Fraction meaning

length 1.6 (= 8/5)

denominator tells us that in how many parts we need to divide equally then we can max to max divide the 1.6 block into 5 parts ...how come 8 which is in the numerator which means we need to consider 8 parts out of 5 parts ..but then again in improper fraction we need to divide 10 blocks out of that we consider 8 parts ..okay u havent shown extra 2 parts right ... if my assumption is true then it is 8/10 but if i convert 8/5 to a mixed fraction 1 3/5
1 whole and 3 pieces ...but still cant digest why it is not 8/10.

Then quotative model tells us that how many groups of each 1/5 unit long of the box that is 1.6 unit in length should be taken so that i make a box i.e 1 unit length .
total 8 pieces * each length 1/5 considering x amt gives me 1 unit length
x=5/8

NOW if i compare the same thing with my apple analogy i.e (2 apples) (3 apples or persons what would be right??)2/3 that is 1/1.5...how many of
the 1.5 apples fit into the 1 apple ...i just make a statement keeping in mind wiht the box analogy ..which is still not clear to me ..in the
I marked fifths to help me scale things.

Now, how many of the 1.6 box fit into the 1 box? That's almost obvious
from the picture: the 1.6 box has been divided into 8 equal parts, and the
1 is equal to 5 of them, so it is 5/8 of the 1.6.

Let's see if I can make a partitive model. You want to find a piece of 1
such that 1.6 of them (that is, 1.6 times its length) will be 1. I'll make
a box of length 1 (this time, dividing it into 8ths, for a reason that
will become clear:
 
PLEASE, PLEASE, PLEASE, make this readable. I've asked you before. I am very tired of trying to figure out what parts are quoting me, what is your question, and what you really think is true. Why in the world are you ending with something I said?? I can't continue trying to help you if you can't help me out.

If you have to quote me, say something like this:

You said, "...".​
I don't understand what you mean by "...".​
I think it should be: ...​

I won't try to answer what you've said here, because I can't make any sense of it.

But really, I think we need to go back to basics. Take an example of a division of whole numbers, maybe 7 divided by 5 or 5 divided by 12, or whatever; do you understand the partitive and quotative models in that case? Once we are clear there, we can move to fraction or decimal problems, which, frankly, most teachers don't even try to model with pictures, so it's not necessary.

Specifically, let's take your apple example. State a specific problem and work through it, without referring to anything I've said, so I can see what you are thinking.
 
thank u to bear patience with me
length 1.6 (= 8/5)

denominator tells us that in how many parts we need to divide equally then we can max to max divide the 1.6 block into 5 parts ...how come 8 which is in the numerator which means we need to consider 8 parts out of 5 parts ..but then again in improper fraction we need to divide 10 blocks out of that we consider 8 parts ..okay u havent shown extra 2 parts right ... if my assumption is true then it is 8/10 but if i convert 8/5 to a mixed fraction 1 3/5
1 whole and 3 pieces ...but still cant digest why it is not 8/10

this is one doubt
 
Saumyojit said:
thank u to bear patience with me
length 1.6 (= 8/5)

denominator tells us that in how many parts we need to divide equally then we can max to max divide the 1.6 block into 5 parts ...how come 8 which is in the numerator which means we need to consider 8 parts out of 5 parts ..but then again in improper fraction we need to divide 10 blocks out of that we consider 8 parts ..okay u havent shown extra 2 parts right ... if my assumption is true then it is 8/10 but if i convert 8/5 to a mixed fraction 1 3/5
1 whole and 3 pieces ...but still cant digest why it is not 8/10

this is one doubt
Click to expand...
I just can't interpret what you are saying. You are mixing too many things together. The fact that a fraction can be written as 8/5 or 16/10 doesn't tell you how anything MUST be divided. It depends on context, which you have removed from the discussion. In fact, a fraction is not a division; there are many divisions that can result in the same fraction.

