mm-433 Subject: Re: Math Methods -- Old Vs. New>I was hoping you would clarify what specific sad things happen. Do you>think it's sad that I never had the privilege of having to compute>logarithms with a slide rule but rather usually always ask a calculator or>some other software when I need such a value (save for trivial logs, of>course) ? I don't. Never lost a wink of sleep over it.>How is the case of long division any fundamentally different wrt the point>we are making here?>-- >Regarding long division, it is classic and everybody should be able to use it;it is not expected to be limited to people with specialized knowledge. Someonewho does not know how to perform long division without a calculator does nothave adequate math skill for their job nor for making some decisions as anordinary consumer. Exceptions could be for people who do not use math for ajob. Also, it is possible for various accountants and salespeople to misleadan unmathematical consumer; such consumer lacking enough math skill to properlyapply math with a calculator. Regarding logarithms on a slide rule or with software, maybe universities madea big mistake many years ago when we were required to use tables of logarithmswhen we studied intermediate algebra, college algebra, and calculus. On theother hand, use of the tables slowed us down a bit because of the time lookingfor the various numbers. If we are without a calclulator, are we then stuck? or could we use a table of logarithms? Does the lack of a calculator suddenlyrender us far less mathematically capable? Apparantly it could. So, severalyears ago, using tables of logarithms was the normal standard of education inmath and science. Some of the reason for having used logarithm tables was formaking calculations easier. Calculators in order to use logarithms because not have skill with the logtables - SAD? Probablyso if the person does not understand logarithms andantilogarithms. A calculator is easier, and requires much less skill.Calculators in order to do long division because not otherwise able to do thelong division? YES, this is sad. What are the universities doing these days with tables of logarithms andtrigonometric values? Have they stopped teaching people how to use thosetables? Characteristic, mantissa? Are these useless now? About graph making,do the math departments no longer deal with curve sketching and such graphing?Students no longer expected to draw curves on paper, make sketches of graphs? Has this skill been abandoned and replaced by software? Such graphing/curvesketching was a major feature of physical science and mathematics courseworkwhen I was a student. G C-- === Subject: Re: Math Methods -- Old Vs. New> Regarding logarithms on a slide rule or with software, maybe universitiesmade> a big mistake many years ago when we were required to use tables oflogarithms> when we studied intermediate algebra, college algebra, and calculus. Onthe> other hand, use of the tables slowed us down a bit because of the timelooking> for the various numbers. If we are without a calclulator, are we thenstuck?> or could we use a table of logarithms? Does the lack of a calculatorsuddenly> render us far less mathematically capable? Apparantly it could. So,several> years ago, using tables of logarithms was the normal standard of educationin> math and science. Some of the reason for having used logarithm tables wasfor> making calculations easier.One of the side benefits of using log tables was that we had to learn linearinterpolation for the in-between values, which is a great application ofproportional thinking, which most definitely is a life skill.Rich.-- === Subject: Re: Math Methods -- Old Vs. New>I was hoping you would clarify what specific sad things happen. Do you>think it's sad that I never had the privilege of having to compute>logarithms with a slide rule but rather usually always ask a calculatoror>some other software when I need such a value (save for trivial logs, of>course) ? I don't. Never lost a wink of sleep over it.How is the case of long division any fundamentally different wrt thepoint>we are making here?-- >Regarding long division, it is classic and everybody should be able to useit;This phrase seems to suggest the mere fact that it is classic knowlege asthe primary reason for learning it.> it is not expected to be limited to people with specialized knowledge.Sure it is. it is expected to be limited to people with the specializedknowlege of being able to divide two numbers with pencil and paper.> Someone> who does not know how to perform long division without a calculator doesnot> have adequate math skill for their job nor for making some decisions as an> ordinary consumer.The exact same thing can be said of finding logs with a slide rule. I wouldwager that even a few old timers (no offense intended) reading this verypost would agree that, once upon a time, a similar impression was left ifone didn't know how to use a slide rule, or at least a similar impressionwould be left on others in similar professions, ones where such knowlege wasnormally commonplace.I guess in Davy Crockett and iel Boone's day, something very similar canbe said regarding the knowlege of how to use a hatchett.> Exceptions could be for people who do not use math for a> job.That would be, hmm, most people :-)> Also, it is possible for various accountants and salespeople to mislead> an unmathematical consumer;Just possible? Surely you jest. A complete range of professions existsthat endeavors (and often succeeds) to mislead not only the mathematicalyunsophisticated, but the mathematically sophisticated as well. It's calledmarketing. such consumer lacking enough math skill to properly> apply math with a calculator.> Regarding logarithms on a slide rule or with software, maybe universitiesmade> a big mistake many years ago when we were required to use tables oflogarithms> when we studied intermediate algebra, college algebra, and calculus. Onthe> other hand, use of the tables slowed us down a bit because of the timelooking> for the various numbers. If we are without a calclulator, are we thenstuck?Well, were you stuck if you didn't have your slide rule or table?> or could we use a table of logarithms? Does the lack of a calculatorsuddenly> render us far less mathematically capable? Apparantly it could. So,several> years ago, using tables of logarithms was the normal standard of educationin> math and science. Some of the reason for having used logarithm tables wasfor> making calculations easier....and so is some of the reason for using calculators today.> Calculators in order to use logarithms because not have skill with the log> tables - SAD? Probablyso if the person does not understand logarithmsand> antilogarithms. A calculator is easier, and requires much less skill.Seems to me a more efficient method for accomplishing something, would beone in which LESS, not more, skill is required. Then even nonskilled peoplecan do it.> Calculators in order to do long division because not otherwise able to dothe> long division? YES, this is sad.I fear you completely miss the point, or if you don't I guess we'll justagree to disagree. Right now if I were asked to round ln(52) to threedecimal places then I, as I suppose most, would pick up a caluclator or somesuch that has that (the ln) function. Do I know how to use a slide rule?No. Did I ever know how to use a slide rule? No. Have I ever even _seen_as slide rule? If so, I don't recall.Is this sad? No, not in my opinion. It just means I'm not as old as some.Just like it means that if a young adult today had never seen a real punchcard, or fixed disk drives the size of a dishwashers with a whopping *32MB*capacity, then they shouldn't feel sad either.> What are the universities doing these days with tables of logarithms and> trigonometric values? Have they stopped teaching people how to use those> tables?What's relevent is that one day they surely *will* stop teaching that, notfor reasons of lack of value wrt classic knowlege, but for reasons of simplynot enough time to cover that PLUS all the utilitarian knowlege they need tocover, such that students are prepared to enter TODAY's real world, notyesteryear's.> Characteristic, mantissa? Are these useless now? About graph making,> do the math departments no longer deal with curve sketching and suchgraphing?> Students no longer expected to draw curves on paper, make sketches ofgraphs?> Has this skill been abandoned and replaced by software? Suchgraphing/curve> sketching was a major feature of physical science and mathematicscoursework> when I was a student...and one day it won't be. Similar arguments can be made towards Euclidianconstructions. At one time, everyoine did them. I don't think that is thecase now.I suppose there was a time when it was necessary to learn how to use a bowand arrow. then guns came along, and that knowledge was less needed. Thenmeat departments came along, and even THAT knowlege (ability to use guns)was less needed, and now we need to know how best to select our cuts ofmeat, whether we're getting ripped off by marketing managers, etc.It would be interesting to see the elective list, or for that matter eventhe required course list, for a typical university student a long time ago.-- http://www.computerhistory.org/ === Subject: Re: Math Methods -- Old Vs. New gives another comment:< gives another comment:> < one in which LESS, not more, skill is required. Then even nonskilledpeople> can do it.>Interesting you make this comment. More can be said of it. I found astudent> learning Math in an integrated course who had used a graphing calculatorto> respond to several questions. His use of the calculator was not much of a> difficulty..... his use of algebra was way off the mark; his responseswere> therefore wrong because of his poor attention and learning of the algebra.His> use of the graphing calculator had nothing to do with his difficulty withthe> algebra.That's probably because the course is intended for the two techniques tocompliment each other. (i'm guessing what you mean by integrated.) Thepoint is, it is most surely stressed that the machine is of little value ifthe underlying principles are not understood. The design of the course, toinclude the textbook, the _machine itself_, the philosophy of the teacher,adminstrators, etc. etc. are such that this is dependence is paramount.Change the philosophy, the design of the book and the machine, andeverything else such that understanding of the basic raw algebraic conceptsis NOT required, and it could be a different story.I have enjoyed the correspondence, but Ithink I've said all I have to say.I don't want to blabor the poijnt anymore. If you reply, I will surely readwith interest but don't expect a return reply.