Equal fractions

[Genady]

How is it explained to school children that or why, e.g., 2/7=4/14=...?

[joigus]

Friends fairly sharing pizza has always worked for me. I seem to remember some explanation with pie when I was a kid. It's possible my brain has edited my memories and it was all more formal than I care or dare to remember. Pies work because rational fractions make immediate intuitive sense, IMO.

I lost my innocence when I had a teacher at university who said something like 'definitions are not to be understood; definitions are definitions!' which, let me say, I think is completely wrong. Definitions should be motivated. For this man definitions were like a thunderbolt from mathematical heaven.

Marcus du Sautoy explains Egyptian, Babilonian, Indian and Chinese mathematics with beans, and peas, and eggs, and things like that, in a wonderful documentary about the history of mathematics.

[Genady]

1 hour ago, joigus said:

I had a teacher at university who said something like 'definitions are not to be understood; definitions are definitions!' which, let me say, I think is completely wrong. Definitions should be motivated. For this man definitions were like a thunderbolt from mathematical heaven.

I also had my share of bad teachers. Fortunately, I had two very good math teachers. They would never say anything like that.

[sethoflagos]

4 hours ago, Genady said:

How is it explained to school children that or why, e.g., 2/7=4/14=...?

We learnt factorisation first as I remember. Then cancellation of common factors.

[studiot]

3 minutes ago, sethoflagos said:

We learnt factorisation first as I remember. Then cancellation of common factors.

 

+1 for remembering your (our) childhood.

[MigL]

whoah !
When you started with

3 hours ago, joigus said:

I lost my innocence when I had a teacher ...

I was thinking something totally different.
My bad !

[joigus]

46 minutes ago, MigL said:

I was thinking something totally different.
My bad !

Oh, that's so Italian! No, com'on, those recollections I'll keep to myself. But you sent me down memory lane now.  

Edited Saturday at 08:36 PM by joigus
minor correction

[Genady]

3 hours ago, joigus said:

I seem to remember some explanation with pie

Me too. But I remember that it bothered me, because how ONE piece of pie can be EQUAL TWO pieces of pie? Sure, they weigh the same, but they are different in so many ways...

[studiot]

3 minutes ago, Genady said:

Me too. But I remember that it bothered me, because how ONE piece of pie can be EQUAL TWO pieces of pie? Sure, they weigh the same, but they are different in so many ways...

Clearly your second childhood has arrived.

Only children will say Mom my piece of pie is smaller than his.

 

😀

[Genady]

3 minutes ago, studiot said:

children will say Mom my piece of pie is smaller than his.

Not in my case. I was an only child.

[joigus]

23 minutes ago, Genady said:

Me too. But I remember that it bothered me, because how ONE piece of pie can be EQUAL TWO pieces of pie? Sure, they weigh the same, but they are different in so many ways...

You mean two sevenths of a pie are different from the 2/7 of the other portion?

21 minutes ago, studiot said:

Only children will say Mom my piece of pie is smaller than his.

Oh, OK, I get it. Children appear to be able to count the exact number of atoms in a portion of pie.  

[Genady]

2 minutes ago, joigus said:

You mean two sevenths of a pie are different from the 2/7 of the other portion?

I mean, ONE third is different from TWO sixths.

[sethoflagos]

1 hour ago, studiot said:

+1 for remembering your (our) childhood.

Why do I remember this more clearly than my twenties and thirties? Scary.

 

[Genady]

3 hours ago, sethoflagos said:

We learnt factorisation first as I remember. Then cancellation of common factors.

Did they explain/justify it? I mean, why this action leaves the numbers unchanged.

[studiot]

17 minutes ago, Genady said:

Did they explain/justify it? I mean, why this action leaves the numbers unchanged.

As I recall it was not called cancellation of common factors it was called 'Do the same thing to top and bottom'

 

So either multiply the 2 and 7 by or divide the 4 and 14 by 2, depending upon which way round the question was asked.

Often it would be simplify or express in simplest terms 4/14

Edited Saturday at 11:50 PM by studiot

[Genady]

14 minutes ago, studiot said:

As I recall it was not called cancellation of common factors it was called 'Do the same thing to top and bottom'

 

So either multiply the 2 and 7 by or divide the 4 and 14 by 2, depending upon which way round the question was asked.

Often it would be simplify or express in simplest terms 4/14

If the same thing which is done to top and bottom is multiply or divide, it is ok. But if this thing is add or subtract, it is not. IOW, 4/14 is certainly not the same as 2/12 (subtracting 2 from top and bottom). How do we know that it is the same as 2/7 (dividing top and bottom by 2)?