I will not talk about this any more until you are sure of the other issues.
Saumyojit said:
@Dr.Peterson please explain both quotative and partitave with 7/5 (improper fraction case) ...u can take any eg of apples ...
Click to expand...
A quotative example of 7 divided by 5 (note: this is about division, not about fractions!) would take 5 as the size of a part, and ask how many parts there are. For example, if I split 7 apples into groups of 5, how many piles can I put them into? (This makes little sense in reality, because piles aren't naturally counted by fractions. But if you can picture 1 2/5 piles, that is, one complete pile, and 2/5 of another, that is the answer.) A more reasonable example would be to split a 7-meter board into 5-meter pieces; how many 5-meter pieces do I get? (Answer: 1, and 2/5 of another.)

A partitive example would take 5 as the number of parts, and ask how big each part is. With apples, I would split 7 apples into 5 piles, and ask how many apples (that is, how much of an apple) is in each pile. This makes sense.
 
just to clear one thing :
i found out in one site Partitive example:
"4 pencils cost 36 cents. How much does one pencil cost?
The cost of 36 cents must be divided among the 4 pencils. (Cost per pencil is one group)"

here it is saying cost per pencil is one group but i guessed that per pencil is the one group not the cost. all the pencils tells us the no of groups
Click to expand...
 
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Saumyojit said:
just to clear one thing :
i found out in one site Partitive example:
Click to expand...
This example is really about ratios (price per pencil), but it is at least close to partitive. If you imagine a pile of 36 pennies and want to assign equal numbers to each pencil (its cost), then you are dividing the whole by the number of parts to find the size of a part, which is partitive.

In general, division is just the inverse of multiplication; the model is not the important thing mathematically. As I see it, the main value of the partitive-quotative distinction is to ensure that teachers do not teach only one kind of application of division, so that students get used to many different ways in which it applies.
 
Dr.Peterson said:
This example is really about ratios (price per pencil), but it is at least close to partitive. If you imagine a pile of 36 pennies and want to assign equal numbers to each pencil (its cost), then you are dividing the whole by the number of parts to find the size of a part, which is partitive.

In general, division is just the inverse of multiplication; the model is not the important thing mathematically. As I see it, the main value of the partitive-quotative distinction is to ensure that teachers do not teach only one kind of application of division, so that students get used to many different ways in which it applies.
Click to expand...


they have stated that this is a division ... now i am really confused between fraction and dvison.
How to identify by reading statemnts that it is ratio or fraction or divison
 
A division is an operation acting on two numbers, whose result is a number.

A ratio is a relationship between two numbers, typically written with a colon and read as "something to something". (A rate is a particular kind of ratio, written as "something per something".)

A fraction is a way of writing a single number, which amounts to a division that is not carried out (so that 2/3 means "the number you would get if you divided 2 by 3").

But what I said about the example being about ratios was not primarily about this distinction. The main point was that you are dividing quantities measured in two different units, with the result being a rate ("this per that"). A purely partitive division would divide a quantity in some unit (apples, meters, ...) by a pure number (4, meaning 4 piles or 4 pieces, say), with the result being in the same unit as the whole (12 apples divided among 3 people yields 4 for each person). It's not really very different, since you could describe that as " 4 per person"; but to me, the "per" in the answer gives a different focus to the problem.

And you may observe that immediately after the example you are asking about in that page, they mention ratios! So their thinking is in line with mine.

Note also that they use "measurement" for what is essentially "quotative". There are a number of different ways to talk about all this, and the categories are not exhaustive. None of this is important mathematically, only pedagogically.
 
Dr.Peterson said:
That is, everything is measured in fifths: 1 is 5/5, 1.6 is 8/5. I could just as well have marked tenths. To show 1.6 on a ruler, it has to be marked in tenths, or at least fifths.
Click to expand...
What i understood that we have to bring 1 and 1.6 in the same unit to compare between them as 1.6 is in fifths so 1 will be 5/5 of 1.6 . the unit is fifths.

See 1/1.6 no of pieces = 1/(8/5)=5/8 size of required piece

Question is framed as: what shall be the size of each piece if 1.6 of them equals 1 whole? okay

So at first i have 8 of 1/5ths where 5 of 1/5ths means 1 box . The rest 3 came from another box of 1/5ths .

As u said
1: "Now I want a piece that I can multiply by 8/5 to get the whole 1"
2: "In order for those 8 parts to equal the one, each of them must be an eighth"

i can do this in 2 ways to convert from fifth to eighth.

I dont know why the case 2 is more easy .