-- -- === Subject: Re: Math Methods -- Old Vs. NewI understand but saw this last message after responding to an earlier post.OK, Done.G C>I have enjoyed the correspondence, but Ithink I've said all I have to say.>I don't want to blabor the poijnt anymore. If you reply, I will surely read>with interest but don't expect a return reply.-- === Subject: Re: Math Methods -- Old Vs. New further comments about long division, etc.:>Just possible? Surely you jest. A complete range of professions exists>that endeavors (and often succeeds) to mislead not only the mathematicaly>unsophisticated, but the mathematically sophisticated as well. It's called>marketing.No jesting meant. I was acquainted with a worker whose parent performed somemoney scheme on him; this worker did not enough sophistication in math tofigure out what was being done.You probably correct about this marketing statement. It is a reason forwhich I have paid very strong attention to exponential growth and decay studiesfrom College Algebra in my recent and ongoing restudy of this subject. G C-- === Subject: Re: Math Methods -- Old Vs. NewDarrel responds to my statements about learning long division:>Sure it is. it is expected to be limited to people with the specialized>knowlege of being able to divide two numbers with pencil and paper.Knowing long division is expected to be common knowledge and skill. G C-- === Subject: Re: Math Methods -- Old Vs. New> Darrel responds to my statements about learning long division:>Sure it is. it is expected to be limited to people with the specialized>knowlege of being able to divide two numbers with pencil and paper.Knowing long division is expected to be common knowledge and skill.Please forgive my deliberate choice of words; I had the specific aim ofgetting you to state yourself the very point I am trying to make. Yes, wecurrently expect long division to be common knowlege and skill. We can goon and on as to the reasons why we expect this, but suffice it to say, fornow, that we do expect it.At one time we were expected to know how to use bow and arrow, and flint andspear. Grocery stores were nonexistent.At one time we were expected to know how to write letters with quill andink, and seal envelopes with wax. Email was nonexistent.At some time in the future (sooner than most care to admit) we (the generalmasses) will no longer be expected to know long division, and those thatdo choose to learn it will be primarily for reasons of learning the classicsand not out of any necessity.I further submit, which I have before, that a strong argument can be madethat even NOW, most people don't NEED long division out of necessity. I'mtalking about the civilized world here as a baseline, not some third worldcontry where one can travel hundreds of miles before getting to a storewhere replacement batteries for your calculator can be bought :-).Times change. We are in the midst of it right now. The simple fact is werely more and more on technology every day, less and less on manual methods.Consider this very correspondence, for example. I have no problem witharguing the value of studying long division from a classical perspective,but many times those who argue for it (and most of pure math, for thatmatter) do so with an unrealized bias that most people will actually NEEDthis stuff.If we really needed it so much, then more people would know it. It's thatsimple. Like I said before, I would be hard pressed (as well as most peopleI know) to remember the time where I actually needed to perform longdivision outside of an educational setting.Those consumers you talk about, that need to know when they are gettingripped off. They have access to technology too. 4-functions calculatorsabound. Those consumers that do a long division BY HAND is probably becausethey are too lazy to dig out the calculator, or to go get one. those thathave the calculator readily available, will most likely use IT for anyneeded long division save for trivial cases.-- -- === Subject: Re: Math Methods -- Old Vs. NewI probably did make and share MOST of your point. My point is that knowing howto perform long division is classic knowledge which will never loose itsvalue. It is still a big part of mathematics instruction for elementary andhigh school education. People can still use it and should not need to rely ona calculator to substitute for understanding division. My earlier characterization of sad was that a common person should not be sounskilled that they be unable to perform a division calculation without theyhave a calculator. Some proficiency tests today utilize a split test section in Math; one partrequiring the skill of arithmatic and pencil-paper calculation and prohibitingthe use of a calculator; and another part permitting the use of a calculator. The two parts emphasize somewhat different purposes. The one thing that everyone understands is that a calculator is a greatcomputation tool for achieving calculations efficiently. We are now startingto overblow ourselves with technology, evolving more and more technology, untilmany things will be complicated to control. Think about Hal, the robot, from2001-A Space Odyssy. Yes, we are in a technological change, but being familiarwith the long division process will likely still be valuable. It still givesus skill that we can do which does not necessarily rely on a calculator, and ithelps us in understanding basic arithmetic; yet, in the ordinary world, mostpeople, most of the time will use a calculator. Quoted message here:My comment:>> Knowing long division is expected to be common knowledge and skill.'s comment:>Please forgive my deliberate choice of words; I had the specific aim of>getting you to state yourself the very point I am trying to make. Yes, we>currently expect long division to be common knowlege and skill. We can go>on and on as to the reasons why we expect this, but suffice it to say, for>now, that we do expect it.>At one time we were expected to know how to use bow and arrow, and flint and>spear. Grocery stores were nonexistent.>At one time we were expected to know how to write letters with quill and>ink, and seal envelopes with wax. Email was nonexistent.>At some time in the future (sooner than most care to admit) we (the general>masses) will no longer be expected to know long division, and those that>do choose to learn it will be primarily for reasons of learning the classics>and not out of any necessity.>I further submit, which I have before, that a strong argument can be made>that even NOW, most people don't NEED long division out of necessity. I'm>talking about the civilized world here as a baseline, not some third world>contry where one can travel hundreds of miles before getting to a store>where replacement batteries for your calculator can be bought :-).>Times change. We are in the midst of it right now. The simple fact is we>rely more and more on technology every day, less and less on manual methods.>Consider this very correspondence, for example. I have no problem with>arguing the value of studying long division from a classical perspective,>but many times those who argue for it (and most of pure math, for that>matter) do so with an unrealized bias that most people will actually NEED>this stuff.>If we really needed it so much, then more people would know it. It's that>simple. Like I said before, I would be hard pressed (as well as most people>I know) to remember the time where I actually needed to perform long>division outside of an educational setting.>Those consumers you talk about, that need to know when they are getting>ripped off. They have access to technology too. 4-functions calculators>abound. Those consumers that do a long division BY HAND is probably because>they are too lazy to dig out the calculator, or to go get one. those that>have the calculator readily available, will most likely use IT for any>needed long division save for trivial cases.>-- >G C-- === Subject: PLEASE HELP! :(Math seems to give me a great amount of trouble...I have this page ofword problems, and I've no clue how to do ANY of them! I don't wantthe answers...I just need someone to explain! PLEASE!-- === Subject: Re: PLEASE HELP! :(Word problems oftern try to confuse us by giving extra informationthat isn't important to us for solving the problem. Try to sort outwhat information is needed for solving the problem and then whatinformation is not needed. Also, sometimes drawing pictures of whatis happening in the problem helps. Any sort of visual representationof the word problem will make it easier to understand.I hope this helps!-- === Subject: Re: PLEASE HELP! :(Look for the action words...is means =more usually means +less usuall means -time means *As you read the problem, write down an equation.> Math seems to give me a great amount of trouble...I have this page of> word problems, and I've no clue how to do ANY of them! I don't want> the answers...I just need someone to explain! PLEASE!-- === Subject: Re: PLEASE HELP! :(> Math seems to give me a great amount of trouble...I have this page of> word problems, and I've no clue how to do ANY of them! I don't want> the answers...I just need someone to explain! PLEASE!Sure.What's an example? and I/We'll guide you through it.-- === Subject: DerivativesHonestly, this is my first time trying to post an answer.Here goes:xy=1y=1/xdy/dx = -1/x^2hence the gradient of the tgt would be -1/x^2((1/x) + 1))/(x-1) = -1/(x^2)(1+x)/x = (x-1)* (-1)/(x^2)x(1+x) = 1-xx+x^2 = 1+xx^2+2x-1=0Solving this quad eqn, you shld get x=-1+root2 or -1-root2There you go!Phew, can hardly believe i manage to solve it. Anyway, i was justthinking how come you've got only one solution?Pls get back to me if you need any clarifications...Indra-- === Subject: so many Fs...I can relate to your plight. A few years back, I was entasked to teacha class of grade 5 pupils. Being new and enthusiastic, I set the testabove their capabilities. Guess what? 3/4 of them failed. I was in adilemna. I didn't know if I should return their scripts withoutaltering the marks. A teacher told me that the class will bedevastated to find out thay had done so badly. So, I took an iniativeand started plotting a Normal distribution for their marks. Hehe, mystats lecturer would have been so proud of me! As a result? Half theclass passed and I managed to raise the lowest marks so he wouldnt bevery depressed.Nonethelss, you could have guessed, we had remedial lessons almostevery day after school since that!Indra-- === Subject: Re: so many Fs...