I'm not sure this is entirely appropriate for this group. But the atmosphere here is fairly relaxed so I thought I'd give it a shot.
I'm looking for an intuitive explanation as to why the repeating decimal .9999.....=1. I know the algebraic trick and the convergence of geometric series explanations. But like I said I'm looking for something more intuitive. Something using manipulatives would be great.
"Dale Henderson" <nil...@hotpop.com> wrote: > I'm not sure this is entirely appropriate for this group.
Way over their heads! They are only amateur DIY experimenters on kids.
"Look, Ma! No brains!"
> the atmosphere here is fairly relaxed ...
... like a flatulent homeskoola after eating 20 cans of baked beans.
> How do the various math curriculums handles this?
1) A Beka Books from the unaccredited Pensacola Christian College think it's a miracle and needs no explanation. "Just believe!"
2) The Robinson Self-Teaching Home School Curriculum doesn't have iot in their 1911 Encyclopedia Britannica, 1913 Webster's Dictionary or Original King James Version of the Bible - therefore it doesn't exist!
Dale Henderson wrote: > I'm not sure this is entirely appropriate for this group. But the > atmosphere here is fairly relaxed so I thought I'd give it a shot.
> I'm looking for an intuitive explanation as to why the repeating > decimal .9999.....=1. I know the algebraic trick and the convergence > of geometric series explanations. But like I said I'm looking for > something more intuitive. Something using manipulatives would be > great.
> How do the various math curriculums handles this?
Dale: I'm looking for an intuitive explanation as to why the repeating decimal .9999.....=1. [...] Scott: The following works if we understand that we are talking about decimals that repeat infinitely, but we can't write them that way using only ASCII.
if .99999 = 1 then .99999 x 10 = 10 then 9.99999 = 10 then 9.99999 - .99999 = 10 - 1 then 9 = 9
I think of this as "the algebraic trick" that Dale said he knew, but wanted something different.
Basically, 0.999999...=9*sum(n=1 to inf) [(1/10)^n] = 9*S. So, 10*S=1.11111....=1+S=>10*S=1+S=>9*S=1=>S=1/9. 0.9999999.....=9*S=9*(1/9)=1.
In general, if S=sum(n=1 to inf) (r^n) and S converges (r<1), then (1/r)*S=1+S => [(1/r)-1]*S=1 => S=1/[(1/r)-1].
But, manipulatives? I can see how to use them to "show" 9*(1/9)=1, but connecting this to the infinite decimal expansion, I don't see. It seems to me you'd have to anchor this with some given equality between the fraction and the decimal expansion, like (1/9)=0.11111.... .
You could start with the finite sum, I guess. So, S= sum (n=1 to m) (r^n). Then, you're on safe ground to multiply by (1/r), and write (1/r)*S=1+S-(r^n). Solve for S: [(1/r)-1]*S=1-(r^m), or S=[1-(r^m)]/[(1/r)-1]. So, imagine a sequence of "manipulative experiments" trying to verify this formula for m=1,2,3, and so on (for r=1/10 we get (9/10)/(9)=(1/10), (99/100)/9=(11/100),(999/1000)/9=(111/1000),...). But for even m=3, I guess what this would mean is you've got a whole area divided up into a thousand equal parts, and you'd be showing that (1/10)+(1/100)+(1/1000)=111*(1/1000) => one "tenth" plus one "hundredth" plus one "thousandth" equals one hundred eleven "thousandths", where you have a supply of tenths, hundredths, and thousandths. That is, it strikes me that already for m=3 we are pretty darn abstract and dealing with impractically large numbers of physical objects.
I know I tend to think algebraically and then, if geometry is required, to think back towards it, translating the algebra into the geometry. So, to me, the algebraic formulas, summing either the finite series or the infinite one, are the intuitive things, and I'm not sure that any "manipulatives" could make them more intuitive than they already are---especially given that there seems to be an unavoidable infinite limit involved in whatever demonstration/proof one tries. But I'm willing to listen to suggestions.