CASE 1: Now for this ; size of the piece needs to have 8 in the denom and 5 in the numerator(5/8) . Now in 8/5 i have 8 "one fifths" whcih means first i need to convert the fifths into eighths , so i divide (each 1/5th) /8=1/40 ths , now for making the numerator '5' i need to multiply 1/40 ths by 5 which produces 5/40 which gives me 1/8 th per piece size of 1 box ( newly created or transformed box) by simpliying.
So 5 parts of 1/5th --> 8 parts of 1/8th .

CASE 2: This time i will multiply first 1/5 by 5 so it gives me 1 whole and then divide that whole into 8 pieces giving me "1/8th" per piece size.
As i know the answer is 5/8 so i will consider 5 pieces out of 8.

NOW U SAID "each of those eighths must be 1/5 of the piece I am making" .

I dont understand this particular line . why 1/5 of the piece? I just cannot relate

As far i have understood --> in 1/1.6 no of pieces = 5/8 is the size of each piece per 1.6 parts or Size is 5/8 of 1 whole box
If i do arithmetic and find out the quotient i.e 5/8 i can see that i have to consider 5 parts out of 8 parts of 1 box .

But did not understood the meaning of the line "each of those eighths must be 1/5 of the piece I am making"
@Dr.Peterson
 
If you read U again for you, most helpers will stop reading your posts. This type of writing is clearly mentioned in the forum rules as unacceptable.
 
I'm not going to search through a 6-month-old thread to see what I was saying, then work through your long, confusing question about it. That's beyond my pay scale.
 
Dr.Peterson said:
I'm not going to search through a 6-month-old thread
Click to expand...
http://mathforum.org/library/drmath/view/77467.html
Thank u for replying
All that i have written in post #50 is in the link "decimal in the denom" and I did not sat down with the link's post until two days ago when i was actually trying to explain it to myself . 6 months ago i did not understood quotative and partitive but in these months i understood what these things are and so decided 2 days ago to go through the Link' s post again; There are few parts that i did not understand so i have written my concern.

U dont have to search anything in this thread it is just that the link was mentioned in this thread and thats why i have stated my recent doubt in this thread .
@Dr.Peterson
I need ur help with post #50
Jomo said:
This type of writing is clearly mentioned in the forum rules as unacceptable.
Click to expand...
There are sometimes(especially with my case) when the post will be actually discussed(in the future also sometimes) i will not understand some parts or words , so i need to clear my basics and go way back to fundamentals which took me atleast 6 months (in this case) and then come back to where we are actually discususing about the post 6 months ago.
I know for the helpers its difficlt to again read the previous comments to bring their mind to what actually we were discususing in this post but I cannot do anything about it . Sorry .
 
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Let me rephrase post no 50
Everything is in the given link
Dr.Peterson said:
That is, everything is measured in fifths:
Click to expand...
What i understood that we have to bring 1 and 1.6 in the same unit to compare between them as 1.6 is in fifths so 1 will be 5/5 of 1.6 . the unit is fifths. RIGHT?

Partitive of 1/1.6
After reading the partitve section of this link written by u
I See 1/1.6 no of pieces = 1/(8/5)=5/8 size of required piece

Question is framed as: what shall be the size of each piece if 1.6 of them equals 1 whole? okay

So at first i have 8 of 1/5ths where 5 of 1/5ths means 1 box . The rest 3 came from another box of 1/5ths .

As u said
1: "Now I want a piece that I can multiply by 8/5 to get the whole 1"
2: "In order for those 8 parts to equal the one, each of them must be an eighth"

objective: to convert from fifth to eighth.
THere are 2 cases.

I dont know why the case 2 is more easy .

CASE 1: size of the piece that i am finding(5/8) needs to have 8 in the denom and 5 in the numerator . Now in 8/5 (original) i have 8 "one fifths" which means first i need to convert the fifths into eighths , so i divide (each 1/5th) /8=1/40 ths ,
Now for making the numerator '5' ; i need to multiply 1/40 ths by 5 which produces 5/40 which gives me by simplyfing "1/8 th per piece size" of 1 box that is newly created or transformed from the fifth box
So 5 parts of 1/5th --> 8 parts of 1/8th .

CASE 2: This time i will multiply first 1/5 by 5 so it gives me 1 whole and then divide that whole into 8 pieces giving me "1/8th" per piece size.
As i know the answer is 5/8 so i will consider 5 pieces out of 8.

