Is division confused with fractions?

[Myuncle]

Why do we teach our kids that 6 ÷ 3 = 2? If I divide 6 oranges in 3, what I get is not 2 oranges, but 2+2+2, without subtracting 4 oranges. So why don't we teach 6 ÷ 3 = 2+2+2, 1÷2 = 0.5+0.5, etc etc? If I say "I would like a quarter of this cake", then I am not dividing, but I am dividing and subtracting, I am doing a fraction, 1/4. Don't you think divisions shouldn't be confused with fractions?

[HalfWit]

Why do we teach our kids that 6 ÷ 3 = 2?

Because 1/3 is the multiplicative inverse of 3 in the real numbers; and 6 * (1/3) = 2. The application to biological entities such as oranges is better left to advanced courses in applied mathematics.

 

I think math pedagogy would be improved if we would refrain from confusing fractions with rational numbers.

[John Cuthber]

Why do we teach our kids that 6 ÷ 3 = 2? If I divide 6 oranges in 3, what I get is not 2 oranges, but 2+2+2, without subtracting 4 oranges. So why don't we teach 6 ÷ 3 = 2+2+2, 1÷2 = 0.5+0.5, etc etc? If I say "I would like a quarter of this cake", then I am not dividing, but I am dividing and subtracting, I am doing a fraction, 1/4. Don't you think divisions shouldn't be confused with fractions?

Division is subtraction.

Q) How many times can you take away 2 oranges from a box initially containing ten oranges?

A) five times

10/2 =5

[studiot]

 

If I divide 6 oranges in 3, what I get is not 2 oranges, but 2+2+2,

 

 

Do you? Are you quite sure?

 

Where was division into equal parts mentioned?

 

Why can you not divide 6 oranges into three (still whole) parts thus:

 

4 + 1 + 1

 

?

 

Are not the meanings of the terms 'division', 'rational' and 'fraction' usually apparent from the context?

 

 

Division is subtraction.

 

Not always.

I understand what Myuncle is saying.

 

Division has different meaning in mathematics, inheritance and physics where dividing a whole into parts is often meant.

 

Strangely you can multiply by dividing.

 

For example :

 

The general divided his force into two.

 

Meaning he now had two forces!

Edited August 4, 2013 by studiot

[michel123456]

Why do we teach our kids that 6 ÷ 3 = 2? If I divide 6 oranges in 3, what I get is not 2 oranges, but 2+2+2, without subtracting 4 oranges. So why don't we teach 6 ÷ 3 = 2+2+2, 1÷2 = 0.5+0.5, etc etc? If I say "I would like a quarter of this cake", then I am not dividing, but I am dividing and subtracting, I am doing a fraction, 1/4. Don't you think divisions shouldn't be confused with fractions?

I like that kind of questions.

Although it is not a question.

It simply means that you must use mathematics with caution and that any operation will have the meaning that you want to put in. And that the correct result of a correct equation will have a lot of wrong meanings and only one correct meaning.

 

 

Do you? Are you quite sure?

 

Where was division into equal parts mentioned?

 

Why can you not divide 6 oranges into three (still whole) parts thus:

 

4 + 1 + 1

 

?

 

Are not the meanings of the terms 'division', 'rational' and 'fraction' usually apparent from the context?

 

 

Not always.

I understand what Myuncle is saying.

 

Division has different meaning in mathematics, inheritance and physics where dividing a whole into parts is often meant.

 

Strangely you can multiply by dividing.

 

For example :

 

The general divided his force into two.

 

Meaning he now had two forces!

(emphasis mine)

IIRC mathematical division is always in equal parts.

[studiot]

 

IIRC mathematical division is always in equal parts.

 

 

Sorry, can't agree old bud.

 

and that goes to the heart of Myuncle's question.

 

 

The quotient formed by dividing 6 by 3 is 2.

It is a single pure number, not a number of parts, each comprising a pure number.

 

However it is perfectly respectable mathematics to divide something into a (not necessarily equal) number of parts, which may or may not be numbers. For example a pie diagram.

 

I note that john cuthber's interpretation of repeated subtraction also end up with three parts, not a single number.

 

 

Myuncle wished to distinguish between these two uses.

Edited August 5, 2013 by studiot

[michel123456]

 

Sorry, can't agree old bud.

 

and that goes to the heart of Myuncle's question.

 

 

The quotient formed by dividing 6 by 3 is 2.

It is a single pure number, not a number of parts, each comprising a pure number.

 

(...)

3 is the number of (equal) parts.

