mm-117 Here I go again. Another rant on a familiar topic.I am old fashioned 7th grade chalk and blackboard, pencil and paper, mathteacher. The kind who likes to teach math the way it was taught back in the50s.I started teaching math, in an urban inner city school district, in 1983. Atthat time, at least in my school district, the old way was still the norm. Calculators were not used. Cooperative learning, with the exception oetting students work in pairs on occasion, was not emphasized. Manipulatives? We used nothing more than those we had when I was taught 7thgrade math. Namely rulers, protractors, compasses, graph paper, and the like. Nothing even remotely resembling these colored blocks we have today.When I rst started math in 1983, the emphasis was upon ARITHMETIC. I used tospend almost an entire week reviewing the mechanics and processes of longdivision, giving my students lots and lots of practice to go with it. This is an example of what the critics and reformers of today call Drill And Kill.Fast forward to today. I am now in my 21st year of teaching math, in the verysame school building I started in.However, I am no longer allowed to teach math the old way. I do so at my ownrisk.Here are some of the guidelines to which I am now required to adhere:***My students must have access to calculators each and every single day.***Cooperative learning strategies must be employed a minimum of three times aweek.***Hands on learning strategies must be employed a minimum of three times aweek.This emphasis has been going on for a number of years now. However, I used tobe able to let it all go in one ear, out the other, shut my classroom door, andthen do my own thing.No more. All the math teachers in my school district are under a microscope tomake sure that we are adhering to these guidelines.Last week, I was called on the carpet for not doing so.I was told: The test scores of our students are extremely low. It isextremely important that you use more hands on strategies. This is what ourstudents need to get our test scores up.The gist of what was being said to me was that the overuse of traditionalmethods for teaching math is the primary cause of our students low test scoresin math.Now then. Let me share an anecdotal observation. The mathematical prociencyof my students when I rst started teaching in 1983 was a whole lot betterthan it is today. Especially in basic arithmetic computation. Back then, I could give a quizconsisting of ten long division problems with a three-digit divisor. Most ofmy students could get at least six of them right.Today, most of my students look at a long division problem with a one-digitdivisor and have no idea what to do. Last year, one of my Honors students asked to use a calculator to gure outhow many times 3 goes into 24. Im serious.I really cant prove the mathematical superiority of my students during thatera because the tests we used then are different than the tests we use now. Back then, prociency in long division, for example, was a testing objective. Not today.I have a different theory on the low math test scores of the students in mydistrict.My theory is that these scores may be attributed to the overemphasis andoveruse of calculators, manipulatives, and cooperative learning in ourmathematics instruction.In other words, I believe that our low test scores may be attributed not to theoveruse of traditional methods -- but not enough of them. Just the opposite ofwhat I was told last week.My superiors blame our test scores on too much of the old from stubborn olddinosaur teachers like myself -- and not enough of the new.In contast, I blame our test scores on too much of the new and not enough ofthe old.All this boils down to what is commonly known as the Math Wars. Whenit comes to the teaching of math, I am a traditionalist. Right down to myheart and soul. Its like it runs through my veins.Yet, I am stuck in a school district whose philosophy of mathematicsinstruction is staunchly in the camp of the constructivists. Heres my question. What am I going to do about it?I feel like a POW. And I think its about time I started planning my escape.DennisDennis-- newsgroup website: http://www.thinkspot.net/k12math/newsgroup charter: http://www.thinkspot.net/k12math/charter.html =I think that it is useful to separate out what is being debated here. Areyou arguing against constructivism and manipulatives, or for the primacyof arithmetic in the mathematics curriculum? I think that I disagree withyou on both counts, but they are *different* counts.It is not a tenet of constructivism that arithmetic should not be taught inthe math curriculum. Constructivism is at root actually a philosophy of*learning* (and understanding) rather than a methodology of teaching anyway.It most certainly is not a curriculum.If it were decided that long division should be the touchstone of mathteaching (perish the thought) then constructivism would still have acontribution to make; the challenge is to teach *with understanding*. To theextent that the long division algorithm is beautiful (clearly that is anattribute that is in the eye of the beholder, or maybe relative to theobservers frame of reference.....) it is only beautiful when understandingthe underlieing principles involved. If it is taught by rote - to the extentthat it *can* be taught successfully by rote; a theme tackled by otherscontributing to this thread - then it is merely a longwinded way ofachieving a result which could be obtained by a calculator. In essence, itmakes no difference whether the answer is obtained by calculator or by longdivision, if the principles are not understood.It is the rote-learning that constructivism is challenging. Androte-teaching.There are other reasons why some of us do not mourn the passing of theemphasis on artithmetic. I know that we like to think back to the golden ageof math teaching, when everyone knew their tables, knew how to perform longmultiplication and long divsion, and nobody really bothered aboutunderstanding them (and math teachers bemoaned the fact that Euclid was nolonger central to the curriculum). It is a little difcult to prove that itever existed however; even harder to date it (1950s? Well, John Holtpublished How Children Fail in 1964, IIRC). And of course, as other postershave pointed out, there have been other changes in schools since then - somepositive, actually, like the fact that we are now actually worried about themathematical of the mass of the population.It can be argued -as most posters have - that calculators makepencil-and-paper methods redundant, but like you I dont buy into theutilitarian view of education. So whether you retain these methods becomes aquestion of really how beautiful you dofind them. In any case, there is,without the utilitarian view, absolutley no reason to teach the methods byrote, nor to make a fetish of them. The problem, in the golden age, wasthat for most students, trying to teach algorithms like the long divisionalgorithm was either ultimately futile, or so time-consuming that it led tolimiting students math curriculum to arithmetic, and thus denied themaccess to other areas of mathematics of great(er) beauty. My own primaryschool (US equivalent?) math education consisted of a diet of times tablesand fractions.The other issue with Golden Ageism is that it does not value what we havegained, only what we have lost. One of the pluses, we would hope, is agreater problem solving ability, which is a generic skill, rather than anitem of specic knowledge. Now how you value a generic skill like this isdependant on how you view education as a whole. The other thing is that Iwould hope that no one is setting this current era as the great GA - wehavent got things right now, and the challenge of education means that weprobably never will.As always, Im glad to see a discussion pop up on this newsgroup onpedagogy; it makes a pleasant change from the usual diet of homework help!-- It can be argued -as most posters have - that calculators make> pencil-and-paper methods redundant, but like you I dont buy into the> utilitarian view of education. So whether you retain these methods becomes a> question of really how beautiful you dofind them. In any case, there is,> without the utilitarian view, absolutley no reason to teach the methods by> rote, nor to make a fetish of them. The problem, in the golden age, was> that for most students, trying to teach algorithms like the long division> algorithm was either ultimately futile, or so time-consuming that it led to> limiting students math curriculum to arithmetic, and thus denied them> access to other areas of mathematics of great(er) beauty. My own primary> school (US equivalent?) math education consisted of a diet of times tables> and fractions.I see little value in massive memory work in math, but I do see valuein learning algorithms---learning how to follow them, learning whythey work, and learning how to construct new ones. The main value Isee in learning long division is that it is a fairly involvedalgorithm for a problem that children can understand and that answerscan be checked fairly easily. The explanation for why it works helpschildren understand how algorithms are constructed. The actual resultof the division is not the point of teaching long division (thoughgetting the answer wrong consistent indicates either an inability tofollow the algorithm or a misunderstanding of it).Do most students need to learn how to read, understand, follow, andmodify algorithms? Absolutely---in cooking, accounting, socialwork, science, engineering, ... almost every eld, people have tolearn to do things according to pre-established procedures and need tolearn when and how to modify the procedures to meet new needs.Is long-division the only way to teach algorithms? Obviously not, butit has advantages over many other algorithms in being easy to checkwhether or not the algorithm is working and being fairly easy to debugcommon mistakes.-- 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, BioinformaticsA§iations for identication only.-- newsgroup website: http://www.thinkspot.net/k12math/newsgroup charter: http://www.thinkspot.net/k12math/charter.html Here I go again. Another rant on a familiar topic.>> I am old fashioned 7th grade chalk and blackboard, pencil and paper, math> teacher. The kind who likes to teach math the way it was taught back inthe> 50s.>> I started teaching math, in an urban inner city school district, in 1983.At> that time, at least in my school district, the old way was still thenorm.>> Calculators were not used. Cooperative learning, with the exception of> letting students work in pairs on occasion, was not emphasized.>> Manipulatives? We used nothing more than those we had when I was taught7th> grade math. Namely rulers, protractors, compasses, graph paper, and thelike.> Nothing even remotely resembling these colored blocks we have today.> etc.....I have heard this before. But quite honestly, I just cant agree that the1950s was the pinnacle of math education.I teach both science and math. When I rst left the old style lecture,read, and test (open their heads, pour in the knowledge) behind, I wasskeptical. But now, I would never go back.There is example after example that the old style methods dont work. Thinkthey did in the past? Then why are so many adults my age, educated in the1950s, now frightened and phobic about mathematics and have retained verylittle? I learned math only because I enjoyed it and went outside class tokeep my interest intact.I tried modern methods for science rst, doing a modied Workshop Physicsapproach. Although it was tough (lecture is so much easier), although somestudents were initially put off by it, the results were phenomenal. Theinformation they retained was magnitudes better than the traditionalapproach. Those students still in old-fashioned classes could not come closeto the active learning students. Putting the responsibility for learning onthe students themselves, seeing them as learners rather than recordingmachines, also increased their self condence and had them thinking I canlearn this! That counts for a lot. The standardized tests conrmed what Iwas seeing.All the science classes are run this way now, the test results have run awayfrom those schools still stuck in the 1950s.I am doing this with math now. Having students actively learn is making bigstrides here as well. Just a few weeks ago an 11th grader and her parentscame to an open house. They thanked me, for the rst time their daughteractually understood math instead of just memorizing rules and doing rote,useless work. Her knowledge of math had jumped off the scale. Thestandardized test results for the active math classes are well above thestudents in traditional classes just as it was for science.There is always a hesitation to move beyond what one is used to. Activeteaching is stressful because you must know your subject inside out, bothconceptually and practically. It is much more intense than a controlledlecture environment. Quite frankly, the few failures Ive seen have all beendue to teachers who are unable or unwilling to do the extra work required,to throw away their yellowed notes from 50 years ago.I dont want to seem mean (thats not my goal, really) but the schools arebeing forced get tough. Active teaching, unlike some other teaching changes,makes a big positive difference. The needs of the students are much moreimportant than preserving the habits of teachers who are stuck in a teachingrut dug for the last half century.-- newsgroup website: http://www.thinkspot.net/k12math/newsgroup charter: http://www.thinkspot.net/k12math/charter.html =I enjoyed your post and thought Id reply to give a differentperspective. Im 45 and have always struggeled with math through outmy years. When I was in 3rd grade I just didnt get it and remained inspecial math classes which continued to slow progress for me andfrustrate me. On occassions, I would be matched to teachers thatwould connect with me and the help they provided was effective in theshort term; however, as I went through the system, math was always anexperimental tool, which changed from season to season. Thus math asa language remains foreign and fragmented with out a foundation, andfewer and fewer people are able to clearly communicate anythingmeaningful.I just returned to school and Im taking elementary college math andtoday, Im re-learning fractions, percents and all the basic math whichhas been the root of my math anxiety. Thus far, all is well, but Imust admit that the book for the class, often times fails to connectand communicate ideas from which I can build on; Im not sure why thatmind that there is a universal frustration with math abstraction. Ithink motivation is a key within the battles related to math wars,i.e., there must be a positive way to build upon foundationinformation and then to form additional (and meaningful)layers in alogical order. As a younger student, I was never motivated because Iwas frustrated, and now that Im older, Im still frustrated, but farmore motivated. As a teacher, perhaps it is your challenge to unlockpotential and to motivate frustrated students (and parents). Notevery kid will adapt to one method, because they each have a differentset of values. I dont envy math teachers, as it must be verydifcult to communicate; none-the-less, keep up the good work!Good luck!Bill-- newsgroup website: http://www.thinkspot.net/k12math/newsgroup charter: http://www.thinkspot.net/k12math/charter.html =Hey Dennis,I am a new fashioned investigation and inquiry-based mathematics teacher,recently moved up to teaching 8th graders after teaching grades 4/5. Mydistrict considers itself cutting edge in many ways, none of which is intechnology. *chuckle* But it does give me an interesting comparativeperspective to some of the causes of mathematical disability you seem to berelaying to us. Mostly, I am absolutely appalled at the scrutiny and lack ofprofessional integrity with which your district is treating its mathematicseducaiton.Oh, those good old days of maths education. Acording to nation-wide mathstest that were administered to students all the way back into the 50s, thegood old days of mathematics education never really did exist. Even then,only about 20% of the students really understood what they were doing andcould apply the mathematical reasoning to novel situations. In my state ofWashington, our adult population has an effective 20% illiteracy rate. Whichmeans that 60% of the students in those traditional math classes, eventhough they could read the material, were being left in the dust.Mathematics education was preparing only a percentage of students equal tothe number that, today, avail themselves of college and university in thiscountry.What certainly _was_ different, and I suspect it plays into what you areexperiencing with your students today, is that the students, from grade 1through grade 10, were taught maths as a series of steps to be memorized andthen applied in specic contexts. So they were trained from an early age tomemorize and repeat; memorize and repeat; memorize and repeat. They couldmimic what the teacher had done, but they could not replicate the kind ofthinking that the teacher had to go through. Today, your students are mostlikely completely untrained in the art of faithfully replicating a series ofmemorized ANYthing, and so they cannot come close to the success of yourstudents from the past.mathematics must be treated as the symbolic form of reasoning that it trulyis. At least, that is the goal.Hearing your story, I cringe and I feel