Is my thinking process right till this?

DOUBT: NOW U SAID "each of those eighths must be 1/5 of the piece I am making" .

I dont understand this particular line . why 1/5 of the piece? I just cannot relate

As far i have understood --> in 1/1.6 no of pieces = 5/8 is the size of each piece per 1.6 parts or Size is 5/8 of 1 whole box
If i do arithmetic and find out the quotient i.e 5/8 i can see that i have to consider 5 parts out of 8 parts of 1 box .

But did not understood the meaning of the line "each of those eighths must be 1/5 of the piece I am making"
@Dr.Peterson
 
I'm sorry, but I simply don't want to try to continue this discussion. What you write here is not self-contained, and is asking about something I said in a context I don't want to figure out again. And your writing, frankly, is just to hard to follow.

If you want to get an answer, please state the problem you are working on at this point, and ask about that. If you don't understand something I wrote back then, just ignore it, and ask your question fresh so I can answer it better. (That's the same advice I've given about Wikipedia and other sources!)

It will be best if you do this in a new thread, as the mere existence of 50 posts makes me feel overwhelmed.
 
Dr.Peterson said:
please state the problem you are working on at this point, and ask about that
Click to expand...
Okay. I am stating the problemn but please read .
Partitive of 1/1.6 In partitive section from the link i saw that
In order for those 8 parts to equal the one, each of them must be an eighth; and each of those eighths must be 1/5 of the piece I am making. So it must consist of 5 of the eighths:
Click to expand...
As far i have understood " we are dividing 1 whole box into 'x' size of a piece of that 1 box such that 1.6 of that piece equals 1 whole box "-->1 whole box /1.6 no of piece = 5/8 is the size of each piece per 1.6 or Size is 5/8 of 1 whole box.
SO acc to answer I got the logic that One WHOLE BOX is divided into 8 parts of 1/8th and i have to consider 5 parts out of 8 parts of 1 box.
In this way i think my interpretation is right.


DOUBT: Now u said "each of those eighths must be 1/5 of the piece I am making. So it must consist of 5 of the eighths"

I dont understand this particular line . why 1/5 of the piece?

If i interpet the line "each of those eighths must be 1/5 of the piece I am making" literally it means 1/5 * 5/8 gives me 1/8th.
 
If you write U again for YOU, most helpers will stop reading your posts. This type of writing is clearly mentioned in the forum rules as unacceptable.
 
Why do you focus so hard on trying to make or understand analogies?

Can I divide something into 1.8 parts? No. It does not correspond to a possible operation in the world of physical experience. I always have a whole number of parts.

[MATH]\dfrac{27}{3} = 9 \iff 3 * 9 = 27.[/MATH]
[MATH]\dfrac{27}{1.8} = x \iff 1.8 * x = 27.[/MATH]
It is the extension of a mathematical idea beyond the purely physical operation that originally gave rise to the mathematical idea. All you need to know to do mathematics is to realize that division is the inverse operation to multiplication. There is no need whatsoever to try to deal with it in physical terms of dividing something into a whole number of exactly equal parts, something that is not always achievable in the physical world.

Can we create a physical analogy? We can. Suppose we are told that the 27 represents 27 kilograms. Then we can always say that what the problem REALLY means is that we are to divide those kilograms into two parts that are not equal but are in the exact ratio of 10 to 8.

Simple. One part will be 15 kilograms and the other will be 12 kilograms, which do add up to 27 kilograms., and are in the ratio of 15 to 12, which is the same as 5 to 4, which is the same as 10 to 8.

How in the world do I get there?

Think in grams. I have 27000 grams. I can in principle divide those into 18 equal parts. Each part will contain 1500 grams. But I don't want eighteen equal parts. I want 2 unequal parts. So I combine 10 of those parts into one, giving me 15000 grams or 15 kilograms, and combine the other eight into one, fiving me 12000 grams or 12 kilograms.

I think this whole boxes thing was to try to explain a method of teaching about fractions. I personally am not enchanted with the method; nor do I think Dr. Peterson was necessarily doing anything but explaining it.

