, giving
"What is the total cost for all books if the bookstore charges $10 for
each of the first 2 books and $6 for each of any subsequent ones."
[I can only hope that I will not be derided as being obfuscatory for
having suggested this change!]
BTW, concerning the response which Chris got from the teacher: It
certainly does not surprise me, alas. I suspect that, despite how
diplomatically Chris may have handled matters, the teacher viewed the
situation as a threat to his/her competence or authority, and shrugging
it all off seemed the best way to deal with that perceived threat.
Unfortunate, to say the least.
David
Darrell
Indeed. My guess is the instructor (being lazy) simply read off the answer
in the instructor's manual--assuming it to be the final word--instead of
utilizing her own mind to answer the student's question. A good instructor
is able to deal with such "imperfections" in the text. A bad instructor
simply reads the text to the students. I fear the latter is true for this
case.
--
Darrell
Doug Magnoli
> [I can only hope that I will not be derided as being obfuscatory for
> having suggested this change!]
You're obfuscatory, you're obfuscatory, you're the obfuscatoriest person I've
never met. ;-)
>
>
> BTW, concerning the response which Chris got from the teacher: It
> certainly does not surprise me, alas. I suspect that, despite how
> diplomatically Chris may have handled matters, the teacher viewed the
> situation as a threat to his/her competence or authority, and shrugging
> it all off seemed the best way to deal with that perceived threat.
> Unfortunate, to say the least.
And _that's_ what I meant when I said your solution was likely to be above the
grade level of the teacher, I certainly intended no suggestion that your
explanation (which I found crystal clear, btw) was murky.
Doug Magnoli
I don't suppose you have any interest in carrying in a pile of these responses
and laying them quietly on the teacher's desk.
I didn't think so.
-Doug Magnoli
[Delete the two and the three for email.]
David W. Cantrell
> "David W. Cantrell" wrote:
>
> > [I can only hope that I will not be derided as being obfuscatory for
> > having suggested this change!]
>
> You're obfuscatory, you're obfuscatory, you're the obfuscatoriest person
> I've never met. ;-)
:-)
My bracketed comment was written with the response of Albert Lai in mind.
:-( It seems that he completely misunderstood my intent, although I tried
to make it clear. In my original response, I began by suggesting that a
solution such as the one which Chris already had would be appropriate. My
opinion in this regard has never changed. It is the type of solution which
Lai also thinks appropriate.
However, Chris had wondered if a solution could be written in a _single
algebraic_ expression, avoiding any explicit appeal to a "choice
function". Preceeded by the important qualifier "just for the heck of it",
I then gave a single algebraic expression, using something in the form
z/Sqrt(z^2). I have _never_ thought it desireable to express the solution
in that way -- certainly not for 7th grade students!, or indeed for anyone
else -- EXCEPT _merely_ to demonstrate (to someone above 7th grade, of
course) that it _can_ be done in a single algebraic expression. I agree
that such an expression is baroque, that writing the solution in that way
is obfuscatory, etc.
> > BTW, concerning the response which Chris got from the teacher: It
> > certainly does not surprise me, alas. I suspect that, despite how
> > diplomatically Chris may have handled matters, the teacher viewed the
> > situation as a threat to his/her competence or authority, and shrugging
> > it all off seemed the best way to deal with that perceived threat.
> > Unfortunate, to say the least.
>
> And _that's_ what I meant when I said your solution was likely to be
> above the grade level of the teacher, I certainly intended no suggestion
> that your explanation (which I found crystal clear, btw) was murky.
Doug, I never thought you suggested my explanation was murky.
I would never have dreamt of showing my single algebraic expression to
a 7th-grade teacher! Rather, had I been Chris, I would simply have asked
whether it would be acceptable to give the answer in an IF...THEN...
format. Then, if the teacher said "No.", I would have asked how he/she
would write the answer so that it would be valid for any (nonnegative
integer) number of books purchased,...
Cheers,
David
Bob Pease
> certainly does not surprise me, alas. I suspect that, despite how
> diplomatically Chris may have handled matters, the teacher viewed the
> situation as a threat to his/her competence or authority, and shrugging
> it all off seemed the best way to deal with that perceived threat.
> Unfortunate, to say the least.
>
> David
In fairness, it I had given the problem as stated in class, and a parent
came to me with a stack of responses as in this thread, I would have been
amazed that any parent would care enough.
However, I would have to thank the parent for the interest, but making the
following observation.
My teaching was usually regarded as having standards that are too high for
"normal " students.
