Open-Ended Math Questions?

[gnome]

The argument I've had the most success converting the wayward with is not a formal proof, or really a proof of any kind, but a question:

For any two real numbers a and b, there is a third number x which is the difference between them. For the numbers 0.999... and 1, then, what is x?

This usually leads to the answer 0.000...1, but once the absurdity of that particular construction is explained, I have found that most people see reason. Most, not all.

Heck, take that further... ask them what's between .999... +x, and 1.
In addition you could point out that between any two RATIONAL numbers there is a third rational number. Repeating decimals are rational. 1 is rational. What is the rational number that is between them?

 

[rockoon]

Two months ago, XYZ corp sold at $15/share. Last month, it sold at $10. Today, it's selling at $7. What number extends the sequence 15, 10, 7, ... --- and should I buy, sell, or hold?

..., 15, 10, 7, ...

Number of n-uniform tilings having n different arrangements of polygons about its vertices.

Yesterday, the patient's temperature was 38. This morning, it was 39. RIght now, it's 38. Is the patient getting better or worse? Do we continue with the current round of medicine or change drugs?

..., 38, 39, 38, ...

Number of square-free integers in closed interval [n,2n-1], i.e. among n consecutive numbers beginning with n.


All toying around aside.. your point is dead-solid. Sequences appear everywhere. Consider physics! Measurement and Prediction.

 

[DeviousB]

Really, can you show me that?

...and...

Ok, that might be the smart way to do it, but what about as a kid working it out with pencil and paper? I mean, can you on your own arbitrarily choose some numbers and generate three formulas for three different sequences? It took me half an hour to get two answers with my Grade 12 level math. Once using 0 and 1, and then 1 and 2. And if a kid comes up with the aforementioned "easy solutions", well that's phenomenal, and goes to show the open-endedness of the question. I think it's one of the best such problems proposed so far.

Well this has been ansered already, but how often am I going to get a chance to use the difference method?

Let's take an example:

I. (2, 4, ?, ...).

You could always assume that the difference between any two successive terms is the same (in this case 2) and get the series...

II. (2, (+2 =) 4, (+2 =) 6, (+2 =) 8, ...)

But of course the difference between one pair might always be one more than the difference of the preceeding pair...

III. (2, (+2 =) 4, (+3 =) 7, (+4 =) 11, ...)

Or one less...

IV. (2, (+2 =) 4, (+1 =) 5, (+0 =) 5, (-1 = ) 4, ...)

Or two more...

V. (2, (+2 =) 4, (+4 =) 8, (+6 =) 14, ...)

Or twice as much as...

VI. (2, (+2 =) 4, (+4 =) 8, (+8 =) 16, ...)

Choices, choices. In this case an infinite number of them.

 

[DeviousB]

Oh, and the same applies to any three number sequence, four number sequence, and so on.

For ReFLeX only, can you see why?

 

[ReFLeX]

I recently realized two conventions I'd forgotten about since my last math class. How to refer to the previous term, and first, second, third differences and so on. Made it a lot harder.

 

[Soapy Sam]

But everyone is famililar with clocks. And those are a number system unto themselves as well. What's five hours before 2a.m.? Not -3 a.m.

Do you know, till this minute , I never thought of clock time as a number system- perhaps because the earliest clock I remember (which I learned to tell time on) had Roman numerals which I read as letters " EX-aye-aye ". Later I thought of time in the clock sense as a solar position . Funny how different minds "see" things quite differently. It's just Mod 24 arithmetic .

You're thinking, again, of "infinity" as a kind of number. It's not. At this level, its a description of a process -- a process that need not end. But at each point in the process, it's identical to our normal finite processes, so our intuitions about the intermediate stages are still legitimate. Things may break down when we try to extrapolate to a hypothetical end state -- but that's because, by definition, there is no "end state" for the process.

Take your point, though I'm uneasy about the end state complications, which I see as crucial. I feel this distinction of "number" from " process" needs to be stressed far more in school texts. It would have helped me greatly about 1970.
I still hold the view that infinite processes do not exist in any real sense. Viewing the infinite part as "temporal" (We just keep dividing 10 by three, forever") helps me mentally model what's going on, but doesn't increase my confidence that such a process can be real (in the general sense of "real").