[studiot]

4 minutes ago, Genady said:

If the same thing which is done to top and bottom is multiply or divide, it is ok. But if this thing is add or subtract, it is not. IOW, 4/14 is certainly not the same as 2/12 (subtracting 2 from top and bottom). How do we know that it is the same as 2/7 (dividing top and bottom by 2)?

Do you prove Pythagoras every time you use it ?

Of course not.

The teacher may well explain that if we multiply by 1, which is the same as 1/1 or 2/2 or 3/3 etc we don't change anything.

But no suggestion of adding or subtracting would be made .

In fact later the same or similar discussion would occur when introducing powers and roots of fractions.

[Genady]

Here (in "The Road to Reality") Roder Penrose asks the same question, but perhaps clearer than me:

Quote

I recall that when at school, at the age of about 11, I was somewhat taken aback when the teacher asked the class what a fraction (such as 3/8) actually is! Various suggestions came forth concerning the dividing up of pieces of pie and the like, but these were rejected by the teacher on the (valid) grounds that they merely referred to imprecise physical situations to which the precise mathematical notion of a fraction was to be applied; they did not tell us what that clear-cut mathematical notion actually is. Other suggestions came forward, such as 3/8 is ‘something with a 3 at the top and 8 at the bottom with a horizontal line in between’ and I was distinctly surprised to find that the teacher seemed to be taking these suggestions seriously! ...
Let us start with my classmate’s ‘something with a 3 at the top and 8 on the bottom’. Basically, this is suggesting to us that a fraction is specified by an ordered pair of whole numbers, in this case the numbers 3 and 8. But we clearly cannot regard the fraction as being such an ordered pair because, for example, the fraction 6/16 is the same number as the fraction 3/8, whereas the pair (6, 16) is certainly not the same as the pair (3, 8). This is only an issue of cancelling; for we can write 6/16 as (3x2)/(8x2) and then cancel the 2 from the top and the bottom to get 3/8. Why are we allowed to do this and thereby, in some sense, ‘equate’ the pair (6, 16) with the pair (3, 8)?
 

 

[Endy0816]

7 hours ago, Genady said:

I mean, ONE third is different from TWO sixths.

Cuts each slice in two, takes 5 sixths of the pizza for services rendered.

 

I think of them in more algebraic terms. The numerator and denominator are together expressing a relationship, rather than being fully defined themselves.

 

[studiot]

7 hours ago, Genady said:

Here (in "The Road to Reality") Roder Penrose asks the same question, but perhaps clearer than me:

 

One thing to remember is that we do not teach any serious subject all in one go.

Mathematics especially requires what I call a 'spiral approach'.

Here a very simplified version is first presented, not the whole nine yards (nice expression in English for you).

This will tie in with what has gone before and what may be presented in the future as the subject is revisited again and again as we work around the spiral.

 

So in the present context fractions will be naturally introduced after addition, subtraction ,multiplication and division.

Note these are treated separately and simply and usually called 'sums' (and in the olden days tables).

So a fraction naturally becomes replacing the dots in the division sign with actual numbers.

This fits in with your Penrose description and emphasises that it is division being talked about.

Which naturally leads to division of pies, apples, sweets whatever.

In turn this leads to finding out the difference between dividing a bag of sweets and a single pie.

This leads back to proper and improper fractions (and perhaps vulgar fractions).

The four arithmetic operations can then be revisited in the light of fractions, leading to the introduction of decimal fractions.

Then we have the return to the four operations to work decimal fractions.

 

 

Looking ahead to secondary school and algebra we are set up to try algebraic fractions and find out why (4-2) / (14-2) is not the same as (2-2) / (7-2.

The next bump comes when the teacher needs to keep emphasising that dy/dx is not a fraction, but a complete entity in itself.

 

 

Having said all this it would be very helpful if you would indicate why you are asking these questions and where you are going with the answers  ?

[Genady]

18 minutes ago, studiot said:

why you are asking these questions and where you are going with the answers  ?

I live with the clear, rigorous construction of rational numbers ("fractions") using equivalence classes for very long; this construction is obvious and intuitive to me. OTOH, I realize that it is not fit to school children. I also know many intelligent adults who think that math is just "following the rules", which I think is the outcome of poor presentation of mathematical concepts in schools. So, I wonder what justifications/explanations of these "rules" for school children are there, if at all.

