Assumptions in Mathematical (Calculus) word problems

[ajb]

The whole point of my issue was how do you know that?

Basically you take it on trust that whoever set the question is not trying to trick you in anyway.

 

Again, is it axiomatic or a theory?

It is not an axiom or theory in mathematics.

[Amad27]

Basically you take it on trust that whoever set the question is not trying to trick you in anyway.

 

 

It is not an axiom or theory in mathematics.

 

Hello,

 

I think you are right here.

 

Trust is the key in talking I suppose. (Like how I am trusting you )

 

The question is, should you trust on no basis?

 

In school they teach of these word problems, and they often teach the least complicated way to go about.

 

They never say why....

 

Thanks @ajb.

 

EDIT

------

They actually teach in school that you implicitly trust the author that he isnt hiding any information.

 

And idea why? OR should I just accept this (no reason)?

 

Also, they implicitly teach you that all information is given,

But again, no reason =(

Edited August 24, 2014 by Amad27

[ajb]

The question is, should you trust on no basis?

At the level you are talking about, you should trust your teachers and the authors of textbook. But beware that they may not be telling you the whole truth and there maybe subtleties that will arise in later study.

[Amad27]

At the level you are talking about, you should trust your teachers and the authors of textbook. But beware that they may not be telling you the whole truth and there maybe subtleties that will arise in later study.

Hello @ajb.

 

Thank you for advice.

 

In which later studies as you point out will the authors not be telling the truth?

 

also,

 

We have learned to assume the questions aren't trick questions since we were in school.

 

In school teaching, why do you take all information given and not assume it is a trick question? I am talking in formal school education. This is true with my peers as well =) Do you know?

Edited August 25, 2014 by Amad27

[pzkpfw]

Are you going to keep asking what's essentially the same question again and again for the rest of the year? Starts to look like trolling to me.

[Amad27]

Are you going to keep asking what's essentially the same question again and again for the rest of the year? Starts to look like trolling to me.

I am not trolling anyone; otherwise I wouldn't have discussed this.

 

Jeez, I thought a forum would be a good idea to consult with other people, but for some reason it is either some down-voting my posts or telling me that I am a troll. In the context of mathematics, I never troll. Never, never, never ever. I can't believe you thought that.

[Unity+]

In school teaching, why do you take all information given and not assume it is a trick question? I am talking in formal school education. This is true with my peers as well =) Do you know?

That is somewhat similar to paranoia if there is no evidence to support the idea, unless there is a pattern of him doing this.

[Amad27]

That is somewhat similar to paranoia if there is no evidence to support the idea, unless there is a pattern of him doing this.

 

Hello @Unity+ Can you be willing to tell me something?

 

When you talked a long while ago about the deductive reasoning of

 

"What is in the set, and what is not in set"

 

"It's less of an assumption and more of a conclusion based on deductive reasoning. It's like asking "Well, we may know what is in the set, but then what is not in the set?" You are asking a question that could have an infinite amount of possibilities. Therefore, the logic is not reasonable to follow."

Do you still believe this is true? (After this while)?

[Unity+]

 

Hello @Unity+ Can you be willing to tell me something?

 

When you talked a long while ago about the deductive reasoning of

 

"What is in the set, and what is not in set"

 

"It's less of an assumption and more of a conclusion based on deductive reasoning. It's like asking "Well, we may know what is in the set, but then what is not in the set?" You are asking a question that could have an infinite amount of possibilities. Therefore, the logic is not reasonable to follow."

Do you still believe this is true? (After this while)?

I still believe it is true, as do a majority of Mathematicians. Deductive reasoning is the core of many proofs of Mathematics that exist.

 

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

[Amad27]

I still believe it is true, as do a majority of Mathematicians. Deductive reasoning is the core of many proofs of Mathematics that exist.

 

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

 

mmmm.... I researched this, but can you tell me how it works as deductive reasoning? I mean how do you conclude that using deduction?

[studiot]

I am going to look at this question of yours from some different perspectives.

 

What is the context of the question?

 

What does the question want you to find?