A "countable'" infinity is an infinite process that can be carried out as a series of discrete, countable, steps. For example, I can "count" the positive integers as follows : 1, 2, 3, 4, .... Although I can never get to the end, I do know that any specific positive integer will be in that sequence somewhere, and that no integer will appear more than once. (In technical terms, this is a 1:1 and onto mapping.) The positive integers are thus a "countably infinite" set.

OK. "Countable " here means "in theory we can see no obstruction to counting them all, given enough time"

The number of real numbers, or the number of functions, are not countable. There's no way to set them up so that I can enumerate them one by one and make sure of getting all of them. (Again, this is provable, but the proof details might be confusing. If you want to read the proof, it's "Cantor's diagonalization.")

Whereas here there is no way even in theory.

My usage of "Uncountable" was a bit more specific, in the sense that if the counting process (or time) required to enumerate the integers is itself infinite, then even the integers are uncountable in reality. (Forgive my inability to drop the real world. It's a lifelong habit).

You're not understanding. The magnitude 0.99999.... is not infinite, nor does it involve an infinity to calculate that total. The expression "0.999...." is of course infinite. But that's back to your point about magnitude vs. quantity (which I admit I didn't fully understand -- your terminology is confusing). The magnitude of 0.999... can be determined fully using a limit process without using any infinite processes.

I understand that the magnitude of .999... is not infinite. We agree that 1 is the upper limit of the series. Still, it appears that to derive the magnitude of 1, you are using an expression which is and must be infinite.

My whole argument is that this is impossible in the real universe. We must cut it short- at a convenient and appropriate value, which is by definition less than the upper limit. I don't doubt we can get the value to within a metric femto gnat's ass of 1. I just don't think that's the same thing as 1.

{On my non standard terminology, for which I apologise,- "Two beans " is a quantity. "Beans" is the object- what the quantity is a quantity "of ".
"Two" is the magnitude of the quantity. The amount of it. That's all.

When we discard "beans" and retain "two", we have a number, as opposed to a magnitude of a quantity.
I don't think these are the same thing. "Two beans" makes sense in a way that "two" does not. (To me.) I suspect your world view may differ sharply from mine here. The distinction may have no real value beyond the psychological. I honestly don't know.}

Basically, what I can do is I can prove that the result of a particular finite process lets me get as close as I like to the actual answer. This means that if I need accuracy to within 1%, I only need to use the first (e.g.) three digits, since the remaining unbounded sequence cannot change the value by more than 1%.

Up to here, I'm fine with the argument.

If I can prove that for any x% accuracy, I need only look at the first y digits (y might be x+1, or 2x, or something like that), then I can calculate the value of the infinite sequence exactly.

Sorry. I just don't see this step at all. It's as if you changed languages.
Something in my head just fails to meaningfully interpret that part.

I'll go look at some examples, see if I can rattle my brain cell into place.

I appreciate the trouble you've taken here . I realise it's very frustrating when you understand something and yet keep failing to get it across to someone else.

 

[gnome]

My whole argument is that this is impossible in the real universe. We must cut it short- at a convenient and appropriate value, which is by definition less than the upper limit. I don't doubt we can get the value to within a metric femto gnat's ass of 1. I just don't think that's the same thing as 1.

In that case you're rejecting the question, not the answer. If .999... is unachievable in the environment you describe, the question does not apply. It's not that you have .9999... and can't quite reach 1... it's that you never had .9999... in the first place. You had something less than that.

 

[GreedyAlgorithm]

I understand that the magnitude of .999... is not infinite. We agree that 1 is the upper limit of the series. Still, it appears that to derive the magnitude of 1, you are using an expression which is and must be infinite.

My whole argument is that this is impossible in the real universe. We must cut it short- at a convenient and appropriate value, which is by definition less than the upper limit. I don't doubt we can get the value to within a metric femto gnat's ass of 1. I just don't think that's the same thing as 1.

You may be interested in the concept of a supertask^WP. It appears you believe that supertasks do not exist, which is a pretty standard belief, but have not been able to view limits except as a supertask. Maybe this will help convince you (not a proof, just trying for the lightbulb):

What is 1+1/2+1/4+1/8? It's 15/8.

How about 1+1/2+1/4+1/8+1/16+1/32+1/64? It's 127/64.

