mm-304
Subject: Re: Can't figure this out...> Common
denominator:>> (sin x)/(1+cos x)+(1+cos x)/(sin x)>> = ((sin
x)^2 + (1 + cos x)^2)/ ((sin x)(1+cos x))>> = ((sin x)^2 + 1
+2(cos x) +(cos x)^2)/ ((sin x)(1+cos x))>> = 2(1+ (cos x)) /
((sin x)(1+cos x))>> = 2 / (sin x)>> = 2 csc x>> Constraint:
cos(x)!=-1, to avoid division by 0.Would you not also need a
constraint: sin x !=0, for the same reason?> When sin(x)=0,
cos(x) is either +1 or -1. If cos(x)=-1, the left hand> side
of the identity to prove is undefined, but the right hand side
is> infinite. When cos(x)=+1 both sides of the identity are
infinite,> which one COULD argue is still an acceptable
identity. That is, one> can allow division by 0, but not 0/0.
I'd be happy, though, with a> student who disallowed all
multiples of pi for x, and not just the odd> multiples. > -- >
Kevin Karplus> K.K. the right hand side is infinite > No.
2csc(x) is *never infinite*.> K.K. one can allow division by
0> No. What algebraic universe are you playing in?> K.K. I'd
be happy, though, with a student who disallowed all multiples
of> pi for x > Certainly, since this would be the most
complete and accurate statement of> the domain of the
identity.> --- Joe
(remember,-we-are-not-doing-one-point-compactification-in-k12)
Sroka> -- > Delete the second o to e-mail me.I am astonished at
the responses, but perhaps I shouldn't be; the trendsto
complicate beyond comprehension or to see all the difficulties
butnot the obvious are far too prevalent.The above is a quote
from a recent posting in sci.electronics.design. The John
Woodgate, OOO - Own Opinions Only.
http://www.jmwa.demon.co.uk.Way to go, John. s.e.d. is a group
which includes a passel of wise old curmudgeons, andyounger
curmudgeons, too. They are much less tolerant of vapid
postingsthan we in k12.e.m seem to be. These folks are willing
to spend lots oftime imparting their wisdom, as long as they
don't suspect you of askingfor homework answers.--- Joe--
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Subject: Stationary pointsHiI was wondering if there was
an easier way to do this problem. if you have a function (x,y)
=(x+y+1)^2----------x^2+y^2+1is there any easy was of finding
and classifying the stationary pointsie. finding the first and
second devrivative along with the mixedderivative. I've tried
it and get a big mess with expanding things outand hardly
anything cancels!any guidance would be much apprecia.Jon--
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Subject: Re: Games> I recently had my principal come
into my room when the children were> playing different math
games. She was rather upset at me saying that> the games were
fluff and that the kids needed to be doing work> instead. Some
of the games they were playing checkers, connect four,> on
these games and their connection to higher level thinking, use
of> strategy, and educational purpose? Or do all of you think I
am off> base for allowing my children to play these games?In
what way is the playing of these games furthering your
curriculumrequirements? As an 8th grade math teacher I am
getting kids that do notknow their basic math. If these games
are taking away time that would bespent on basic math then I
can see the principal's point. A big issue ishow often you are
doing this. If it is something you do once a monththen I think
it is relatively harmless and may have some slighteducational
benefit. If you are doing this once a week then I think
theprincipal has a point. You are not going to be able to
cover yourcurriculum in the 80% of the time you are
allotting.Another problem I see is that none of these games
has explicit mathcontent. There is a version of connect 4 that
does have some mathcontent and that might be a better choice.1
2 3 4 5 67 8 9 10 12 1415 16 18 20 21 2425 27 28 30 32 3536 40
42 45 48 4954 56 63 64 72 813 4 5 6 7 8 9Each player has
colored counters and there are two neutral pieces (I
usepennies). The first player places each penny on one of the
number of thestrip of numbers at the bottom (the 3 to 9
strip). They then put acolored counter on the number equal to
the product of those numbers. Soif they put the two pennies on
the and 7 they would then put one oftheir colored counters on
the 35. Then the next player can move one ofthe pennies to
another number and put one of their colored counters onthe
product of the 2 numbers. So the second player might move a
pennyfrom the 7 to the 8 and then put one of their counters on
the 40. Theycontinue like this, each time moving only one
penny, until a player gets4 in a row. This is more strategic
than regular 4 in a row and itinvolves reviewing and
practicing basic math multiplication facts. Youcould also
devise a similar version for addition.But the bottom line for
me is that students tend to come to meunder-prepared in basics
so I would like to see elementary teachers doingtheir higher
order critical thinking in a context that involves,develops,
and reinforces those basic skills. If they have the skills
Ican deal with their higher order processes. But if they have
somelimi higher order processes but extremely deficient
skills, it becomesvery difficult to get them to master
prealgebra and algebra conceptsbecause they are constantly
getting bogged down in the skill part of theproblem. If they
