Open-Ended Math Questions?

[roger]

But that's my complaint! Ambiguity doesn't make for a good open-ended problem, the way I see it. If people are going to debate over an answer, they should at least be responding to the same question!

I agree.

I actually find the fact that people got so many different answers interesting, though. Surely a lot of that was due to poor math skills. But I think there is a bit of a psychological aspect to it, too - in how somebody responds to the ambiguity. Science types, dominant in this thread, conclude the question is asking about profit, and see no other answer than $20. I'm often a 'glass half empty' person, and if I had executed that transaction in real life, I'd walk away thinking 'Stupid. I should have $30 now, but I only have 20. Dang, I lost 10 bucks in this deal!" Whereas a glass half full person would be thinking "Yay. I made 20 bucks. Ya, I should have held out and got the full 30, but I'm still up $20!!!"

But yes, the problem statement is ambiguous - define it better, and the problem is not 'open ended'.

 

[drkitten]

But that's my complaint! Ambiguity doesn't make for a good open-ended problem, the way I see it.

There's no other way for a question of fact to be "open-ended."

This doesn't just apply to mathematics, by the way. It applies to any scholarly discipline.

The question "Who was America's first President of the United States" is equally ambiguous, since the technical title of the presiding officer of the Continental Congress after the Articles of Confederation were adopted was "President of the United States in Congress Assembled." So depending upon how you interpret the phrase, the answer is either George Washington (under the Constitution), Samuel Huntingdon (under the Articles of Confederation) or Peyton Randoph (under the first Continental Congress).

What's the largest country? Russia has the largest area, but China the largest population?

How many people live in New York -- the city, the metropolitan area, or the state?

If people are going to debate over an answer, they should at least be responding to the same question!

From an educational standpoint, the value of an open-ended question is exactly the debate and the opportunity to analyze the underlying framework and assumptions. It's the "teachable moment" where the students learn that there is more than one way to interpret a given question and that the same question can yield different correct answers if the interpretations vary.

 

[ReFLeX]

There's no other way for a question of fact to be "open-ended."

...

From an educational standpoint, the value of an open-ended question is exactly the debate and the opportunity to analyze the underlying framework and assumptions. It's the "teachable moment" where the students learn that there is more than one way to interpret a given question and that the same question can yield different correct answers if the interpretations vary.

Ok. My idea of open-ended was one that people will use a variety of strategies to solve it, but all have a chance of arriving at the correct answer. Even if it's a question of fact. I think there are infinite such examples where no interpretation is necessary but there will have been several different methods employed. While no math problems are so really stimulating to me as the non-intuitive fallacies mentioned, I can see your point.

 

[fribble]

Mathematical questions are often actually questions about definitions- the interesting number question is a clear example, as is the 0.999 = 1 question, which hinges on how we define the terms "number" and "infinity". To anyone who accepts the mathematical definitions of those terms , the answer is "yes". To anyone who thinks (as I do) that the term "infinity" is meaningless, the answer is "No".
Math questions are no different from any linguistic questions. The answer depends on what the words mean. You might express the question in mathematical symbols, but that simply adds an extra layer of translation.

You don't need to use the word infinity at all when talking about 0.999... . 0.999... is usually defined as the limit of the sequence 0.9, 0.99, 0.999, 0.9999, ... i.e. the sequence where term n is 1 - 10^-n. And, using a bit of first-year uni maths, the limit of that sequence is 1. Absolutely no question about it. I'm not sure what complaint you have about infinity...?

 

[AmateurScientist]

You don't need to use the word infinity at all when talking about 0.999... . 0.999... is usually defined as the limit of the sequence 0.9, 0.99, 0.999, 0.9999, ... i.e. the sequence where term n is 1 - 10^-n. And, using a bit of first-year uni maths, the limit of that sequence is 1. Absolutely no question about it. I'm not sure what complaint you have about infinity...?

I was taught a much simpler proof in 7th grade pre-algebra (which actually included some algebra).

Let N = .999... (I don't know how to make a repeating symbol on my keyboard)

10 N = 9.999... (multiple both sides by 10)

10N - N = 9.999... - .999... (subtract N from each side)

9N = 9

N = 1

.999... = 1

 

[Zombified]

I was taught a much simpler proof in 7th grade pre-algebra (which actually included some algebra).