[latex]$$\sum_{i=0}^{100}\frac{1}{2^i}=\frac{2535301200456458802993406410751}{1267650600228229401496703205376}$$[/latex]

How do we get those numbers? You could say that for the sum to 100, we start with 1, then add 1/2, then add 1/4, etc. until we've added 100 terms to our initial 1. But there's a shortcut.

[latex]$$\sum_{i=0}^N\frac{1}{2^i}=\frac{2^{i+1}-1}{2^i}$$[/latex]

Suddenly we don't have to add up 101 terms, we can just apply this formula. Now, if you like, you can assume that computing the sum with infinitely many terms would be a supertask and thus impossible. But we don't have to actually do it. We can figure out what the result would have to be, even if we can't do the brute force computations. That's an intuitive way of looking at a limit that might work for you: what the result of looking at infinitely many (not arbitrarily many, infinitely many) terms in a sequence would be if we could do it. We can't, okay, but if we could there is only one answer we could get. And if there is more than one answer, or if the terms get arbitrarily large, then the limit doesn't exist. Even if supertasks existed the sequence wouldn't get anywhere.

 

[69dodge]

In answer to your puzzle, I'm afraid all I can do is add the two numbers to the number of significant places shown , which (mental arithmetic here) gives 1.323231
I presume correctly this would repeat(... 323232323... )until one chose to stop, but if the two numbers are themselves meaningless (which remains my supposition), I doubt the answer has meaning either.

Not meaningless. Meaningful. Full of meaning, I say.

Well, a deal's a deal: I hereby apologise for asking the question.

(I needn't actually mean it, right?)

Here's something by Hamming I came across that seems sort of on-topic.

 

[drkitten]

Take your point, though I'm uneasy about the end state complications, which I see as crucial.

You're right, end state complications are crucial, and they're one of the major areas where mathematicians muck this stuff up. That's part of what the formalization as "limits" is supposed to do, to define rigourously the areas where end-state complications arise and where they don't. (E.g., as GreedyAlgorithm put it, "if there is more than one answer, or if the terms get arbitrarily large, then the limit doesn't exist" and you can't use this style of reasoning. That's one way an end-state complication can arise....)

I feel this distinction of "number" from " process" needs to be stressed far more in school texts. It would have helped me greatly about 1970.

It might not have. This kind of highly-theoretical view of mathematics education was popular in the United States in the late 60s and early 70s (the infamous "New Math"), but it is now considered to be a failure. Either elementary school students lacked the ability to view the material with the necessary degree of abstraction, or else the teachers lacked the theoretical understanding to present the material well. I suspect it was a combination of both.

I have seen this "New Math" approach be very successful at the college level -- but unless you were in college in 1970 and majoring in mathematics, it wouldn't have helped you as much as you think.

I still hold the view that infinite processes do not exist in any real sense. Viewing the infinite part as "temporal" (We just keep dividing 10 by three, forever") helps me mentally model what's going on, but doesn't increase my confidence that such a process can be real (in the general sense of "real").

I'm still not sure what you consider to be "real," then.

Infinite processes only exist in potentia -- but that doesn't make them "unreal." Just as an example -- how long can a motor race be? I believe the Indianapolis 500 is five hundred miles long and takes a few hours. There are some "endurance races" that are 24 hours long (or more). How long can a race be. Is there any physical limit that prevents a race from lasting for 48 hours? 96? A week? A month? A year? Where do you draw the line?

I submit that for any length of time you suggest, a race that lasts ten minutes more would be physically possible. Thus, the maximum possible length of a race is unbounded -- "infinite." Any particular race that you run will be of finite length, but there's no maximum. And there's nothing unreal about this observation.

Now, let's imagine for a moment that we will have an endurance-style motor race. We both get all the pit crews, fuel, tires, etc. that we want, and we can even change drivers on our team, so there are no physical limitations we need to worry about. Let's also suppose that you're driving a high-performance race car with a top speed of 500 kph or so, and I'm driving a Geo Metro with a top speed of 100kph. Allowing for pit stops and the like, you can cover 200km per hour, while I would be lucky to cover 70.

How long will the race have to be to let me win?