>I can relate to your plight. A few years back, I was entasked to teach>a class of grade 5 pupils. Being new and enthusiastic, I set the test>above their capabilities. Guess what? 3/4 of them failed. I was in a>dilemna. I didn't know if I should return their scripts without>altering the marks. A teacher told me that the class will be>devastated to find out thay had done so badly. So, I took an iniative>and started plotting a Normal distribution for their marks. Hehe, my>stats lecturer would have been so proud of me! As a result? Half the>class passed and I managed to raise the lowest marks so he wouldnt be>very depressed.>Nonethelss, you could have guessed, we had remedial lessons almost>every day after school since that!so finish your story... how did they do???-- === Subject: so many Fs...Well, they did complain that math was difficult, but over the months,after going through the concepts slowly, they got the hang of it andmiraculously all of them passed the final yr exams.-- === Subject: Re: so many Fs...>Well, they did complain that math was difficult, but over the months,>after going through the concepts slowly, they got the hang of it and>miraculously all of them passed the final yr exams.Wow! Sounds like you did a great job!-- === Subject: Re: Magic Circles> Hello! I am in the 5th grade and for school, I am learning about Magic> Circles! It says: Use the first 11 positive multiples of 5, and the> magic sum is: 90! They are saying: Add the first 11 multiples of five> to get 90. Example: 55+5+30=90. Get it! So can someone please help me> find some more than just the problem above!If I'm understanding this correctly, you need to take the first 11multiples of 5 (5, 10, 15, 20, 25, 30, 35, 40, 45, 50, 55) and adddifferent combinations to make 90. The first example you gave was55+5+30. I would approach this problem by breaking it down. forexample, I would take the first number, 5, and subtract it from 90,leaving us with 85. Then I would ask myself, of the remaining numberswhich two add up to 85? Well, 40 and 45 do, so those are possiblesolutions. 50 and 35 do also. 55 and 30 (which was given) and 25 and60 (which is not a possible answer given our set). You can then do thesame operation with the next number in the set, 10. 90-10=80. What 2numbers add us up to get 80? Well, there is 50 +30, etc. Hope thathelps. --- === Subject: multiplying negative numbersAll answers to this question have been with using words so it wouldmake sense, but what about the mathematical way of multiplyingnegative numbers. my question is WHY? Don't want to see answer withwords.... Tell me why it is they way and who decided it was this way.tks-- === Subject: Re: multiplying negative numbers> All answers to this question have been with using words so it would> make sense, but what about the mathematical way of multiplying> negative numbers. my question is WHY? Don't want to see answer with> words.... Tell me why it is they way and who decided it was this way.a= -1b= 1c= -1(-1)(1) = -1(-1)(-1) = -1a(b+c) = ab+ac-1(1 + -1) = (-1)(1)+(-1)(-1)-1(0) = -1 + -10 = -2-- -- === Subject: multiplying negative numbersMario sent the following message:------------------------------------------ === Subject: multiplying negative numbersAuthor: Mario All answers to this question have been with using words so it wouldmake sense, but what about the mathematical way of multiplyingnegative numbers. my question is WHY? Don't want to see answer withwords.... Tell me why it is they way and who decided it was this way.-------------------------------------------First, many previous messages on this subject have been sent to emaillists having to do with math education, so you might want to lookthese up in the archives. (I and many others have responded before.)Second, Tanner gave an answer which seems to respond to Mario'sfirst sentence, but which I think misses the point of his questions --because it avoids the question of WHY? and WHO decided it should workin the usual way. shows that if we first accept that we want our number system tosatisfy all the usual general rules of arithmetic that make it eitheran integral domain or a field, then it necessarily follows that theusual rules for multiplying the additive inverses of elements hold.But in fact most of these rules about additive inverses undermultiplication, or at least variants of them using subtraction, werefirst discovered by shop keepers and merchants and tax collectorsbefore the invention of negative numbers and many years before anyonethought to isolate and write down what we now call the axioms for integral domains or fields. For example, suppose that A,B,C,D are positive numbers and AD one can think of situations where it would benatural to discover that B-(D-C) in fact gave the same result as(B-D)+C. So this became one of the known facts: B-(D-C)=(B-D)+C.Some situations lead naturally to discovering that in the product B*D,if B is reduced by A, then the product is reduced by A*D. That is, wehave the rule (B-A)*D = B*D - A*D. And of course everyone knew thatY*W=W*Y. So then if in the product B*D we reduce B by A and D by C,then (B-A)*(D-C)=[B*(D-C)]-[A*(D-C)]=[B*D-B*C]-[A*D-A*C]=([B*D-B*C] -A*D)+A*C. When people invented negative numbers anddefined addition for them, they noticed that subtraction could also beextended naturally and expressions such as B-A could now be thought ofas B+(-A) and there was no longer any need to check first that A < B,and A and B also need not be positive. Then the known rules abovethat connect subtraction and multiplying certainly stongly suggested(well ok, forced) how to extend multiplication to negative numbers sothat all the old rules still worked the same for positive numbers asabove. In Europe this was pretty much sorted out (by a few people) bythe 1400-1500's, I think. -- === Subject: Re: multiplying negative numbers> Mario sent the following message:> ------------------------------------------ === > Subject: multiplying negative numbers> Author: Mario All answers to this question have been with using words so it would> make sense, but what about the mathematical way of multiplying> negative numbers. my question is WHY? Don't want to see answer with> words.... Tell me why it is they way and who decided it was this way.> -------------------------------------------> First, many previous messages on this subject have been sent to email> lists having to do with math education, so you might want to look> these up in the archives. (I and many others have responded before.)Second, Tanner gave an answer which seems to respond to Mario's> first sentence, but which I think misses the point of his questions --> because it avoids the question of WHY? and WHO decided it should work> in the usual way. > shows that if we first accept that we want our number system to> satisfy all the usual general rules of arithmetic that make it either> an integral domain or a field, then it necessarily follows that the> usual rules for multiplying the additive inverses of elements hold.But in fact most of these rules about additive inverses under> multiplication, or at least variants of them using subtraction, were> first discovered by shop keepers and merchants and tax collectors> before the invention of negative numbers and many years before anyone> thought to isolate and write down what we now call the axioms for > integral domains or fields. answer to why these rules are the way they are.But why is quite a subjective term, and so for the question of whythey are the way they are, why they are universally true, I'dpersonally disagree that the theory doesn't give an answer as to why.(Because why is so subjective a term, I'd say that any proof givesan answer to why.)We have a why: These familiar number systems and their rules have beendiscovered to be a subset of a larger universe out there, just waitingto be discovered more and more fully (just like the physicaluniverse). It is thus forced upon us - we have to accept that atleast these familiar number systems satisfy all the general rules ofarithmetic that make them either rings, integral domains, or fields.We don't have a choice. (Although we can use Peano arithmetic toderive these rules along with the number systems themselves, beforeany talk of rings or integral domains or fields, these rules applyalso to a larger universe than just these number systems, which is whyI like the former approach.)(Please note that I say an answer.) As for who, these merchants are not the ones who decided that theserules for our familiar number systems be the way they are. No onedecided. The rules have been forced upon us.These merchants discovered some of these seeming rules, but, likethe ancient astronomers who discovered some things, really had no clueas to what was really out there in the universe waiting to bediscovered. By the nature of these rules, they admit no exception, nocounterexample, in the given domain. So I'd rather say that thesemerchants discovered what seemed to be rules. For me, you really can'tsay that you've discovered a universally applicable rule unless youcan prove that there isn't a counterexample lurking out theresomewhere. This is why I believe in the need for, at a minimum, aproof to answer why.-- === Subject: Re: multiplying negative numbers,While you have completely answered this question, it might help othersto focus on the single fact that:(-2).(2) =(2).(-2)That is, we can comfortably interpret a negative number like (-2) asbeing in the opposite direction along the number line as the number(2). Then (positive) multiples of that negative number are naturallyinterpreted as being further out in that same direction (if we arethinking of integer multiples).The difficulty arises when we try to interpret multiplying by anegative number; our desire to stick with the commutative law commitsus to intrepreting (-2) times 2 as equal to (2) times (-2); so letus interpret multiplication by a negative number as reversing thedirection of the number we're multiplying it against, so that-2times becomes reverse direction and multiply by 2.Then -2 times (-2) becomes reverse the direction of (-2), i.e.make it +2, and multiply by 2, giving us (-2).(-2)=4.This becomes awkward to say, because in order to correct any confusionwe need to distinguish between the order in which we are multiplyingtwo different numbers, when we have previously defined this order tobe irrelevant. Thus we lack an agreed - upon vocabulary to discussthis with.This argument is quite easily turned into picture by the way.