Let N = .999... (I don't know how to make a repeating symbol on my keyboard)

10 N = 9.999... (multiple both sides by 10)

10N - N = 9.999... - .999... (subtract N from each side)

9N = 9

N = 1

.999... = 1

Good enough for pre-algebra, but to make that proof formal, you have to define operations on infinitely expanded digit strings, again in terms of sequences, and by the time you've done that you've covered more than enough math to make the limit argument.

 

[Pidge]

What's the largest country? Russia has the largest area, but China the largest population?

How many people live in New York -- the city, the metropolitan area, or the state?

Sorry, I can't help this, but:

BRIDGEKEEPER: Hee hee heh. Stop! What is your name?
ARTHUR: It is Arthur, King of the Britons.
BRIDGEKEEPER: What is your quest?
ARTHUR: To seek the Holy Grail.
BRIDGEKEEPER: What is the air-speed velocity of an unladen swallow?
ARTHUR: What do you mean? An African or European swallow?
BRIDGEKEEPER: Huh? I-- I don't know that! Auuuuuuuugh! [ flung into the chasm]
BEDEVERE: How do know so much about swallows?
ARTHUR: Well, you have to know these things when you're a king, you know.

 

[AmateurScientist]

Good enough for pre-algebra, but to make that proof formal, you have to define operations on infinitely expanded digit strings, again in terms of sequences, and by the time you've done that you've covered more than enough math to make the limit argument.

No, you don't. That is a formal proof.

You don't have to define any operations on an infinitely repeating decimal. It's quite simple. Multiplying by 10 means you move the decimal point one place to the right. Done. Everything else remains exactly as it was.

You're covered no more math than the most basic of functions on algebraic equations and simple multiplication by 10 and subtraction, which are basic arithmetic functions.

Limits are not required to prove that .999... = 1. You can even do it using fractions. Take 1/9, or 1/3, for example. 1/9 = .111... Multiply both sides by 9. You get 1 = .999... 1/3 = .333... Multiply each side by 3 and you get 1 = .999... yet again.

Simple.

AS

 

[NobbyNobbs]

I don't think so,

If you take the decimal expansion of pi and score out all the 9s, you'd get
3.141526535873.......

This would still have an infinite number of places and not repeat, yet would never be capable of producing the equivalent of pi_hat.

Infinite places and non-repetition are insufficient to prove the presence of 100 9s in a row.

I think.

Forgive me for possibly missing the obvious, but...

Should I be asking you for proof that striking out the 9's results in an infinite, non-repeating decimal? I mean, the infinite part is obvious, but how do we know that it still doesn't repeat? Am I missing something?

 

[Zombified]

To play devil's advocate for a second, while the original problem as stated is not well posed as either a math problem or an arithmetic exercise, recognizing it as ambiguous and open to interpretation is a useful skill to acquire. Imagine if it were in the context of a debate on markets, and you could interpret it as a gain or loss depending on how you meant to make your point.

 

[Zombified]

You don't have to define any operations on an infinitely repeating decimal. It's quite simple. Multiplying by 10 means you move the decimal point one place to the right. Done. Everything else remains exactly as it was.

No, you really do have to define what multiplying decimals means, and you even gave an informal definition of a special case. If you want a formal definition of how to deal with infinite strings of decimal digits, however, you really do need to handle sequences that converge to reals. It isn't that difficult, it just deals with the fiddly details.

Edit to add: given the subject of the thread my nerdish obsession with the fiddly details is certainly a bit much, and I apologize for the derail.

 

[AmateurScientist]

No, you really do have to define what multiplying decimals means, and you even gave an informal definition of a special case.

Yes, you do, but multiplying decimals in the special case of multiplying by 10 is usually the first case we learn, and we learn that it means moving the decimal point one place to the right. It is the decimal system -- base 10 -- after all. When I was in school, we had already learned that before 7th grade. I hope that's still true in most decent schools.

If you want to be really pedantic, you have to define what addition and subtraction mean, and what multiplication means, and so forth. Or course that's true, but as I already stated, those are simple arithmetic functions. Students learn those (or at least they're supposed to) well before 7th grade. How to perform very basic functions on a simple algebraic equation are usually learned in pre-algebra, when I first learned that proof.

If you want a formal definition of how to deal with infinite strings of decimal digits, however, you really do need to handle sequences that converge to reals. It isn't that difficult, it just deals with the fiddly details.