>I know I tend to think algebraically and then, if geometry is >required, to think back towards it, translating the algebra into >the geometry. So, to me, the algebraic formulas, summing >either the finite series or the infinite one, are the >intuitive things, and I'm not sure that any "manipulatives" >could make them more intuitive than they already are---especially >given that there seems to be an unavoidable infinite limit >involved in whatever demonstration/proof one tries. But >I'm willing to listen to suggestions.
That was kind of my reaction too. I think it already is intuitive. However I'm wondering if using a graph with an asymptotic line approaching a limit might help someone to grasp the idea.
Jayne Kulikauskas wrote: > On Tue, 18 Apr 2006 11:56:38 -0400, "Michael S. Morris" > <msmor...@netdirect.net> wrote:
> [...] >> I know I tend to think algebraically and then, if geometry is >> required, to think back towards it, translating the algebra into >> the geometry. So, to me, the algebraic formulas, summing >> either the finite series or the infinite one, are the >> intuitive things, and I'm not sure that any "manipulatives" >> could make them more intuitive than they already are---especially >> given that there seems to be an unavoidable infinite limit >> involved in whatever demonstration/proof one tries. But >> I'm willing to listen to suggestions.
> That was kind of my reaction too. I think it already is intuitive. > However I'm wondering if using a graph with an asymptotic line > approaching a limit might help someone to grasp the idea.
> Jayne
Isn't this missing the inherent difference in the *infinite* series? An asymptotic approach treats the series as a constructed number, whereas the arithmetical treatment emphasises the identity.
<paul.dana...@watwinc.com> wrote: >Jayne Kulikauskas wrote: >> On Tue, 18 Apr 2006 11:56:38 -0400, "Michael S. Morris" >> <msmor...@netdirect.net> wrote:
>> [...] >>> I know I tend to think algebraically and then, if geometry is >>> required, to think back towards it, translating the algebra into >>> the geometry. So, to me, the algebraic formulas, summing >>> either the finite series or the infinite one, are the >>> intuitive things, and I'm not sure that any "manipulatives" >>> could make them more intuitive than they already are---especially >>> given that there seems to be an unavoidable infinite limit >>> involved in whatever demonstration/proof one tries. But >>> I'm willing to listen to suggestions.
>> That was kind of my reaction too. I think it already is intuitive. >> However I'm wondering if using a graph with an asymptotic line >> approaching a limit might help someone to grasp the idea.
>> Jayne
>Isn't this missing the inherent difference in the *infinite* series? An >asymptotic approach treats the series as a constructed number, whereas the >arithmetical treatment emphasises the identity.
Theoretically, yes, but in practice the thickness of the pencil line as it approached the limit would make it look like an identity.
>> Jayne Kulikauskas wrote: >>> On Tue, 18 Apr 2006 11:56:38 -0400, "Michael S. Morris" >>> <msmor...@netdirect.net> wrote:
>>> [...] >>>> I know I tend to think algebraically and then, if geometry is >>>> required, to think back towards it, translating the algebra into >>>> the geometry. So, to me, the algebraic formulas, summing >>>> either the finite series or the infinite one, are the >>>> intuitive things, and I'm not sure that any "manipulatives" >>>> could make them more intuitive than they already are---especially >>>> given that there seems to be an unavoidable infinite limit >>>> involved in whatever demonstration/proof one tries. But >>>> I'm willing to listen to suggestions.
>>> That was kind of my reaction too. I think it already is intuitive. >>> However I'm wondering if using a graph with an asymptotic line >>> approaching a limit might help someone to grasp the idea.
>>> Jayne
>> Isn't this missing the inherent difference in the *infinite* series? >> An asymptotic approach treats the series as a constructed number, >> whereas the arithmetical treatment emphasises the identity.
> Theoretically, yes, but in practice the thickness of the pencil line > as it approached the limit would make it look like an identity.
> Jayne
I put that badly, perhaps - the infinite series 0.999... isn't a very long finite series which approaches 1 at the limit, it is something intrinsically different.