-- === Subject: Re: multiplying negative numbersThe more intuitive ways of seeing things the better. Why? Because thisand other intuitive ways of understanding are justifiable by thetheory. They have to be.A good exercise for teachers and students is to try to see how thegiven intuitive method can be mathematically justified. It's good toask, What are we actually doing in terms of the mathematical theory?For example, for this method, where we have (-2)(-2), when we say wereverse the direction of -2 and then multiply by 2, we are actuallytransferring the negative sign from one element to the other. So howdo we know this is mathematically OK? Because, generally:(-a)b = a(-b). So if a = m and b = -n, we just replace, as in (-m)(-n)= m(-(-n)).In the example it would be: (-a)b = a(-b). So if a = 2 and b = -2, we just replace, as in (-2)(-2)= 2(-(-2)).In the example, I don't know whether you think that the order shouldbe reversed from the above. (Either way, it works, but it's abouttrying to make the words in the intuitive way match the theory asclose as possible.)In my opinion, we should never leave the mathematical theory if wereally want to understand. It should at least always be lurking in thebackground, ready to surface when called upon. And in my opinion,Liping Ma's book shows that this is the direction in which thingsshould go (the actual theory at least being in the background, readyto surface).> ,While you have completely answered this question, it might help others> to focus on the single fact that:(-2).(2) =(2).(-2)That is, we can comfortably interpret a negative number like (-2) as> being in the opposite direction along the number line as the number> (2). Then (positive) multiples of that negative number are naturally> interpreted as being further out in that same direction (if we are> thinking of integer multiples).> The difficulty arises when we try to interpret multiplying by a> negative number; our desire to stick with the commutative law commits> us to intrepreting (-2) times 2 as equal to (2) times (-2); so let> us interpret multiplication by a negative number as reversing the> direction of the number we're multiplying it against, so that> -2times becomes reverse direction and multiply by 2.> Then -2 times (-2) becomes reverse the direction of (-2), i.e.> make it +2, and multiply by 2, giving us (-2).(-2)=4.This becomes awkward to say, because in order to correct any confusion> we need to distinguish between the order in which we are multiplying> two different numbers, when we have previously defined this order to> be irrelevant. Thus we lack an agreed - upon vocabulary to discuss> this with.This argument is quite easily turned into picture by the way.-- === Subject: multiplying negative numbersAll answers to this question have been with using words so it wouldmake sense, but what about the mathematical way of multiplyingnegative numbers. my question is WHY? Don't want to see answer withwords.... Tell me why it is they way and who decided it was this way.I am waiting for an answer to this question which does not use words,as requested by the poster. I'm not holding my breath, though.mattsmom-- Subject: Help please w/ completing the squaresFor some strange reason I can't figure out these particular questions.I was able to solve other questions similar to it. But for these twoquestions I'm not coming up with the right answer. Please HELP!Solve by completing the square:2x^2-3x+1=0 ANDx^2-x-1=0 . === Subject: Re: Help please w/ completing the squares (kinda long)> Solve by completing the square:> 2x^2-3x+1=0> AND> x^2-x-1=0Here's how it works: 2x^2-3x+1=0Divide by 2 to get the first coefficient to 1. x^2-(3/2)x+(1/2)=0Subtract the last number (1/2) on both sides x^2-(3/2)x=-(1/2)Take the second coef. (-3/2), halve it (-3/4) and square it (9/16),then add it on x^2-(3/2)x+(9/16)=(1/16)Now what you can do is factor the left side to (x-(that number we gotwhen we halved it just before) )^2, and since -(-3/4)=+3/4, then (x+(3/4) )^2=(1/16)Now square root (remember positive and negative answers) x+(3/4)=1/4 OR x+(3/4)=-(1/4)Subtract 3/4 and you have two answers x=-.5 OR x=1Now do the same thing for the second equation x^2-x-1=0x^2-x-1=0 (since the first coef. is 1, skip that step)x^2-x-1=0 (subtract -1 -- or add 1)x^2-x=1 (halve, square, and add -- (-1)/2 = -1/2 and) ( (-1/2)^2 = 1/4 )x^2-x+(1/4)=(5/4) (factor)(x-(1/2) )^2=(5/4) (square root)x-(1/2)=sqrt(5)/2 ORx-(1/2)=-sqrt(5)/2 (add that -1/2)x=sqrt(5)/2+1/2= 1.618... OR x=-sqrt(5)/2+1/2= -.618...Summary:Solution to 2x^2-3x+1=0 : x = -.5 or 1 x^2-x-1=0 : x = -.618... or 1.618...Those are you two (rather, four) answers. Do you see what went wrong?Good luck-- === Subject: Re: Math Methods -- Old Vs. New> such that has that (the ln) function. Do I know how to use a slide rul=e?> No. Did I ever know how to use a slide rule? No. Have I ever even=20> seen> as slide rule? If so, I don't recall.That's a real shame, as the slide rule is the lawlog(xy) = log(x) + log(y)made material.--Robin Chapman, www.maths.ex.ac.uk/~rjc/rjc.hNeedless to say, I had the last laugh. Partridge, _Bouncing Back_ (14 times)-- === Subject: Re: Math Methods -- Old Vs. New> such that has that (the ln) function. Do I know how to use a slide rul=> e?> No. Did I ever know how to use a slide rule? No. Have I ever even=20> seen> as slide rule? If so, I don't recall.> That's a real shame, as the slide rule is the law> log(xy) = log(x) + log(y)> made material.A purely mathematical statement is not material. A slide rule ismaterial. It's just a machine. It is not a shame that I have never usedone no more than it is a shame for my never having to kill meat with a spearand flint.-- -- === Subject: Multidigit subtraction: An alternativeConsider 657,202-219,204=437,998We all know the regular way. I'll write what we can put on paper forclarity and comparison to the alternative way: 4 6 1 9 6 5 7, 2 0 2- 2 1 9, 2 0 4= 4 3 7, 9 9 8I'm sure everyone knows where the digits 4,6,1,9 came from. My French education again, as in multi-digit division, suggests thatthe regular way is not universal. Here is how we were taught to dosubtraction. The 1s would be written in small before the digit toindicate place value: 1 1 1 6 5 7. 2 0 2- 2 1 9. 2 0 4= 4 3 7. 9 9 8mattsmom-- === Subject: Re: Multidigit subtraction: An alternative> Some of you may already know about this and why it works, but I'd like> to share it with those who don't. And if you teach multidigit> subtraction, some students might find it a blessing. > Apologies if any of that was confusing. I have since learned that somecall this method the payback method. I wonder why it's not taughtmuch. We always hear about the borrowing method, where we modify theminuend by reducing the place values by one when necessary. But it hasthe conceptual complications that I mention that are not present inthis payback method, where we modify the subtrahend by increasing theplace values by one when necessary.-- === Subject: Numerical value for a GoogleMy son is in grade 4 and has come home with homework to find anumerical value for a Google. We can't find anything. Anyone know?-- === Subject: Re: Numerical value for a Google> My son is in grade 4 and has come home with homework to find a> numerical value for a Google. We can't find anything. Anyone know?I suspect a slight communication failure somewhere along he line, as you will have much better luck searching with the correct spelling of the word: googol. Googling googol (!) leads to an answer fairly quickly.And, yeah, I know, I could've just _told_ you what a googol is, but why cheat you out of the satisfaction of discovery? :)Matt T-- === Subject: Re: Numerical value for a Google> My son is in grade 4 and has come home with homework to find a> numerical value for a Google. We can't find anything. Anyone know?> googol: 10^100 (i.e. 1 followed by 100 zeros), a term coined by the US mathematician Edward Kasner.googolplex: 10^googol-- === Subject: DerivativesI need to use the product rule for finding derivatives. My question is(3x-2)(2x + 7)^(-7). I'm not sure how to distribute the power -7 into2x + 7. After I get that I know I am suppose to expand and then usethe power rule. Please help. -- === Subject: Re: DerivativesI need to use the product rule for finding derivatives. My question is> (3x-2)(2x + 7)^(-7). I'm not sure how to distribute the power -7 into> 2x + 7. After I get that I know I am suppose to expand and then use> the power rule. Please help.(3x - 2)(2x + 7)^(-7) = (3x - 2)/(2x + 7)^7Then from Pascal's triangle(2x + 7)^7 = (2x)^7 + 7(2x)^6*7 + 21(2x)^5*7^2 + 35(2x)^4*7^3 +35(2x)^3*7^4 + 21(2x)^2*7^5 + 7(2x)7^6 + 7^7But now you haven't got a product for applying the product rule to;you've got a ghastly quotient.Surely you should proceed as follows:(d/dx){(3x - 2)(2x + 7)^(-7)} = (3x - 2)(d/dx){(2x + 7)^(-7)} + (2x + 7)^(-7)(d/dx)(3x - 2)(That applies the product rule.)(d/dx){(3x - 2)(2x + 7)^(-7)} = (3x - 2){(-7)(2x + 7)^(-8)*(2)} + (2x + 7)^(-7)*3(That does the differentiation, where I've used (d/dx)(f^n) = n*f^(n-1)*(df/dx) )(d/dx){(3x - 2)(2x + 7)^(-7)} = (2x + 7)^(-7) * (3 - 14(3x - 2)/(2x + 7))(That tidies up by factoring.)If any of this is interesting to you, you should check it; especiallythe Pascal's triangle stuff.-- G.C.-- === Subject: why do we teach long division of a polynomials?I teach two 9th grade classes and 10th grade math lab basically I'mthere to help them with remedial concepts and to re-enforce what theylearn in their main math class.The main math teacher taught them long division of a polynomial lastweek and I'm preparing my review lessons now. Here is a problem fromtheir test last week:Divide x^2-13x+40 by (x-5)Now I would just factor it and then check my workbut she wants themto long division.But I'm having trouble understanding why they need to know how to dothis. I honestly never understood why *I* learned it either. I knowit's a way to factor a polynomialbut, when would you know one factorof a polynomial and need to find the other? Most of the time, youdon't know what any of the factors are. It seems to make factoringoverly complicated.It's a confusing topic to teach and I hope someone her can give mesome ideas to make it relevant I always tell my kids why I'mteaching them things. So, I need to be convinced that it's not just auseless math t with limited applications.-- === Subject: Re: why do we teach long division of a polynomials?