 

What is an assumption?

 

Why are any assumptions needed and under what circumstances do you need to make them?

 

Are there any other sources of information?

 

What sort of questions are out there and what is their purpose?

 

How do you become good at extracting information and supplying what’s missing?

 

OK so last two perspectives first (the easy ones together). You become good by lots of practice.

It also helps if you can enjoy it.

There are several ways to get this practice.

Many people enjoy doing puzzles, brainteasers and the like.

You can find these in magazines, books of puzzles, online and quiz programs.

Older maths books aimed at the 12-15 age range often had many word based questions for pupils to practice with along with advice on how to do them.

For example:

54 pots of jam were tested and it was found that some were 1oz overweight and the remainder 1/2oz underweight. The total weight was correct however. How many jars were underweight?

 

A tank X contains 1000 gallons of water and water flows out at a rate of 5 gallons per minute.

Another tank Y is empty and water is flowing in at 10 gallons per minute.

(1) How much water is in each tank after the water has been running for t minutes?

(2) How much more water is there now in tank X than in tank Y? (answer in terms of t)

(3) Find this amount when t= 10; t=60; t=100. What does the last answer mean?

 

The second question shows an important point that often part (1) of such a question calculates something you will need in the second or later parts.

 

For the rest of the perspectives I will use your example.

 

 

"A spherical balloon is inflated with gas at the
rate of 800 cubic centimeters per minute. How fast is the radius of the balloon increasing at the instant the radius is (a) 30 centimeters and (b) 60 centimeters?"(Larson Calculus P 153)

 

So context

Had this been from a physics book then physics considerations might have been important. Such as temperature, pressure, Boyles or Charles law, the elasticity of the balloon.

 

 

But this was from a mathematics book in the mathematics section so mathematics is important and such physics considerations are unimportant and can be discarded.

But what mathematical considerations?

 

Well first we extract the information given.

This information is only the information given.

Nothing else.

No assumptions, guesses, inspirations or whatever.

The balloon is a sphere.

 

Then we supply relevant (mathematical) information that we know.

This information is not an assumption.

It is a hard fact or formula that we have learned/been taught.

 

In this case the volume of a sphere is [math]V = \frac{{4\pi {r^3}}}{3}[/math]

 

Do we need to assume that the balloon does not change shape?

Not really since the volume of any boxy or blocky object is still proportional to the cube of it’s ‘radius’.

 

What is the context?

 

Well calculus and the books want us to differentiate an equation connecting volume with radius.

 

So do we need any assumptions about holes in the balloon?

 

Well no because holes are not included in the standard formula connecting volume and radius.

 

So long as we extract the facts given and add facts that the book expects us to know as part of our course, we should not need to make any assumptions.

[Unity+]

 

mmmm.... I researched this, but can you tell me how it works as deductive reasoning? I mean how do you conclude that using deduction?

Well, here is a simple proof.

 

We are trying to find what x is equal to. The information provided states that x is a whole number and that it lies between 1 and 3. Since it is a whole numbers and is between these two numbers, it must be 2(of course you can state that the assumption is we are using the standard set of axioms, but even that can be stated in the problem).

[Amad27]

I am going to look at this question of yours from some different perspectives.

 

What is the context of the question?

 

What does the question want you to find?

 

What is an assumption?

 

Why are any assumptions needed and under what circumstances do you need to make them?

 

Are there any other sources of information?

 

What sort of questions are out there and what is their purpose?

 

How do you become good at extracting information and supplying what’s missing?

 

OK so last two perspectives first (the easy ones together). You become good by lots of practice.

It also helps if you can enjoy it.

There are several ways to get this practice.

Many people enjoy doing puzzles, brainteasers and the like.

You can find these in magazines, books of puzzles, online and quiz programs.

Older maths books aimed at the 12-15 age range often had many word based questions for pupils to practice with along with advice on how to do them.

For example:

54 pots of jam were tested and it was found that some were 1oz overweight and the remainder 1/2oz underweight. The total weight was correct however. How many jars were underweight?