[Myuncle]

Yes, all I would like, is to help kids (and myself...) to understand better. Kids can be our best teachers simply because they have no prejudices, and their questions can seriously make you feel uncomfortable. Does the definition of division implies equal parts and subtraction? I don't know, it would be useful to clarify this . Math is all based on the concept that a unity is identical to another unity, 1=1, that's all theory, but in practice it might not even exist, since an orange is not exactly identical to an other orange, an atom might not be exactly identical to an other atom, a quark might not be identical to a quark, etc. So 1=1? Only in theory, only in our fantasy and imagination, but not in reality. If we use math, it means that, anyway we all agree that in theory a unity is identical to another unity, that's a very useful agreement between humans, very convenient, that allows us to agree on measurements and quantities.

[swansont]

6 divided by 3 is not the right example to use. What is 7 divided by 3?

[ajb]

Math is all based on the concept that a unity is identical to another unity, 1=1, that's all theory, but in practice it might not even exist...

Well, if you have a nice mathematical structure like a group, then you can show that you have a unique identity element, which acts in the same way from the left and the right. So, with the real numbers, with the group action of multiplication we have a group structure and so the identity is unique.

 

This is not true in generality. We can have structures on sets that have no identity or more than one identity. So mathematics does allow much more "strange rules" than just numbers and multiplication.

 

...since an orange is not exactly identical to an other orange, an atom might not be exactly identical to an other atom, a quark might not be identical to a quark, etc. So 1=1?

Okay, but mathematics is an abstraction of nature and a very useful one at that.

 

Only in theory, only in our fantasy and imagination, but not in reality. If we use math, it means that, anyway we all agree that in theory a unity is identical to another unity, that's a very useful agreement between humans, very convenient, that allows us to agree on measurements and quantities.

You are starting to think about how mathematics applies to nature. Whenever we model something we are forced to use mathematical abstraction. By doing so we have to use the rules of mathematics, this is where we all agree that (given what I said before) that "1=1".

 

Anyway, with the opening question. Yes divison and fractions are confused a lot.

 

Division is the process of finding the inverse of invertiable elements. For the real numbers, with the binary operation of standard multiplication, division is looking for the unique "a" in ab = ba =1, for all numbers b, except for zero. The number zero is noninvertiable.

 

Fractions are just notation for elements of the real numbers. That is a/b (b not equal to zero) represents a real number.

 

Both these concepts are useful in describing what one often means by divison and fractions in school. Basically the idea of splitting a set of objects into smaller sets; which is much easier to visualise for "nice sets".

 

 

What is 7 divided by 3?

The real number that I will represent by 7/3.

 

Edited August 5, 2013 by ajb

[overtone]

All confusions in the student are legitimate - grownups often overlook the variety, the complexity, possible in the approach to new mathematical concepts.

 

Almost all new things in mathematics are difficult to grasp, but simple in hindsight. Almost all the notation appears chosen from a very large collection of possibilities at first, but appears simple and almost inevitable afterwards. The words used have special and different meanings which badly confuse at first, maintain clarity afterwards ( teach math in a language other than the student's primary one, if you can). This hindsight impression of simplicity is partly illusion - division, say, is far from intrinsically simple, no matter how automatic one's routine handling of it has become (quite probably you, typical grownup, have never actually figured out what was going on.) - and partly amnesia, as a grateful mind seizes on the "right answer" and prunes frustration.

 

Sometimes I have had good luck with the language used at the supposed "higher" levels of mathematics and rhetoric - using active verbs such as "now use 3 to divide 7 - what do you get?", remembering to include the word "if" explicitly, describing multiplication by a negative number as a rotation or flip - it seems to help some people a lot. The better mathematicians tend to write well in general, which hints.

[swansont]

The real number that I will represent by 7/3.

 

Quite.

 

I am interested on how Myuncle presents this as subtraction of integers.

[Myuncle]

Quite.

 

I am interested on how Myuncle presents this as subtraction of integers.

To kids I would teach directly 7÷3=2.3333333333333333333333333333333+2.3333333333333333333333333333333+2.3333333333333333333333333333333 or simply 7÷3=2.3333333333333333333333333333333×3

[swansont]

So kids understand decimals before fractions? That's not what I recall.

[overtone]

 

 

To kids I would teach directly 7÷3=2.3333333333333333333333333333333+2.3333333333333333333333333333333+2.3333333333333333333333333333333 or simply 7÷3=2.3333333333333333333333333333333×3

Please don't - this is the tutor talking, the guy who will have to fix this kind of problem after habituation runs into the wall of college calculus.

 

It's very important that you not encourage the placing of "=" signs between unequal entities. It's very important that you not encourage the early conversion of fractions (or square roots, exponential/logarithmic and trig expressions, etc) to decimals, before indicated algebraic manipulations have been done.