terrible for you. As far as mydistrict is concerned, they took the approach that all these manipulativesand calculators and everything are _tools_ that we have at our disposal.They trust, ultimately, that the teacher will be making conscientiousdecisions about what to do, and they provide us with data and assessmenttools to help us determine what really is working. A teacher who could showthat their traditional method of instruction is working for his students aseffectively as the new methods do for other students, would not be pushed tofarther investigate and hopefully adopt the new methods. But you know, if Iworked in a district where they came into my classroom and said, show mehow many cooperative things your students have done this week, I wouldprobably quit.YOU are the mathematics education specialist in your building. I think thatyou guys might want to approach your administrators from that perspective.Your administrators are, statistically speaking in our society, most likelyNOT well-versed in mathematics. You can probably appeal to their innumeracy(i.e. show them how little they know) and easily prove to them that YOU, notthey, understand the scope of mathematics. But then appeal to what pressure_they_ are feeling. Address their concerns and allay their fears. But thishas to be done with as a unied group and with a condence and calmnessabout you and your colleagues that says: we know what our job is (teachingthe kids to think mathematically), we know how to get them there, and weCARE about these kids enough that we have purposefully chosen this routebecause... (evidence inserted here).Look, the mathematically ILLITERATE are in charge of our country! How elsecould they expect that by 2012 100% of kids will meet or exceed grade-levelexpections? I think the only way to do that really would be to say that thestudent will be able to breathe in my mathematics classroom. Your districthas adopted these draconian methods in an attempt to reverse 80 years ofstatistically elitist mathematics-focus in a few months. They are in panicmode. Theyve done too little for too long and now they are foisting theresponsibility upon those in the front line. If we, as teachers, ACCEPT theblame, then we will get the associated scorn and lack of respect. But if we,as teachers, stand up and say: NO, then they will have to face some harshrealities and accept some blame themselves or at higher levels. Or perhaps,just perhaps, they will come to realize that mathematics is not aone-size-ts-all thing for all kids at age X.Oh, who knows.Good luck! Sorry for the length of the reply.BJ MacNevin-- newsgroup website: http://www.thinkspot.net/k12math/newsgroup charter: http://www.thinkspot.net/k12math/charter.html =support of some of the points of my earlier post.> Hey Dennis,>> I am a new fashioned investigation and inquiry-based mathematicsteacher,> recently moved up to teaching 8th graders after teaching grades 4/5. My> district considers itself cutting edge in many ways, none of which is in> technology. *chuckle* But it does give me an interesting comparative> perspective to some of the causes of mathematical disability you seem tobe> relaying to us. Mostly, I am absolutely appalled at the scrutiny and lackof> professional integrity with which your district is treating itsmathematics> educaiton.>> Oh, those good old days of maths education. Acording to nation-wide maths> test that were administered to students all the way back into the 50s,the> good old days of mathematics education never really did exist. Eventhen,> only about 20% of the students really understood what they were doing and> could apply the mathematical reasoning to novel situations. In my state of> Washington, our adult population has an effective 20% illiteracy rate.Which> means that 60% of the students in those traditional math classes, even> though they could read the material, were being left in the dust.> Mathematics education was preparing only a percentage of students equal to> the number that, today, avail themselves of college and university in this> country.>> What certainly _was_ different, and I suspect it plays into what you are> experiencing with your students today, is that the students, from grade 1> through grade 10, were taught maths as a series of steps to be memorizedand> then applied in specic contexts. So they were trained from an early ageto> memorize and repeat; memorize and repeat; memorize and repeat. They could> mimic what the teacher had done, but they could not replicate the kind of> thinking that the teacher had to go through. Today, your students are most> likely completely untrained in the art of faithfully replicating a seriesof> memorized ANYthing, and so they cannot come close to the success of your> students from the past.>> calculators a lot, I dont know that I would necessarily blame the> calculators. We do a lot of mental math in elementary, and of talkingabout> numbers and their relationships. Not all schools are as far along asothers> in this endeavour, but those that are see tremendous improvements fortheir> students. But our reform-elementary school students would also have a VERY> hard time replicating in any meaningful fashion the long-division steps> unless they had a thorough understanding of why it works. But that is the> primary thrust of mathematics reform in education. Beyond all else> (calculators, computers, manipulatives), the reformists believe that> mathematics must be treated as the symbolic form of reasoning that ittruly> is. At least, that is the goal.>> Hearing your story, I cringe and I feel terrible for you. As far as my> district is concerned, they took the approach that all these manipulatives> and calculators and everything are _tools_ that we have at our disposal.> They trust, ultimately, that the teacher will be making conscientious> decisions about what to do, and they provide us with data and assessment> tools to help us determine what really is working. A teacher who couldshow> that their traditional method of instruction is working for his studentsas> effectively as the new methods do for other students, would not be pushedto> farther investigate and hopefully adopt the new methods. But you know, ifI> worked in a district where they came into my classroom and said, show me> how many cooperative things your students have done this week, I would> probably quit.>> YOU are the mathematics education specialist in your building. I thinkthat> you guys might want to approach your administrators from that perspective.> Your administrators are, statistically speaking in our society, mostlikely> NOT well-versed in mathematics. You can probably appeal to theirinnumeracy> (i.e. show them how little they know) and easily prove to them that YOU,not> they, understand the scope of mathematics.This is analogous to a 4 star commanding general proving to the President ofthe United States that he (the pres) knows very little in comparison abouthow to wage and win a war. That would be very easy to do in most cases(prove that point) with the obvious exception of presidents that are retiredcommanding generals themselves. The point would, however, be irrelevent.The President commands the forces; the generals follow the orders.You cant escape the fact that, despite even considerably lessermathematical knowledge than the math teacher, the administrator has madequite well for himself in life. Hes an _administrator_, after all, andeven in charge of the teacher in this case. ther is certainly somethingvery valid and relative to be said about that.Reality is, we (your average Joe) *demonstrably* do not need to know most ofthis stuff (mathematics). Else, your average Joe WOULD now this stuff. Buthe doesnt. Yet, Joe gets by, he makes ends meet, he goes on to have littleJoes, society functions. Even some of these Joes become our bosses.> But then appeal to what pressure> _they_ are feeling. Address their concerns and allay their fears. But this> has to be done with as a unied group and with a condence and calmness> about you and your colleagues that says: we know what our job is (teaching> the kids to think mathematically), we know how to get them there, and we> CARE about these kids enough that we have purposefully chosen this route> because... (evidence inserted here).The pressures that they (the admins) are probably feeling are mandates fromon high that things be done this certain way. Or, they could be the onesdirectly responsible for the mandates, although everyone has _someone_ toanswer to. In the long run, we either conform or we dont. Those thatconform are team players; those that dont are labelled as troublemakers,placing their own agenda ahead of the teams.>> Look, the mathematically ILLITERATE are in charge of our country!My very point exactly! ..but the tone in which that statement was madesuggests you believe this to be a bad thing. How so? is there somethingintrinsic about mathematics that implies that we MUST know it? Ifmathematical literacy was really necessary, then those in charge would bemathematically literate out of neccessity. But theyre not, therefore it isnot necessary to be mathematicaly literate to succeed in life (ie be incharge) although it is supposed to be a good thing to instill in our youthto make a good life for themselves, do what it takes to get a good job (iebeing in charge).-- Darrell-- newsgroup website: http://www.thinkspot.net/k12math/newsgroup charter: http://www.thinkspot.net/k12math/charter.html =<...> Here are some of the guidelines to which I am now required to adhere:>> ***My students must have access to calculators each and every single day.>> ***Cooperative learning strategies must be employed a minimum of threetimes a> week.>> ***Hands on learning strategies must be employed a minimum of three timesa> week.>> This emphasis has been going on for a number of years now. However, Iused to> be able to let it all go in one ear, out the other, shut my classroomdoor, and> then do my own thing.>> No more. All the math teachers in my school district are under amicroscope to> make sure that we are adhering to these guidelines.>> Last week, I was called on the carpet for not doing so.There was a time when bread came unsliced, thus people sliced their ownbread out of necessity. You will surely agree the need for bread slicers,as a necessity in everyones possesion, is not nearly as great today as itwas then. Yeah, those slicing the bread need them, but those are arelatively few, so there exists special bread slicing training for those.The time of most other mortals, not needing to know how to slice bread, isbetter spent learning something else.There was a time when most mortals determimed results of arithmeticoperations by hand *out of necessity*, ie the old way, before such handygizmos as calculators, etc. were widely available. Please try to addressthe next question free of any and all bias towards or against any old wayor new way or what not of teaching math. I know its impossible tocompletely discard all bias (we are all biased one way or the other) butoften times being reminded to put forth the effort nonetheless, results in amore objective analysis of ones position (ie it can demonstrate just howbiased we really *are*, which is quite often much more than we thought...)Would you say the need for most people today to be able to perform the samearithmetic operations by hand is as great today as it was *then*?Are you sure about that? I mean, are you REALLY sure? Considering that acalculator is in just about everyones reach (and by mandated denitionWILL be within your students reach at all times), do you really--honestlyand truthfully--believe that it is just as important today for the averagemortal to know how to do long division, as it was before such technology wasnot widely available?Yeah, those programming these machines need to know the math, but again,those are a relatively few (unless you live in India).>> I was told: The test scores of our students are extremely low. It is> extremely important that you use more hands on strategies. This is whatour> students need to get our test scores up.>> The gist of what was being said to me was that the overuse of traditional> methods for teaching math is the primary cause of our students low testscores> in math.>> Now then. Let me share an anecdotal observation. The mathematicalprociency> of my students when I rst started teaching in 1983 was a whole lotbetter> than it is today.>> Especially in basic arithmetic computation. Back then, I could give aquiz> consisting of ten long division problems with a three-digit divisor.Most of> my students could get at least six of them right.Because they knew how to do long division *out of necessity*.>> Today, most of my students look at a long division problem with aone-digit> divisor and have no idea what to do.Could this possibly be because it is no longer really necessary to know longdivision by hand? Hmmm...>> Last year, one of my Honors students asked to use a calculator to gureout> how many times 3 goes into 24. Im serious....and I also sometimes use a calculator (provided one is handy) to answerthe very same question. ...and I count on my ngers. So what.>> I really cant prove the mathematical superiority of my students duringthat> era because the tests we used then are different than the tests we usenow....and thats the key to understanding what is being asked of you. Societyis *redening* what the necessary level of mathematical knowlege is. donttake this the wrong way, but entrusting mathematics teachers with decidingwhat level of mathematics knowledge society need posses, makes about as muchsense as entrusting generals to decide who and when we go to war. Thereinput is invaluable, of course, but the nger on the button ALWAYS needs tobe that of a civilian, unless we like living under martial law.By all means, keep the bread slicer as an heirloom, passing it down fromgeneration to generation to be proudly displayed, but never really used muchmore. Much like punch cards in data processing.> Back then, prociency in long division, for example, was a testingobjective.> Not today.Whats so wrong with that? I mean, if we simply dont need to know it, wedont need to KNOW it. Theres a reason why most people, when asked atrandom, cant accuratly quote the Pythagoren Theorem, or the QudraticFormula. Its simply because most dont need to know it. a similarargument can be made towards the need of most to know how to perform longdivision.>> I have a different theory on the low math test scores of the students inmy> district.>> My theory is that these scores may be attributed to the overemphasis and> overuse of calculators, manipulatives, and cooperative learning in our> mathematics instruction.>> In other words, I believe that our low test scores may be attributed notto the> overuse of traditional methods -- but not enough of them. Just theopposite of> what I was told last week.>> My superiors blame our test scores on too much of the old from stubbornold> dinosaur teachers like myself -- and not enough of the new.>> In contast, I blame our test scores on too much of the new and notenough of> the old.I sincerly believe your admirable love, dedication, and obvious enthusiasmfor the teaching of mathematics is skewing your view of the big picture.There will exists such a world where fewer and fewer people will actuallyneed to *know* much of the mathematics taught today. Were already in thetransition. Even many engineers will tell you they use technology mostlyfor their mathematics needs.>> All this boils down to what is commonly known as the Math Wars. When> it comes to the teaching of math, I am a traditionalist. Right down to my> heart and soul. Its like it runs through my veins.Think of the analogy of Bread Slicer Wars, where only those having avested interest in the production of bread slicers for the masses are theones demanding more, not less, be produced today. All others, free of suchbias, clearly see that most today do not need bread slicers.with respect-- Darrell-- newsgroup website: http://www.thinkspot.net/k12math/newsgroup charter: http://www.thinkspot.net/k12math/charter.html Think of the analogy of Bread Slicer Wars, where only those having a>vested interest in the production of bread slicers for the masses are the>ones demanding more, not less, be produced today. All others, free of such>bias, clearly see that most today do not need bread slicers.>>with respect>-- >DarrellBread today, especially the best bread, still comes unsliced.G C-- newsgroup website: http://www.thinkspot.net/k12math/newsgroup charter: http://www.thinkspot.net/k12math/charter.html Think of the analogy of Bread Slicer Wars, where only those having a>vested interest in the production of bread slicers for the masses are the>ones demanding more, not less, be produced today. All others, free ofsuch>bias, clearly see that most today do not need bread slicers.>>with respect>-- >Darrell>> Bread today, especially the best bread, still comes unsliced.In the U.S., where the masses swarm to Wal-Mart, most bread is _sliced_.Im talking quantity here. No one is arguing against fresh baked bread (orlong division) tasting better... so if youre one to savor the avor oong division, be my guest. If youre like me, youd rather cut to thechase so you can spend time savoring other things....Until such time even _those_ things dont taste as good as they once did,then wefind yet other things to savor.