Wikipedia:Reference desk/Archives/Mathematics/2013 November 18

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November 18

Maths

From the following information, calculate tie passenger kilometers. Number of buses 5,number of days operated in the month 25,number of trips made by each bus per day 4,distance of route 25 km(one side),capacity of bus 50 passengers(actual passengers travelling 90% of capacity) — Preceding unsigned comment added by 112.79.41.150 (talk) 13:43, 18 November 2013 (UTC)

Please see 'How can I get my question answered?' at the top of this page in particular "We don't do your homework for you, though we’ll help you past the stuck point." Dmcq (talk) 13:52, 18 November 2013 (UTC)

Agreed. Just multiply all those numbers together. The only tricky parts are that you multiply by 0.9 for 90% and you may need to multiply by 2 if you assume the buses have passengers going both ways. Do your work and show us the results, and we will tell you if you made any mistakes. StuRat (talk) 20:37, 18 November 2013 (UTC)

Implicit differentiation of 3 variables

Have I some this the correct way?

(x^3)*y-sinz+z^2=0

(3x^2)y=cosz(dz/dx)-2z(dz/dx)

dz/dx=(3x^2)y/(cosz-2u)

x^3-cosz(dz/dy)-2z(dz/dy)=0

dz/dy=x^3/(cosz+2z)

— Preceding unsigned comment added by 82.132.238.92 (talk) 15:45, 18 November 2013 (UTC)

No, unless ${\frac {dy}{dx}}=0$. In general ${\frac {d}{dx}}(x^{3}y)=3x^{2}y+x^{3}{\frac {dy}{dx}}$ by the product rule. 150.203.188.53 (talk) 20:16, 18 November 2013 (UTC)

Thanks but I'm not trying to find dy/dx. I'm trying to find dz/dx. Clover345 (talk) 17:55, 19 November 2013 (UTC)

You have the the condition w=0 with the function definition w=x³y-sin(z)+z². Differentiate w to get dw=3x²y dx + x³dy - cos(z)dz + 2zdz. Now set dw=0 because w=0, and dy=0 because you are interested in dz/dx for constant y. 0=3x²y dx - (cos(z)-2z)dz. Solve to get dz/dx=3x²y / (cos(z)-2z). Bo Jacoby (talk) 05:09, 23 November 2013 (UTC).

Do we still have to know how to process an algebraic structure?

Do schoolchildren still have to learn that how to transform (x - 3)(x - 4) into x² + -7x + 12? Do they still teach this at school? If yes, wouldn't it be wiser just to teach how to input that into the a program and let the computer take care of it? (and concentrate in making sure that the students really understand math) OsmanRF34 (talk) 22:46, 18 November 2013 (UTC)

It's better you understand why things are the way they are, lest you forget the origins of reasonings and begin to worship them as some sort of divine moral code. — Preceding unsigned comment added by 144.82.114.106 (talk) 23:44, 18 November 2013 (UTC)

But incidentally, it is the teaching of this same mechanically ability (processing a structure, not knowing why) what I believe that leads us to a kind of religion, where people don't know why they are doing what they are doing. Imagine that a teacher wouldn't care much if the students manage to transform it from a to b, but would ask why they are doing this or that, or how to solve this or that problem, leaving the mechanical part to the computer. OsmanRF34 (talk) 00:10, 19 November 2013 (UTC)

There's a lot of things wrong with this idea. The major problem is that you are assuming that there is some clean line between "mechanical" and "abstract" or "mechanical" and "problem solving", but there isn't. If you don't have a firm handle on how binomials multiply, looking at how polynomials factor seems either pointless and unmotivated, or difficult and tedious- it also makes factoring into binomials seem artificial since you don't really understand the basis. It's almost like saying students should just use calculators to do multidigit addition and multiplication, that they should be taught to "problem solve" with addition and multiplication instead. But what would this be? What kind of problems would they be solving? It's like saying higschool German classes should spend most of their time focusing on linguistic theory and that vocabulary lessons and the specific grammar should take a back seat, but you don't end up with people who can speak German that way.

This isn't to say that there isn't a problem with how math is taught, and what math, today; but demphasizing basic skills isn't the solution (which, by the way, sounds a lot like New math). The better solution would be to integrate the purpose of the various things studied into the lesson, nobody wants to learn something that seems purposeless (I spend around 4-5 hours a day studying math, I have a hard time slogging through material that's presented as existing in a vacuum). Another big problem is that teachers usually don't know the subject that well themselves, hence, they lack the means to make the subject interesting. Thus, you end up with teachers that spend the class writing formulas on the board while they drone, expecting the students to just plod along- generally happy if they can keep everyone awake through the lesson.