[studiot]

40 minutes ago, Genady said:

I live with the clear, rigorous construction of rational numbers ("fractions") using equivalence classes for very long; this construction is obvious and intuitive to me. OTOH, I realize that it is not fit to school children. I also know many intelligent adults who think that math is just "following the rules", which I think is the outcome of poor presentation of mathematical concepts in schools. So, I wonder what justifications/explanations of these "rules" for school children are there, if at all.

OK personal interest. That's great. Thanks.

 

Even equivalence classes have to start with a bunch (I won't say set) of rules.

And these rules define what can and cannot be done with the resulting numbers and expressions.
Furthermore these rules have to be learned and accepted.

 

It is also worth noting that some number systems are not susceptible to this analysis.

The number systems employed in Polynesian and Australian Aboriginal tribes for instance.

Also the Systeme Internationale organisation has recently added 'number' as a fundamental physical  property base to add to the original 5  ( mass, length, time, electric current density and illumination).

[studiot]

When I first started arithmetic (we put sums on the fron of our exercise books) we didn't write mathematical expressions like or do brackets or equivalence classes

567 + 123 =

We laid out our 'sums' like this (sorry I can't get this site to represent a continuous line under the sum)

 

[math]\begin{array}{*{20}{c}}
   5 & 6 & 7 & {}  \\
   1 & 2 & 3 & {}  \\
    -  &  -  &  -  &  +   \\
   6 & 9 & 0 & {}  \\
\end{array}[/math]

 

and

[math]\begin{array}{*{20}{c}}
   5 & 6 & 7 & {}  \\
   1 & 2 & 3 & {}  \\
    -  &  -  &  -  &  -   \\
   4 & 4 & 4 & {}  \\
\end{array}[/math]

 

Muliplication and division was laid out similarly.

After the idea of fractions was introduced as already noted in a previous post we found that fractions as the ratio of two numbers became messy and this led on to the idea of decimal fractions, laid out in the same way but now with a whole number part a decimal part and a decimal point.

 

I don't know if they still do this in school but I do worry about loss of the development from arithmetic into algebra in these days where no one actually needs to do calculations this way.

Edited 21 hours ago by studiot

[studiot]

Some official current practice sites

 

https://mathshub.thirdspacelearning.com/

 

https://assets.publishing.service.gov.uk/media/5a7da548ed915d2ac884cb07/PRIMARY_national_curriculum_-_Mathematics_220714.pdf

 

factorise 2⁴⁸⁴  +  1

Edited 19 hours ago by studiot

[Genady]

5 hours ago, studiot said:

When I first started arithmetic (we put sums on the fron of our exercise books) we didn't write mathematical expressions like or do brackets or equivalence classes

567 + 123 =

We laid out our 'sums' like this (sorry I can't get this site to represent a continuous line under the sum)

 

51−662−973−0+

 

and

51−462−473−4−

 

Muliplication and division was laid out similarly.

After the idea of fractions was introduced as already noted in a previous post we found that fractions as the ratio of two numbers became messy and this led on to the idea of decimal fractions, laid out in the same way but now with a whole number part a decimal part and a decimal point.

 

I don't know if they still do this in school but I do worry about loss of the development from arithmetic into algebra in these days where no one actually needs to do calculations this way.

Your recollections reminded me of this Feynman's parable (from "QED: The Strange Theory of Light and Matter"):

Quote

To make calculations, the Maya had invented a system of bars and dots to represent numbers (including zero) and had rules by which to calculate and predict not only the risings and settings of Venus, but other celestial phenomena, such as lunar eclipses.

In those days, only a few Maya priests could do such elaborate calculations. Now, suppose we were to ask one of them how to do just one step in the process of predicting when Venus will next rise as a morning star—subtracting two numbers. And let’s assume that, unlike today, we had not gone to school and did not know how to subtract. How would the priest explain to us what subtraction is?

He could either teach us the numbers represented by the bars and dots and the rules for “subtracting” them, or he could tell us what he was really doing: “Suppose we want to subtract 236 from 584. First, count out 584 beans and put them in a pot. Then take out 236 beans and put them to one side. Finally, count the beans left in the pot. That number is the result of subtracting 236 from 584.”

You might say, “My Quetzalcoatl! What tedium—counting beans, putting them in, taking them out—what a job!”

To which the priest would reply, “That’s why we have the rules for the bars and dots. The rules are tricky, but they are a much more efficient way of getting the answer than by counting beans. The important thing is, it makes no difference as far as the answer is concerned: we can predict the appearance of Venus by counting beans (which is slow, but easy to understand) or by using the tricky rules (which is much faster, but you must spend years in school to learn them).”