If I were to word problems with precision, I could expect ridicule from the
coaches for using "Fancy-Pants" language and replaced with a coach that was
somewhat dimmer than the teacher mentioned in this thread.
This is one more of my indictments of the sorry state of Math Education in
the U.S. but it is showing up in the in ability of all but a small cadre to
deal with any level of abstraction beyond listening to a parson rave on
Sunday morning.
original statement of the problem.
*****
"What is the total cost for
all books if the bookstore charges $10 for the first 2 books and $6 for any
subsequent ones." The requirement is that she use only one variable ( X )
for the number of books.
*****
the details have been discussed in other posts.
I would have been surprised if the teacher would have reacted otherwise.
The original question as stated is ambiguous in at least two respects.
this is a surefire indication that the teacher is expecting a mediocre
popular response, actually as a treat from the expected answer which would
be
"Sixteen dollars because you jest add"
Thus she would jubilantly mark CORRECT answers as "Wrong" because she didn't
understand the entire problem.
Is sad that this class is labeled a "Accelerated" when analysis and
originality is brutally repressed.
Personally , I like " if the symbol "X" is meant to mean " the number of
books taken Book", then the cost is $10 if X = 1 or 2, otherwise it is
$16 if you take more than two"
But if a kid had written this as an answer, you would get a grade penalty
and probably detention for being a wise guy.
Actually a better answer is " Screw "X", Unless you're stupid, it is $16
plus the cost of renting a truck to load as many books as you can get in
it!!"
It DOES make the teacher look especially dim because the answer she gave
doesn't even follow her own vague instructions.
(20 + (X - 2) * 6).
Even the question were... "Write an algebraic expression for..."
Just as an afterthought, In High School
I would have got detention if I had given my answer instead of the one
supplied by this teacher.
Bob Pease
Barb Knox
> My teaching was usually regarded as having standards that are too high for
> "normal " students.
> If I were to word problems with precision, I could expect ridicule from the
> coaches for using "Fancy-Pants" language and replaced with a coach that was
> somewhat dimmer than the teacher mentioned in this thread.
<boggle> Huh?!? There are several possible readings of this, all of which
I find appalling:
1. "Coach" is used in that institution as a sensitive new-age euphemism
for "teacher", in which case this is an appalling abuse of language, even
apart from the fact that the teachers (yclept "coaches") have such low
aspirations that they would actively discourage accuracy and precision in
mathematics.
2. "Coach" means sports or PE coach, in which case it's appalling that
the maths teachers care what the PE teachers think about teaching maths.
What, will they beat you up for your lunch money if they don't like your
teaching style? <g>
3. "Coach" means sports or PE coach, and they are in fact drafted in to
teach maths classes between football practices. That can't be true, can
it?
> This is one more of my indictments of the sorry state of Math Education in
> the U.S. but it is showing up in the in ability of all but a small cadre to
> deal with any level of abstraction beyond listening to a parson rave on
> Sunday morning.
Personally, I lay a lot of responsibility for that on the teachers of the
early-grade teachers, who do nothing to help those early-grade teachers
overcome their personal maths traumas and so those attitudes are passed on
to yet another generation. As my namesake said, "Math class is hard."
And IMO the level of abstraction of the average sermon is really, really
low. Maybe that was your point?
Also IMO, the inability to cope with abstraction is a symptom of a more
general failure to cultivate imagination. I expect that the vast amount
of time spent with TV and video games has a lot to do with this, but like
all of us whinging geezers I have no clue as to a possible solution.
And regarding maths standards in the US versus other countries, I grew up
in the US and now live in New Zealand. I recently did some senior high
school maths teaching and was very pleased to see that the calculus class
uses a book that actually contains *proofs*, not some dumbed-down "Harvard
Calculus"-like waffle. I was less pleased to see that virtually all the
calculus students were foreign students, not native NZers.
[snip]
--
---------------------------
| BBB b \ barbara minus knox at iname stop com
| B B aa rrr b |
| BBB a a r bbb | "Don't change until the empire falls --
| B B a a r b b | You'd laugh so hard you'll crack the walls."
| BBB aa a r bbb | -- G. Slick
-----------------------------
Bob Pease
"Barb Knox" <s...@sig.below> wrote in message
news:see-130103...@192.168.1.2...
> [snip]
> > My teaching was usually regarded as having standards that are too high
for
> > "normal " students.
> > If I were to word problems with precision, I could expect ridicule from
the
> > coaches for using "Fancy-Pants" language and replaced with a coach that
was
> > somewhat dimmer than the teacher mentioned in this thread.