Jayne: However I'm wondering if using a graph with an asymptotic line approaching a limit might help someone to grasp the idea. Scott: No. One concept of asymptotes is that the graphed curve never actually reaches the limit.
That would demonstrate that .9999... approaches 1 as the number of decimal places increases, but that it never actually equals 1.
So, I'm not exactly understanding why you say "No." here. It seems to me Jayne's suggestion would amount to we graph a function, f(m), where f(m)=9*[sum (n=1 to m) (1/10)^n]. m would be the abscissa, and f(m) the ordinate. f(m) would have f=1 as an asymptote for m->inf. f(1)=9/10, f(2)=99/100, f(3)=999/1000, and so on. For any finite value of m, f(m)<1, but we visually "see" the limit, which I guess is what a "manipulative" is all about.
>> However I'm wondering if using a graph with an asymptotic line >> approaching a limit might help someone to grasp the idea.
> No. One concept of asymptotes is that the graphed curve never actually > reaches the limit.
> That would demonstrate that .9999... approaches 1 as the number of > decimal places increases, but that it never actually equals 1.
Which is the difference between a limit and an infinite series. The wretched expression "tends to infinity" ought to have been left in the 19th century when they came up with the "delta-epsilon" answer to the philosophical problems of the "infinitesimal" calculus. It just confuses people, because it tempts them to think that an infinite number is the same kind of animal as a very, very big number. "Infinite" entities in mathematics have strange and wonderful properties which a constructivist approach obscures. The ghost of Kronecker lives on ...
J.Pascal" wrote: >> What contact with the outside world do ALL homeschooled children have? >> I am not asking about YOUR particular circumstances but what is UNIVERSAL >> to ALL homeschoolers.
> Do you realize how *political* your insistance on a common > experience for all people is?
I'm not being "political" - I'm being honest. That is why I was asked by the Australian Baptist Union to be the first person in Australia to speak on "the pros and CONS of Christian schooling" at their annual national celebration at Hahndorf, SA, in the early 90s. No-one at the time wanted to discuss the cons of Christian schooling.
If homeschooling is universally good it must be good for EVERYONE ... without exception.
If homeschooling is not universally good then what are the problems associated with it?
NO-ONE on misc.education.home-school.christian wants to discuss the obvious problems of homeschooling. How is that helpful for anyone deciding upon homeschooling? Do you just tell them the good stuff and ignore the bad?
Because I dare to raise the problems I am called a troll.
If everyone thinks the same then you aren't thinking at all.
> It's political, Mark. It's not educational.
Education is also political. Homeschooling is political. Ever read "Teaching as a Subversive Activity?" or the work of Paulo Friere?????
It is also political on misc.education.home-school.christian in that it is (for most people on this ng) a fearful irrational reaction to state control. Such fear paints anyone who deviates from the official line as a troll. The fear related groupings on this ng would be a good area for a sociology PhD thesis.
> The choice between a common authority vs. personal autonomy is a political > choice > made for reasons of political philosophy.
As a trained philosopher, I must disagree with you. It is based on one's world view not on one's political philosophy. Furthermore the choice is not an essentail choice - common authority and personal autonomy can co-exist together. There are also many blends of the two ideas.
> Everyone having everything all the same is a political value held by many > people.
I haven't heard that philosophy espoused anywhere in philosophy in the last 50 years ... except in communism. Who are the "many people" who espouse this philosophy recently?
>> This is a problem with homeschooling. There is NO universal quality >> control in homeschooling. Would you like to discuss this problem ... or >> do >> you just want to continue with your immature gossip about me?
> What you call a problem, I call a strength because I see severe > weakness inherent in the philosophy that demands anything be > "universal."
Quality control is about ensuring quality education. This is an essentail part of state schools, Christian schools, Catholic private schools and distance education. It is NOT an essentail part (apparently) in homeschooling.
