Help with understanding dividing fractions

apple2357

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Working with my 9 year old son on multiplying and dividing fractions.

Without any understanding but playing with a calculator he worked out that to multiply fractions you multiply across:
E.g. 2/7 x 3/5 = 6/35.

Then he started with dividing fractions and typed in

12/24 divide 2/4 ,

He decided you do this in the same way but divide top and bottom so (12 divide 2 ) / ( 24 divide 4) = 6/6 =1
This works and a bit of algebra and you can see why.
And it is consistent with the multiplication algorithm, however the standard way of dividing fractions is to invert the second one and multiply.

Does my son's approach which doesn't work as easily for all fractions offer anything in terms of understanding dividing fractions?
 
Working with my 9 year old son on multiplying and dividing fractions.

Without any understanding but playing with a calculator he worked out that to multiply fractions you multiply across:
E.g. 2/7 x 3/5 = 6/35.

Then he started with dividing fractions and typed in

12/24 divide 2/4 ,

He decided you do this in the same way but divide top and bottom so (12 divide 2 ) / ( 24 divide 4) = 6/6 =1
This works and a bit of algebra and you can see why.
And it is consistent with the multiplication algorithm, however the standard way of dividing fractions is to invert the second one and multiply.

Does my son's approach which doesn't work as easily for all fractions offer anything in terms of understanding dividing fractions?
I would ask your son if he understands why it works. If this method does have a better understanding than other methods but your son does not see it, then it doesn't matter.

I think that your son should understand why you multiply by the reciprocal of the denominator.

(12/24)/(2/4) = (12/24)/(2/4)*(4/2)/(4/2) = (12/24)(4/2)
 
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Working with my 9 year old son on multiplying and dividing fractions.

Without any understanding but playing with a calculator he worked out that to multiply fractions you multiply across:
E.g. 2/7 x 3/5 = 6/35.

Then he started with dividing fractions and typed in

12/24 divide 2/4 ,

He decided you do this in the same way but divide top and bottom so (12 divide 2 ) / ( 24 divide 4) = 6/6 =1
This works and a bit of algebra and you can see why.
And it is consistent with the multiplication algorithm, however the standard way of dividing fractions is to invert the second one and multiply.

Does my son's approach which doesn't work as easily for all fractions offer anything in terms of understanding dividing fractions?
It is of course exciting that your son made an arithmetic discovery through experimentation. Good for him. But aside from being very simple to explain by analogy to multiplication of fractions, I do not see any conceptual advantages. Of course it is true that

\(\displaystyle \dfrac{2}{7} \div \dfrac{3}{8} = \dfrac{\dfrac{2}{7}}{\dfrac{3}{8}}.\)

But that expression is hardly convenient.

\(\displaystyle \dfrac{2}{7} \div \dfrac{3}{8} = \dfrac{2}{7} * \dfrac{8}{3} = \dfrac{2 * 8}{7 * 3} = \dfrac{16}{21}\)

is a more compact and more comprehensible expression.

There may be some pedagogical advantage to starting with your son's method, but teaching young children arithmetic is not something I know anything about.
 
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Working with my 9 year old son on multiplying and dividing fractions.

Without any understanding but playing with a calculator he worked out that to multiply fractions you multiply across:
E.g. 2/7 x 3/5 = 6/35.

Then he started with dividing fractions and typed in

12/24 divide 2/4 ,

He decided you do this in the same way but divide top and bottom so (12 divide 2 ) / ( 24 divide 4) = 6/6 =1
This works and a bit of algebra and you can see why.
And it is consistent with the multiplication algorithm, however the standard way of dividing fractions is to invert the second one and multiply.

Does my son's approach which doesn't work as easily for all fractions offer anything in terms of understanding dividing fractions?

I agree with others that discovering that one can discover ideas in math by oneself is great, especially if there is some sort of understanding of the reason it works.

It happens that I have mentioned this method and some related ones in my blog; see the second section, where a student asked why we can't just divide fractions. There are several useful lessons to be learned there.
 
Many thanks for your thoughts!

For my son's p.o.v he doesn't understand why it works as he only understands equivalent fractions and i don't wish to rush him by teaching him multiplying and dividing fractions, mechanically i could but there is no value in that. Besides i am not sure how i would even begin with it, pictures presumably.