>
> <boggle> Huh?!? There are several possible readings of this, all of which
> I find appalling:
>
> 1. "Coach" is used in that institution as a sensitive new-age euphemism
> for "teacher", in which case this is an appalling abuse of language, even
> apart from the fact that the teachers (yclept "coaches") have such low
> aspirations that they would actively discourage accuracy and precision in
> mathematics.
It is my experience that almost no Math teachers could even pass a Junior
Math Major college level course in the subjects they teach.
A typical example was the Dept. Chairperson at a Private School in CA where
I taught for trhee years.
Her... wisdom.
"Tell them a lot about Chakras and alternative medicine, then give them
every problem in the book that has a teacher's manual solution and have them
put them on the board.'
2. "Coach" means sports or PE coach, in which case it's appalling that
> the maths teachers care what the PE teachers think about teaching maths.
> What, will they beat you up for your lunch money if they don't like your
> teaching style? <g>
Although this SEEMS like the least important of the three, it is actually
the MOST.
Instead of beating you up for your lunch money, they make snide comments to
the Athletic Boosters Club, who control the purse strings and have a
tremendous representation at School Board meetings.
> 3. "Coach" means sports or PE coach, and they are in fact drafted in to
> teach maths classes between football practices. That can't be true, can
> it?
Surely you jest.!!!
In Sterling CO about a decade ago the Superintendent of Schools decalred to
me that he would close the schools rather than hire a teacher who did not
coach.
He said that a MS and 20 years experience were secondary considerations.
At Arapahoe High School in Denver in 1967 I was told by the Principal that
"We need guys with a common touch, and not anyone who teaches this
"High-Powered stuff"
>
> > This is one more of my indictments of the sorry state of Math Education
in
> > the U.S. but it is showing up in the in ability of all but a small cadre
to
> > deal with any level of abstraction beyond listening to a parson rave on
> > Sunday morning.
>
> Personally, I lay a lot of responsibility for that on the teachers of the
> early-grade teachers, who do nothing to help those early-grade teachers
> overcome their personal maths traumas and so those attitudes are passed on
> to yet another generation. As my namesake said, "Math class is hard."
It's also a corrupt world wiew that reagrds academics as a poor second
choice to Athletic prowess.
A good deal of the problem at Columbine was due to the "Jox Rule" climate
which allowed bullying to be a ritual of passage.I know personally a girl
whose boyfriend was one of the "Trench Coat Mafia".
They were nice kids who needed to make a stand in self defense.
The Columbine shooters were not active members.
> And IMO the level of abstraction of the average sermon is really, really
> low. Maybe that was your point?
yup
> Also IMO, the inability to cope with abstraction is a symptom of a more
> general failure to cultivate imagination. I expect that the vast amount
> of time spent with TV and video games has a lot to do with this, but like
> all of us whinging geezers I have no clue as to a possible solution.
Nor I.
Sometimes a system becomes so dysfunctional that catastrophic failure is the
only solution.
> And regarding maths standards in the US versus other countries, I grew up
> in the US and now live in New Zealand. I recently did some senior high
> school maths teaching and was very pleased to see that the calculus class
> uses a book that actually contains *proofs*, not some dumbed-down "Harvard
> Calculus"-like waffle. I was less pleased to see that virtually all the
> calculus students were foreign students, not native NZers.
I taught in Australia at Moorleigh for a year.
Not a lot of difference, perhaps the Aussie are too laid back, but the
teachers actually spun gravel to beat the kids out of the parking lot to get
to the Pub..
and then the kids who aren't hopelessly behind try to get busy in Fourth
Form.
In a recent survey in Geographical Knowledge, US, Brits, and Aussies ranked
in the lowest quartile consistently.
I would expect to win money on a bet that the majority of U.S. TEACHERS of
( non-AP) Calculus neither know the Fundamental Theorem Of Calculus nor
cloud they prove it if stated.
Darrell
news:see-130103...@192.168.1.2...
<...>
> 2. "Coach" means sports or PE coach, in which case it's appalling that
> the maths teachers care what the PE teachers think about teaching maths.
> What, will they beat you up for your lunch money if they don't like your
> teaching style? <g>
In this neck of the woods, many of the math teachers *are* sports coaches!
--
Darrell
Darrell
news:avtcg1$1...@dispatch.concentric.net...
Bob's right. There are schools where *nothing* is more important than
winning ball games. I have personally known people who could not easily
read their own diploma (I mean they _could_ read it but it was quite
difficult.)