are facing the problem 6x = 132 I can teach them to usethe
inverse operation and divide both sides by 6. But if they
can't dothe division they won't get the problem right and they
will becomefrustra, which makes it that much harder for them to
learn.Rich-- tml
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Subject: Re: Games>I recently had my
principal come into my room when the children were>playing
different math games. She was rather upset at me saying
that>the games were fluff and that the kids needed to be doing
work>instead. Some of the games they were playing checkers,
connect four,>on these games and their connection to higher
level thinking, use of>strategy, and educational purpose? Or
do all of you think I am off>base for allowing my children to
play these games?We can't tell from the limi info you have
given. What percentage ofthe math instruction time uses these
games? How do these gamesintegrate into your overall lesson
plan? Are you teaching them gamestrategies? What are your
goals for the part of the instruction usingthe games? Are they
being met?bob-- tml
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Subject: Re: Games> I recently had my
principal come into my room when the children were> playing
different math games. She was rather upset at me saying that>
the games were fluff and that the kids needed to be doing
work> instead. Some of the games they were playing checkers,
connect four,> on these games and their connection to higher
level thinking, use of> strategy, and educational purpose? Or
do all of you think I am off> base for allowing my children to
play these games?The principal is under pressure to have
students meet performancestandards. Other teachers in your
school have the principal's ear.What do they think? Maybe some
of them also use games. If youcan get one of them to support
your approach, that may do the trick.Otherwise, it's an ul
struggle. I don't have any references tojournal that strongly
suppor games as pedagogical strategies, I doubtit would carry
much weight with your principal, who might view it as awar
club instead.-- tml
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Subject: Re: Games>apparently has made)
a very bad impression. As a parent, I for one would be>left
with the same impression should I choose to walk in and check
up on my>son, and this was going on.As a parent, I'd be deligh
if my third-grader were playing chess in class.I wouldn't want
to evaluate the quality of a classroom based on one
snapshot,either. I was surprised that the original poster's
principal did, accordingto the story. As a spuervisor, if I
walked into a classroom and the kidswere running around
grunting like gorillas and scratching under their armpits,I
still wouldn't condemn the teacher before finding out why, and
how thisactivity fit into the bigger picture.So, my question
would be, is game-playing the *only* mathematical activity?Is
it the *main* activity (in terms of time spent at it, or in
terms ofthe techer's effort, or whatever)? Imho there should
be a mix of activities.Yes, they should spend some time
practicing arithmetic facts and skills.I'd say they should
spend a lot of time on estimation -- how many grains ofsand
could we fit in this classroom? How many red-haired people
live inthis town? And I'd have them spend some time on logic
puzzles -- who livesin the yellow house?P.S. All else being
equal, if the kids look like they're happy to be atschool,
that counts for a lot, too. Obviously not if the reason
they'rehappy is that they're never asked to learn anything.
But isn't thatsupposed to be the point of all those tests we
have these days? If thekids are passing the tests, then
everyone should get off the teacher's case.It's not fair if we
only believe the tests when the kids fail them.--
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Subject: Re: Games> I don't have any proof, but I'd have
to say that chess is the king of all> logic games. If children
in third grade know how to play it, by all means> let them.
The plan of attack that is nessesary in the game is a great>
skill!Chess is a game of tactics, not logic. While the
development oftactical thinking and tree search are useful
skills, it is not clearthat 3rd graders are doing much of that
when they play chess (absentextensive coaching), and it is not
at all clear that math classes arethe right place for
it.Having kids playing tactical games without a clear
connection to thecourse objectives does look like a lack of
skill on the teacher'spart. I would be irrita if my son's
teacher were doing that(though the very low level of 2nd grade
math makes the time he spendson it pretty much was
anyway---he'd be better off if the teacherwould just have him
work on his Singapore 3rd grade math workbook, anddidn't
require him to do all the silly time-wasters in
theHarcourt-Brace 2nd grade workbook).If the class is far
enough ahead on the 3rd grade math objectives, I'dsee no
problem with allowing chess as a reward activity (just as
freereading is a suitable reward activity in literacy or
literatureclasses), but not as a replacement for the
arithmetic and wordproblems that should form the basis for 3rd
grade math.There are other games that more directly involve
math skills that onecould justify more easily to a principal.