-- newsgroup website: http://www.thinkspot.net/k12math/newsgroup charter: http://www.thinkspot.net/k12math/charter.html =Darrel at dr6583@msn.com quotes and comments:>>Think of the analogy of Bread Slicer Wars, where only those having a>>vested interest in the production of bread slicers for the masses are the>>ones demanding more, not less, be produced today. All others, free of>such>>bias, clearly see that most today do not need bread slicers.>>with respect>>-- >>Darrell>> Bread today, especially the best bread, still comes unsliced.>>In the U.S., where the masses swarm to Wal-Mart, most bread is _sliced_.>Im talking quantity here. No one is arguing against fresh baked bread (or>long division) tasting better... so if youre one to savor the avor of>long division, be my guest. If youre like me, youd rather cut to the>chase so you can spend time savoring other things.>>...Until such time even _those_ things dont taste as good as they once did,>then wefind yet other things to savor.My treatment of the metaphor comes approximately to accepting long division asclassic knowledge which will never lose its value. G C-- newsgroup website: http://www.thinkspot.net/k12math/newsgroup charter: http://www.thinkspot.net/k12math/charter.html My treatment of the metaphor comes approximately to accepting longdivision as> classic knowledge which will never lose its value.I see. So you acknowledge that studying long division for necessaryutilitarian need is not nearly as important now as it was long ago?-- Darrell-- newsgroup website: http://www.thinkspot.net/k12math/newsgroup charter: http://www.thinkspot.net/k12math/charter.html =Chergarj commented :> My treatment of the metaphor comes approximately to accepting long>division as>> classic knowledge which will never lose its value.>Darrel then commented and asked:>I see. So you acknowledge that studying long division for necessary>utilitarian need is not nearly as important now as it was long ago?>>-- >Darrell>No.G C-- newsgroup website: http://www.thinkspot.net/k12math/newsgroup charter: http://www.thinkspot.net/k12math/charter.html Chergarj commented :>> My treatment of the metaphor comes approximately to accepting long>division as>> classic knowledge which will never lose its value.> Darrel then commented and asked:>>I see. So you acknowledge that studying long division for necessary>utilitarian need is not nearly as important now as it was long ago?>>-- >Darrell> No.not everyone is so kind as you may know from reading some other recentthread! Personally, I cant recall anytime in the recent past where I havefound myself in a position where I actually NEEDED to do a long division byhand, instead of by some other means, save for cases in direct support ofmathematics newsgroups such as this, tutoring someone, or some such.but thats just me...-- Darrell-- newsgroup website: http://www.thinkspot.net/k12math/newsgroup charter: http://www.thinkspot.net/k12math/charter.html not everyone is so kind as you may know from reading some other recent>thread! Personally, I cant recall anytime in the recent past where I have>found myself in a position where I actually NEEDED to do a long division by>hand, instead of by some other means, save for cases in direct support of>mathematics newsgroups such as this, tutoring someone, or some such.>>but thats just me...>>-- >DarrellExactly. Most of the time, I have a calculator nearby, either on a platform in my reach,or in my pocket. Sometimes, when Im too lazy to get the calculator, I mayperform long division for certain tasks (usually one or two digits by one ortwo digits). Many years ago in a labor job, I often needed to perform some simple divisionand since it was not the practice to carry a calculator, I would perform thedivision on a cardboard box. One of the saddest things that can happen to someone who could have occasionsto do mathematical calculatoions is to rst run for a calculator in order tocompute something which they wouldnt know how to do without a calculator.G C-- newsgroup website: http://www.thinkspot.net/k12math/newsgroup charter: http://www.thinkspot.net/k12math/charter.html Most of the time, I have a calculator nearby, either on a platform in myreach,> or in my pocket. Sometimes, when Im too lazy to get the calculator, Imay> perform long division for certain tasks (usually one or two digits by oneor> two digits).> Many years ago in a labor job, I often needed to perform some simpledivision> and since it was not the practice to carry a calculator, I would performthe> division on a cardboard box.>> One of the saddest things that can happen to someone who could haveoccasions> to do mathematical calculatoions is to rst run for a calculator in orderto> compute something which they wouldnt know how to do without a calculator.I was hoping you would clarify what specic sad things happen. Do youthink its sad that I never had the privilege of having to computelogarithms with a slide rule but rather usually always ask a calculator orsome other software when I need such a value (save for trivial logs, ofcourse) ? I dont. Never lost a wink of sleep over it.How is the case of long division any fundamentally different wrt the pointwe are making here?-- Darrell-- newsgroup website: http://www.thinkspot.net/k12math/newsgroup charter: http://www.thinkspot.net/k12math/charter.html not everyone is so kind as you may know from reading some other recent>thread! Personally, I cant recall anytime in the recent past where I have>found myself in a position where I actually NEEDED to do a long division by>hand, instead of by some other means, save for cases in direct support of>mathematics newsgroups such as this, tutoring someone, or some such.>>but thats just me...>>-- >Darrell> Exactly. > Most of the time, I have a calculator nearby, either on a platform in my reach,> or in my pocket. Sometimes, when Im too lazy to get the calculator, I may> perform long division for certain tasks (usually one or two digits by one or> two digits). > Many years ago in a labor job, I often needed to perform some simple division> and since it was not the practice to carry a calculator, I would perform the> division on a cardboard box. > One of the saddest things that can happen to someone who could have occasions> to do mathematical calculatoions is to rst run for a calculator in order to> compute something which they wouldnt know how to do without a calculator.> G CI know what its like to work in general unskilled employmentsituations, and Ive seen and heard of what employment specialistswould probably tell you: That sometimes, to weed out some during theunskilled labor hiring process, the employer gives a basic skillsscreening test on basic grammar and arithmetic. Years ago I took sucha test when applying to get a factory job at a UPS dock loadingtrucks.Also, to become certied plumbers, applicants have to pass a mathtest of basic skills, including very basic plane trigonometry.Whats terribly ironic about all this is this: Those who are probablyleast skilled in these basic skills are the high school dropouts orthose who barely got through high school, with no job training inparticular or who have job skills as laborers. Those who are probablymost skilled in the basic skills are the college educatedprofessionals. But its the former rather than the latter who are mostlikely tofind themselves having to take these basic skills screeningtests for employment or job skill certication.Paul-- newsgroup website: http://www.thinkspot.net/k12math/newsgroup charter: http://www.thinkspot.net/k12math/charter.html =Through discussing use of calculators for arithmetic division, Paul ofuprho@yahoo.com comments:>Those who are probably>least skilled in these basic skills are the high school dropouts or>those who barely got through high school, with no job training in>particular or who have job skills as laborers. Those who are probably>most skilled in the basic skills are the college educated>professionals. But its the former rather than the latter who are most>likely tofind themselves having to take these basic skills screening>tests for employment or job skill certication.>>Paul>Some of this varies from one company to another. Some companies want to checkan applicants written language and mathematical skills by using a formal testor quiz; some companies may use an oral (not necessarily written) test or quiz. Especially applicants to many professional positions are required to perform asubject matter quiz combined with a mathematics quiz. The use of a calculatoris often permitted for some parts of these tests or quizes, but somespoken-interactive quizes may not permit calculators because the interviewer ortest-giver wants tofind out if applicant can understand the arithmetic. Thecandidate has the options of explaning exactly what to do through speech, orsetup an expression on paper, and estimation may be allowed, and evenpractical. In many cases of using math, a calculator is not necessary. The calculationsare too simple; simple enough often that a person could do them in their head.G C-- newsgroup website: http://www.thinkspot.net/k12math/newsgroup charter: http://www.thinkspot.net/k12math/charter.html =strengthen the mind.Which serves to develop and strengthen the mind more?A) Performing all long division problems on a calculatorB) Learning to perform long division problems with pencil and paper withoutthe use of a calculator.My answer is Choice B.Therefore, in accordance with my basic philosophy of education, the attainmentof prociency in solving arithmetic computations with pencil and paper is aworthy educational objective. It serves to develop and strengthen the mind.>>Are you sure about that? I mean, are you REALLY sure? Considering that acalculator is in just about everyones reach (and by mandated denition WILLbe within your students reach at all times), do you really--honestly andtruthfully--believe that it is just as important today for the averagemortal to know how to do long division, as it was before such technology wasnot widely available?<>Could this possibly be because it is no longer really necessary to know longdivision by hand? Hmmm...<>Whats so wrong with that? I mean, if we simply dont need to know it, wedont need to KNOW it.< arithmetic operations by hand is as great today as it was *then*?<<>> Your question is loaded with the word need. I do not base my valuesupon> what I feel is right or wrong with math education upon utilitarian need.>> Of course, if everybody has a hand held calculator within reach, then the> utilitarian need for knowing how to do arithmetic operations by hand isnot> as great as it was *then*.question: Do you think it would be a good idea for a C.S. instructor todemand (tactfully of course) that much, much more time be spent todaylearning how to use punch cards?>> My basic philosophy of education is not rooted in need. Instead, my> philosophy of education is that the primary aim of education is to developand> strengthen the mind.Lots of things can strengthen the mind. Yoga, ballet, prayer, etc., etc.etc. However, the goal of (public) education should be to prepare our youthfor adulthood; to be positive contributors to society. Thats rooted in_need_.>> Which serves to develop and strengthen the mind more?>> A) Performing all long division problems on a calculator>> B) Learning to perform long division problems with pencil and paperwithout> the use of a calculator.>> My answer is Choice B.With respect, I believe you are straying from the subject somewhat. Thequestion is very subjective. Valid arguments can be made for either case,depending on what you treasure. A very strong argment can be made that A)is better because it exposes one to technological means of addressing thetask, thus freeing the minds to focus on other things that not only developand strengthen it, but may very well be rooted in actual need. Two birdswith one stone.>> Therefore, in accordance with my basic philosophy of education, theattainment> of prociency in solving arithmetic computations with pencil and paper isa> worthy educational objective. It serves to develop and strengthen themind.Thats certainly a valid argument according to *your* philosophy ofeducation, however, if the only standard in place is devlopment andstrengthening of the mind, then like I said before there are many thingsthat devsopl and strengthen the mind. Only of very few of these are theones chosen by the powers that be to be dealt with from 8am to 3pm , Mon-Fri(where such power is rooted in the _general public_, not the instructors...)>Are you sure about that? I mean, are you REALLY sure? Considering thata> calculator is in just about everyones reach (and by mandated denitionWILL> be within your students reach at all times), do you really--honestly and> truthfully--believe that it is just as important today for the average> mortal to know how to do long division, as it was before such technologywas> not widely available?<<>> Yes, I do. For the purpose of developing and strengthening the mind.Again, your argument is valid but somewhat misplaced. LSD has been known todevelop and strengthen the mind, too, though I have no personal experiencein such matters :-).>> And also for the sheer beauty and integrity of it.Lots of things are beautiful and have integrity. If youve seen the movieAmerican Beauty, so is a plastic bag twirling around in a devilwind.>> Yes, I see great beauty in a students ability to solve a long division> problem.So do I. But I would not disobey taxpayers by insisting that time from 8amto 3pm be taken teaching this, if they so hapened to tell me that theybelieve I should be spendinmg time on something else.> By discarding this arithmetic task as an educational objective, I feelthat> something very precious and of great beauty is being lost.Same can be said for punch cards in data processing. Lets move on to the21st century, shall we?>Could this possibly be because it is no longer really necessary to knowlong> division by hand? Hmmm...<<>> Once again, you are basing your argument upon what is necessary. Myphilosophy> of education isnt based upon need or what society decides has decidedhas> become or not become necessary.Then with all due respect, you should not be a public school teacher (if infact you are, which I dont know). If youre in a private school, thats upto THOSE powers that be, but in a public school, of which I am a concernedparent having a child in that system, then yeah, you better believe that thePUBLIC ultimately decides (by way of elected ofcials, etc.) what is or isnot neccessary to be taught in public schools.Although I truly believe you have the best of intentions, your attitudeof... to hell with everyone else who pays my salary and tells me what to do,Im going to implement *my* educational philophy.... is a veryinnapropriate attitude. Just like that commanding general telling thePresident to go to hell, were moving in anyway...>Whats so wrong with that? I mean, if we simply dont need to know it,we> dont need to KNOW it.<<>> Theres that need word again.Yes, NEED. Its all based on need, really.>> Its like youre on one planet and Im on another.Cmon, lets stay civilized shall we?>> I dont base my educational values upon sheer need. I base myeducational> values on what serves to devleop and strengthen the mind.> This is something I feel deeply, right down to the core of my soul. Its> almost like a religion with me.thats the key to understanding what is being asked of you and how to dealwith complicance. You have a very strong emotional interest vested in thematter, thus are very biased against anything that seems to contradict yourphilosophy. What you should understand, which will helkp you greatly indealing with how to tackle the situation, is that it is not YOIUR philosophythat youy have been entrusted in portraying (and being paid to portray bytaxpayers).Three guesses whose philosophy you ARE being entrusted and paid to portray.-- Darrell-- newsgroup website: http://www.thinkspot.net/k12math/newsgroup charter: http://www.thinkspot.net/k12math/charter.html Which serves to develop and strengthen the mind more?>>A) Performing all long division problems on a calculator>>B) Learning to perform long division problems with pencil and paper without>the use of a calculator.Better foci: When do you divide, which number goes rst or on top, andwhat does the answer mean?-- charlie dickThe right to be left alone -- the most comprehensive of rights, and the right most valued by a free people. - Justice Louis Brandeis, Olmstead v. U.S. (1928).-- newsgroup website: http://www.thinkspot.net/k12math/newsgroup charter: http://www.thinkspot.net/k12math/charter.html This is something I feel deeply, right down to the core of my soul. Its>almost like a religion with me.>>Call me a fanatic or a dinosaur or whatever, but thats how I feel.>Hey, Dennis:Stop worrying about feel. You have a perfectly valid opinion about how andwhat to teach. You dont really need to FEEL one way or some other. Yourobjective is to help your students learn concepts and skills. The calculatoris a tool, and it should be used for efciency of computation ONLY AFTER yourstudents have adequately studied arithmetic and arithmetic compuation. Certainly you feel condent about this. G C-- newsgroup website: http://www.thinkspot.net/k12math/newsgroup charter: http://www.thinkspot.net/k12math/charter.html =My philosophy of education is that the primary aim of education is todevelop and strengthen the mind.- In this sentence, who is the person who is aiming?You are complaining that the School Administration is compelling youto teach in a way that you dont like, while simultaneously seeking tocompel your students to learn in a way that YOU like.