Well, after the first hour, you're 130km ahead of me. After the second, you're 260km ahead. In general, after the Nth hour, you're 130N km ahead of me. It should be obvious that the race is unwinnable (by me); that there is no length of the race that would let me win.

But that's a statement about infinity, too. The maximum length of the race that you are almost guaranteed to win is also unbounded, because I just keep getting farther and farther behind you.

That's a limit process. I can prove that as the race gets longer, you win by more and more.

And we don't need to actually run the race to establish that....

 

[gnome]

Heck, give drkitten a head start and zeno's paradox-style, start computing at what point Soapy Sam will overtake drkitten.

Expressed as an infinite series, there will be a specific result at the limit. Pick ANY number less than the limit, and it will be too soon. So the limit IS the answer, IN THE REAL WORLD as well as on paper.

 

[ReFLeX]

http://en.wikipedia.org/wiki/Main_Page

Today's featured article:

http://en.wikipedia.org/wiki/0.999...

 

[aries]

HI

@ the OP:

I remember when my sister's oldest was in 9th grade (I think). It is several
years back, but I still remember the, imho, perfect math assignment.

The problem given was this:
A man wants to buy car. He checks out varius dealers and such. He thinks he wants a car driven by gasoline, but he also likes another car which runs on diesel. Given that the gasoline prices are higher than the diesel prices, the students should do a range of mathemathical solving problems, including, but not limited to: calculate which car would be cheapest in the long run.
IIRC, the diesel car did get a little worse mileage than the gasoline car, but the gasloine car did infact, IIRC, cost a little more in insurance than the diesel car.
Now, IIRC, the students should in fact solve this problem as well as they were asked to make a graphical representation of the costs of driving these two cars. (or a pie chart of another sort of graphical mathematichal representation).

Of course, you could also substitute one of these cars with a hybrid.
Or maybe your students like to ski ? Slopes, maths and stuffs,
calucalting the body weight with the height of the slope and stuff.

In the movie 'the mirror has two faces' the math professor is about to give up. Then Barbara Streisand's character takes him to a (base) ball game, and gives him pointers as to how he could get the students interested again:
He asks them how hard a certain player should have hit the ball if it were to a certain distance, he also asks them why a certain player didn't hit the ball and at what angle he should have hit it and such.

In fact, in Num3ber, the FBI math guy recently (in the episode 5, i think) explained that all is math, all is numbers. Try connecting the math to the student's lives. If they have car, try to get them to focus on some of the math issues, mileages and such. Or if they want a certain Ipod nad have a job, try making them work out how much they money they have to make to earn enough money for buying an Ipod.

Also, you could try making cases and get the student to solve them.
One of my favorite cases come from the movie 'race the sun' in which
Halle Berry stars as a teacher who encourages the students in the high school to work together in making a sun driven car. Lots of calculations and maths right there.

I'm not saying that you should go as far as they did in the movie, but maybe you get your students to find out which energy form is the cheapest one - or how much energy the are using at home etc. etc.

I have seen a math exam in high school on Danish TV not long ago. The students should explain the maths behind a Danish flag or they were asked
to explain the concept of 'areal' (area, space ? = you know you multply say
2mX2m and get 4m2 ??) . IIRC, the questions were all open ended in that
they provided multiple solutions as to how the students did in fact come to the correct result.

Airplane tickets and math could be another great openend assignment.
(I would write trains, but I think the OP is in the USA).

In my mind, math should learn students to think logical and reason clearly.

 

[aries]

That's missing the point. The problem itself is designed for younger students, not undergraduates or graduate students. Discussing the problem and how to solve it as we are doing is intended for teachers in training.

Nevertheless, problems of this sort appear on standardized tests like the SAT (although this one might be on the easy side). Any student trying to read more into the problem than is there, as you are doing with your suppositions about its being applied in other contexts, will lead one to solve the problem in an incorrect manner, and likely result in an incorrect response (one different from the official correct response as the test makers deem it).

I agree that the problem is poorly phrased. It is not ambiguous, however. It's rather straightforward, actually.

Those persons reaching a different answer from the simple +$20 response are introducing facts into their analysis that are not presented in the problem. We don't do want our younger students doing that. We want them learning to apply clear, rigorous analysis to problems like this. It is only much later, after mastering basic skills like those required for this problem, and after mastering more complex thought and analysis, that we can introduce placing this kind of problem into other contexts like markets. That doesn't come until much later, probably at the university level for most students.