There are math equivalentsof scrabble ('Smath?) and dominos
(MathSmart), as well as set theory games like Set and logic
games (is Wff'and'Proof still around?).The Math'n'Stuff store
in Seattle has a good collection of math gamesand toys (I
believe they have a web site also).-- Kevin Karplus
karplus@soe.ucsc.edu http://www.soe.ucsc.edu/~karpluslife
member (LAB, Adventure Cycling, American Youth
Hostels)Effective Cycling Instructor #218-ck (lapsed)Professor
of Computer Engineering, University of California, Santa
CruzUndergraduate and Graduate Director,
BioinformaticsAffiliations for identification only.--
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Subject: Math Education Sequence from 7th to 12thWhat
will be the ideal sequence of Math education from 7th to 12th
grade??Is it PreAlgebra, Algebra, Geometry, Algebra II,
Trigonometry and thenCalculus??-- tml
===
Subject: Reply to Math
Education Sequence from 7th to 12thDepends on your level of
capability, some people go to college at theage of 9, some
can't even graduate from high school.-- tml
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Subject: Re:
Math Education Sequence from 7th to 12th> What will be the
ideal sequence of Math education from 7th to 12th grade??> Is
it PreAlgebra, Algebra, Geometry, Algebra II, Trigonometry and
then> Calculus??I don't know about ideal, but that's certainly
the typical sequence.-- tml
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Subject: Re: Math Education
Sequence from 7th to 12thaskjinu@earthlink.net asks about
sequence of courses:>What will be the ideal sequence of Math
education from 7th to 12th grade??>Is it PreAlgebra, Algebra,
Geometry, Algebra II, Trigonometry and then>Calculus??That is
a good sequence of courses for those grade level ranges. Some
studentsmay be underprepared for PreAlgebra, but that can be
remedied during 7 and 8grade to allow such students to study
Algebra I in grade 9. For minimalresults, you want the
students to at least be able to study Algebra 1,Geometry,
Algebra 2, and Trigonometry all during high school. As a
reminderagain, the trick is getting the lower performing
students ready to studyAlgebra by beginning of grade 9. For
myself, I found great use of doing review study of fractions
and receivinginstruction on number sense before studying
Algebra 1. G C -- tml
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Subject: LogarithmsI have recently
star a home course in mathamatics and I am struggling with this
question. Using common logarithms find x when. 10*4^x = 5^(x+4)If
you could show me the working and the answer I would be
very greatfull Thank youLes Harper-- tml
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Subject: Re:
Logarithms> I have recently star a home course in mathamatics
and I am> struggling with this question> Using common
logarithms find x when. 10*4^x = 5^(x+4)Logs makes short work
of exponentiation, which is what you have going on here.The
log of the left side is log 10 + x*log 4The log of the right
side is (x+4)*log 5so we have x*log(4) + log(10) =
(x+4)*log(5)we solve for x x*log(4) - x*log(5) = 4*log(5) -
log(10) 4*log(5) - log(10) x = ------------------ log(4) -
log(5)x = -18.53141859752694938001199246...-- tml
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Subject:
angels[moderator's note: I believe the poster means angles and
not angels.]Hi there my name is zeryab and i do angels I am
having alot of trublewith angels and need someone to help me
so you would not mind to helpme would you thanks. REGARDS
ZERYAB-- tml
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Subject: LD Software Appraisals?I have
developed some basic arithmetic programs that may be
downlaodedat:http://www.geocities.com/ldprograms/
arithmetic101I would appreciate hearing any feedback regarding
their efficacy in anLD/SN/SEd environment.All comments and
criticism welcomed to the e-mail on the web page.Thanks in
advance,Brian-- tml
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Subject: responding to your questionYes
ma'am I think you have every right to say that what your
studentshad nothing to do with learning. Yes chess does
improve strategyskills, but if going to school was all about
learning strategyskills then why dont they teach chess at
school? if i were a teacher,and my students depended on me to
learn, i too would feel discouragedif i felt they were
spending their time in school wasting away theIS FOR
LEARNING!!!-- tml
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Subject: integralsFind a formula for the
integral x^n e^x dx, where n is a naturalnumber and prove the
formula is correct.I have tried this many times and get an
answer, but I was justwondering if someone else could try it
to see if I am doing thingsright.Thanks-- tml
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Subject: Re:
integrals> Find a formula for the integral x^n e^x dx, where n
is a natural> number and prove the formula is correct.> I have
tried this many times and get an answer, but I was just>
wondering if someone else could try it to see if I am doing
things> right.So where's your proof? How are we supposed to
know if you are doing thingsright if you don't even tell us
what you are doing?-- tml
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Subject: fraction minus whole
numbermy daughter has this equation and we don't know how to
solve, pleasehelp: 1/6 minus 2 = ? -- tml
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Subject: Re:
fraction minus whole number<<my daughter has this equation and
we don't know how to solve, pleasehelp: 1/6 minus 2 = ? >>1/6 -
2 = 1/6 - 12/6 = -11/6 = -1 5/6-- tml
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Subject: Number
combinationsCan you give me a formula for calculating the no
of possiblecombinations possible from a quantity of options.
To clarify myrequirement here are the the possibilies for
three options123121323123Therefore the answer is 7-- tml