-- newsgroup website: http://www.thinkspot.net/k12math/newsgroup charter: http://www.thinkspot.net/k12math/charter.html All this boils down to what is commonly known as the Math Wars. When> it comes to the teaching of math, I am a traditionalist. Right down to my> heart and soul. Its like it runs through my veins.> Yet, I am stuck in a school district whose philosophy of mathematics> instruction is staunchly in the camp of the constructivists. > Heres my question. What am I going to do about it?> I feel like a POW. And I think its about time I started planning my escape.I hear you.In my view, most measurable increases in learning occur due to whatgoes on outside the classroom, not in it. And please dontmisunderstand me here. Im just talking about the absolute need forserious individual study. For me anyway, if I learn the material sowell that I can handle any test matched to said material thrown myway, then its mostly because of what I did outside of class, inindividual study mastering the material, not because of what I did inclass. This is because I just cant retain what I learned in-class allthat well, no matter what the pedagogy.In this light, there are some things you can do in order to try getyour students to measure well regardless of what goes on in class.Design the homework assignments so that by your reckoning, if theywere to truly and fundamentally master all parts of all assignments,then they would be well prepared for any matching test thrown at themby you or the state. To try to get them to master this material, trytwo things:1) Provide them with student solutions manuals for their assignments.This would be totally worked out step-by-step solutions for all theassigned problems, done by you of course. (I know it seems like work,but much of it is what we do anyway, since so many of them want to beshown how to do their assignments the next day(s) anyway.) You cantake up their work for a grade or not, and if you do, of course dontgive the relevant solutions until afterwards. But to get them to usethese solutions manuals:2) Regularly give them homework quizzes (tests), which are just randomselections of the homework - no changes to numbers are necessary(unless you want to), just a random selection. Of course, nocalculators are allowed, since these arerecall/reproduction/re-creation tests, where the students have to showall steps to the solution. Such tests are a direct measure of whatthey know well enough to apply - one cant apply it if one cantrecall/re-create it.For instance, if there were 50 problems for the week, you couldrandomly choose 10. They have to master all 50, obviously, since theydont know which 10. You could have some of these quizzes becumulative all the way back to the beginning of the semester. Thatwould be a way to get and keep them prepared for big tests that arecumulative all the way back - including the district or state tests.And its a great way to grade homework - they cant get away with notmastering it. In addition, parents like it. I had a lot of parentsconcerned this past year about their childs grade. But all of themchanged their minds when I explained that the reason their childrenwere failing was because in part, their children were not passingthose tests where they were given all possible test questions andtheir worked out solutions well before any such test. Becauseremember, this is what a homework quiz/test actually is. I explainedthis and the rationale for such tests. They all agreed it was a goodidea. They realized then that their children were just not applyingthemselves to take advantage of the deal of the century (being givenall possible test questions and their worked out solutions well beforeany such test).But regarding 2), there may be some real rebellion from the studentsin spite of parent support, since they the students may not be used tobeing forced to really master and maintain mastery of all theirhomework.Note that 1) and 2) have nothing to do with these math warsregarding what goes on in the classroom during instruction time. Soits a way to try to offset any damage that could be done by somein-class thing, like being forced to let them use calculators in classin non-testing situations on stuff like 24/3.Paul-- newsgroup website: http://www.thinkspot.net/k12math/newsgroup charter: http://www.thinkspot.net/k12math/charter.html =At 3:20 pm a jeweler set three antique clocks to the correct time. Thenext afternoon at 3:20,she found that one clock was correct, one clockwas two minutes fast and the other was 2 minutes slow. At those rates,how long will it take for all three clocks to show 3:20 againlove Jayda-- newsgroup website: http://www.thinkspot.net/k12math/newsgroup charter: http://www.thinkspot.net/k12math/charter.html =15 days> At 3:20 pm a jeweler set three antique clocks to the correct time. The> next afternoon at 3:20,she found that one clock was correct, one clock> was two minutes fast and the other was 2 minutes slow. At those rates,> how long will it take for all three clocks to show 3:20 again>> love Jayda-- newsgroup website: http://www.thinkspot.net/k12math/newsgroup charter: http://www.thinkspot.net/k12math/charter.html At 3:20 pm a jeweler set three antique clocks to the correct time. The>next afternoon at 3:20,she found that one clock was correct, one clock>was two minutes fast and the other was 2 minutes slow. At those rates,>how long will it take for all three clocks to show 3:20 again>>love Jayda12 hours and 2 minutes.G C-- newsgroup website: http://www.thinkspot.net/k12math/newsgroup charter: http://www.thinkspot.net/k12math/charter.html =I just ran across a book that I just realized embodies my idea really quiteclosely of what the non-specialist should learn about mathematics. The bookis Introducing Mathematics by Ziauddin Sardar, Jerry Ravetz and Borin VanLoon. It is 171 pages with by far more space taken up by pictures thanwords, but nonetheless gives a far better idea idea of what mathematics isabout than any curriculum anyone has dared to present.SectionsWhy Maths?, Counting, Written Numbers, The Zero, Special Numbers, LargeNumbers, Powers, Logarithms, Calculation, Equations, Measurement, GreekMathematics, Pythagoras, Zenos Paradoxes, Euclid, Chinese Mathematics, TheChu Chang, Four Chinese Mathematicians, Vedic Geometry, Brahmagupta, JainNumbers, Vedic and Jain Combinations, Mathematical Verse, Ramanujan, IslamicMathematics, Al-Khwarazmi, Development of Algebra, The Discovery ofTrigonometry, Al-Battani, Abu Wafa, Ibn Yunu and Thabit Ibn Qurra, Al-Tusi,Solutions of Problems Involving Integers, Emergence of European Mathematics,Rene Descartes, Analytic Geometry, Functions, The Calculus, Differentiation,Integration, Berkeleys Questions, Eulers God, Non-Euclidean Geometries,N-Dimension Spaces, Evariste Galois, Groups, Boolean Algebra, Cantor andSets, Crisis in Mathematics, Russel and Mathematical Truth, Godels Theorem,The Turing Machine, Fractals, Chaos Theory, Topology, Number Theory,Statistics, P-Values and Outliers, Probability, Uncertainty, Policy Numbers,Mathematics and Eurocentrism, Ethnomathematics, Mathematics and Gender,Where Now?, Further ReadingNow as you may guess, the book came from the history of mathematics sectionof the bookstore rather than the mathematics section. But nonetheless thereis really a lot of mathematics taught in it, and the mathematics that youcan learn from a book like this is, I think, of a far more important kindthat the boring stuff we teach in required mathematics general educationcourses, and denitely far more interesting, important, and possibly evenmore mental strengthening--whatever that is--than the drilling of longdivision. And, by the way, there is absolutely nothing in that book, even inthe nal pages that can be any better understood by someone who can dodivision by pencil and paper than by one who needs a calculator for it.QuotesThe best way to systematize naming and counting is to have a base, anumber that marks the beginning of couting again. The simplest base is justtwo. For example, the Gumulgal, an Australian indigenous people, countedlike this:... This may seem primitive and tedious. But the base two, inform of 0s and 1s is built into digital computers as the foundation of alltheir calculations.Just how easily we can reach large numbers can be well illustrated by thatold evil, the chain letter.In order to multiply or divide two logarithmic expressions, we use the factthat multiplication and division of powers of a number corresponds toaddition and subtraction of these powers.Counting and calculation concern separate, discrete quantities, involvingexact numbers. Measurement, by contrast, concerns continuous magnitudes. Nomeasurement is exact. When we compare the object being measured against astandard, we always interpolate between the points on the nest scale. Andevery report of a complex measurement has (or should have!) an error barto indicate the fringe of uncertainty associated with it.once curves were perceived as graphs of functions, then the problems ofareas could be seen in a double perspective. On the one hand, areas could beexhausted by thin vertical strips; and the other, the area as a newfunction is just the one whose derivative equals the original function.[Berkeley callout] I observe that forming a quotient with the incrementsmakes sense only if it is not zero; otherwise we are dividing by zero, andthat is illegitimate. Is the increment always non-zero, or is it the ghostof a vanished quantity? And apart from that, sirrah, Mr. Newton is naked.Eulers formula is a mysterious, transcendent expression that connects theve more fundamental numbers in the universe:[uncountability of real numbers] How could we possibly construct a numberthat is not on that list? Well suppose we have one that is different fromthe rst number in the rst place, different from the second number in thesecond place, third in the third, fourth in the fourth, and son on, and on.When one is talking about sets in such a general way, there is nothing tostop one from referring to the set of all sets--it makes grammaticalsense, doesnt it? Now, that must be the biggest set of all, and its sizewill be a certain Aleph, lets call it Aleph F for nal. But, like anyother set, it will have a power set, whose number can be dened as 2 to theAleph F. So what we dened as truly biggest set, the set of all sets, cangenerate an even bigger one.[Godel callout] My theorem proved that any consistent mathematical systemmust by incomplete...-- newsgroup website: http://www.thinkspot.net/k12math/newsgroup charter: http://www.thinkspot.net/k12math/charter.html =(1*1)+(2*2)+ ....(999*999)thanx-- newsgroup website: http://www.thinkspot.net/k12math/newsgroup charter: http://www.thinkspot.net/k12math/charter.html (1*1)+(2*2)+....(999*999)> thanxLooks like the sum of the squares of integers from 1 to 999. The formulais ne as is, although the same thing can be stated a little more conciselyin Sigma notation. Plus you may know some formulas concerning certainforms of Sigma sums that may facilitate easy evaluation of the sum (ie soyou dont have to manually, one at a time, square all 999 numbers and addthem, which would be laborious to say the least.)-- Darrell-- newsgroup website: http://www.thinkspot.net/k12math/newsgroup charter: http://www.thinkspot.net/k12math/charter.html > (1*1)+(2*2)+....(999*999)>> thanx> Looks like the sum of the squares of integers from 1 to 999. The formula> is ne as is, although the same thing can be stated a little more concisely> in Sigma notation. Ellipsis (...) is never ne in a a formula. What the ellipsismeans is you, the reader, can guess what I mean here. Thats okwhen you are giving an intuitive feel for a problem, but notsufcient for proofs, algebraic manipulation, or programming into acomputer or calculator.A mathematical formula is explicit, not relying on guesswork by thereader. Indeed Darrell started out with looks like indicating that(in some unquantied way) consistent with the information given. Thesigma formula is much more explicit, so is generally preferred fordoing further manipulation.-- 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, BioinformaticsA§iations for identication only.-- newsgroup website: http://www.thinkspot.net/k12math/newsgroup charter: http://www.thinkspot.net/k12math/charter.html > (1*1)+(2*2)+....(999*999)>> thanx>> Looks like the sum of the squares of integers from 1 to 999. Theformula> is ne as is, although the same thing can be stated a little moreconcisely> in Sigma notation.>> Ellipsis (...) is never ne in a a formula. What the ellipsis> means is you, the reader, can guess what I mean here. Thats ok> when you are giving an intuitive feel for a problem, but not> sufcient for proofs, algebraic manipulation, or programming into a> computer or calculator.>> A mathematical formula is explicit, not relying on guesswork by the> reader. Indeed Darrell started out with looks like indicating that> (in some unquantied way) consistent with the information given. The> sigma formula is much more explicit, so is generally preferred for> doing further manipulation.Ellipses have a common meaning in such a context. They mean continue thepattern. Theguesswork involved is in assuming what exactly IS the pattern in question.In this case, the rst two terms are the squares of 1 and 2, then ..., thenthe square of 999. Therefore a reasonable assumption is that the pattern tobe repeated that is denoted by ..., is the sum of the squares of 3,4,5, andevery other integer up to and including 998.Ellipses are FINE in this formula depending on what is meant by formulawhich is why I purposely surrounded the word in quotation marks not onlyhere but in the previous post. If you consider a formula as simply anestablished form of symbols used in some procedure, then 1^1+2^2+...+999^2is perfectly ne as a formula. It tells us to square 1, then add that tothat the square of 2, then (unless were just really thick) add to that thesquare of 3, so on and so forth, with the last term being the square of 999.This evaluates to a certain number, under a certain reasonable assumption ofwhat pattern is represented by the ellipses....and a myriad of other assumptions concerning what the meanings of everyother symbol used in the expression are, limited only by our pedanticdesires.ALL mathematical expressions are ambiguous to some degree until rules arelaid out how to evaluate them, what number systems they apply to, etc. Asimpleexample: Consider that someone posts a message to this n.g. asking to solvefor x in:x^2 = 256Without SOME assumptions being made, the question is completely ambiguous.It could mean any of a great deal of things. Some of the more commonassumptions most would make are:1. x is real2. what the meaning of the symbols ^ and = are3. what the meaning of the word solve means in this context4. for that matter, what the meanings of every symbol in the expressionare.etc.At some point (most all of the time actually) in order to effectivelycommunicate we constantly trade off rigor for convenience, else we wouldnever really get anywhere since we would constantly be trying explaining tooneanother the very meaning of our correspondence. In the silly equationexample above, we usually assume (the writer of the question, that is) thatthe reader will infer from thecontext what is really intended. In this case, the intent is most probablythat we seek to identify a mathematical object known as a real number suchthat when substituted into the expression in place of x and evaluatedaccording to previously agreed upon rules and meanings of the symbols andoperations implied within (thats a few MORE assumptions), that we get atruestatement as a result (whatever THAT means, so enter yet anotherassumption).Thats a good deal of assumptions. They are very reasonable assumptionsconsidering the context, but they are assumptions nonetheless.It is also a very reasonable assumption that(1*1)+(2*2)+...+(999*999)andSigma(i=1 to 999) i^2are equivalent expressions. IOW, the ellipses represent the sum of thesquares of the integers from 3 to 998 inclusive.Is it logically IMPLIED thats what the intent denitely is? No, no morethan we knowwithout a shadow of a doubt that 16 and -16 are the desired answers to theproblem above. We simply dont know that for a fact at face value. Weapply common sense, eg this appears on a n.g. called k12.ed.math so it isextremely likely that 16 and -16 are the numbers we are looking for since itis extremely likely the assumptions made concerning the question arecorrect assumptions.The Sigma notation for the same number as the aforementioned sum does *not*set in stone what the intent is. It just makes a few of the otherassumptions moot (while introducing a few others). We Specically, we dont have to assume what... means because ... simply does not appear within the expression Sigma(i=1to 999) i^2. We do, however, have to make some other assumptions, eg whatSigma means, and what i=1 to 999 means.Indeed for most any mathematical statement, expression, or what not, we thereaders indeed are guessing what is really meant. Most of the time theseare very educated guesses, although many times they really are shots in thedark and often times we even see someone beginning a reply to a post withsomething like If you really mean...When I said above that it looks like the sum of squares of integers from 1to 999, yeahlooks like it. On aslightly deeper level, however, what I was really getting to is it IS justthat, ie just a different way of expressing the exact same thing, as in 1/2looks like 2/4. Probably a poor choice of words considering that at leastone reader apparently misunderstood my intent.10. What number is represented by that expression? You dont know until Itell you the base. Without further clarication, most would assume base 10though, and very effective communication on this and other forums occurevery day under such an assumption.Back to formula. If you take formula to mean some mathematical STATEMENT(often an equation), ie a complete sentence and not just words or phrasesto use that analogy, then what I MEANT when I originally referred to theOPs expression as a formula is that you could write something like:x = (1*1)+(2*2)+...