That then begs the question: "How do you you really know if your kids are getting a quality edcation?" (because that in essence is what quality control is about)
By this I mean:
How is quality education in homeschooling measured? Does it use BOTH assessment and evaluation? What is the criteria against which assessments are made? Who decides the criteria? What are their qualifications for setting such criteria?
It is of no use to assert that homeschooling is quality education without giving any substantial verifiable proof.
By quality education I mean a minimum of a consistent liberal education with minimum standards and criteria (e.g. literacy, basic maths, knowledge of one's country and political systems and it's place ion the world, general art appreciation) that is available to all children regardless of socio-economic status or the educational level of the parents.
By quality education I do not mean isolated examples of the very best students of a systenm performing well. Every system has those!
Stated in plainer English, how do you really know that homeschooling is good for ALL children? What is the verifiable evidence for your opinion?
> Doesn't matter if it's nutrition, health care or education.
So you wouldn't mind if: - nutritional products had no quality control and were basically useless and without nutrition ... causing people to die from malnutrition - health care had no quality control and anyone could be a doctor, prescribe medication and do surgery ... and hospitals could be located in the local sewers ... causing people to die by visiting a doctor or hospital
Is that the type of state that homeschoolers envisage?
Quality control in nutrition and health care ensures the quality of nutrition and health care that everyone receives.
If there is NO quality control in homeschooling, how can you ensure that you have quality education in homeschooling? Please explain.
> So while I can discuss the issue of "isolation" or I can discuss the > philosophical differences defining an individualist or this sort of > "universal" preference, I can't frame those discussions as a discussion > of a "problem."
Then you are not thinking enough.
If everyone thinks the same then you aren't thinking at all.
> Isn't that what's called "begging the question" when the question > itself is stated such that the answer isn't open to debate or > discussion?
> Nothing about homeschooling is "universal"
Therefore it is without definition? I am therefore a homeschooler because you cannot define homeschooling.
The definition of "homeschooler", in reality, is based on universal attributes of homeschoolers.
What are those attributes?
....
Is isolation of a child from the rest of society a part of quality education?
Is isolation of a child from the rest of society a possibility in homeschooling?
If it is possible, do homeschoolers value isolationism? Why?
If it is NOT possible, please explain how it is not possible with reference to the Branch Davidians at Waco and the People's Temple at Jonestown ...both of whom homeschooled.
Did the the Branch Davidians at Waco and the People's Temple at Jonestown provide quality education for their children?
Why are homeschoolers on misc.education.home-school.christian unable to answer such basic questions?
>>>>> "PD" == Paul Danaher <paul.dana...@watwinc.com> writes:
PD> Which is the difference between a limit and an infinite PD> series. The wretched expression "tends to infinity" ought to have PD> been left in the 19th century when they came up with the PD> "delta-epsilon" answer to the philosophical problems of the PD> "infinitesimal" calculus. It just confuses people, because it PD> tempts them to think that an infinite number is the same kind of PD> animal as a very, very big number. "Infinite" entities in PD> mathematics have strange and wonderful properties which a PD> constructivist approach obscures. The ghost of Kronecker lives on PD> ...
This is the idea I'm trying to get across. That .9999... actually IS 1. And not just a very good approximation. When asked if .9999...=1 students often say that .9999.... is really close but not actually 1.
The algebraic approach you posted in your other post works. But students often think there's something fishy about it and don't fully accept it. (Oh and I found the link you posted with it to be very useful. Thanks)
So what I'm looking for is a way to make this seem clear, without being overly technical.
Oh in the interest of honesty, I'm not doing this for the purpose of homeshcooling. (At least not yet. My kids are too young.) I'm looking for this information for a freshman level university class I'm teaching. But I thought a group of homeschoolers would have more diverse views on this topic. And so far I've recieved some very insightful responses.
MT> "Dale Henderson" <nil...@hotpop.com> wrote: MT> Way over their heads! They are only amateur DIY experimenters on MT> kids.
>> And your professional advice is?
MT> If you are going to homeschool then do it properly with some MT> professional oversight.
Very helpful. And in this case irrelevant. Since I'm actually interested in this question for a university level class I'm teaching.