AS

HI

I wonder why it is then, that most Danish students I know, (the best of them anyway), in College as well as in the senior year at High School, are able to understand and do this reasoning. (sometimes with a little help, but they do understand the whole 'more than one valid response' theorem etc.)

Luckily (yet) we don't have SAT tests and such (yet), making ONLY one
and ONLY response or answer valid for such a question like the one, the OP posted in this thread. However, if the teacher and SAT have decided that the answer is +20 (dollars), then I agree that the wording of the problem is horrendous.

I would suggest that students trying to do the math questions (like this one)
in an 'independent' manner, trying to find out what exactly is meant by
"financial outcome" and such has, in fact, a better grasp of what solving math problems is like.

I have, respectfully, to disagree with the person, I've qouted, if by 'basic skills' he means formulas and stuff like a2xb2=c2. IMHO, learning this (like a skill or something you have to memorize) will get you nothing. Yes, you will as I have done, learned a formula, but I can't remember how to use it or when I'm supposed to use it. And the students will, imho, have learned - nothing.

However, it is also clear to me that without knowledge of the methods you would be using to say build a pyramid or a castle etc. etc. the structure in the building will fail. And the methods do include knowledge of the formulas ?
needed to build said pyramid or castle.

[I apologize for the bad math English, but I hope you understand what I mean].

More generally, speaken, I think a lot of people, teachers included, use
the terms 'data,' 'information', 'knowledge', 'methods' etc. as substitutes.
In my mind they are not.

Data is well Data (no, not the Robot from Star Trek). However, Data holds a lot of data, meaning facts and such. Information is then putting this data into form so that it gives meaning, i.e. interpreting data. Knowledge is when you are able to set the information into perspective and able to use the data and information, while method is the way you do this.

You can't analyse a text or write a job application before someone has taught you (or shown you) how to do this i.e. some methods of doing this.

Likewise you can't solve a math problem, before someome has taught you
methods (ways of doing this) to solve the problem.

 

[Beerina]

It is a bit odd this "had no solution".

Girl starts with $1000

Buys horse, now has horse and $950

Sells horse, now has $1010

Buys horse again, now has $940

Sells horse again, now has $1020

Net profit: $20

 

[EBU]

I've used the horse problem in a course on math teaching. The point is not that there are several solutions; obviously, as previously noted, there is one right answer. The point (or my point, anyhow) is that there is not one right way to get the correct; you could add up the money coming in and money going out and take the difference, or use algebra, or act it out (depending on how old you are), etc. Another point I make with this problem is that you could be really good at arithmetic and still get the wrong answer -- not because you can't add or subtract but because you don't know when to add or subtract.

The teacher who used that to illustrate something wrong (that a problem can have more than one answer) is a classic example of teachers misunderstanding what they are taught and then mis-teaching -- one of the big problems in math education, in my opinion. Very few middle school math teachers understand fractions all that well, for example.

 

[DeviousB]

Very few middle school math teachers understand fractions all that well, for example.

I read somewhere that out of 114 secondary school teachers questioned, only 61 could give the correct answer to a simple problem involving fractions.

That's nearly half!


[Joke.]

 

[DeviousB]

Here's an interesting little puzzle (paraphrased from Chris Maslanka's puzzle column in last Saturday's Guardian). Kinda reminded me of the Monty Hall problem.

You are visiting an island far away. On this island the natives are twice as likely to tell you lie as speak the truth (they don't like foreigners much).

One afternoon you reach a fork in the road, where two natives are sat, and ask them which road leads to the beach.

"That one." says the first native, pointing out one of the paths.

"Yeah, what he said." says the second.

What are the chances that the path pointed to goes to the beach?

 

[EBU]

Here's an interesting little puzzle (paraphrased from Chris Maslanka's puzzle column in last Saturday's Guardian). Kinda reminded me of the Monty Hall problem.

Would it be 1 in 5?

 

[brodski]

Here's an interesting little puzzle (paraphrased from Chris Maslanka's puzzle column in last Saturday's Guardian). Kinda reminded me of the Monty Hall problem.

1 in 9?