+(999*999)...and that would be ne as a formula. To argue it isnt on the basis ofambiguity over what the ellipses represent, is no more valid to argue itisnt over ambiguity over what * represents, or what 2 represents, or what +represents, etc. Point is, we make reasonable assumptions what all thesethings mean. Its just that some of these assumptions may appear safer thanothers. In this case it is pretty safe (relatively speaking) to say thatmost peoples assumption of what 2 means would be correct while, hmm, theremay be some people that may incorrectly assume what ... means. Butassumptionsare like pregnancies. We either have them or we dont, regardless of howsafe an assumption we believe it to be.All that said, we really do need to be careful many times when interpreting... within such an expression. an example is the question: What is thenext number (ll in the blank) in the sequence:1,4,27,_____There are many answers, no one more correct than another, though the authormay have a specic one in mind. The expression in _this_ thread however(from the OP), is not a case where AI would think this type of ambiguityexists, although *technically* such ambiguity does exist. My point issimply, ambiguity exists everywhere anyway...-- Darrell-- newsgroup website: http://www.thinkspot.net/k12math/newsgroup charter: http://www.thinkspot.net/k12math/charter.html (1*1)+(2*2)+....(999*999)> Looks like the sum of the squares of integers from 1 to 999. The > formula is ne as is, although the same thing can be stated a > little more concisely in Sigma notation. > Ellipsis (...) is never ne in a a formula. What the ellipsis> means is you, the reader, can guess what I mean here. Thats ok> when you are giving an intuitive feel for a problem, but not> sufcient for proofsIts insufcient for proofs? What sort of use would it have in proofs for which its insufcient? Its certainly sufcient in the following proof:#> Proof that the sum of integers from 1 to n is n(n+1)/2 for each integer #> n>0:#> #> By induction on n.#> #> Case, n=1. [Omitted.]#> #> Assume that the claim is true for n=N. For n=N+1, then, we have#> 1+2+...+N+(N+1)=(N(N+1)/2)+(N+1) (by inductive hyp.), so#> 1+2+...+N+(N+1)=(N(N+1)+2N+2)/2 ^^^ There is the ellipsis.#> =(N+1)(N+2)/2, so the claim is true for n=N+1.#> #> Thus, by induction, were done.Kevin Karplus, again:> A mathematical formula is explicit, not relying on guesswork by the> reader.Pick up any journal (or textbook, for that matter), and youllfind many a formula whose meaning is not explicit. E.g., from the nearest book at hand (R. Goldberg, _Methods of Real Analysis_, 2/e, Wiley), pg. 255,> | f_n(x) - f(x) | < epsilon ( n >= N )-- where Im mixing TeX with non-TeX notation. This is far from explicit! What does the parenthetical notation on the right mean? is it a statement of fact? A condition on n in the left-hand statement? Moreover, what do those vertical bars mean? Are they just some fancy sort of parentheses? And what are > and <=? Those are naive questions, but even *with* mathematical background, a priori, the statement in the book can conceivably mean the size of the set f_n(x)-f(x) (where x is a set) is less than epsilon for each n>=N. (Thats not what it means in the book.)So, no, mathmatical formulae are not explicit -- they rely on context. if youd read the rest of the book until there, youd know what Goldberg meant. Likewise, the OP relied on context. Anyone with sufcient background in math would know from context what his expression meant, so ellipsis were, indeed, ne in his formula.BA scl Math, PBK, NYU Ive been erasing too much UBE.msh210@math.wustl.edu Of a reply, then, if you have been cheated,-- newsgroup website: http://www.thinkspot.net/k12math/newsgroup charter: http://www.thinkspot.net/k12math/charter.html (1*1)+(2*2)+....(999*999)well, looks like addition of squares...1 + 4 + 9 + 16 + 25 + .... + 998001the formula is going to involve a cubic of n(1) Sum = an^3 + bn^2 + cn + dfor n = 1..999We have 1 = a + b + c + d 5 = 8a + 4b + 2c + d 14 = 27a + 9b + 3c + d 30 = 64a + 16b + 4c + dThis is a matrix equation of the form MX=BIt looks like a = 1/3 b = 1/2 c = 1/6 d = 0Sum = 999^3/3 + 999^2/2 + 999/6 + 0Ill let you gure out what this number is.-- newsgroup website: http://www.thinkspot.net/k12math/newsgroup charter: http://www.thinkspot.net/k12math/charter.html (1*1)+(2*2)+....(999*999)>> well, looks like addition of squares...>> 1 + 4 + 9 + 16 + 25 + .... + 998001>> the formula is going to involve a cubic of n>> (1) Sum = an^3 + bn^2 + cn + d>> for n = 1..999>> We have>> 1 = a + b + c + d> 5 = 8a + 4b + 2c + d> 14 = 27a + 9b + 3c + d> 30 = 64a + 16b + 4c + d>> This is a matrix equation of the form MX=B>> It looks like a = 1/3> b = 1/2> c = 1/6> d = 0>> Sum = 999^3/3 + 999^2/2 + 999/6 + 0>> Ill let you gure out what this number is.?Its just an addition of the squares of integers from 1 to 999. Thats all.Dont overcomplicate things with introducing cubes, etc. Just let n runfrom 1 to 999, squaring each term, and summing them all. There is indeed aquick and easy formula for such a sum. thats what was asked for, aformula.SPOILER||||||||||||||||||||||||||Sigma(i=1 to n) i^2 = n(n+1)(2n+1) /6In this case n=999= 999(1000)(1999) /6= 332,833,500-- Darrell-- newsgroup website: http://www.thinkspot.net/k12math/newsgroup charter: http://www.thinkspot.net/k12math/charter.html =Darrell:I was trying to stress 2 points.Sums of any polynomial equations of power n always involves a polynomialequation of power n+1.Secondly I was trying to illustrate the power of using matrices tofind thecoefcients of any polynomial equation.We both came to the same answer.GP-Pari has a neat expression sum(x=1,999,x^2)I just didnt want to just write 332,833,500 and let it go at that.-- newsgroup website: http://www.thinkspot.net/k12math/newsgroup charter: http://www.thinkspot.net/k12math/charter.html Darrell:>> I was trying to stress 2 points.>> Sums of any polynomial equations of power n always involves a polynomial> equation of power n+1.x + x involves a quadratic?>> Secondly I was trying to illustrate the power of using matrices tofindthe> coefcients of any polynomial equation.>> We both came to the same answer.>> GP-Pari has a neat expression sum(x=1,999,x^2)>> I just didnt want to just write 332,833,500 and let it go at that.Why does removal of the entire context (usually quoted) appear to becommonplace on this n.g. :-)At any rate, I refreshed my memory what you are talking about and mustdeclare that Ifind your choice of formula quite inconvenient. I wouldguess that most would assume the intent of the OP to be something like:give me a quick way to rewrite this expression concisely, and/or give recipefor evaluation.Whats GP-Pari out of curiosity, cause the expression you listed (which Ialso listed and also gave formula of evaluation) is a very well-knownsummation formula ?-- Darrell-- newsgroup website: http://www.thinkspot.net/k12math/newsgroup charter: http://www.thinkspot.net/k12math/charter.html Sums of any polynomial equations of power n always involves a polynomial> equation of power n+1.>> x + x involves a quadratic?The sum of the naturals to the 1st power certainly is a quadratic:1^1 + 2^1 + 3^1 + . . . + n^1 = n(n+1)/2And in general if we have a polynomial:P(x) = a0 + a1x^1 + a2x^2 + . . . + anx^n and we form the sumP(1) + P(2) + P(3) + . . . + P(k)there will be a closed form formula for the sum that will be in the form ofa polynomial of power n+1. And the method of undetermined coefcients canbe used to determine what the coefcients of that polynomial are. I thinkthat is what the poster meant when they said Sums of any polynomialequations of power n always involves a polynomial equation of power n+1.Rich-- newsgroup website: http://www.thinkspot.net/k12math/newsgroup charter: http://www.thinkspot.net/k12math/charter.html > Sums of any polynomial equations of power n always involves apolynomial> equation of power n+1.>> x + x involves a quadratic?>> The sum of the naturals to the 1st power certainly is a quadratic:>> 1^1 + 2^1 + 3^1 + . . . + n^1 = n(n+1)/2>> And in general if we have a polynomial:>> P(x) = a0 + a1x^1 + a2x^2 + . . . + anx^n and we form the sum>> P(1) + P(2) + P(3) + . . . + P(k)>> there will be a closed form formula for the sum that will be in the formof> a polynomial of power n+1. And the method of undetermined coefcientscan> be used to determine what the coefcients of that polynomial are. Ithink> that is what the poster meant when they said Sums of any polynomial> equations of power n always involves a polynomial equation of power n+1.rather long way to approach the problem assuming we haveSigma(i=1,n)[i^2]=[n(n+1)(2n+1)]/6 at our disposal. But such an assumptionmay be unwarranted, or even if it is warranted there is certainly value inlearning to address the problem as anonymous suggested. At any rate,yeah, even that sum formula obviously involves a cubic.-- Darrell-- newsgroup website: http://www.thinkspot.net 7ï WEcf !.85Ãèÿÿ Many people never give it any thought, and may evenerroneously think that the measured speed is the *average* speed of theball from pitcher to batter or from server to the returner of serve.Lots of people, like coaches and some parents, have radar guns to measurethe speed of the ball thats pitched (or served). Lots of kids even haveGlove Radar(tm), a small radar attached to their basebal glove whichmeasures the speed of the ball at a short distance from the glove.The point is, accurate measurement of the speed of a baseball is fairlycommonplace. Forcing an assumption of average speed or negligible airresistance ies in the face of these nice measurements. (Sorry, Icouldnt resist. :) )In physics we are given the formula d = (1/2)at^2 for the distance that anobject will fall under the inuence of gravity. Any good teacher ofphysics *or of mathematics* who covers this topic will quickly add thatthis formula is made inaccurate very quickly due to air resistance on thefalling object. The point need not be belabored, but it *should be made*.No one had mentioned that the proportion method would give an answer thatsimilarly would be inaccurate, due primarily to air resistance.That was my point. *It should be mentioned*.And even more to the point, the interaction of the air with the pitchedball cant be ignored by anyone even mildly familiar with baseball. Itsthe air that gives the pitcher the variety of pitches in his bag oftricks. Try throwing a curve ball on the moon, Darrell.-- newsgroup website: http://www.thinkspot.net/k12math/newsgroup charter: http://www.thinkspot.net/k12math/charter.html In physics we are given the formula d = (1/2)at^2 for the distance that an> object will fall under the inuence of gravity. Any good teacher of> physics *or of mathematics* who covers this topic will quickly add that> this formula is made inaccurate very quickly due to air resistance on the> falling object.Uh... That formula is derived from an assumption of constant acceleration(and not just the acceleration of gravity). Any good teacher of physics *orof mathematics* who covers this topic will ACTUALLY say that this formula ismade inaccurate if you use it for accelerations that are not constant, aswith air resistance.-- newsgroup website: http://www.thinkspot.net/k12math/newsgroup charter: http://www.thinkspot.net/k12math/charter.html > Lets get real... a simple introductory algebra question implying nothingmore> than a simple proportion? This interpretation is not obvious. On the> face of it, it looks like an honest question, not a textbook homework> problem. If it is a textbook homework problem, must there be a> dumbing-down of the physics of baseball merely to justify some> oversimplication? There are plenty of non-defective textbook problems> on proportions... from mixing a batch of concrete to adjusting a recipe> for home-made fudge.Whether a problem from some textbook or worksheet, or an actual questionfrom an actual little leaguer curious how his pitching speed measures up8realtively speaking* to that of a big leagure, is irelevent. What _is_relevent, is the question requires no more than a simple proprtion to answerunder certain very reasonable assumptions.Under certain other assumptions, the question can involve considerably moresophisticated techniques to answer. The question remains: Do you reallythink were supposed to take all the factors you mention into account?>> Radar guns are used at baseball games and tennis matches to measure the> speed of the ball. Many people never give it any thought, and may even> erroneously think that the measured speed is the *average* speed of the> ball from pitcher to batter or from server to the returner of serve....and some may think its the instanteous speed at the precise intersectionof radar beam and baseball. So what. The means by which the speed ismeasured is never addressed in this problem.>> Lots of people, like coaches and some parents, have radar guns to measure> the speed of the ball thats pitched (or served). Lots of kids even have> Glove Radar(tm), a small radar attached to their basebal glove which> measures the speed of the ball at a short distance from the glove.How is this relevant to the question?>> The point is, accurate measurement of the speed of a baseball is fairly> commonplace. Forcing an assumption of average speed or negligible air> resistance ies in the face of these nice measurements. (Sorry, I> couldnt resist. :) )Again, whats the relevancy here?>> In physics we are given the formula d = (1/2)at^2 for the distance that an> object will fall under the inuence of gravity. Any good teacher of> physics *or of mathematics* who covers this topic will quickly add that> this formula is made inaccurate very quickly due to air resistance on the> falling object.>> The point need not be belabored, but it *should be made*...but youre doing exactly that, belaboring the point. why not also accointfor the error in measurement of the speed? For that matter, why not clarifyexactly WHAT speed weer talking about (instantenous speed at time ballleaves pitcher, average speed from pitcher to batter, speed as measured bysome radar or laser device, etc.)>> No one had mentioned that the proportion method would give an answer that> similarly would be inaccurate, due primarily to air resistance.>> That was my point. *It should be mentioned*.It is also true that the path from pitcher to catcher (or bat) is not astraight line, the baseball has gyroscopic properties (it may be rotating),and a MYRIAD of other equally interesting physical phenominae at work herethat effect the answer to the question, or make the question ambiguous. Atsome point we need to make some reasonable assumptions what is intended bythe question and what is implied by the given information. Of course, atany time (and not just in this thread but in ANY thread) we run the risk ofmisinterpreting the queston, or incorrectly assuming what was intended.Again, the question remains: Do you really think all you mentioned wasintended to be taken into account? I dont, not without furtherclarication anyway...and I dont appear to be alone in my feelings towardsthat matter.-- Darrell-- newsgroup website: http://www.thinkspot.net/k12math/newsgroup charter: http://www.thinkspot.net/k12math/charter.html =>the question requires no more than a simple proprtion to answer> under certain very reasonable assumptions.*You* are assuming that there is no air on earth. *I* am assuming that the baseball is being pitched on earth.> The question remains: Do you really> think were supposed to take all the factors you mention into account?*I* mentioned *one* factor only -- the ball ies through air. *You* are the only one mentioning other factors, probably in the hopethat this will hide your no-air reasonable assumption... here areyour words:>>? Do you really think the intent of the problem is to consider all these>>factors? Youre reading too much into the problem. Why not also consider>>that the mound is of higher elevation than the plate, air resistance,>>gravity, the pitch may not necessarily be straight, the position of the>>sun in the sky, etc. etc. etc.More of your words:> It is also true that the path from pitcher to catcher (or bat) is not a> straight line, the baseball has gyroscopic properties (it may be rotating),> and a MYRIAD of other equally interesting physical phenominae at work here> that effect the answer to the question, or make the question ambiguous. At> some point we need to make some reasonable assumptions what is intended by> the question and what is implied by the given information. Of course, at> any time (and not just in this thread but in ANY thread) we run the risk of> misinterpreting the queston, or incorrectly assuming what was intended.> Again, the question remains: Do you really think all you mentioned was> intended to be taken into account?Hey, if the air resistance factor alone makes your analysis andmethodology wrong, why bother? >I dont, not without further clarication anyway...Gosh, do you think that clarication will justify assuming *no air*?