I would think that a professional educator would have plenty of suggestions for this particular question. Its a concept many students problems with. And since what works with one student will not work with another, the professional educator would need a arsenal of explainations to help the students that aren't getting it.
So Mark, do you have anything constructive to add? I'd be glad to hear it.
Dale Henderson wrote: >>>>>> "PD" == Paul Danaher <paul.dana...@watwinc.com> writes:
>> Which is the difference between a limit and an infinite >> series. The wretched expression "tends to infinity" ought to have >> been left in the 19th century when they came up with the >> "delta-epsilon" answer to the philosophical problems of the >> "infinitesimal" calculus. It just confuses people, because it >> tempts them to think that an infinite number is the same kind of >> animal as a very, very big number. "Infinite" entities in >> mathematics have strange and wonderful properties which a >> constructivist approach obscures. The ghost of Kronecker lives on >> ...
> This is the idea I'm trying to get across. That .9999... actually IS > 1. And not just a very good approximation. When asked if .9999...=1 > students often say that .9999.... is really close but not actually 1.
> The algebraic approach you posted in your other post works. But > students often think there's something fishy about it and don't fully > accept it. (Oh and I found the link you posted with it to be very > useful. Thanks)
It isn't algebraic - it uses the basic operations of arithmetic on an *infinite* series.
> So what I'm looking for is a way to make this seem clear, without > being overly technical.
> Oh in the interest of honesty, I'm not doing this for the purpose of > homeshcooling. (At least not yet. My kids are too young.) I'm looking > for this information for a freshman level university class I'm > teaching. But I thought a group of homeschoolers would have more > diverse views on this topic. And so far I've recieved some very > insightful responses.
Keep focusing on the difference between "incrementally infinite" and *infinite* ...
"Dale Henderson" <nil...@hotpop.com> wrote: >>> And your professional advice is? > MT> If you are going to homeschool then do it properly with some > MT> professional oversight.
> Very helpful. And in this case irrelevant.
Education is always irrelevant to amateur homeskoolas.
MT> If you are going to homeschool then do it properly with some MT> professional oversight.
>> Very helpful. And in this case irrelevant.
MT> Education is always irrelevant to amateur homeskoolas.
I'll take your continued ridicule to mean that you have nothing constructive to add. Further I'll assume that you do not even understand the question at hand.
If you actually care about the education that homeschool students receive, it would be prudent to post helpful, constructive advice when questions like "How do I teach...?" arise in the hopes that someone might take your well-informed advice. In doing so, you could wind up helping home-educated students. You might even offer some educational theory for the questions at hand. In the current case, you might discuss the problems students have grasping concepts involving infinity.
But you don't do any of this. Instead you ridicule and it has become blindinlgly obvious that your goal isn't to help home-educated children but to ridicule home-schoolers, christians and americans.
> ........it has become > blindinlgly obvious that your goal isn't to help home-educated > children but to ridicule home-schoolers, christians and americans.
Do we have a nominee yet for "understatement of the year?" LOL
-- -- Dalene Barnes AOL IM: TX Dalene
"You shall teach them (God's commands) diligently to your children, and shall talk of them when you sit in your house, when you walk by the way, when you lie down, and when you rise up." - Deuteronomy 6:7
What sort of morons are 'oxymorons'? They're figures of speech in which words with an opposite meaning are used together - wise fool, making haste slowly, etc. The Port Arthur tragedy got headlines around the world; as did our PM's swift response in reining in the gun lobby.
Now these 'sporting shooters' (I'm not referring to farmers and others who shoot to survive) are beginning to mobilize. They range in opinion from the dangerous redneck far-right to those who keep a gun for 'protection', and others who shoot for sport. It's the latter group I'm concerned with here.