It was simply interesting that he discovered this by accident. I don't think it is uncommon when we are learning mathematics to over-generalise and build up knowledge in this way. Sometimes this is correct and helpful and can be explained later but at other times it can lead to deep misconceptions.

I was simply curious as i had never thought about dividing fractions in the way he thought, possibly because i was told there was a way to do it ( i.e. turn the second one upside down and multiply) but he had the freedom to explore on the calculator!
 
I agree with others that discovering that one can discover ideas in math by oneself is great, especially if there is some sort of understanding of the reason it works.

It happens that I have mentioned this method and some related ones in my blog; see the second section, where a student asked why we can't just divide fractions. There are several useful lessons to be learned there.


Thanks, we posted at the same time but i will look at your link. Always an interesting read!
 
Many thanks for your thoughts!

For my son's p.o.v he doesn't understand why it works as he only understands equivalent fractions and i don't wish to rush him by teaching him multiplying and dividing fractions, mechanically i could but there is no value in that. Besides i am not sure how i would even begin with it, pictures presumably.

It was simply interesting that he discovered this by accident. I don't think it is uncommon when we are learning mathematics to over-generalise and build up knowledge in this way. Sometimes this is correct and helpful and can be explained later but at other times it can lead to deep misconceptions.

I was simply curious as i had never thought about dividing fractions in the way he thought, possibly because i was told there was a way to do it ( i.e. turn the second one upside down and multiply) but he had the freedom to explore on the calculator!
Calculators are good to explore and check your work but NOT for doing simple arithmetic. Please keep that in mind but it does seem that you have that under control!
 
I agree with others that discovering that one can discover ideas in math by oneself is great, especially if there is some sort of understanding of the reason it works.

It happens that I have mentioned this method and some related ones in my blog; see the second section, where a student asked why we can't just divide fractions. There are several useful lessons to be learned there.


Really good blog! Exactly the issue i have been thinking about.

By the way, the link below seems to be broken. I curious what this picture would look like!

(9/2)/(3/4): A Picture
 
Really good blog! Exactly the issue i have been thinking about.

By the way, the link below seems to be broken. I curious what this picture would look like!

(9/2)/(3/4): A Picture

Earlier today, I had trouble getting to the site, but just now I got this page with no trouble, then I couldn't get to another page. Hopefully it's just temporary.

In any case, pretty much the whole page is quoted in the blog. The picture is what you saw.
 
Working with my 9 year old son on multiplying and dividing fractions.

Without any understanding but playing with a calculator he worked out that to multiply fractions you multiply across:
E.g. 2/7 x 3/5 = 6/35.

Then he started with dividing fractions and typed in

12/24 divide 2/4 ,

He decided you do this in the same way but divide top and bottom so (12 divide 2 ) / ( 24 divide 4) = 6/6 =1
This works and a bit of algebra and you can see why.
And it is consistent with the multiplication algorithm, however the standard way of dividing fractions is to invert the second one and multiply.

Does my son's approach which doesn't work as easily for all fractions offer anything in terms of understanding dividing fractions?
I do see a connection between the usual method and your son's discovery.

\(\displaystyle \frac{12}{24}\div\frac{2}{4} = \frac{12}{24} * \frac{4}{2}\)

When you do the cancelling (across the multiplication sign), you're effectively doing \(\displaystyle 12\div 4\) which goes on the numerator and \(\displaystyle 24\div 4\) which goes on the denominator. When your son does start learning division of fractions, you can point this out as being the same as what he discovered himself.
 
I do see a connection between the usual method and your son's discovery.

\(\displaystyle \frac{12}{24}\div\frac{2}{4} = \frac{12}{24} * \frac{4}{2}\)

When you do the cancelling (across the multiplication sign), you're effectively doing \(\displaystyle 12\div 4\) which goes on the numerator and \(\displaystyle 24\div 4\) which goes on the denominator. When your son does start learning division of fractions, you can point this out as being the same as what he discovered himself.
I suspect that you have a typo as you have too many 4s in \(\displaystyle 12\div 4\) and \(\displaystyle 24\div 4\)
 
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