Finally, you're expecting a lot out of people. It's almost impossible to appreciate something abstractly without working through the boring bits (like monkeying around with formulas, etc.). After you fully grasp something, it's very easy to feel like it's all just common sense, but that's rarely the case. When you're learning something new, concrete examples make a world of difference- I would imagine that most highschool graduates, if motivated enough, could "learn" what a compact topological space is in an hour, the definition is not very hard; however, without a useful base of examples to draw upon, it would be absurd to pretend they actually understand the concepts. Mathematics is cumulative, things don't just pop out of the ether, if you want to actually be good at it, you need to work through all the nasty boring mechanical bits, you need to master the concrete examples, you need to master the abstract theoretical notions, etc. etc. etc. The problem with your idea is that you are creating some false dilemma of "focus solely on processing formulas" or "let the computer do the grunt work", but there's a world of in between, and that "in between" is exactly what learning mathematics is.Phoenixia1177 (talk) 01:28, 19 November 2013 (UTC)

Shouldn't students just enter their questions on the Wikipedia reference desk and put the title 'Math' on top? In the future even that will be unnecessary as they'll just say the question to their smartphone and it'll answer. It won't matter that they garble what they say because they don't understand, the smartphone will correct for that. They'll also compose tunes and pick friends and, well, the world will be a much simpler place! ;-) Dmcq (talk) 13:11, 19 November 2013 (UTC)

There is a classic Fritz Leiber novella "The Creature from Cleveland Depths" which basically asks: When machines can do our thinking for us, what will they need us for? --RDBury (talk) 14:32, 19 November 2013 (UTC)

It's not about letting machines do the thinking work for us, but just letting machines do the crunching work for us. I have no doubt that many people can 'solve' mathematical problems without understanding what they are doing. They are doing the job of a computer, not realizing that (x - 3)(x - 4) are the factors of x² + -7x + 12 or even worse, not even knowing why polynomials are good for. The work students could be concentrating in, is to transform real life problems into a formal mathematical structure, or translating word problems into mathematical equations. However, I have the impression that modern day education is as insightful today as it was in the past. What is the point of asking ten times the same question in the form (x - 6)(x - 8) or (x + 3)(x - 2) or (x - 13)(x + 4)? We could be asking the student just once~to solve by hand one of these and ask on the top of that what are these elements the base from. OsmanRF34 (talk) 19:46, 19 November 2013 (UTC)

There seems to be a problem between what you are saying and what you seem to want to be saying- the result of successful education should lead to insightful students with problem solving skills, but nothing you are suggesting will accomplish that. As I mentioned above, this notion of "let computers do the crunching" doesn't make any sense, what is the "crunching"? What you seem to be objecting to is making students spend all their time doing drill exercises without context, which is reasonable, but, like I said, what you are suggesting isn't a solution. This isn't really the place to debate what the solution is, but I can tell you that without a major overhaul, you aren't going to have students capable of learning to translate real life problems into mathematical formalism- to be honest, most nonmath major adults seem to have trouble in that area, it is unlikely that there is going to be a simple fix.Phoenixia1177 (talk) 05:59, 20 November 2013 (UTC)

If you were wanting a literal answer to your original question then yes, in Scotland anyway, pupils are taught that sort of algebra at about the age of 13. More than fifty years earlier I was taught that too at about the same age. Log tables and slide rules have gone, scientific calculators are ubiquitous (except when not allowed) and computer use (including some programming) is used over a wide range of subjects. I think there is less slog (and homework) and more logical thought. Thincat (talk) 16:50, 19 November 2013 (UTC)

Transforming (x – 3)(x – 4) into x² – 7x + 12 requires applying rules of commutativity, associativity and distributivity of operations. Also transforming repeated multiplication into exponentiation and operating on exponents. Of course one can perform it all mechanically, by a written routine. However these routines ale usually taught after, and on a basis of, numbers' and operators' properties. Learning them lets us understand assumptions that were made and rules of reasoning we use, and that is important part of 'understanding math' IMO.
On the other hand, 'inputting that into the a program and letting the computer take care of it' requires no knowledge of what is being done but a pure belief that the program can do anything usable and it will do that right. That's not understanding math at all. --CiaPan (talk) 11:58, 20 November 2013 (UTC)

Paste this: 4-2/2= into the Windows Calculator and 'let the program take care of it'. The result may be 1 or 3, depending on the mode. Which one is correct? And what does really 'correct' mean?
Finally, how would you 'make sure that the students really understand math' if they have to rely on the Calculator's reply but they have no practice in, and consequently no understanding of, processing such a simple arithmetic expressions? --CiaPan (talk) 12:13, 20 November 2013 (UTC)

You can access a series of textbooks based on New York State's high school math curriculum here. —Nelson Ricardo (talk) 20:53, 21 November 2013 (UTC)