> and I dont appear to be alone in my feelings towards> that matter.Wow! That sure settles it. Maybe we can all vote on whether there is airon earth.Next week, should we vote on whether the earth is at, too?Darrell, you keep using the assumptions and clarication argument asthough you are saying if its a problem in an algebra book, lets use amethod that is consistent with the material in that book. Im saying that if its a problem in an algebra book, and if the materialin that book clearly does not apply to the problem, and if the technique(proportions) gives a really wrong answer, then the problem isinappropriate and defective for that level.In other threads people argue about the relevance and necessity ofmathematics for the average Joe. Promoting junk science, merely becauseit makes a junk mathematical method appropriate, really encouragesridicule for mathematics.--- Joe-- newsgroup website: http://www.thinkspot.net/k12math/newsgroup charter: http://www.thinkspot.net/k12math/charter.html > >the question requires no more than a simple proprtion to answer> under certain very reasonable assumptions.>> *You* are assuming that there is no air on earth.How silly of you. Of course there is air on the Earth. all I am assumingis were not supposed to factor in air resistance in THIS problem. Have younever seen physical problems in math books where such instructons are given(or even IMPLIED?)>> *I* am assuming that the baseball is being pitched on earth.I can see you are being overly pedantic here. by your standard, there issimply not enough info to answer the question. I suspect you would justthrow your hands in the air and state not enough info.> The question remains: Do you really> think were supposed to take all the factors you mention into account?> *I* mentioned *one* factor only -- the ball ies through air.Thats the point. You seem to be very insistent that certain real-worldfactors be considered but for some reason are very choosy over WHICH factorsshould be considered. Hey, if youre going to be pedentic then go all theway! You should also consider every otehr possible factor, and like I saidby that standard there is certainly not near enough info given to answer thequestion.>> *You* are the only one mentioning other factors, probably in the hope> that this will hide your no-air reasonable assumption... here are> your words:We dont need to requote prior posts. If you, I, or anyone else need referto them, we can look them up. That said, the record is very clear that itwas no one other than yourself that insisted on mentioning other factorsto be considered for this problem.>>? Do you really think the intent of the problem is to consider allthese>>factors? Youre reading too much into the problem. Why not alsoconsider>>that the mound is of higher elevation than the plate, air resistance,>>gravity, the pitch may not necessarily be straight, the position ofthe>>sun in the sky, etc. etc. etc.>> More of your words:Apparently you are avoiding the question.<...>> Darrell, you keep using the assumptions and clarication argument as> though you are saying if its a problem in an algebra book, lets use a> method that is consistent with the material in that book.nothing wrong with that if that is indeed the case. Many such fallingobject problems not only in algebra books but even in calculus book arepresented under the assumption of negligible air resistance, if not statedEXPLICITLY as many times it is.>> Im saying that if its a problem in an algebra book, and if the material> in that book clearly does not apply to the problem, and if the technique> (proportions) gives a really wrong answer, then the problem is> inappropriate and defective for that level.>> In other threads people argue about the relevance and necessity of> mathematics for the average Joe. Promoting junk science, merely because> it makes a junk mathematical method appropriate, really encourages> ridicule for mathematics.You change the subject. I simply assisted, as did otehrs, with the answerto this very simple question. Whether Joe little leaguer really NEEDS towork this out by hand is an entirely different matter.-- Darrell-- newsgroup website: http://www.thinkspot.net/k12math/newsgroup charter: http://www.thinkspot.net/k12math/charter.html = Of course there is air on the Earth.Phew, we found something to agree upon.> ... were not supposed to factor in air resistance in THIS problem. Here we go again. Admit it, you have no clue *how* to factor in air resistance.> by your standard, there is simply not enough info to answer the question.Nah... by your standard the problem is too complicated, so *poof!* youmade the air disappear.> You seem to be very insistent that certain real-world> factors be considered but for some reason are very choosy over WHICH factors> should be considered.ONE factor, which you cant cope with. Air resistance. Learn somephysics, and you will understand why it is more important than some of theJUNK that you have been throwing around.> We dont need to requote prior posts. If you, I, or anyone else need refer> to them, we can look them up.If only you would... > Apparently you are avoiding the question.You ignore an important factor, and want me to address the JUNK that youthrow around, so I am ignoring your little debaters tricks that you useto attempt to conceal your ignorance.> Whether Joe little leaguer really NEEDS to work this out by hand is anentirely different matter.Wow! Another little trick from the debater. No one needs to work thisout by hand... oh wait, maybe *you* do.Darrell, stick to trying to beat people up in an argument about long division: I see. So you acknowledge that studying long division for necessaryutilitarian need is not nearly as important now as it was long ago? You have a necessary need (heheh) to study physics.-- -- newsgroup website: http://www.thinkspot.net/k12math/newsgroup charter: http://www.thinkspot.net/k12math/charter.html Of course there is air on the Earth.>> Phew, we found something to agree upon.>> ... were not supposed to factor in air resistance in THIS problem.>> Here we go again. Admit it, you have no clue *how* to factor in airresistance.>> by your standard, there is simply not enough info to answer thequestion.>> Nah... by your standard the problem is too complicated, so *poof!* you> made the air disappear.I fail to see the benet of such an exchange. Your increasing rudeness isdistracting from your point.> You seem to be very insistent that certain real-world> factors be considered but for some reason are very choosy over WHICHfactors> should be considered.> ONE factor, which you cant cope with. Air resistance.So were supposed to take from this that you chose a single factor tostress, being air resistance, among a myriad of factors available and*applicable* in the real world, based solely on some believe that Ive neverworked a problem where air resistance is NOT neglible?What ind of basis is that for such an argument? Seems like nothing morethan one upmanship to me.> Learn some> physics, and you will understand why it is more important than some of the> JUNK that you have been throwing around.Lean smm-115 Here is a pb I'd like to submit, which I guess is close to the Metric > TSP and wonder whether some of you may know a solution (exact or approx).> Given an interdistance matrix (Aij=d(Xi,Xj) for some XiOs), what is > the best permutation so that given r>0 the sum, for every line of the > matrix, of the r terms around the diagonal is minimum?Could you be a bit more explicit about what the problem is?What are you permuting? What are you minimizing?One possibility that comes to mind:Find P, a permuttion. I also dont recall being givenany information regarding the height of the mound and the height of theplate.There are a myriad of things in addition to air resistance that effect thepath of a real projectiles motion. But for some reason, its OK not todwell on anything OTHER than air resistance, and if we dont swell on airresistance (to include a neglible air resistance) then we have committedsome sin or something.I leave the rest of the thread to you as apparantly ypou ust dont get itand never will. I leave one last thought to consider. Did it ever occur toyou that if this really IS a real kid with a real curiousity about hisrelative pitching speed compared to a big leagure, that for all intents andpurposes the comparison shuld be made dont need to requote prior posts. If you, I, or anyone else needrefer> to them, we can look them up.> If only you would...> Apparently you are avoiding the question.> You ignore an important factor,I, as many reasonable people, also ignore several other important factorsbecuase no data was given pertinent to any such factors therefore areasonable person may choose to assume any and all such factors pitches.Look back at the problem, J.J. the ONLY thing asked for was the speed theball need be travelling in order to go a specic longer distance in thesame amount of time. thats the only thing that was asked for, NOT theactual precise equations of motion.> Darrell, stick to trying to beat people up in an argument about longdivision:>> I see. So you acknowledge that studying long division for necessary> utilitarian need is not nearly as important now as it was long ago?? this is from anotehr thread entirely. J.J, it is very obvious that youhave some type of problem with me and I really don;t think the problem is rooted in anything that has been written in THIS thread. If you havesomething to say regarding another thread, then please do so within thatthread, or start your own thread.-- Darrell-- newsgroup website: http://www.thinkspot.net/k12math/newsgroup charter: http://www.thinkspot.net/k12math/charter.html if a little league pitcher can pitch 72 mph and the distance from the> mound to home plate is 54 feet, how many miles per would it have to to> be going if it were to go 60.6 feet in the same amount of time?> Is there a formula that I can use?It looks like all of the replies so far have been mathematically simplistic. Simple proportions *could* be used if the speed of a baseball wereconstant in its ight from pitcher to batter.But its not.A typical major league fastball loses about 9 mph from the moment itleaves the pitchers hand until comprehensive in answering theOPs simple question.In the Physics ofBaseball, which *may* answer the question. I dont own a copy.Coming up with *accurate* equations for the motion of a pitched baseballwould be fun, I suppose. Its really a question of exteriorballistics... the ight of a projectile through the atmosphere.--- Joe-- -- newsgroup website: http://www.thinkspot.net/k12math/newsgroup charter: http://www.thinkspot.net/k12math/charter.html > if a little league pitcher can pitch 72 mph and the distance from the> mound to home plate is 54 feet, how many miles per would it have to to> be going if it were to go 60.6 feet in the same amount of time?>> Is there a formula that I can use?> It looks like all of the replies so far have been mathematicallysimplistic.>> Simple proportions *could* be used if the speed of a baseball were> constant in its ight from pitcher to batter.? Do you really think the intent of the problem is to consider all thesefactors? Youre reading too much into the problem. Why not also considerthat the mound is of higher elevation than the plate, air resistance,gravity, the pitch may not necessarily be straight, the position of thesun in the sky, etc. etc. etc.>> But its not.>> A typical major league fastball loses about 9 mph from the moment it> leaves the pitchers hand until it crosses the plate.Obvsiously, we are talking about average comprehensive in answering the> OPs simple question.>> In Physics of> Baseball, which *may* answer the question. I dont own a copy.>> Coming up with *accurate* equations for the motion of a pitched baseball> would be fun, I suppose.We would need much more information than what was given in order to do so.But, thats not what was asked for (the equations of motion).> Its really a question of exterior> ballistics... the ight of a projectile through the atmosphere.No, its just a simple introductory algebra question implying nothing morethan a simple proportion. You are overcomplicating the question. Thethings you discuss are certainly interesting, but are not the point of_this_ problem, obviously.-- Darrell-- newsgroup website: http://www.thinkspot.net/k12math/newsgroup charter: http://www.thinkspot.net/k12math/charter.html =hi. how many possible combinations will i get if there are 7 numbersto be drawn from a total of 27 balls? this is like a lotto game http://www.thinkspot.net/k12math/newsgroup charter: http://www.thinkspot.net/k12math/charter.html hi. how many possible combinations will i get if there are 7 numbers> to be drawn from a total of 27 balls? this is like a lotto 888030. -- G.C.-- newsgroup website: http://www.thinkspot.net/k12math/newsgroup charter: http://www.thinkspot.net/k12math/charter.html hi. how many possible combinations will i get if there are 7 numbers> to be drawn from a total of 27 balls? this is like a lotto balls are drawn without replacement (you can only pick each ball once) and order doesnt matter, then there are 27 choose 7 ways to pick the balls (that is, 27!/(20!*7!))-- newsgroup website: http://www.thinkspot.net/k12math/newsgroup charter: http://www.thinkspot.net/k12math/charter.html =I have developed Basically Math as a result of working with Title Istudents. It is a small structured booklet for students to learnfrom. Basically Math contains addition, subtraction, multiplicationand division. In addition to original learning of the facts, it is agreat way to review. The second book is tests and a graph. I havetried this method, in fact, it was developed as a result of my work asa substitute teacher. I also worked as a volunteer during summerschool so that I could use my materials and as a result, my method hasimproved. I am just nishing up (after eight months)and plan tosubmit it to a publisher. I hope to run this as a home business. Weare trying to keep this cost effective so schools will be able to useit and perhaps PTOs can sell copies so students have one at home. Ifyou are still interested, I can send you a copy after it is printed. Let me know if you are interested. Hope you are having a good schoolyear. I am subbing again as usual. I retired after thirty-sevenyears of teaching, however, I never left the school. The bigdifference now is that I have a choice in whether or not I go to website: http://www.thinkspot.net/k12math/newsgroup charter: http://www.thinkspot.net/k12math/charter.html =school district (a different one than my daughter attends). Ourdistrict uses a different math program than my daughters. My daughtercame home distraught on Monday because she didnt feel like she haddone well on her timed math test. There were 73 problems (additionwith answers up to 12). She only completed 20 problems. I think thisis unrealistic for a 1st grader in the rst month of school. They didmath of course in kindergarten but nothing like these problems. I wasable to see the teachers manual and it stated that this page is atimed test for 6 minutes. On the next page it gave directions on howto do the back page. My understanding is that the rst page (24problems) was what was suppposed to be completed in the six minutes.Not the full 73 (there were 49 on the back side). Please help meunderstand this. Am I misinterpreting the directions? My daughter isa letter to the teacher to give me a call when she had a chance. Shedid and immediately got a huge attitude with me. She kept repeatingthat 2 children (there are 22 in the class) that nished all 73problems and got them all right. My response to her was I am happyfor those chidlren but theya rent mine and I am rally concerned aboutthem. She just kept repeating that remark. Am I being stubborn onthis? We do ash carbs for 15-30 mintues a night-- 6 nites a week..Am I being unrealistic or in denial? What steps can I take to ensureher success in this? Please any feedback..criticism or suggestions,-- newsgroup website: http://www.thinkspot.net/k12math/newsgroup charter: http://www.thinkspot.net/k12math/charter.html =Hey! I was hoping someone could help me with these two problems. Ineed to simplify. Im going to use the word RAD for the radical sign(because I dont know what else to use) : )Problem 1:RAD2 + RAD2/49 (For RAD2/49,the radical sign is over the entire problem)Problem 2: 3 RAD2 + RAD50~Poppy~-- newsgroup website: http://www.thinkspot.net/k12math/newsgroup charter: http://www.thinkspot.net/k12math/charter.html Hey! I was hoping someone could help me with these two problems. I> need to simplify. Im going to use the word RAD for the radical sign> (because I dont know what else to use) : )> Problem 1:> RAD2 + RAD2/49 (For RAD2/49,> the radical sign is over the entire problem)> Problem 2: > 3 RAD2 + RAD50without parentheses, Ifind these formulas incomprehensible.A common way to write the radical sign, when it is used for squareroot, is as sqrt(x).So was your rst question sqrt(2) + sqrt(2/49) or sqrt(2+sqrt(2)/49) or sqrt(2 + sqrt(2/49)) or ...To solve your problems, remember that sqrt(x * y^2) = sqrt(x) * abs(y)(for real values of x and y).-- 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, BioinformaticsA§iations for identication only.