> .... Here is a problem from their test last week:Divide x^2-13x+40 by (x-5)Now I would just factor it and then check my workbut she wants them> to long division.But I'm having trouble understanding why they need to know how to do> this. I honestly never understood why *I* learned it either. I know> it's a way to factor a polynomialbut, when would you know one factor> of a polynomial and need to find the other?.... You've had some good replies, but there's also a misunderstanding inwhat you've written. Any polynomial can be divided by any other of lowerdegree, giving a quotient and remainder. If the second polynomial happensto be a factor of the first then the remainder will be 0, but that's aspecial case. There's a strong analogy with the behaviour of natural numbers. Forsoalgorithm for finding the greatest common divisor of two integers, andlong division of polynomials is used in a similar process. This fact maynot help much at the level you're teaching, but the analogy with divisionof integers may help in understanding what's going on. -- === Subject: Re: why do we teach long division of a polynomials?susan@futurebird.com (Susan Murray) wonders:>But I'm having trouble understanding why they need to know how to do>this. I honestly never understood why *I* learned it either. I know>it's a way to factor a polynomial but, when would you know one factor>of a polynomial and need to find the other? Most of the time, you>don't know what any of the factors are. It seems to make factoring>overly complicated.>It's a confusing topic to teach and I hope someone her can give me>some ideas to make it relevant I always tell my kids why I'm>teaching them things. So, I need to be convinced that it's not just a>useless math t with limited applications.Here is a modern application. All data transferred across theInternet travels in chunks called packets. Broadly speaking, allthe bits in a packet are treated as the coefficients of apolynomial (whose coefficients thus have value 0 or 1 only) andmatters are arranged so that the polynomial is a multiple of a afixed polynomial called a CRC polynomial. The receiver of apacket checks if the received polynomial is divisible (perfectlywithout any remainder) by the CRC polynomial by ***using long division***If the remainder is zero, the receiver accepts the packet ashaving been received correctly (without any transmissionerrors). If the remainder is not zero, the receiver knows thatthe some of the bits in the packet were changed duringtransmission, and asks for a re-transmission of the packet.You could motivate this by your example of x^2 -13x + 40 beingdivisible perfectly by x-5 but x^2 -12x + 40 leaving a nonzeroremainder etc.You would be surprised at how many undergraduate and graduatestudents in college have no clue as to how long division ofpolynomials works when CRC techniques are being explained inclasses on computer networks...---- === Subject: Re: why do we teach long division of a polynomials?Isn't one use partial fraction expansions?Casey-- === Subject: Re: why do we teach long division of a polynomials?> I teach two 9th grade classes and 10th grade math lab basically I'm> there to help them with remedial concepts and to re-enforce what they> learn in their main math class.> The main math teacher taught them long division of a polynomial last> week and I'm preparing my review lessons now. Here is a problem from> their test last week:> Divide x^2-13x+40 by (x-5)> Now I would just factor it and then check my work-but she wants them> to long division.> But I'm having trouble understanding why they need to know how to do> this. I honestly never understood why *I* learned it either. I know> it's a way to factor a polynomial-but,The teacher is not stressing this just for the heck of it. There are times(at least in one's math education) where it is useful to perform polynomiallong division. An example right off the top of my head is finding slant(oblique) asymptotes. For rational functions in *simplified form*, ie afteryou have already factored and cancelled common factors if applicable, theseoccur when the numerator is exactly one degree higher than the denominator.Do the polynomial division to get a quotient Q which will be linear. Theequation of the slant asymptote will be y=Q (the remainder is disregardedfor this purpose).This is but a single application of polynomial division. There are severalothers. Another example: In calculus, some integration techniques applywhen an integrand is not improper, ie when the numerator is not of higherdegree than the denominator. In such cases it is often useful to performthe division then the expression can be integrated.Forgive my curiosity, but do math teachers not have to take calculusanymore???-- -- === Subject: Re: why do we teach long division of a polynomials?>But I'm having trouble understanding why they need to know how to do>this. I honestly never understood why *I* learned it either. I know>it's a way to factor a polynomial.89but, when would you know one factor>of a polynomial and need to find the other? Most of the time, you>don't know what any of the factors are. It seems to make factoring>overly complicated.>It's a confusing topic to teach and I hope someone her can give me>some ideas to make it relevant .89 I always tell my kids why I'm>teaching them things. So, I need to be convinced that it's not just a>useless math t with limited applications.>-- You may want to find some or all of the zeros of a polynomial function. SChoolexercises will often deal with quadratic functions because they are less workto solve, permitting the students to concentrate on the skills being taught . At times, higher order polynomial functions may be factored or simplified tolinear and quadratic factors and using either long division or syntheticdivision may be needed to render the polynomial into factored form. For simpler examples, volumes of boxes according to certain descriptions can bedescribed with polynomial equations or functions. Check a college algebratextbook for further instruction and examples. In any of the good ones, youshould find a section which includes a rational zeros test; although the methodof examining potential zeros is with synthetic division, long division is stillvalid.G C-- === Subject: Re: why do we teach long division of a polynomials?P.S.Here are a few links that talk about CRCs:http://www.relisoft.com/Science/CrcMath.hhttp:// www.hyperdictionary.com/dictionary/cyclic+reduncy+checkhttp:// www.wvu.edu/~lawfac/mmcdiarmid/digital%20signatures.htmA Google search will yield many, many more; and you might find onethat you prefer to those above.To reply directly to me, replace all 'z' with 'a' in email address.-- === Subject: Re: why do we teach long division of a polynomials?This is quite advanced, but the theory behind the CRCs (cyclicalrenduncy checks) that are used to check for data integrity invirtually are computer systems is based on polynomial division. To reply directly to me, replace all 'z' with 'a' in email address.>I teach two 9th grade classes and 10th grade math lab basically I'm>there to help them with remedial concepts and to re-enforce what they>learn in their main math class.>The main math teacher taught them long division of a polynomial last>week and I'm preparing my review lessons now. Here is a problem from>their test last week:>Divide x^2-13x+40 by (x-5)>Now I would just factor it and then check my workbut she wants them>to long division.>But I'm having trouble understanding why they need to know how to do>this. I honestly never understood why *I* learned it either. I know>it's a way to factor a polynomialbut, when would you know one factor>of a polynomial and need to find the other? Most of the time, you>don't know what any of the factors are. It seems to make factoring>overly complicated.>It's a confusing topic to teach and I hope someone her can give me>some ideas to make it relevant I always tell my kids why I'm>teaching them things. So, I need to be convinced that it's not just a>useless math t with limited applications.-- === Subject: Re: why do we teach long division of a polynomials?> I teach two 9th grade classes and 10th grade math lab basically I'm> there to help them with remedial concepts and to re-enforce what they> learn in their main math class.> The main math teacher taught them long division of a polynomial last> week and I'm preparing my review lessons now. Here is a problem from> their test last week:> Divide x^2-13x+40 by (x-5)> Now I would just factor it and then check my work-but she wants them> to long division.> But I'm having trouble understanding why they need to know how to do> this. I honestly never understood why *I* learned it either. I know> it's a way to factor a polynomial-but, when would you know one factor> of a polynomial and need to find the other? Most of the time, you> don't know what any of the factors are. It seems to make factoring> overly complicated.> It's a confusing topic to teach and I hope someone her can give me> some ideas to make it relevant - I always tell my kids why I'm> teaching them things. So, I need to be convinced that it's not just a> useless math t with limited applications.Actually, there's only two times when I've actually used the long division,but always until I got higher in math. You see long division when you findslant asymptotes for rational functions. Way down the road when thesestudents hit calculus, they'll use it to integrate.-- === Subject: Re: why do we teach long division of a polynomials?> Actually, there's only two times when I've actually used the long division,> but always until I got higher in math. You see long division when you find> slant asymptotes for rational functions. Way down the road when these> students hit calculus, they'll use it to integrate.When I learned to integrate, I learned the long division. I had it beforethen, but only as an aside in an advanced high school math course. It waseasy to learn since I understood (by then) how to manipulate variables-- itjust seems annoying to teach it to 9th graders who still confuse x+x withx*x-- Never discourage anyone...who continually makes progress, no matter howslow. - Plato-- === Subject: Re: so many Fs ... : (> -3-8=11 is a goodie. How does it happen? Well, two negatives make a> positive, don't they?!?!? Teaching math by rule of thumb.Yup. And they LOVE rules. They can multiply sign numbers so quickly--but adding and subtracting confuses them. This tells me that theydon't really know what the numbers mean. It scares me.I keep trying to remember how I learned these concepts-- but, some howI just can't remember a time when it was not obvious to me.-S-- === Subject: Re: so many Fs ... : (I had a similar experience this year. I teach 9th grade Algebra. School starts in mid-August for us so the students and I have had a while to recover from the first couple of tests :-).I spent two weeks on fraction and percent REVIEW. At least I thought it was review. I gave the first test and was surprised how many students flunked it. I felt like I had done something wrong. The second test was on percents. They did a little better but still too many of them flunked for something that was supposed to be a review. I have instituted the following and it seems to have helped (I'm most concerned that they get the material...ultimately): 1. I allow one retest a quarter. Pick your lowest test grade and you can take a re-test. The problems are not exactly the same on the re-test (I change the numbers). 