I guess there are some good reasons why men (they're mostly males) should inflict pain/death on animals for fun, but I haven't heard them yet. And, speaking as a Christian, I find any sport where people set out to maim or kill a violation of the 'Creation mandate.' The Creator, says Genesis 1-3, has given humans responsibility to be good stewards of creation. All creation praises God (Psalm 148:9-10). God watches over the doe as it gives birth to the fawn, and God-as-sustainer provides food for the young ravens (Job 39:1, 38:41). Jesus said his Father cares for sparrows...
Sure, Christians have had different views here. Generally, Catholic countries are less humane than Protestant nations. In a conference to discuss ending the practice of transporting animals for 22 hours without a break for water or food (October 1994) Spain, Portugal and Italy resisted moves to treat animals more humanely. Britain and Germany voted on the opposite side. In Spain the Catholic hierarchy supports bull-fighting, in Canada fur trapping and seal hunting, in Norway whaling, and in Ireland many clergy enjoy hare coursing. Rome and Canterbury are more divided on issues of animal welfare than artificial contraception.
The saints disagree too. St Francis is well known for befriending animals. But St. Thomas Aquinas wrote 'It is not wrong for man to make use of [animals], either by killing or in any way whatever.' Unfortunately Aquinas has carried more weight in Catholic officialdom. Catholic theology bears a heavy responsibility for much of the cruelty to animals in the Western world.
Was it Voltaire who wrote, 'Those who believe in absurdities will commit atrocities'? Calvin said we owe animals justice. Quaker George Fox encouraged his followers to 'do good' to them. John Wesley believed in animals' immortality.
It's ironic (moronic?) that our gung-ho culture should pay over $300 million (U.S.) to watch 'Jurassic Park': a movie about large, fast, and better-armed animals hunting humans. Shades of 'Hollywood versus America'! On the other hand there's 'Babe' - a pig that can act but refuses to be a ham... :-)
Talk about a schizophrenic culture: we care about our pet dog or cat or horse, then eat turkey for Christmas. The killing of turkeys is often brutal: these heavy birds are fully conscious when hung upside down by their legs on the conveyer belt before their throats are cut.
Now some argue that as animals are not spiritual/moral - or thoughtful - beings 'made in the image of God' then they are simply commodities to be exploited. But the relevant question is not 'can animals think?' but 'can animals suffer?' If they can, inflicting pain on them creates enormous moral problems.
The recent book by Jeffrey Masson & Susan McCarthy, 'When Elephants Weep', will make you weep, if you have any compassion. Animals, these authors point out, have emotional lives at least as complex as our own. Of course you don't have to be 'Christian' to be an animal liberationist. The world's expert there is a humanist - Professor Peter Singer, who edited 'The Great Ape Project'. He argues that apes have the intelligence of a two-year-old child and have three basic rights - life, liberty and prohibition of torture. So let the silent majority on this issue become more vocal.
Shooters are not sportsmen, they're vandals.
-- Shalom! Rowland Croucher
"If only it were so simple! If only there were evil people somewhere insidiously committing evil deeds and it were necessary to separate them from the rest of us and destroy them. But the line dividing good and evil cuts through the heart of every human being. And who is willing to destroy a piece of his own heart?" Aleksandr Solzhenitsyn
>> ........it has become >> blindinlgly obvious that your goal isn't to help home-educated >> children but to ridicule home-schoolers, christians and americans.
> Do we have a nominee yet for "understatement of the year?" LOL
To me, it was more the remberance of a song.
I saw the light I saw the light No more darkness No more night
approaches 1 at the limit, it is something intrinsically different.
I know we aren't talking, but I don't get this idea at all. The *limit* implies infinite. And the infinite series is the limit of a sequence of finite series. Moreover, this emphasis on the difference-in-kind---well, something in me really rebels at that. I mean, .999=999/10,000 can perfectly well be understood as an infinite series in its own right. One way to do this would be 0.999=9/10 + 9/100 + 9/1000 + 0/10000 + 0/100000 + .... Another would be 0.999= 0.998999999999... .
I mean, one can write any finite number as an infinite series in this way. And, the point of 0.999...=1 is the very opposite of "they are different animals". Rather, it is that they are exactly the same animal.
|