-- newsgroup website: http://www.thinkspot.net/k12math/newsgroup charter: http://www.thinkspot.net/k12math/charter.html =Poppy,For the rst one RAD2 is already the simplest, however for the fraction youcan take the square root of 49 and get 7. So that would give you RAD2 +(RAD2)/7 notice that now the radical is only in the numerator. Them you canmultiply the rst one by (7/7) since that equals one and then you have likefractions and you can add them... 7RAD2/7 + (RAD2)/7 = 8RAD2/7.the second one 3RAD2 is already simplest....but for the second part you canrewrite it as RAD(2 * 25), then take the square root of the 25 and take itoutside the RAD, giving 5RAD2. Then since you have the same radical for bothparts: 3RAD2 + 5RAD2 you can add them...to get 8RAD2.If you just had me do your homework, I hope understand what I did or when itcomes up on the test youre out of luck.Good Luck,John> Hey! I was hoping someone could help me with these two problems. I> need to simplify. Im going to use the word RAD for the radical sign> (because I dont know what else to use) : )>> Problem 1:> RAD2 + RAD2/49 (For RAD2/49,> the radical sign is over the entire problem)> Problem 2:> 3 RAD2 + RAD50>> ~Poppy~>-- newsgroup website: http://www.thinkspot.net/k12math/newsgroup charter: http://www.thinkspot.net/k12math/charter.html =OK, Im stumped.The general formula for compound interest where you want a lump sumfuture value from your present value is:FV = PV (1+(r/m))^mN(where FV is future value, pv is todays value, r is annual/simpleinterest rate, m is number of compounding periods per year, N isnumber of years)I cant seem to convert this to an eqn that solves for m. I cantget m to not be on both sides of the equation. There must be someidentity, simplication or substitution.In fact, I cant evenfind the end formula - I assume I CAN solve form, right?-- newsgroup website: http://www.thinkspot.net/k12math/newsgroup charter: http://www.thinkspot.net/k12math/charter.html The general formula for compound interest where you want a lump sum> future value from your present value is:> FV = PV (1+(r/m))^mN> (where FV is future value, pv is todays value, r is annual/simple> interest rate, m is number of compounding periods per year, N is> number of years)> I cant seem to convert this to an eqn that solves for m. I cant> get m to not be on both sides of the equation. There must be some> identity, simplication or substitution.> In fact, I cant evenfind the end formula - I assume I CAN solve for> m, right?I dont believe that there is a simple closed-form solution for this problem.You will have to use numerical approximation methods.You should be aware that for large m you may not have enough precisionin numbers rounded to the nearest penny to be able to distinguish onem from another, since the limit as m->innity of (1+ r/m)^(mN) ise^(rN) (sometimes called continuous compounding). Daily compoundingis very close to continuous compounding, so I often use continuouscompounding as an approximation method to the more complicated dailycompounding technique that banks prefer.-- 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, BioinformaticsA§iations for identication only.-- newsgroup website: http://www.thinkspot.net/k12math/newsgroup charter: http://www.thinkspot.net/k12math/charter.html The general formula for compound interest where you want a lump sum> future value from your present value is:> FV = PV (1+(r/m))^mN> (where FV is future value, pv is todays value, r is annual/simple> interest rate, m is number of compounding periods per year, N is> number of years)> I cant seem to convert this to an eqn that solves for m. I cant> get m to not be on both sides of the equation. There must be some> identity, simplication or substitution.> In fact, I cant evenfind the end formula - I assume I CAN solve for> m, right?> I dont believe that there is a simple closed-form solution for this problem.> You will have to use numerical approximation methods.> You should be aware that for large m you may not have enough precision> in numbers rounded to the nearest penny to be able to distinguish one> m from another, since the limit as m->innity of (1+ r/m)^(mN) is> e^(rN) (sometimes called continuous compounding). Daily compounding> is very close to continuous compounding, so I often use continuous> compounding as an approximation method to the more complicated daily> compounding technique that banks prefer.> -- > Kevin Karplus karplus@soe.ucsc.edu http://www.soe.ucsc.edu/~karplus> life member (LAB, Adventure Cycling, American Youth Hostels)> Effective Cycling Instructor #218-ck (lapsed)> Professor of Computer Engineering, University of California, Santa Cruz> Undergraduate and Graduate Director, Bioinformatics> A§iations for identication only.And of course, theres not always a solution, period - ie, thesituation where no amount of compounding (unless you you believe inquantum compounding more frequent than continuous?) would be enoughto get to the future value.-- newsgroup website: http://www.thinkspot.net/k12math/newsgroup charter: http://www.thinkspot.net/k12math/charter.html =A linear system contains two equations. The graph of the rstequation is a line that passes through the points A(2,3) and B(-2,7). Determine the equation of the second line if it isPERPENDICULAR to the rst line and the solution to the linear systemis (0,5).I found the equation of the line running through the points A and Band got y=-x+5. Now I know that perpendicular to this is m=1, but whatelse do I do? What do I put on a graph? Help me out PLEASE! ASAP!-- newsgroup website: http://www.thinkspot.net/k12math/newsgroup charter: http://www.thinkspot.net/k12math/charter.html A linear system contains two equations. The graph of the rst> equation is a line that passes through the points A(2,3) and> B(-2,7). Determine the equation of the second line if it is> PERPENDICULAR to the rst line and the solution to the linear system> is (0,5).>> I found the equation of the line running through the points A and B> and got y=-x+5. Now I know that perpendicular to this is m=1, but what> else do I do?The system, as determined by you thusfar, is:y = -x + 5y = x + b...for a certain constant bAsk yourself how you determined the constant term for the rst equation was5. Because you were given (0,5) as the solution, know the slope y-interceptform, and 5=(-0)+5, right?So, whats b? Is not also the 2nd equation in slope y-intercept form and isnot the solution (0,5) the y-intercept for BOTH equations?> What do I put on a graph?Nothing, unless you are told to graph something or just want to.-- Darrell-- newsgroup website: http://www.thinkspot.net/k12math/newsgroup charter: http://www.thinkspot.net/k12math/charter.html A linear system contains two equations. The graph of the rst>equation is a line that passes through the points A(2,3) and >B(-2,7). Determine the equation of the second line if it is>PERPENDICULAR to the rst line and the solution to the linear system>is (0,5).>>I found the equation of the line running through the points A and B>and got y=-x+5. Now I know that perpendicular to this is m=1, but what>else do I do? >What do I put on a graph? Help me out PLEASE! ASAP!>>-- Your rst equation is correct.m=1 is correct for the second line.You then have y=x + b and you want a value for b.What to put on a graph? For second line, you are given a point on this line,(0,5). Graph this point. graphicallyfind other points on this line using theslope m=+1; connect the points to represent the line. Obviously, they-intercept for this second line is given as point (0,5)....G C-- newsgroup website: http://www.thinkspot.net/k12math/newsgroup charter: http://www.thinkspot.net/k12math/charter.html >A linear system contains two equations. The graph of the rst>>equation is a line that passes through the points A(2,3) and >>B(-2,7). Determine the equation of the second line if it is>>PERPENDICULAR to the rst line and the solution to the linear system>>is (0,5).>>I found the equation of the line running through the points A and B>>and got y=-x+5. Now I know that perpendicular to this is m=1, but what>>else do I do? >>What do I put on a graph? Help me out PLEASE! ASAP!>>-- > Your rst equation is correct.> m=1 is correct for the second line.> You then have y=x + b and you want a value for b.> What to put on a graph? For second line, you are given a point on this line,> (0,5). Graph this point. graphicallyfind other points on this line using the> slope m=+1; connect the points to represent the line. Obviously, the> y-intercept for this second line is given as point (0,5)....> G C> In general, if you know the slope of the second line (in your case, m = 5), and you know a point on that line (you know that (0, 5) is on the line), you can substitute x, y, and m into the slope-intercept form of a linear equation y = mx + b tofind b.In your case, its simpler - youre given a point thats on the y-axis, which means that the y-coordinate of that point is your y-intercept; that is, b = 5. So your slope is m = 1, y-intercept is b = 5, and your line is y = 1x + 5.-- newsgroup website: http://www.thinkspot.net/k12math/newsgroup charter: http://www.thinkspot.net/k12math/charter.html =problem reads Valley middle school is having a canned food driveSixthgrade students have collected 150 more cans than the seventhgrade students. Together the students have collected a total of 530cans How many more cans did each grade level collect? is there aneasy way to explain how to gure this problem out?-- newsgroup website: http://www.thinkspot.net/k12math/newsgroup charter: http://www.thinkspot.net/k12math/charter.html problem reads Valley middle school is having a canned food drive>Sixthgrade students have collected 150 more cans than the seventh>grade students. Together the students have collected a total of 530>cans How many more cans did each grade level collect? is there an>easy way to explain how to gure this problem out?>Yes.You have an error in the question, but the correction is probably to remove theword more.G C-- newsgroup website: http://www.thinkspot.net/k12math/newsgroup charter: http://www.thinkspot.net/k12math/charter.html >problem reads Valley middle school is having a canned food drive>>Sixthgrade students have collected 150 more cans than the seventh>>grade students. Together the students have collected a total of 530>>cans How many more cans did each grade level collect? is there an>>easy way to explain how to gure this problem out?> Yes.> You have an error in the question, but the correction is probably to> remove the word more.Why do you think there is an error in the question? The questionmakes sense the way it is, and removing the word more changes thequestion into a totally different one.-- 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, BioinformaticsA§iations for identication only.-- newsgroup website: http://www.thinkspot.net/k12math/newsgroup charter: http://www.thinkspot.net/k12math/charter.html problem reads Valley middle school is having a canned food drive>Sixthgrade students have collected 150 more cans than the seventh>grade students. Together the students have collected a total of 530>cans How many more cans did each grade level collect? is there an>easy way to explain how to gure this problem out?>> Yes.>> You have an error in the question, but the correction is probably to>> remove the word more.>>Why do you think there is an error in the question? The question>makes sense the way it is, and removing the word more changes the>question into a totally different one.>-- >Kevin Karplus tThe difculty is this: How many more cans did each grade level collect?Each grade level more than what? The rst use of more seems reasonable. The second more seems out of place. Grade six class collected 190 cans more than grade 7 class. Grade six thenactually collected 190 + 150 cans. Grade 7 collected 190 cans. The moredoes not apply to grade 7 class.Obviously I did not rst solve the original given problem; only gave a commentof something that needed adjustment. G C -- newsgroup website: http://www.thinkspot.net/k12math/newsgroup charter: http://www.thinkspot.net/k12math/charter.html problem reads Valley middle school is having a canned food drive>Sixthgrade students have collected 150 more cans than the seventh>grade students. Together the students have collected a total of 530>cans How many more cans did each grade level collect? is there an>easy way to explain how to gure this problem out?>> Yes.>> You have an error in the question, but the correction is probably to>> remove the word more.>>Why do you think there is an error in the question? The question>makes sense the way it is, and removing the word more changes the>question into a totally different one.>There are two more in the original question. Removing the rst onemakes no sense, as you suggest. But the second one doesnt belongthere.bob-- newsgroup website: http://www.thinkspot.net/k12math/newsgroup charter: http://www.thinkspot.net/k12math/charter.html problem reads Valley middle school is having a canned food drive> Sixthgrade students have collected 150 more cans than the seventh> grade students. Together the students have collected a total of 530> cans How many more cans did each grade level collect? is there an> easy way to explain how to gure this problem out?>If we call x the cans collected by 6th grade, x+x-150=530. Solve for x.2x-150=5302x=680x=340The 6th grade collected 340 cans and 7th grade collected 340-150=190 cans.David Moran-- newsgroup website: http://www.thinkspot.net/k12math/newsgroup charter: http://www.thinkspot.net/k12math/charter.html =Find the tangets from the point (1,-1) intersectthe curve.-plz give a step solution, I know what the answeris(from back of book), but I dont know how to do it.answer = -1+-root2-- newsgroup website: http://www.thinkspot.net/k12math/newsgroup charter: http://www.thinkspot.net/k12math/charter.html Find the tangets from the point (1,-1) intersect> the curve.> -plz give a step solution, I know what the answer> is(from back of book), but I dont know how to do it.> answer = -1+-root2Rather than ask for a solution to be just dumped into your lap, why notget your moneys worth from the school where you are taking the course? Try one of the following.1. Ask the teacher how to do this problem.2. Ask the TA how to do this problem.3. Go to the Math Dept.s tutoring center and ask how to do this problem.or, (4) ask a knowledgeable friend how to do this problem.In any of the four ways above, you can then show what you know and whereyou are confused. The person helping you can then give you the mosteffective help in solving this particular problem, as well as giving yousome insight into solving other problems like this.-- -- newsgroup website: http://www.thinkspot.net/k12math/newsgroup charter: http://www.thinkspot.net/k12math/charter.html =Im developing a Word add in that will print a graph, x-axis, y-axis,points, line segments and lines. Are there any Mathematics/Algebra teacherson this list that would like a free copy in exchange for comments,criticisms, and suggestions?Professor Martin WeissmanEssex County CollegeNewark, NJ 07102Cellphone: 347-528-7837-- newsgroup website: http://www.thinkspot.net/k12math/newsgroup charter: http://www.thinkspot.net/k12math/charter.html =Just curious, how is your add-in different than an excel chart object in aJ.> Im developing a Word add in that will print a graph, x-axis, y-axis,> points, line segments and lines. Are there any Mathematics/Algebrateachers> on this list that would like a free copy in exchange for comments,> criticisms, and suggestions?> Professor Martin Weissman> Essex County College> Newark, NJ 07102> Cellphone: 347-528-7837>-- newsgroup website: http://www.thinkspot.net/k12math/newsgroup charter: http://www.thinkspot.net/k12math/charter.html colleagues,Does anybody know how many math teachers are in USA advance for your help.--Aliaksandr Murauski, Cradle Fields, http:///www.cradleelds.com-- newsgroup website: http://www.thinkspot.net/k12math/newsgroup charter: http://www.thinkspot.net/k12math/charter.html =That seems to be a composition of functions that evaluates in thefollowing way: (f*g)(x) = f(g(x))--Aliaksandr Murauski,Cradle Fields - Software for Mathematics Study and Research Workhttp://www.cradleelds.com>f(x) = x-3>g(x) = 1> -------> x^2 -9>>nd: (f*g)(x)>>Could anyone please show me step by step how to solve this problem?>I have a math test later on today, and forgot how to multiply>functions (I know that there is some factoring, etc. involved, just>ont know how to do it). -- newsgroup website: http://www.thinkspot.net/k12math/newsgroup charter: http://www.thinkspot.net/k12math/charter.html =Some of you may already know about this and why it works, but Id liketo share it with those who dont. And if you teach multidigitsubtraction, some students mightfind it a blessing. Nancy Clifton, afriend of mine, shared it with me years ago. She didnt know why itworked, but it worked every time. She was afraid offinding acounterexample, but taught it as a permanent sub, praying that it wasjustiable, because it was a blessing for so many of her struggling(high school) students. I assured her it was ne.I know that there are ways to use the fact that numbers arepolynomials in base 10 and that we can decompose the numbers, but weneed a fast, easy method, one that is mathematically justiable,universally applicable, and at least as easy as the standard method.So Im sharing the below from this standpoint.Ill sometimes use place value where we could use digit becauseafter modication, a place value may take on a value greater than asingle digit. (This can happen in either the regular way or thealternative way.)Ill discuss rst the regular way, then a partial algebraicexplanation of the regular way and an algebraic derivation of thisalternative way, and then this alternative way. Ill end by giving myhunches as to why this alternative way is easier for some. (And pleaseforgive any typo that may or may not exist.)Consider 657,202-219,204=437,998We all know the regular way. Ill 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 8Im sure everyone knows where the digits 4,6,1,9 came from. Algebraically, sticking to the transformed columns, what we do isthis:For each transformed column, where digit m is in the minuend and digits is in the subtrahend:(m-1)-s if m-1 >= sand(m-1+10)-s if m-1 < s (we can do the +10 mentally, so sometimes itsnot written out).But notice that(m-1)-s = m-1-s = m-s-1 = m+(-s)+(-1) = m+(-(s+1)) = m-(s+1) and(m-1+10)-s = m-1+10-s = m+10-s-1 = m+10+(-s)+(-1) = m+10+(-(s+1)) =m+10-(s+1).The last expressions in these equality sequences denote theaforementioned alternative way, shown below (notice that we haveessentially the same inequality as before after each if):m-(s+1) if m >= s+1andm+10-(s+1) if m < s+1.For the alternative way, Ill write it as wed sometimes see it onpaper, and then further explain. 