When I was in high school (middle TN), some math classes were regularly
conducted *at* football practice. AFAIK a policy still exists there that
states a certain GPA be maintained by athletes, which I'm sure is fairly
common nationwide. This was the subject of a recent discussion with a
relative, whos daughter is a cheerleader. She had unfortunatly received a D
and was suspended from the squad for two games. A discussion ensued as to
the reasons why none of the football team seem to ever get suspended for
poor grades. Let me put it this way. Unlike cheerleaders, football players
never recieved poor grades. Football players win ball games, not
cheerleaders. Coaches regularly taught many of the hard subjects (like
math) for a reason, and non-coaches who demonstrated a pattern of giving
failing marks to athletes soon became non-teachers as well, or at the very
least non-teachers of athletes. The really good teachers usually taught
everyone else (and there _were_ some really good ones, BTW). Of course,
there was always the occasional excellent student athlete, who would usually
insist on the "good" teachers. The common coaching philosophy in these
cases was, "That's fine, as long as it doesn't interfere with playing
football."
On a similar issue, I do admire though how the NCAA seems to be upholding
standards. I mean, it's not that uncommon now for a ranked team to suspend
a key player for reasons having nothing to do with performance on the field.
This happend most recently this season at FSU, IIRC. Action like that would
*never* be considered in these parts. Unless they were in jail or dead,
they were on the field. Correction: If in jail they would be released for
games and practice. I think that actually happend here a few years ago.
And no, it's not just a redneck issue--things like this happen throughout
the nation.
--
Darrell
Greg
"Barb Knox" <s...@sig.below> wrote in message
news:see-130103...@192.168.1.2...
[...]
> And regarding maths standards in the US versus other countries, I grew up
> in the US and now live in New Zealand. I recently did some senior high
> school maths teaching and was very pleased to see that the calculus class
> uses a book that actually contains *proofs*, not some dumbed-down "Harvard
> Calculus"-like waffle. I was less pleased to see that virtually all the
> calculus students were foreign students, not native NZers.
>
[...]
Kia Ora, Barb,
We must be near neighbours. I find your observation interesting. Firstly you
need to define your terms. Who are the 'native NZers' of whom you speak? Are
they tangata whenua, tau iwi, Maori or pakeha? What school were you teaching
at? What was the proportion of foreign students? Was it the International
School?
As a counter example the small NZ school I taught in last year had a class
of 8 taking year 13 Maths with Calculus. One was a Portugese exchange
student. The others were local (local means they and their families are
normally resident in NZ). There was also four students taking year 13 Maths
with Statistics - all were local.
--
Greg
(remove BALL from address to reply)
"All generalizations are dangerous, even this one."
Alexandre Dumas
Barb Knox
> "Barb Knox" <s...@sig.below> wrote in message
> news:see-130103...@192.168.1.2...
>
> [...]
> > And regarding maths standards in the US versus other countries, I grew up
> > in the US and now live in New Zealand. I recently did some senior high
> > school maths teaching and was very pleased to see that the calculus class
> > uses a book that actually contains *proofs*, not some dumbed-down "Harvard
> > Calculus"-like waffle. I was less pleased to see that virtually all the
> > calculus students were foreign students, not native NZers.
> [...]
>
> Kia Ora, Barb,
Tena koe, Greg.
> We must be near neighbours. I find your observation interesting. Firstly you
> need to define your terms. Who are the 'native NZers' of whom you speak? Are
> they tangata whenua, tau iwi, Maori or pakeha?
Yes <g>. By "native NZer" I mean someone who was born in NZ, which is (in
my dialect at least) the common definition of "native", so I didn't see
any need to define it. I didn't expect it to be a contentious term.
> What school were you teaching at?
A public girls' high school.
> What was the proportion of foreign students?
I don't know overall -- maybe around twenty percent. The calculus classes
were nearly one hundred percent.
> Was it the International School?
No. Until now I had not heard of that school.
> As a counter example the small NZ school I taught in last year had a class
> of 8 taking year 13 Maths with Calculus. One was a Portugese exchange
> student. The others were local (local means they and their families are
> normally resident in NZ). There was also four students taking year 13 Maths
> with Statistics - all were local.
Speaking of Statistics, I didn't impute any universality or statistical
significance to my experience; it's just something that discomfited me and
seemed plausibly relevant. So your experience doesn't "counter" mine; it
was just different.
Having answered your questions I'd now like to ask you one. What did you
take "native NZer" to mean? I'm aware that my US dialect differers from
NZ ones, and I'm always on the lookout to refine my usage.