 

A tank X contains 1000 gallons of water and water flows out at a rate of 5 gallons per minute.

Another tank Y is empty and water is flowing in at 10 gallons per minute.

(1) How much water is in each tank after the water has been running for t minutes?

(2) How much more water is there now in tank X than in tank Y? (answer in terms of t)

(3) Find this amount when t= 10; t=60; t=100. What does the last answer mean?

 

The second question shows an important point that often part (1) of such a question calculates something you will need in the second or later parts.

 

For the rest of the perspectives I will use your example.

 

 

 

So context

Had this been from a physics book then physics considerations might have been important. Such as temperature, pressure, Boyles or Charles law, the elasticity of the balloon.

 

 

But this was from a mathematics book in the mathematics section so mathematics is important and such physics considerations are unimportant and can be discarded.

But what mathematical considerations?

 

Well first we extract the information given.

This information is only the information given.

Nothing else.

No assumptions, guesses, inspirations or whatever.

The balloon is a sphere.

 

Then we supply relevant (mathematical) information that we know.

This information is not an assumption.

It is a hard fact or formula that we have learned/been taught.

 

In this case the volume of a sphere is [math]V = \frac{{4\pi {r^3}}}{3}[/math]

 

Do we need to assume that the balloon does not change shape?

Not really since the volume of any boxy or blocky object is still proportional to the cube of it’s ‘radius’.

 

What is the context?

 

Well calculus and the books want us to differentiate an equation connecting volume with radius.

 

So do we need any assumptions about holes in the balloon?

 

Well no because holes are not included in the standard formula connecting volume and radius.

 

So long as we extract the facts given and add facts that the book expects us to know as part of our course, we should not need to make any assumptions.

 

So really,

 

Are we making any assumptions when we state that there is no hole in the balloon?

 

What is a mathematical (word problem) assumption??

[studiot]

 

Are we making any assumptions when we state that there is no hole in the balloon?

 

Yes of course you are.

 

But you don't need to make that assumption, any more than you dont need to make other assumptions like

 

The balloon is not rising through the air and cooling and so shrinking.

 

The expansion of the balloon is not sufficient to burst it.

 

and so on.

 

Each of these possibilities can be calculated by more complicated physics, leading to more complicated maths, but we ignore them.

 

We ignore them because the original question has not mentioned them so we don't either.

 

We concentrate on what the question does mention.

 

If you look, all these assumptions contain the word 'not', they are about what the question is not.

 

Try to avoid negative assumptions about what questions are not.

They don't help solve the real question itself.

Edited September 8, 2014 by studiot

[Amad27]

 

Yes of course you are.

 

But you don't need to make that assumption, any more than you dont need to make other assumptions like

 

The balloon is not rising through the air and cooling and so shrinking.

 

The expansion of the balloon is not sufficient to burst it.

 

and so on.

 

Each of these possibilities can be calculated by more complicated physics, leading to more complicated maths, but we ignore them.

 

We ignore them because the original question has not mentioned them so we don't either.

 

We concentrate on what the question does mention.

 

If you look, all these assumptions contain the word 'not', they are about what the question is not.

 

Try to avoid negative assumptions about what questions are not.

They don't help solve the real question itself.

 

I was talking about this elsewhere as well,

 

"I'm afraid that without excluding unstated hypotheses and unstated possibilities, we'd never find any satisfying answers, and math could certainly not be applied to any purpose. The list of "what ifs" could persist indefinitely. Answers can be fine-tuned and/or adjusted if we do get more information provided. But we'd never have an answer to fine-tune if we start from the beginning by contemplating what possible complications might exist, and what if this or what if that, and what about...? "

 

What do you think?

[studiot]

Well yes it is true that you have to start somewhere to develop a theory or whatever.

 

This is exactly what you will do when you leave college and work in the outside world.

There you often have to decide for yourself what level of accuracy, what parameters want to calculate, what level of sophistication your calculation (remembering that in general the more sophisticated the more it costs) ans finally if your answer or result is sufficient.