2. I have started testing my students more frequently. They seem to do better on very focused quiz/tests. 3. I do five minute checks (3 or 4 problems) on the material that was presented the day (or two...we are on block scheduling too) before. They put the problems and answers in their notebooks. We go over them in class and I can see where they are still having problems with new concepts. 4. The day before the test I invite any of my students to come in at lunch and do a review with me. This has helped immensely.I have seen quite a bit of improvement on the last three tests. I don't feel like I have compromised my teaching or grading and the students are getting the material. I would also like to echo what others have said about using the human number line as a great way to teach positive and negative numbers. It has worked very well for me in the past.Good luck and hang in there. If you are bothering to ask these questions of yourself then you are a good teacher.> 10th grade math. I need some adviceI'm feeling a bit overwhelmed. Nearly all of my students failed their> first math test. There are a variety of reasons for this I think, some> of my students don't do homework or studysome have entered the 9th> grade at a 5th grade or 4th grade levela few have learning> disabilities. Many of them do not read or write English terribly well.> But, most are brightand could do well, I think. If only if I knew how> to show them how.-- Bernadette Calhounremove 'lovely' to reply-- === Subject: Re: so many Fs ... : (> 10th grade math. I need some advice?I'm feeling a bit overwhelmed. Nearly all of my students failed their> first math test. There are a variety of reasons for this I think, some> of my students don't do homework or study?some have entered the 9th> grade at a 5th grade or 4th grade level?a few have learning> disabilities. Many of them do not read or write English terribly well.> But, most are bright?and could do well, I think. If only if I knew how> to show them how.Out of 40 or so students I have in 9th grade, two scored Bs , three> Cs, 8 Ds and the rest answered less than half of the questions> correct. I would venture that the exam was simply too hard, but I used> questions from the regent's identical to many we did in class and as> homework.I'm depressed about giving them the tests back?I don't want them to> give up hope. Especially the ones who do their homework and who I> know try hard. But, they still have not grasped the concepts of our> first unit. I can't lie to them?Besides. I didn't think the test was> too hard?I timed myself taking it and it took me about 4 minutes?so, I> gave them 40?many were still working when the bell rang. How can it> take so long to answer a question like: -3-8 ? We did dozens of> examples in class. They did OK on the homework. Maybe they were just> stressed by the test? I don't know.I teach 90 minute periods so I spent the first half of class working> with them on example problems?maybe I'm just a bad teacher.I don't know what to do.I hear you, and know what you're up against. By the time they get into high school and can't do -3-8, then you knowthey've been passed along, perhaps by grading methods that essentiallyprovide good grades in spite of them demonstrably knowing nothing ontests.I failed all of my tests, and yet I got a C in Spanish! This is whatone of my last year's geometry students exclaimed in response to myexplanation as to why a high percentage of his final grade comes fromtests (and a low percentage come from everything else put together -classwork, notebook checks, whatever). He further went on to say thatbeing able to fail tests but pass courses is the way it ought to be.I think that maybe you have a lot of students like that. They may tryto wait you out to see whether you crumble and pass them along withbare passing grades in spite of having a low or even moderate failingaverage on tests. Try not to give in to this. But even if you stick toyour standards, they may wait for summer school, where part-timers canbe notorious for handing out A's like candy for nothing.There are things you can do to alleviate parent concern if you stickto your standards. I'll share about this in a minute. But first:If the percentage of the final grade that comes from tests is nothigh, then let's see what happens.Suppose that the final grade is 50% tests. The weighting formula thatI use is then 0.5x + 0.5y = G, where G is the final numerical grade, xis the final average for tests, and y is the final average foreverything else. Note how low the final test average can be to obtaina 70% final numerical grade, assuming a 10-point letter grade scalewhere 70 is the lowest C:0.5x + 0.5(100) = 70 yields x = 40 as the lowest average a student canhave from the tests but still get a C in the course.Consider: 1) 80% is close to the theoretical limit as to how low I can gowithout letting the students get away with bumping their letter gradeup by more than one letter grade, like from an F to a C. This limitwould be 75%. With 75%, a final test average of anything below a 60would still yield a final grade below a 70.2) 80% is the theoretical limit as to where the final test average isno lower than 50% but where the student can still obtain 60% for afinal grade, the lowest D on the 10-point scale. Thus 0.8x + 0.2(100)= 60 yields x = 50. I won't accept a situation where test average isbelow 50% and where the student can still pass with a D.To all: I'd like to see your take on all this. Is the grading schemesof some educators the main reason why we see students obtaining goodgrades in their courses but obtaining low scores on their tests aswell as those tests out there, the state tests? To the point ofgraduating functionally illiterate? To the point of not being able todo -3-8?Here's how you can stick to your guns and still make tests ameaningfully high percentage of the grade: Make around 50% of theirtests what I call homework/classwork tests. This is where you choose asubset of the homework and in-class examples as test items for thewhole test. Don't even change the numbers. If they've mastered theirhomework and classwork, then they can reproduce the solutions. If not,not. I'm not talking about final answers only. I'm talking about allthe steps. Parents will be quite understanding when you tell them thattheir children are being given a great opportunity to score well butnot applying themselves to do so. And don't be shocked when those whoare failing now will still fail when given even this opportunity toscore well. Of course the point of such tests is that real mastery ofhomework and such is required for real mastery of the subject: If theycan't produce homework solutions on tests, then something is wrong.But like I said, they'll try to wait you out and see if you crumbleand pass them along like all their former teachers. (I'm talking aboutthose who are in or any close to the category of not being able to dosuch as -3-8.) Even if you stick to your standards, they may wait forsummer school, where part-timers can be notorious for handing out A'slike candy for nothing.-- === Subject: Re: so many Fs ... : (> But like I said, they'll try to wait you out and see if you crumble> and pass them along like all their former teachers. (I'm talking about> those who are in or any close to the category of not being able to do> such as -3-8.) Even if you stick to your standards, they may wait for> summer school, where part-timers can be notorious for handing out A's> like candy for nothing.I think signed numbers are just about the hardest thing to teach, I'veused number lines, analogies to money, manipulative and even flashcards (I thought if they simply memorized enough of the simpleproblems the patterns would become apparent*) but some of my studentskeep making the same mistakes.My 10th graders who are doing work with combining like terms andfactoring get answers wrong mostly because they have trouble adding*4x and 1x and getting *3x*I think back on my own education in maths and I don't remember signednumbers being such a big stumbling blockThey keep asking me for rules to solve the signed number addition andsubtraction problems I worked out what the rules would be and theyare quite complex and intricateit just seems so much simpler to learnthe properties of the numbers than memorize:When adding a negative and a positive number choose the sign value ofthe number with the larger absolute value will be the sign of youranswer. If the positive number is larger subtract the negative numberfrom it and assign a positive sign. If the negative number is largersubtract the positive number from the absolute value of the negativenumber and assign a negative sign. If both numbers are negative addthe absolute values of the numbers and * blah blah blah*It's just too much, and memorizing such rules is NOT understandingmathematics, IMNSHO.By the way my school mandates that tests and quizzes may only be 30percent of the grade 30 percent must be classroom participation 20 isnotebooks, 10 percent projects and 10 percent homework. This is theofficial grading policy of all under performing schools in NewYork City.Even with such an easy grading policy most of my students are stillfailing the course since they do not hand in their notebooks, fail toparticipate in class or have excessive (and unexcused) absences. Atfirst I thought the policy was silly, but now I'm OK with itthe onlystudents who are doing well in my class are learning the materialsince they take notes, do homework and come to class without fail.-- === Subject: so many fsone thing you might keep in mind is that integers are a hard topic forstudents to comprehend. for some reason their brains just dontprocess it, like a foreign language. i have two things that i haveused that have worked for me.one thing is to use your few b grade students that are getting it toexplain it to those who are not getting it. often teachers say groupsare impossible in middle school, but i find they are useful in thissituation as the ones who get it will get frustrated with those thatdo not and they will explain it in a way they can understand. i dontknow yet why it works, i just know it seems to work. maybe they speaka special language, or maybe just not coming from a teacher it becomeseasier.the other thing i often do is to realize the most important thing isfor the kids to understand the topic, not just to be tested. let themtake home the quiz/test they failed and see what mistakes they made. allow them to fix their errors at home with the book or help ofparents or siblings and more often than not they will beat themselvesup for the stupid mistakes and never make them again. i give themhalf credit for all the ones they take the time to fix, which usuallybrings them up from an 'f' to a 'c' grade.