--
---------------------------
| BBB b \ barbara minus knox at iname stop com
| B B aa rrr b |
| BBB a a r bbb |
| B B a a r b b |
| BBB aa a r bbb |
-----------------------------
Greg
"Barb Knox" <s...@sig.below> wrote in message
Thank you for clarifying your earlier post.
The word 'native' can have different meanings. Sometimes the context makes
it clear which meaning was intended. When James Cook in his journals spoke
of the 'consent of the Natives' I have no doubt he was speaking of the
Maori people. Your context was not clear. I read 'native NZers' as being a
subset of 'NZers'. Apparently that was not your intention. In NZ the word
'native' is now more likely to be used describing flora than people.
I had read your sentence, that referred to 'virtually all the calculus
students', as a generalisation rather than the statistic from one school. I
was concerned that others, who respect your opinions, might read it as a
generalisation too.
Barb Knox
> "Barb Knox" <s...@sig.below> wrote in message
> news:see-140103...@192.168.1.2...
> > In article <3e23...@clear.net.nz>, "Greg" <greg...@contact.net.nz>
> wrote:
> >
> > > "Barb Knox" <s...@sig.below> wrote in message
> > > news:see-130103...@192.168.1.2...
> > > [...]
> > > > And regarding maths standards in the US versus other countries,
> > > > I grew up in the US and now live in New Zealand. I recently did
> > > > some senior high school maths teaching and was very pleased to see
> > > > that the calculus class uses a book that actually contains *proofs*,
> > > > not some dumbed-down "Harvard Calculus"-like waffle. I was less
> > > > pleased to see that virtually all the calculus students were foreign
> > > > students, not native NZers.
> > > [...]
> > >
> > > Kia Ora, Barb,
> >
> > Tena koe, Greg.
> >
> > > We must be near neighbours. I find your observation interesting. Firstly
> > > you need to define your terms. Who are the 'native NZers' of whom
> > > you speak? Are they tangata whenua, tau iwi, Maori or pakeha?
> >
> > Yes <g>. By "native NZer" I mean someone who was born in NZ, which is (in
> > my dialect at least) the common definition of "native", so I didn't see
> > any need to define it. I didn't expect it to be a contentious term.
[snip]
> > > As a counter example the small NZ school I taught in last year had a
> > > class of 8 taking year 13 Maths with Calculus. One was a Portugese
> > > exchange student. The others were local (local means they and their
> > > families are normally resident in NZ). There was also four students
> > > taking year 13 Maths with Statistics - all were local.
> >
> > Speaking of Statistics, I didn't impute any universality or statistical
> > significance to my experience; it's just something that discomfited me and
> > seemed plausibly relevant. So your experience doesn't "counter" mine; it
> > was just different.
> >
> > Having answered your questions I'd now like to ask you one. What did you
> > take "native NZer" to mean? I'm aware that my US dialect differers from
> > NZ ones, and I'm always on the lookout to refine my usage.
> >
> Kia Ora, Barb,
>
> Thank you for clarifying your earlier post.
>
> The word 'native' can have different meanings. Sometimes the context makes
> it clear which meaning was intended. When James Cook in his journals spoke
> of the 'consent of the Natives' I have no doubt he was speaking of the
> Maori people.
Reasonably so, since they were the people who had been born here.
Admittedly, in common British colonial parlance, "natives" had a lot of
the connotations that "wogs" has today; but the term was also used without
racist or ethnocentric overtones.
> Your context was not clear. I read 'native NZers' as being a
> subset of 'NZers'.
I think my context was clear. I did mean "native NZers" to designate a
proper subset of "NZers" -- the ones who were born here.
> Apparently that was not your intention.
But it was. Being an immigrant NZer I am aware of differences between me
and you natives (assuming you are one). Maybe I should use the redundant
phrase "native born" instead, to avoid tripping over colonial baggage.
> In NZ the word 'native' is now more likely to be used describing flora
> than people.
Thanks for the usage tip. When in Rome...
> I had read your sentence, that referred to 'virtually all the calculus
> students', as a generalisation rather than the statistic from one school. I
> was concerned that others, who respect your opinions, might read it as a
> generalisation too.
Interesting. How should I rephrase "I was less pleased to see that ..."
to indicate I was reacting to what I personally saw rather than making a
generalisation?
Albert Lai
> > > [I can only hope that I will not be derided as being obfuscatory for
> > > having suggested this change!]
> My bracketed comment was written with the response of Albert Lai in mind.
> :-(
I meant that the expression, not the person who designed it, was
obfuscatory. We are violently agreeing with each other on this.
That said, since the audience consists not just of you and me in the
know, it was important to make and explain the point that clever
obfuscation is not a desirable way of communication.