That is absolutely not required for students to learn and demonstrate the algebraic proof I gave. Convergence of a series is a concept learned in calculus, or perhaps even in pre-calculus. You simply don't need to understand convergence in order to understand the proof or why it is true.

In order to analyze why .999... = 1, then surely a comprehensive explanation would include a discussion of converging series and limits. Do 7th graders need that in order to follow the proof and be able to demonstrate it? Nope. I remembered it from 7th grade on, and I confused and befuddled some of my fellow math students in college with it.

Edit to add: given the subject of the thread my nerdish obsession with the fiddly details is certainly a bit much, and I apologize for the derail.

No apologies are necessary, in my opinion, but then I didn't start the thread. Rigorous attention to details is necessary when applying mathematics.

AS

 

[AmateurScientist]

To play devil's advocate for a second, while the original problem as stated is not well posed as either a math problem or an arithmetic exercise, recognizing it as ambiguous and open to interpretation is a useful skill to acquire. Imagine if it were in the context of a debate on markets, and you could interpret it as a gain or loss depending on how you meant to make your point.

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

 

[RenaissanceBiker]

Okay, I thought of one. What is the next number in this series?

2, 4, ...

is it a) 5, b) 6, c) 7 or d) 8

I do remember a problem like this on a college test. There are two correct answers. The next number is 6 if the series is just adding 2 to the previous number. The next number is 8 if the series is multiplying the previous number by 2.

 

[DeviousB]

Okay, I thought of one. What is the next number in this series?

2, 4, ...

is it a) 5, b) 6, c) 7 or d) 8

I do remember a problem like this on a college test. There are two correct answers. The next number is 6 if the series is just adding 2 to the previous number. The next number is 8 if the series is multiplying the previous number by 2.

And it is 32 if you take half of the cube of the previous term,
and it is 5.65685... if you multiply the square root of the previous term by sqrt(8),
and it is, well you get the idea...

 

[AmateurScientist]

Okay, I thought of one. What is the next number in this series?

2, 4, ...

is it a) 5, b) 6, c) 7 or d) 8

I do remember a problem like this on a college test. There are two correct answers. The next number is 6 if the series is just adding 2 to the previous number. The next number is 8 if the series is multiplying the previous number by 2.

That problem is indeed ambiguous. Are you sure you weren't given 3 or even 4 numbers in the series? If I recall correctly, that's usually how those questions are posed?

AS

 

[RenaissanceBiker]

It was quite a few years ago, but I remember there being more than one correct answer given. Can anyone else think of problems of this nature?

 

[drkitten]

It was quite a few years ago, but I remember there being more than one correct answer given. Can anyone else think of problems of this nature?

Any sequence problem admits of multiple solutions.

 

[fribble]

No, you don't. That is a formal proof.

You don't have to define any operations on an infinitely repeating decimal. It's quite simple. Multiplying by 10 means you move the decimal point one place to the right. Done. Everything else remains exactly as it was.

You're covered no more math than the most basic of functions on algebraic equations and simple multiplication by 10 and subtraction, which are basic arithmetic functions.

Limits are not required to prove that .999... = 1. You can even do it using fractions. Take 1/9, or 1/3, for example. 1/9 = .111... Multiply both sides by 9. You get 1 = .999... 1/3 = .333... Multiply each side by 3 and you get 1 = .999... yet again.

Simple.

AS

It is NOT a formal proof! The reason being that you're assuming that you know what it means to subtract infinite decimal expansions from each other. Mathematicians tend not to be very happy when you invoke infinity, because it can be very counterintuitive.

Take 1 - 1/2 + 1/3 - 1/4 + 1/5 - ... . (Ok, first I should prove that this sequence converges, but I'm too lazy to do it right now ) It has a certain value. But, by rearranging the terms, I can make this sequence converge to whatever the heck I like. (see end of article "Alternating series" on Wikipedia.) And it's knowing about infinity-related loopholes like this that would make a mathematician lose sleep at night having seen your first "proof". You don't know what precisely this 0.999... object is.

So, for that reason, you have to clarify exactly what you mean by 0.999..., without using infinity, to formalise the proof.

 

[fribble]

Any sequence problem admits of multiple solutions.

Bingo! More than that, any such sequence problem admits almost ANY solution... Here is a sequence that starts 2, 4, 42:

nth term: 18n^2 - 52n + 36