-- === Subject: Re: so many Fs ... : (I'm not sure; did I respond to this one?Show the students adding and subtracting with WHOLE, positive, numbers on anumber line. Show students how this works. Then do the same using number linewhich includes negativa AND positive whole numbers. Establishing the code forsigned number addition or subtraction can be done after this first stuff isexamined and practiced.Was this response in line with the origianal question? I lost track of theoriginal.G C-- === Subject: Re: so many Fs ... : (> I'm not sure; did I respond to this one?> Show the students adding and subtracting with WHOLE, positive, numbers ona> number line. Show students how this works. Then do the same using numberline> which includes negativa AND positive whole numbers. Establishing the codefor> signed number addition or subtraction can be done after this first stuffis> examined and practiced.Coincidentally we had an in-service today which was on this point. Onemethod that was suggested for low achievers was to have them walk thenumber line.You start with a large laminated number line with the numbers perhaps 1 footapart that you put down on the floor. The walker starts at the position ofthe first number of the problem. The operation in the expression determineswhether the walker faces to the right (+) or to the left (-). Then the signof the 2nd operand determines whether the walker goes forward (+) (i.e., inthe direction they are facing) or backwards (-).For example in -3 - 8 <- X<------------------------------------------------------------ --------------------------> -11 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 67We start at -3, the first operand. We face to the left because theoperation is subtraction. And now we go forward (forward because the secondoperand is positive) and arrive at -11.Another example:-2 - (-7) <- X<------------------------------------------------------------ --------------------------> -11 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 67This time we start at -2 and again face left because the operation is againsubtraction. But now, since the 2nd operand is a negative number we walkbackwards seven units, which takes us to +5.You start with the students walking the number line and then you transfer itto doing the exact same thing on paper and pencil. Once the students arecomfortable with doing these with the number line the next step is to use itto teach them the rules of adding integers and showing them that subtractingis the same as adding the additive inverse.The advantage of this method is it starts with something very concrete andkinesthetic and introduces the abstract analog at a manageable pace.Another method that was brought up involved some simple manipulatives thatcan also be modeled with just paper and pencil. This involves littleplastic +'s and -'s. We used plastic tile spacers for the +'s and tilespacers with the two opposite ends cut off for the -'s. You divide yourpaper into two areas the work area and the bank area. In the bank area youput pairs of symbols consisting on one + and 1 -. You prep the activity byconvincing the kids that a + and - together make 0.Start with an easy first problem like 3 + 5.You start by putting 3 +'s in the work area. You now add 5 +'s to the workarea giving you 8.Now you kick it up a notch and tackle 3- 5.Again you start by putting 3 +'s in the work area. Now you need to subtract5 +'s but you don;t have 5 +'s. So you add 2 0's each of which consists ofa +- pair.+ + + ---> + + + (+ -) (+ -)Now in your work area you have 5 +'s and 2 -'s and you haven't change thevalue in your work area because all you did was add some 0's. But now youcan subtract the 5 +'s you need to and when you do you are left with 2-'s soyour answer is -2.Now one more: -3 - (-7)Start with 3 -'s in the work area. You can't subtract 7 -'s from that soagain you need to add some 0's. Let's say the student miscalculates anddecides to add 7 0's to the work area. You have:- - - (+ -) (+ -) (+ -) (+ -) (+ -) (+ -) (+ -) or- - - - - - - - - - + + + + + + +Now we can subtract 7 -'s and when we do we are left with:- - - + + + + + + +We can rearrange these as(+ -) (+ -) (+ -) + + + +Now the (+ -)'s can be dropped since they are each equal to 0, leaving uswith- - - - in the work area so our answer is -4.It is much more cumbersome on paper than it is using the little + and -symbols on an overhead or, for the brave, having the students do it with thelittle + and - symbols.Again, the starting point is something concrete that is much more accessibleto the students than just giving them rules of addition and subtraction.Hope you find some useful ideas here.Rich-- === Subject: Re: so many Fs ... : (> Coincidentally we had an in-service today which was on this point. One> method that was suggested for low achievers was to have them walk the> number line.You start with a large laminated number line with the numbers perhaps 1 foot> apart that you put down on the floor. The walker starts at the position of> the first number of the problem. The operation in the expression determines> whether the walker faces to the right (+) or to the left (-). Then the sign> of the 2nd operand determines whether the walker goes forward (+) (i.e., in> the direction they are facing) or backwards (-).I did this exercise with first graders last year (not the whole class,just the top 1/4), but I had them measure out the number line outsideusing 1' rulers, putting in the numbers every foot. I then had themstand on the number line and I rolled dice to (including a +/- die)tell one how far to move. He or she counted off the steps and reportedwhere they ended up. It was great fun for first graders, but I don'tknow if it would work with 7th graders! None of them had any troublewith the plus or minus numbers. There was some emotional difficulty for one student who somehowthought that negative nubmers were bad, and when his random rollsended up with him at minus 10 he thought one of the other students waslaughing at him. This was cleared up, but it did delay things for afew minutes. The other student was not laughing at him, by the way,he was just having a good time moving with the numbers.-- 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.-- === Subject: Re: so many Fs ... : ( know if it would work with 7th graders!Good point, since we all know that 7th graders aren't as mature as firstgraders (g).Rich-- === Subject: Re: so many Fs ... : (In the Algebra Project, which is introduced to 6th graders and isspread over two months, Moses illustrates the number line withsubway stops (in MA) and bus stops (Mississipi). Starting with astation as zero (CEntral Squarae in Cambridge, MA), he has students goforward or outbound (+) and backward or outbound (-). He also usesAfrican drums to demonstrate intervals and fractions. The reason ittakes 2 months is that math is integrated with the rest of thecurriculum. So after a trip, students not only work on math (byplotting their itinerary on a number line), but also write essaysdescribing what they saw at different stations.Reaction to the Algebra Project in Matt's school, where it originated,has been mixed. Some dislike the integration of humanities/socialstudies/math; others feel that the concepts are too simple and itshould not take two months tot each them Still others feel that theycould have been introduced earlier. And there are some who feel thatit does the job well. All students take Algebra 1 in 8th grade.mattsmom-- === Subject: OT Threads in k12.ed.mathThe threading in k12.ed.math doesn't seem to work. Often I see replieswithout the Re: and often they are not subordinate to the posts thatthey are a reply to. Is it my news reader?-- G.C. === ====== The moderator responds: === ==============Although this thread is off-topic for this group, I will post this singlemessage on it, for the general information of all the participants.As you have given no specific examples of messages, with subject lines anddates and/or Message-IDs, I can only guess at what you are seeing.However, I strongly suspect that you are referring to messages that are posted by persons using the MathForum.org web site for access to this newsgroup.The MathForum web site is an excellent site for mathematics education,and they post archives, and sometimes also portals, to many newsgroupsand mailing lists that are about math education. The moderated newsgroupk12.ed.math can be accessed from the Math Forum's site, here:http://mathforum.org/epigone/k12.ed.mathAs you can see, the archives available at the Math Forum, make it look likesome kind of web discussion board or archive, and not like a newsgroup asyou and I see it. Furthermore, many newbies or those with very little Internet experience access the group from that site.It is possible to post also to the newsgroup from the Math Forum site, butthe participants often change the subject, fail to quote messages theyare responding to and this may be what is breaking the threaded display inyour newsreader. It varies from newsreader to newsreader and may also beaffected by your particular configuration settings in your newsreader.Because of the format presented on the MathForum web site, many of the participants accessing the newsgroup via that portal believe they arereally participating in a web discussion board. They do not realize thatother readers are not also accessing it via the same web interface. So,they do not realize how important it is to maintain the same subject andto also quote a portion of the message to which they are responding, asthey figure anyone who reads their message on the Math Form web site willhave immediate access to the message preceding their own.I have previously discussed possible improvements to the posting interface from the Math Forum site with the administrators of the Math Forum. It would help a great deal if they would default to the same subject with aRe: in front of it on replies, and also include quoted text by default.However, this is apparently not very high on their priority list, andtherefore I do not have any expectation of this changing any time in the near future.So, the choices are:(1) To reject any and all posts from the Math Forum that do not conformto the usual practices of Re: and some modicum ofquoted text. This option seems much to harsh to me, and so I do not follow this policy.