 

But in school or college you are looking to achieve a predetermined result, set by the curriculum, so you don't have the luxury of trying an approximation and refining it.

 

The nearest I can think of in a college setting would be to check if a formula you think is correct by substituting some simple easy to calculate numbers to see if it yields a known result.

 

Which all still comes back to

 

Don't add things they haven't asked for.

Edited September 8, 2014 by studiot

[Amad27]

Well yes it is true that you have to start somewhere to develop a theory or whatever.

 

This is exactly what you will do when you leave college and work in the outside world.

There you often have to decide for yourself what level of accuracy, what parameters want to calculate, what level of sophistication your calculation (remembering that in general the more sophisticated the more it costs) ans finally if your answer or result is sufficient.

 

But in school or college you are looking to achieve a predetermined result, set by the curriculum, so you don't have the luxury of trying an approximation and refining it.

 

The nearest I can think of in a college setting would be to check if a formula you think is correct by substituting some simple easy to calculate numbers to see if it yields a known result.

 

Which all still comes back to

 

Don't add things they haven't asked for.

 

Yes, I think we are finally getting somewhere

 

I just want to ask one thing here.

 

So as you said it is important to develop a theory etc.

 

Are you agreeing with what I pasted before?

 

Thanks

[studiot]

Yes innovation can be important, but there is a time and a place for everything.

 

There is a time to innovate and a time to be straightforward, boring and conventional.

 

I give you the words of the song "Turn Turn Turn"

 

http://en.wikipedia.org/wiki/Turn!_Turn!_Turn!

[harshgoel1975]

My way.. as usual is different.. I won't make any assumptions... but for unknown or unspecified information.. i will make cases... like in your question .. there will be cases.. case 1 : there is no hole... and solution is........ case 2 ...... there is hole.. and solution is..........case 3..... and solution is...... so no need to make assumption.... at the same time with the given time ... we need to limit the cases and align with the question.... hope this help to understanding what it mean by making assumption in maths... these are not assumption but rather cases or solution for which that condition is met...

Edited September 9, 2014 by harshgoel1975

[studiot]

Do you prefer this semi verbal, semi pictorial question?

 

 

 

 

Edit I also like harshgoel's idea of separating into cases.

 

Unfortunately that would not be acceptable in an exam, even if you had time.

Further the effects in each case would have to be independent to be thus separable.

Edited September 9, 2014 by studiot

[imatfaal]

Do you prefer this semi verbal, semi pictorial question?

 

 

 

Question8.jpg

 

 

Took me more than 20 seconds.

[Amad27]

@studiot I have no clue as to how to answer that car question, seems to be very difficult. But now that I think about it, I suppose what the people first said here was correct. That the word problem is supposed to provide you with all information about the existing objects in the scenario Of the "story ."

 

But the only regard to be everyone is that how you would know that the word problem must provide all information to you?

 

Thanks

[studiot]

 

how you would know that the word problem must provide all information to you

 

The same way I know that seven eights are fifty six or thirteen thirteens are one hundred and sixty nine, although I've neve calculated them, or that Lagos is the capital of Nigeria, although I've never been there.

 

My teacher told me. And I believed her.

 

It is very difficult to progress unless you have enough confidence in your teacher to believe her.

Edited September 9, 2014 by studiot

[harshgoel1975]

Do you prefer this semi verbal, semi pictorial question?

 

 

 

Question8.jpg

 

Edit I also like harshgoel's idea of separating into cases.

 

Unfortunately that would not be acceptable in an exam, even if you had time.

Further the effects in each case would have to be independent to be thus separable.

Ans is 87, just see from the driver seat.....

[Amad27]

 

The same way I know that seven eights are fifty six or thirteen thirteens are one hundred and sixty nine, although I've neve calculated them, or that Lagos is the capital of Nigeria, although I've never been there.

 

My teacher told me. And I believed her.

 

It is very difficult to progress unless you have enough confidence in your teacher to believe her.

 

Hello,

 

In the context of a textbook though, how would you justify that?

 

I remember you saying it has always been that way (since the Ancient Greeks)??