[Greg H.]

simply 7÷3=2.3333333333333333333333333333333×3

Except that's wrong because

[math] 7 \div 3 > 2.3333333333333333333333333333333333 \times 3[/math]

 

You could say

[math] 7 \div 3 = (2 \times 3) + 1[/math]

Which is why kids get introduced to the concept of remainder division first. It's not pretty, but it's at least mathematically accurate.

[timo]

1/3 is the multiplicative inverse of 3 in the real numbers; and 6 * (1/3) = 2. The application to biological entities such as oranges is better left to advanced courses in applied mathematics.

Not "+1" but "<thumb up> like"

Edited August 5, 2013 by timo

[studiot]

 

I am interested on how Myuncle presents this as subtraction of integers.

 

 

Why is division to be restricted to integers?

 

Even more interesting is to consider the dual propositions:

 

Take 6 oranges and divide them by 2, which come out at 3 as noted by others.

 

But compare

 

Take 6 oranges and divide them by 1/2. This comes out at??

 

Or what about [math]\int {xydx} [/math] divided by [math]\int {ydx} [/math]

 

How do these examples play with the idea of parts, equal or otherwise?

 

Edited August 5, 2013 by studiot

[Myuncle]

It's very important that you not encourage the placing of "=" signs between unequal entities.

You are right. Instead of "=", we could use the sign "≈" ? That's how I would have liked to be taught when I was a kid, even before algebra. That would make things easier maybe, if taught very early.

So kids understand decimals before fractions? That's not what I recall.

They can. Show them 7 tennis balls, ask them to divide them by 3 in equal parts. They will tell you immediately that it's impossible. Now show them 7 biscuits, ask them to divide them by 3 in equal parts, they don't have any problem to tell you that you need to brake 1 biscuit in 3 equal parts. I would insist with biscuits in class, teaching that division is the art of separating things in equal parts, without discarding any. Once they learn that, they would learn fractions much better, later in life, a fraction is much more than a division, you are keeping a portion and discarding the rest, this portion is magic, it reveals a ratio, a proportion, a constant relation between two things.

[John Cuthber]

 

 

 

Take 6 oranges and divide them by 1/2. This comes out at??

 

Q) How many times can you take half an orange from a box containing 6 oranges.

A) 12

 

Divide in half isn't the same as divide by half.

Confusing those is just sloppy English.

[studiot]

 

Confusing those is just sloppy English

 

 

Pity you have started mud slinging again.

 

I am quite well aware of the difference between dividing in half and by half.

 

That is the whole crux of the question, that even in mathematics, it (edit : dividing ) has at least two meanings.

 

I notice, incidentally, that you have avoided my questions about dividing pies or integrals.

Edited August 5, 2013 by studiot

[swansont]

You are right. Instead of "=", we could use the sign "≈" ? That's how I would have liked to be taught when I was a kid, even before algebra. That would make things easier maybe, if taught very early.

They can. Show them 7 tennis balls, ask them to divide them by 3 in equal parts. They will tell you immediately that it's impossible. Now show them 7 biscuits, ask them to divide them by 3 in equal parts, they don't have any problem to tell you that you need to brake 1 biscuit in 3 equal parts. I would insist with biscuits in class, teaching that division is the art of separating things in equal parts, without discarding any. Once they learn that, they would learn fractions much better, later in life, a fraction is much more than a division, you are keeping a portion and discarding the rest, this portion is magic, it reveals a ratio, a proportion, a constant relation between two things.

 

That's not decimals, that's fractions, which supports my point.

[Myuncle]

 

That's not decimals, that's fractions, which supports my point.

Why are you so concerned about your point, have a look at this

http://www.mathsisfun.com/decimal-fraction-percentage.html

[ajb]

Why are you so concerned about your point, have a look at this

http://www.mathsisfun.com/decimal-fraction-percentage.html

The example values are all very neat, appart from 1/3. As a decimal this does not have a very nice representation.

[math] 7 \div 3 = (2 \times 3) + 1[/math]

Which is why kids get introduced to the concept of remainder division first. It's not pretty, but it's at least mathematically accurate.

You mean [math]7 = (2 \times 3) +1[/math]. Which we can then pretty much use as the definition of dividing

 

[math]7 \div 3 := \frac{7}{3} = 2 + \frac{1}{3}[/math].

 

As you have said, this leads to the notion of a remainder.

Edited August 6, 2013 by ajb

[studiot]

Consider the following

 

[math]\frac{3}{{10}}[/math]

What does it say?

 

Well it could say I have divided something into 10 (equal) parts, but that I only have three of them,

 

So do the other seven parts exist, did they ever exist?

 

Can I consider 3/10 as an operator that carrys out the operation of dividing into 10 and discarding seven of the parts?