(2) The moderator could edit the posts to bring them into conformance.However, I do not have the time to do this, not to mention that I preferto have a mostly hands-off policy as far as editing goes. I usually I reject them and send them back to the author asking the author of(3) I could choose a path somewhere in between, where most posts fromthe Math Forum are posted, and only the really far-out submissionsthat would not make sense in the newsgroup due to their being competely out of context and/or replies to much much too old posts arerejected. It is this option that I have chosen.I hope this clarifies the phenomenon that you are seeing, and if anyonehas further comments please send them to the moderator's privateemail address at kem-moderator@k12groups.org as such meta-discussionabout the newsgroup is off-topic and is usually not posted here.Any follow-ups to this post that are sent to the newsgroup as a submittedemail to the moderator as described above.k12.ed.math moderatorkem-moderator@k12groups.orghttp://www.thinkspot.net/ k12math/ === Subject: ProbabilityIn a group of seven people, what is the probability, to the nearestpercent, that at least two have their birthdays in the same month?-- === Subject: Re: Probability> In a group of seven people, what is the probability, to the nearest> percent, that at least two have their birthdays in the same month?Tell us what you've tried so far so we don't feel like we are just doingyour homework for you. If you need a hint to get started, the probabilitythat at least two have the same month of birth is 1 minus the probabilitythat none have the same month of birth.Rich-- === Subject: sign function-- ===Subject: Re: sign functionf(x) = abs(x) / x-- === Subject: Re: sign function> f(x) = abs(x) / xwhich is undefined for x=0.The most common way to do the sign function is { 1 x>0sign(x) = { 0 x=0 { -1 x<0though some people prefer a definition in which sign(0) = 1.-- 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.-- === Subject: Re: sign function> f(x) = abs(x) / x> which is undefined for x=0.> The most common way to do the sign function is> { 1 x>0> sign(x) = { 0 x=0> { -1 x<0Yes. But I would guess that the OP is already familiar with thatdefinition, and that instead, he wants, as he said, an algebraic formulafor the signum function.In a response to the OP's thread of the same title in sci.math,I suggested sgn(x) = 2/(1 + 0^x) - 1but I have no idea whether that fulfills the OP's idea of an algebraicformula or not. Interested readers might look at the sci.math thread formore details.BTW, I submitted a response to the OP in this newsgroup yesterday. Forsome reason unknown to me, that response has never appeared here.l-- === Subject: Re: sign functionWhat is OP? and why are you calling me OP?> f(x) = abs(x) / xwhich is undefined for x=0.The most common way to do the sign function is { 1 x>0> sign(x) = { 0 x=0> { -1 x<0> Yes. But I would guess that the OP is already familiar with that> definition, and that instead, he wants, as he said, an algebraic formula> for the signum function.> In a response to the OP's thread of the same title in sci.math,> I suggested> sgn(x) = 2/(1 + 0^x) - 1> but I have no idea whether that fulfills the OP's idea of an algebraic> formula or not. Interested readers might look at the sci.math thread for> more details.> BTW, I submitted a response to the OP in this newsgroup yesterday. For> some reason unknown to me, that response has never appeared here.> l-- === Subject: Re: sign function> What is OP?Original Poster (i.e., the person who started the thread)> and why are you calling me OP?Why not? After all, you did start the thread.OP is a common and convenient abbreviation used in newsgroups.David-- === Subject: Re: sign function>What is OP? and why are you calling me OP?O=OriginalP=PosterOP = Original Posterjust an abbreviation-- === Subject: In need of curriculum writers for grades K-5HelloI am looking for both Math and Language Arts curriculum developers forelementary, grades K-5, to help us develop online curriculum. Thiswould be stly virtual work. If you know of anyone, please havethem write to me @ jill@learningtoday.comAny help would be greatly appreciated.Best ,JillJill Carr, Ed.DVice President of EducationLearning Today jill@learningtoday.com-- === Subject: First grade troubles (was: Calling all first grade teachers!! Need feedback or suggestions!)I am sorry to hear of your troubles. I understand how upset you canbe. I am a kindergarten teacher and we don't learn addition andsubtraction using equations until the end of the year. The only thingthat a can think of that the teacher's purpose was to set a highstandard for the children to acheive. Just continue to practice withthe flash cards and use positive reinforcement with your daughter, soas to encourage her self esteem. She can do it!!!-- === Subject: Difficulty with exponentsI need clarification onthese type of sums:1. x^1/2 Y^1/2 (2X^2 Y^2 - 4XY)10^1/2 X 10^1/3(a^2/3 b^3/4)^3/2a^1/2 (a^1/2 + b^1/2) + b^1/2 (a^1/2 - b^1/2)P19, 2.4 (c), 8, (vii)Also I have difficulty with sums such as:2^n+2 X 2^n-3 / 2^2n-5I have no trouble with any other algebraarea but this area may hold me up in future areas.Thank youJason-- === Subject: Re: Difficulty with exponentsYou are probably concerned with laws of exponents. a^m * a^n = a^(m+n)a^(-m) = 1/(a^m) Transcribe those above onto standard notation as would typically be done onpaper manually written and they may seem to more clearly make sense. G C>I need clarification on>these type of sums:>1. x^1/2 Y^1/2 (2X^2 Y^2 - 4XY)>10^1/2 X 10^1/3>(a^2/3 b^3/4)^3/2>a^1/2 (a^1/2 + b^1/2) + b^1/2 (a^1/2 - b^1/2)>P19, 2.4 (c), 8, (vii)>Also I have difficulty with sums such as:>2^n+2 X 2^n-3 / 2^2n-5>I have no trouble with any other algebra>area but this area may hold me up in future areas.>Thank you>Jason>-- -- === Subject: Math History PostersHello!I am a math teacher in Colorado Springs, CO. Three years ago I wanted mathhistory posters to share with my students - such topics as, whocame up with algebra and when? Or, what did Descartes contribute besides thecoordinate plane? However, I could not find any such itemsat the teacher stores. That is when I made my own. Other math teachers havesince shown an interest in my posters. I am still creating moreafter school each day. There is so much to cover!Some categories that I have are:Regions - Egyptian, Greek, Mayan, and more.Eras - Medieval, Renaissance, 20th Century, and more.Topics - Algebra, Women in Mathematics, Calculating Machines.Mathematicians - Descartes, Euler, Turing and more.If you would like to see what I have done so far, please visit my onlineMath History Catalog. The URL is athttp://www.mathisradical.com/Catalog.htmI'm not a professional sales person, just a math teacher who came up with aproduct that fits a need in a mathematics classroom. They areinexpensive and useful. I also have a free newsletter that includes lessonplans, biographies of mathematicians and Black Line Masters foryour classroom.Sincerely,Kavon Rueterkavon@mathisradical.comhttp://www.mathisradical.com-- === Subject: Re: Math History PostersJan Gullberg!The Oxford: World Encyclopedia has timelines near the front of thebook - which have a Science and Technology column.On the subject of timelines: The Oxford: Nature Encyclopedia hastimelines in the back for Biology, Chemistry, and Physics, amongothers.Casey-- === Subject: Re: Math History Posters>Topics - Algebra, Women in Mathematics, Calculating Machines.Remember the slide rule; that counts as some kind of calculating machine. >Or, what did Descartes contribute besides the>coordinate plane?There is the Descartes Rule of Signs for Polynomials helping to determinenumbers of real zeros for those functions. G C-- === Subject: Help me understand Euclid's algorithmI am having trouble understanding Euclid's algorith. Why is it that ifm and n are positive integers and m = qn + r then any number thatdivides both m and n must also divide m - qn = r, and any number thatdivides both n and r must also divide qn + r = m?0 <= r < n, so r could be any number. But r depends on both m and n.If m and n are divisible by x then qn is also divisible by x. Butsubtracting one number divisible by x by another number divisible by xdoesn't always make a third number divisible by x. Does it? ax - bx =c, x(a - b) = c. Oh, never mind.Is my logic sound?-- === Subject: Re: Help me understand Euclid's algorithm> I am having trouble understanding Euclid's algorith. Why is it that if> m and n are positive integers and m = qn + r then any number that> divides both m and n must also divide m - qn = r, Say that a divides both m and n. Then m=ab and n=ac. We now have:ab - qac = rRewriting:r = a(b-qc)Thus, since a divides m and n, a also divides r.and any number that> divides both n and r must also divide qn + r = m?Same thing here...Letting n=de and r=df, we get:qde + df = mm = d(qe+f)So d divides m> 0 <= r < n, so r could be any number. But r depends on both m and n.> If m and n are divisible by x then qn is also divisible by x. But> subtracting one number divisible by x by another number divisible by x> doesn't always make a third number divisible by x. Does it? ax - bx => c, x(a - b) = c. Oh, never mind.Is my logic sound?NOW I read this part yup, that's it....adding or subtracting multiples of x will always give you another multiple of x.-- === Subject: transforming formulaeI understand the processes and I can do most questions required of me;However, is there an 'order of operations' for transposing formulae? Iunderstand that you must 'keep the bce' do to one side that youwould do to the other, but in what order. Is there an order? does itmatter?T = 2pi sqrt L/g^3I get stuck where I am required to make x the subject and there is anx on both sides.x-y / z = x + 3yThank youJason(PS thanks & Chergarj - for difficulty with exponents)-- === Subject: Re: transforming formulae> I understand the processes and I can do most questions required of me;> However, is there an 'order of operations' for transposing formulae? I> understand that you must 'keep the bce' do to one side that you> would do to the other, but in what order. Is there an order? does it> matter?T = 2pi sqrt L/g^3Since all you'll be doing is multiplying/dividing, the order doesn't matter. If it was, say, T=2L+5, you would want to do the add/subtract before the mult/div.> I get stuck where I am required to make x the subject and there is an> x on both sides.> x-y / z = x + 3yNow, is this (x-y)/z or x-(y/z)? ;-) In any case, the thing to do is get all the x on one side, then separate it out. Assuming you meant the first case, I'll start by multiplying both sides by z:x-y = zx + 3zyx - zx = y + 3zyx * (1-z) = y + 3zyx = (y+3zy)/(1-z)--