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sumpm1

Member

Registered: 2007-03-05

Posts: 42

Is this function discrete or continuous?

Hello, had a simple discrete or continuous function question posed in my 9th grade algebra class. Here is the question:

Q: A recipe for brownies reads that 1 cup of milk will make 6 brownies. Is this function continuous or discrete?

What say you?

Bob

Administrator

Registered: 2010-06-20

Posts: 10,143

Re: Is this function discrete or continuous?

hi sumpm1

It all depends on whether you can make, say, half a brownie.  If it is only possible to make whole brownies, then the function is discrete but (and I should confess, I am no cook!) I had a look at a recipe

http://allrecipes.co.uk/recipe/6023/qui … wnies.aspx

and the key phrase is:

3.Cool, and slice into equal square portions.

Clearly, that means you can divide up your mix into any shape and size.

Then you are in the realms of philosophy.

For if you take a brownie and cut it in two, have you got two half brownies or just two brownies (albeit smaller)?

So it begins to look like you can argue this either way.  I think your innocent question is going to run and run. 

EDIT:  If you go on subdividing a brownie into smaller and smaller pieces, eventually you will have to stop when you reach the atomic level.  That would make it discrete.

Oh darn!  That means this thread will also get side-tracked into whether it is possible to 'split the brownie atom'.

According to Dictionary.com

a small, chewy, cakelike cookie, usually made with chocolate and containing nuts.

Small seems to imply that a split brownie becomes two brownies => a discrete function.

If you teacher fails to grasp this argument, say you made some brownies to bring in and, to make sure there were enough to feed the whole school, you divided them up into tiny slices.  Hold up an empty hand and offer the teacher his / her share.

Bob

Last edited by Bob (2011-11-04 21:25:36)

---

Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

Au101

Member

Registered: 2010-12-01

Posts: 353

Re: Is this function discrete or continuous?

That is really quite some advice from a retired teacher, bob bundy You must have been popular with the students

Bob

Administrator

Registered: 2010-06-20

Posts: 10,143

Re: Is this function discrete or continuous?

hi Au101,

Thank you.  But why the past tense?  Just because they've stopped paying me, doesn't stop me going to school.  I went in yesterday to help set up for the fireworks night and tomorrow I'm back in doing the technical rehearsing for a dance show.

Well it keeps me off the streets. 

Bob

---

Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

Au101

Member

Registered: 2010-12-01

Posts: 353

Re: Is this function discrete or continuous?

Well, all I meant was that they've stopped paying you, but all the same - if you don't mind me saying so - it's nice to hear that

Bob

Administrator

Registered: 2010-06-20

Posts: 10,143

Re: Is this function discrete or continuous?

hi Au101

OK.  Nice to hear from you again.  It's become a hobby now, and I've still got lots of friends there.

And I enjoy helping people here too.

Bob

---

Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

sumpm1

Member

Registered: 2007-03-05

Posts: 42

Re: Is this function discrete or continuous?

hi sumpm1

It all depends on whether you can make, say, half a brownie.  If it is only possible to make whole brownies, then the function is discrete but (and I should confess, I am no cook!) I had a look at a recipe

http://allrecipes.co.uk/recipe/6023/qui … wnies.aspx

and the key phrase is:

3.Cool, and slice into equal square portions.

Clearly, that means you can divide up your mix into any shape and size.

Then you are in the realms of philosophy.

For if you take a brownie and cut it in two, have you got two half brownies or just two brownies (albeit smaller)?

So it begins to look like you can argue this either way.  I think your innocent question is going to run and run. 

EDIT:  If you go on subdividing a brownie into smaller and smaller pieces, eventually you will have to stop when you reach the atomic level.  That would make it discrete.

Oh darn!  That means this thread will also get side-tracked into whether it is possible to 'split the brownie atom'.

According to Dictionary.com

a small, chewy, cakelike cookie, usually made with chocolate and containing nuts.

Small seems to imply that a split brownie becomes two brownies => a discrete function.

If you teacher fails to grasp this argument, say you made some brownies to bring in and, to make sure there were enough to feed the whole school, you divided them up into tiny slices.  Hold up an empty hand and offer the teacher his / her share.