6 5 7, 2 0 2- 2 1 9, 2 0 4 2 10 3 1= 4 3 7, 9 9 8Some may already see whats going on. Heres what we did:First, we go through and make the column transformations, then we goback and subtract, mentally adding 10 as in the regular way to theminuend place value where needed. Note that for the regular way or forthis alternative way, if we make the column transformations rst, wecan go back through and subtract from left to right or from right toleft - we dont have to start with the rightmost column when we get tothe subtracting. (When rst learning, I recommend that the columntransformations be done rst for both the regular way and alternativeway until theres enough condence to do it mentally. Then we donthave to go through the columns twice. But theres nothing wrong withgoing through them twice, to avoid working memory mistakes. My workingmemory is weak, so Ifind myself scheming like this to tax it less.)In the rightmost column, 2,4, we see that 2 < 4, so we assign 0+1 = 1for the next columns subtrahend place value, giving 0,1 as the newcolumn pair. 0 < 1, so we have 2+1 = 3 for the next columnssubtrahend place value, giving 2,3 as the new column pair. 2 < 3, sowe have 9+1 = 10 for the next columns subtrahend place value, giving7,10 as the new column pair. 7 < 10, so we have 1+1 = 2 for the nextcolumns subtrahend place value, giving 5,2 as the new column pair. 5is not less than 2, so we leave the next column alone, leaving 6,2 asthat column pair. We then go through again and subtract, mentallyadding 10 to the minuend place value as in the regular way whennecessary.Why is this alternative way easier for some? In the columntransformations, its conceptually less complicated.In the regular way, suppose that in a given column, such as in thesecond column from the right in the example, the top digit (in theminuend) is 0. Suppose that we need to reduce the top digit 0 by 1,replacing 0 with 9, by using modular reasoning: In mod 10, 0-1 = -1 =9, and also 0 = 10, meaning we could let 0-1 mod 10 denote 10-1 = 9mod 10. Now we have to decide whether to reduce the top digit in thenext column to the left. But in our given column, the new top digit 9is not less than the bottom digit (in the subtrahend). And, when wehave a situation such as in our example in the second from rightcolumn where 0 is the bottom digit, the old top digit 0 is also notless than this bottom digit. So here, neither the old top digit northe new top digit is less than the bottom digit. But we still have toreduce the top digit in the next column. Why? To see why, we rememberthat 9 = -1 mod 10, and in the integers, -1 is always less than thebottom digit in our given column. This can be confusing. No wondersome struggle.But in the alternative way, there is no need to use this modularreasoning. 0 is always 0, 9 is always 9, and 10 is always 10. Note: Ineither the regular way or the alternative way, when we get to thesubtracting, we mentally add 10 to the minuend place value whennecessary. So in the alternative way, regardless of whether 0 is theold or new minuend place value, 0 is still 0 and 10 is still 10.If one were to say that the regular way is preferable because itteaches modular reasoning, then Id say that if it were preferable,Id disagree that this would be a good reason. This is because in amodular context, there can be zero divisors, meaning that we can haveab = 0 even when a and b are both nonzero (2*5 = 10 = 0 in mod 10),and this means that we cannot have the type of order that allows forgreater than or less than. Before we can have this type of order, wehave to disallow zero divisors. So with the regular way, werewa'ng between two worlds, where in one we have greater than orless than and where in the other one we dont. No wonder some getdazed and confused.Try the two approaches on something like 500,000-123,456. For thereasons I outlined above, you mightfind that some who struggle in theregular way on examples like this (with lots of 0s in the minuend,especially lots of consecutive 0s) willfind the alternative easier.Paul-- newsgroup website: http://www.thinkspot.net/k12math/newsgroup charter: http://www.thinkspot.net/k12math/charter.html =10th grade math. I need some adviceIm feeling a bit overwhelmed. Nearly all of my students failed theirrst math test. There are a variety of reasons for this I think, someof my students dont do homework or studysome have entered the 9thgrade at a 5th grade or 4th grade levela few have learningdisabilities. 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 howto show them how.Out of 40 or so students I have in 9th grade, two scored Bs , threeCs, 8 Ds and the rest answered less than half of the questionscorrect. I would venture that the exam was simply too hard, but I usedquestions from the regents identical to many we did in class and ashomework.Im depressed about giving them the tests backI dont want them togive up hope. Especially the ones who do their homework and who Iknow try hard. But, they still have not grasped the concepts of ourrst unit. I cant lie to themBesides. I didnt think the test wastoo hardI timed myself taking it and it took me about 4 minutesso, Igave them 40many were still working when the bell rang. How can ittake so long to answer a question like: -3-8 ? We did dozens ofexamples in class. They did OK on the homework. Maybe they were juststressed by the test? I dont know.I teach 90 minute periods so I spent the rst half of class workingwith them on example problemsmaybe Im just a bad teacher.I dont know what to do.-- newsgroup website: http://www.thinkspot.net/k12math/newsgroup charter: http://www.thinkspot.net/k12math/charter.html I teach 9th and 10th grade math. I need some advice.> Im feeling a bit overwhelmed....Region 9; District 7.Hand back the test.Go over the test. Be supportive.They are used to doing badly.Give a makeup with the same questions,changing some or all of the numbers.Average their scores on the two tests ordiscount the rst test entirely.Its not you. Hang in there.-- newsgroup website: http://www.thinkspot.net/k12math/newsgroup charter: http://www.thinkspot.net/k12math/charter.html => 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 dont know.>> I teach 90 minute periods so I spent the rst half of class working> with them on example problems-maybe Im just a bad teacher.>> I dont know what to do.>Isnt it depressing? And it happens to all of us. You wonder just whatlanguage youve been teaching in. I doubt youre a bad teacher; youreasking the right questions of yourself for a start. Answering them is not soeasy, of course!I think the main thing is this; if they didnt get it when you taught itthe rst time, then they wont get it if you just repeat the same way ofteaching it again. Theres a story about English people (I am English, BTW!)travelling in France. Whenever they were not understood by the French,theyd very deliberately, very patiently, repeat themselves. Just louder. Ifyoure not careful, teaching can be like this. Just *how* did you teach itthe rst time through - then change that approach. Particularly if yourteaching was of the lecture/explain-do example-class exercise pattern (akatraditional). Doing dozens of examples does not always equate to learning,or understanding. I dont know - maybe that isnt youre preferred teachingstyle? Anyway, whatever you did, try a different approach if you go over thetopic again.And negative numbers are not an easy concept (I dont recall the US grades -9th/10th? What age is that?). Big clues can be gained from looking at whattheir wrong answers were. Then design a lesson focussing on theirmisconceptions; cognitive conict is the aim, in jargonese I think. Donttry to avoid their misconceptions by running a lesson on how it should bedone. Focus on what they *are* doing. Did they have consistent wronganswers? Or did they put no answers?-3-8=11 is a goodie. How does it happen? Well, two negatives make apositive, dont they?!?!? Teaching math by rule of thumb.Is 90 minutes usual in US classrooms BTW? Thats a load of time at astretch - great if you have some good activities to do, but a hell of a longtime to lecture for (or to concentrate in).M.-- =I need help on solving a system of equations using the Gaussianelimination method. Many of the help available uses three equations,and three unknowns, but Im stuck on problems such as:3x - 2y = -53x - 4y = -7A step by step explanation of how to solve a problem like this wouldbe greatly appreciated.-- newsgroup website: http://www.thinkspot.net/k12math/newsgroup charter: http://www.thinkspot.net/k12math/charter.html I need help on solving a system of equations using the Gaussian> elimination method. Many of the help available uses three equations,> and three unknowns, but Im stuck on problems such as:> 3x - 2y = -5> 3x - 4y = -7> A step by step explanation of how to solve a problem like this would> be greatly appreciated.To solve a system of equations using the Gaussian elimination methodwe must multiply second, third, ... equations so rst non-zerocoefcient would be equal to that of rst equation and would accordwith the same unknown variable (rst equation must have non-zerocoefcient before rst independent variable).Example:3*x-2*y=-5x-y=5So we must multiply second equation by 33*x-2*y=-53*x-3*y=15In case when coefcient before rst unknown variable is zero forsecond or third or ... equation we must skip this equation.After that we must take away rst equation from all other equations.New system will consist of this differences and rst equation.In case when coefcient before rst unknown variable is zero forsecond or third or ... equation we must again skip this equation.Example:(3*x-3*y)-(3*x-2*y)=15-(-5) -y=20New system:3*x-2*y=-5-y=20Example for zero coefcient:3*x-2*y+5*z=152*x-4*y+6*z=205*y+3*z=2Multiply second equation by 3/2 and skip third equation:3*x-2*y+5*z=153*x-6*y+9*z=305*y+3*z=2Take away rst equation from second equation and skip third:(3*x-6*y+9*z)-(3*x-2*y+5*z)=30-15 -4*y+4*z=15New system:3*x-2*y+5*z=15 (will be left on next stage)-4*y+4*z=155*y+3*z=2After that, we must leave the rst equation aside and repeat thisoperation counting that the rst equation is next (in example,-4*y+4*z=15).Example:-4*y+4*z=155*y+3*z=2Multiply second equation by -4/5:-4*y+4*z=15-4*y-12/5*z=-8/5Take away rst from second:(-4*y-12/5*z)-(-4*y+4*z)=-8/5-15 -32/5*z=-83/5 32*z=83New system:-4*y+4*z=1532*z=83We must repeat this operations until we will have in the last equationonly one variable (we have it in this case). Then we must write allequation that we have left and last view of the system. Then we mustsolve this system from last to rst equation.Example:3*x-2*y+5*z=15-4*y+4*z=1532*z=83Third z=83/32;Second: -4*y+4*(83/32)=15 -4*y+83/8=15 -32*y+83=120 -32*y =37 y=-37/32;First: 3*x-2*(-37/32)+5*(83/32)=15 3*x-74/32+415/32=15 3*x+341/32=15 96*x+341=480 x=131/96.Solving of your case:Multiplying (not need)Taking away:(3x - 4y) - (3x - 2y) = -7 - (-5) -2y = -2New system:3x - 2y = -5-2y = -2Second y = -2/(-2) = 1;First: 3x - 2*1 = -5 3x - 2 = -5 3x = -3 x = -3/3 = -1.Ivan.-- newsgroup website: http://www.thinkspot.net/k12math/newsgroup charter: http://www.thinkspot.net/k12math/charter.html I need help on solving a system of equations using the Gaussian> elimination method. Many of the help available uses three equations,> and three unknowns, but Im stuck on problems such as:>> 3x - 2y = -5> 3x - 4y = -7>> A step by step explanation of how to solve a problem like this would> be greatly appreciated.This is a square system, meaning the number of variables is the same asthe number of equations, ie the coefcient matrix is square. Heres asample 3x3 system Ill work out for you. Use a monospaced font to keepeverything aligned. 2x + y + z = 1 6x + 2y + z = -1-2x + 2y + z = 7In augmented matrix form this is:[ 2 1 1 | 1][ 6 2 1 | -1][-2 2 1 | 7]The rst pivot position is usually the rst element in row 1. If thiselement were 0, we would interchange this row with one below it. Pivots arealways nonzero, and whenever a zero occurs as a pivot element we interchangethe row with a row *below* it.To pivot on this element, in the context of Gaussian elimination, means tomake all elements 0 below the pivot, ie in the same column but below it,using elementary row operations. In this case, we can subtract three timesthe rst row from the second row:[ 2 1 1 | 1][ 0 -1 -2 | -4] R2 - 3*R1[-2 2 1 | 7]...and replace row 3 with the sum of it and row 1:[ 2 1 1 | 1][ 0 -1 -2 | -4][ 0 3 2 | 8] R3 + R1We are nished with the rst pivot, as all elements below the pivotelement are 0.Next pivot element is on the diagonal to the right and below, which is -1.(this is not necessarily set in stone, but do this for now.) We see that wecan make the 3 below that -1 a zero by replacing row 3 with the sum of itand 3 times row 2:[ 2 1 1 | 1][ 0 -1 -2 | -4][ 0 0 -4 | -4] R3 + 3*R2The next pivot element is -4, but since there is nothing below this elementthe pivoting process is complete and we say the coefcient matrix istriangularized. It is evident why we call it this by inspection of thecoefcient matrix.Remember, this augmented matrix represents the system (which is isequivelent to the original system):2x + y + z = 1 - y - 2z = -4 - 4z = -4To get the solution, begin by solving the last equation:-4z = -4z = 1...then substitute into the previous equation and solve:-y - 2z = -4-y - 2(1) = -4-y - 2 = -4-y = -2y = 2...then substitute into the previous equation and solve:2x + y + z = 12x + 2 + 1 = 12x = -2x = -1This method of obtaining the solution once traingularized is called backsubstitution, and the overall method of solving this system is calledGaussian elimination with back substitution.There is a similar method called Gauss-Jordan elimination where the goal isto get all 1s across the diagonal of the coefcient matrix, and all 0severywhere else in the coefcient matrix, such that the solution can bedirectly read off of the constant matrix.The biggest hurdle of those Ive had to help with both of these types ofeliminations (perhaps your hurdle is similar) is not any problem recognizingwhat needs to be done. Thats easy, determine the pivot and get 0severywhere below the pivot. Rather, most problems seem to be _how_ to goabout doing that. Its really quite simple once you do a couple of them.For example, if I were pivoting on some element and the element below thepivot was a 7 and some element above the pivot was a 4/9, I know I need toturn that 7 into a 0 and one way to go about that is to multiply the entirerow containing 4/9 by whatever is necessary to get -7 and add this to therow containing 7, replacing the third row with this new sum, thus makingthat 7 that was there originally, a 0 as desired.What do I need to multiply 4/9 by to get -7?(4/9)x = -74x = -63x = -63/4So if, for example, my pivot position was R2C2 in a 3x3 coeff. matrix, withthe 4/9 occuring directly above the pivot position in R1C2 and the 7directly below the pivot position in R3C2, I could multiply row 1 by -63/4then replace row 3 with the sum of rows 1 and 3, ie replace row 3 withR3+(-63/4)R1 per elementary row operation #3.Practive makes perfect on these.HTH-- Darrell-- newsgroup website: http://www.thinkspot.net/k12math/newsgroup charter: http://www.thinkspot.net/k12math/charter.html I need help on solving a system of equations using the Gaussian> elimination method. Many of the help available uses three equations,> and three unknowns, but Im stuck on problems such as:>> 3x - 2y = -5> 3x - 4y = -7>> A step by step explanation of how to solve a problem like this would> be greatly appreciated.>First thing that you want to ask is What can I multiply one or bothequations to get rid of a variable?Looking at the system, both x coefcients are 3, so that will be theeasiest to get rid of. But how?Remember -a+a=0; so we need one of the equations to have an x coefcientof -3. If I multiply the rst equation by -1, I get -3x+2y=5.If I add the equation in the last step to the second equation, x is goneso -2y=-2 so y=1.Now substitute back into one of the equations and solve for x.3x-2(1)=-53x-2=-53x=-3x=-1So the solution is (-1,1).The check is left to you.David Moran-- newsgroup website: http://www.thinkspot.net/k12math/newsgroup charter: http://www.thinkspot.net/k12math/charter.html =Circles! It says: Use the rst 11 positive multiples of 5, and themagic sum is: 90! They are saying: Add the rst 11 multiples of veto get 90. Example: 55+5+30=90. Get it! So can someone please help mend some more than just the problem above!-- =