I am not enamored of the analogy that I gave, but if you must have an analogy of whether a denominator that is not a positive whole number can correspond to real operations in the physical world then try thinking of it that way.
 
I'd entirely forgotten that this discussion was about something I wrote in 2013. That's what happens when a thread gets too long, and that's why I don't want to continue such a thread.

Let me quote more from that, as your quoting snippets with no context is part of the reason I find these discussions so annoying:

Code: 
Let's see if I can make a partitive model. You want to find a piece of 1
such that 1.6 of them (that is, 1.6 times its length) will be 1. I'll make
a box of length 1 (this time, dividing it into 8ths, for a reason that
will become clear:

                   1
   +-------------------------------+
   |   :   :   :   :   :   :   :   |
   +-------------------------------+

Now I want a piece that I can multiply by 8/5 to get the whole 1.
Multiplying by 8/5 amounts to making 8 parts, each of which is 1/5 of the
original. In order for those 8 parts to equal the one, each of them must
be an eighth; and each of those eighths must be 1/5 of the piece I am
making. So it must consist of 5 of the eighths:

                   1
   +-------------------------------+
   |XXX:XXX:XXX:XXX:XXX:   :   :   |
   +-------------------------------+

So my answer is 5/8.
Now your comment is:
Saumyojit said:
As far i have understood " we are dividing 1 whole box into 'x' size of a piece of that 1 box such that 1.6 of that piece equals 1 whole box "-->1 whole box /1.6 no of piece = 5/8 is the size of each piece per 1.6 or Size is 5/8 of 1 whole box.
SO acc to answer I got the logic that One WHOLE BOX is divided into 8 parts of 1/8th and i have to consider 5 parts out of 8 parts of 1 box.
In this way i think my interpretation is right.

DOUBT: Now u said "each of those eighths must be 1/5 of the piece I am making. So it must consist of 5 of the eighths"

I dont understand this particular line . why 1/5 of the piece?

If i interpret the line "each of those eighths must be 1/5 of the piece I am making" literally it means 1/5 * 5/8 gives me 1/8th.
Click to expand...
We want a quantity such that 8/5 times the quantity equals 1. We've found that therefore 8 times 1/5 of the quantity equals 1, so 1/5 of the quantity must be 1/8 [of the whole]. Do you follow that far?

Then, since one of my quantity is 5 times 1/5 of my quantity, my quantity is therefore 5 times 1/8.

But if you understand what you say numerically, then you don't need to understand what I said there. It was meant to help someone who wanted to understand fractions visually, and that is not the best way to understand fractions in the first place. As I've said, if you don't understand what someone says in an attempt to make it understandable, skip it, and look for something that you do understand.

Here is what it's really all about, without talking about pieces: To divide 1 by 8/5, we are looking for a number x such that x*8/5 = 1; that is, x*8*1/5 = 1. Then x*8 = 5 (multiplying both sides by 5), and x = 5/8 (dividing both sides by 8). That's all it takes.
 
@Dr.Peterson

Dr.Peterson said:
you don't need to understand what I said there. It was meant to help someone who wanted to understand fractions visually
Click to expand...
No i need to understand
Dr.Peterson said:
We've found that therefore 8 times 1/5 of the quantity equals 1, so 1/5 of the quantity must be 1/8 [of the whole]. Do you follow that far?
Click to expand...
yes I understood --> 1/5 of the quantity =1/8 of the whole --> 1/5 *x=1/8th


Dr.Peterson said:
my quantity is therefore 5 times 1/8
Click to expand...

yes it can be seen from the equation 1/5*x=1/8th --> taking 5 from lhs to rhs gives me x=5/8 th of 1 whole
SO my quantity is therefore 5 times 1/8 of the whole box

But why did u said
Dr.Peterson said:
since one of my quantity is 5 times 1/5 of my quantity
Click to expand...

As i was following ur post ur equation stands like this "1/5 *x=1/8th" i see that 1/5 of my quantity is 1/8th . I dont get it where is the "5 times of 1/5 "

OKAY What i observed just now 1/5 *x =1/8th --> then u multiplyed both sides by 5 i get

EQUATION: 5 *1/8=5 *1/5*x --> which means one of my quantity (5*1/8 lhs part ) is 5 times 1/5 of my quantity(x)

This is what u are trying to mean
 
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