Bob

Haha, thanks for going so in depth. I must confess, I am a 9th grade teacher and this was a problem on one of my warm-ups. Some of my colleagues saw the problem and argued for discrete, I argued for continuous. Here is my argument:

If you were to use 2 cups per the instructions, you would have 12 brownies, and 3 to 18 and so on; likewise 1/2 a cup would seemingly result in 3 brownies per the instructions, and 1/4 cup would then seem to make 1.5 of the brownies that the instructions are describing. Since we are making these brownies, we may end up with 3.5 of these brownies described in the instructions.

I compare the situation to the following:

1.  The directions on an ice cube tray say that one cup of water will make 6 cubes. (After you use 1/6 cup, the first cube slot is full, and at some point the second slot is 1/3 full and so on.)

2. The directions on a bag of concrete say that one pound of mix will make 6 bricks. (After you fill the form for the first brick, you begin filling the second, and so on.)

3. Your printer takes 1 minute to print 6 copies of a flyer. (At some point the second flyer is 5/6 of the way printed, and so on...)

Bob

Administrator

Registered: 2010-06-20

Posts: 10,143

Re: Is this function discrete or continuous?

hi sumpm1

Oh!  You are the teacher!  In that case it looks like you are making brownies for the whole class.  Got to experiment after all!

Your analysis gets to the crux of the problem.  Is a fraction of a brownie in fact just one (smaller) brownie?

But, on the other hand, does it matter?  If you put this one to the class and they can see that point, then you've achieved the objective in getting them to understand the difference between discrete and continuous.  If they've grasped that, then what does the actual answer matter?  The discussion is much more important than the outcome.

But I'd still argue for discrete:  Half an ice cube is still an ice 'cube'; half a brick is no use to anyone; half a flyer ditto.

Keep thinking! 

Bob

---

Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

Au101

Member

Registered: 2010-12-01

Posts: 353

Re: Is this function discrete or continuous?

I would actually agree with bobbym here. In mathematics and science, discrete objects, such as the integers, do not vary smoothly, they have distinct values, little blocks, as it were. With the real numbers, which are continuous, you can take two numbers which are as close together as you like, for example 3.111 and 3.112 and still have many numbers between these two. With your brownie recipe, I think that you are making individual chunks and so I would argue that it is discrete, it can't vary arbitrarily.

But that's just my take on the issue

sumpm1

Member

Registered: 2007-03-05

Posts: 42

Re: Is this function discrete or continuous?

I would actually agree with bobbym here. In mathematics and science, discrete objects, such as the integers, do not vary smoothly, they have distinct values, little blocks, as it were. With the real numbers, which are continuous, you can take two numbers which are as close together as you like, for example 3.111 and 3.112 and still have many numbers between these two. With your brownie recipe, I think that you are making individual chunks and so I would argue that it is discrete, it can't vary arbitrarily.

But that's just my take on the issue

As far as what you guys are saying about ice cube, brick, or brownie "being useful" is not the story though.

My point is this. If you printer takes 1 minute to print 6 flyers... Well half of a flyer is not useful, but you MUST agree that at SOME POINT a flyer was halfway done printing, that data STILL EXISTED, whether you find it useful or not!

So I have raised the point to my colleagues: Either SOMEONE or SOMETHING in the problem must disctate the discrete nature of the input or the output.

In those cases stated "an ice cube, a brick, a printed flyer..." If we are creating them, then we must admit that at some point they were "half way created" and all of the continuous data exists. But if the problem says "you are buying PRE MADE bricks..." then SOMEONE (the store employees) won't let you you break a brick in half or buy half of a brick. The store won't let you buy half of a PRE MADE discrete unit of brownie.

So I argue that SOMEONE (the people or the situation), or something (nature dictates that humans and apples exist as discrete growing organisms) must dictate this discrete nature.

And as far as going so in depth to say, could we actually use "one molecule of milk?" in our example, this is beyond the discussion i think. But others have brought this up and said that any data that is being measured or recorded in reality is discrete because we can only measure to certain approximations of values of time, weight... and so on. But I only see that as being true when the data is actually being measured. If I put on a blindfold and pour some milk into a container, even though I haven't determined a measurement for what I poured, I can create infinite values of milk. But if you asked me to measure 3.14... cups of milk, no I couldn't... so a big difference on "measurement vs possible values."