Math Is Fun Forum

paulb203

Member

Registered: 2023-02-24

Posts: 314

What is the best way to understand division

Does, for example, 10÷2 mean 10 split into 2 equal parts?

Or does it mean how many 2s are in 10?

If we split 10 into 2 equal parts we get 2 parts of 5.

If we ask how many 2s are in 10 we get 5 2s.

Almost the same answer for both questions. Well, the same answer if we simply say, ‘5’.

This question never occurred to me when I was solving simple problems like 10÷2, or 1÷4, etc. But when faced with, for example, 10÷1/2, I had to think about it more.

Does 10÷1/2 mean 10 split into... No, that doesn’t seem to work.

Does 10÷1/2 mean how many 1/2s are in 10?

10÷1/2=20. There are 20 1/2s in 10. That seems to work.

Is that my answer; the best way to understand division is by asking how many x’s are in y?

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Prioritise. Persevere. No pain, no gain.

Jai Ganesh

Administrator

Registered: 2005-06-28

Posts: 48,398

Re: What is the best way to understand division

When the parts are closest to e, the natural logarithm base, that value is calculated.

For example, 3.33... to the power of 3 is approximately 37 and 2.5 raised to the power 4 is 39.0625.

Hence, the maximum value of product of all products is 39.0625.

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It appears to me that if one wants to make progress in mathematics, one should study the masters and not the pupils. - Niels Henrik Abel.

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

Bob

Administrator

Registered: 2010-06-20

Posts: 10,621

Re: What is the best way to understand division

hi paulb203

I think Ganesh is answering someone else's question here. 

Another way to define divsion is to make it the inverse process from multiplication.

So if a times b = c the c divided by a is b.

So so if x = 10  ÷  ½ then  ½   times x = 10 .... double both sides and x = 20

Or what do you have to multiply  ½ by in order to get 10.  answer   ½ of 20 = 10.

But all these definitions amount to the same thing so it is up to  you which you prefer.

Bob

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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

Jai Ganesh

Administrator

Registered: 2005-06-28

Posts: 48,398

Re: What is the best way to understand division

When the parts are closest to e, the natural logarithm base, that value is calculated.

For example, 3.33... to the power of 3 is approximately 37 and 2.5 raised to the power 4 is 39.0625.

Hence, the maximum value of product of all products is 39.0625.

This is a piece of information, not intended to be a solution.

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It appears to me that if one wants to make progress in mathematics, one should study the masters and not the pupils. - Niels Henrik Abel.

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

paulb203

Member

Registered: 2023-02-24

Posts: 314

Re: What is the best way to understand division

Thanks, Ganesh.

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Prioritise. Persevere. No pain, no gain.

paulb203

Member

Registered: 2023-02-24

Posts: 314

Re: What is the best way to understand division

Thanks, Bob.
I do find looking at the inverse process helpful when checking my answer.
“But all these definitions amount to the same thing so it is up to  you which you prefer.”
One of the first Google results I got for my question was;
“Division in maths is the process of breaking a number up into equal parts, and finding out how many equal parts can be made. For example, dividing 15 by 3 means splitting 15 into 3 equal groups of 5.”
Is this answer wrong, or at least incomplete? Should it read, ‘One way of understanding division is... etc’? It’s not said that another way of looking at 15÷3 is to ask, ‘How many 3s are there in 15?’
Either way, I do find it interesting. With some questions I find it helpful to think of a physical object (often a cake) being shared amongst children. 1÷2, 1÷4, etc. With others  a different approach cab be more helpful.
What about negative numbers, when it comes to division?
If I’m faced with -10÷2 I think of a debt of £10 being split into 2 debts of £5 (-10÷2=-5) with maybe 2 people involved, each taking on a debt of £5
When faced with -5x2 I think of someone having 2 separate debts of £5 and therefore being £10 in debt (-5x2=-10)
But when it comes to, for example, -10÷-5, it seems less straightforward. I know the rule (a negative divided by a negative = a positive)so I know it goes as follows; -10÷-5=2; but I can’t think of a ‘debt’ example like above (because there is no such as, ‘-5 people’. Although could I think of it as, ‘How many -5s are there in -10, in the same way as I might think of 10÷5 as, ‘How many 5s are there in 10’? So a debt of £10 is the same as 2 debts of £5

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Prioritise. Persevere. No pain, no gain.

paulb203

Member

Registered: 2023-02-24

Posts: 314

Re: What is the best way to understand division

There are some problems I can’t seem to come up with real life examples for, e.g;-5x-2=10
If I think of the -5 as a debt of £5 I can’t make sense of multiplying this by -2 as there is no such thing as -2 people (2 people with a debt of £5 each makes sense, but not negative two people etc).
And the inverse process doesn’t help me either (10÷-2=-5). I can strive to remember the rules, and use the inverse process to check my answers, but to really understand this I think I need to be able to come up with real life examples.

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Prioritise. Persevere. No pain, no gain.

Bob

Administrator

Registered: 2010-06-20

Posts: 10,621

Re: What is the best way to understand division

hi paulb203

That google quote is fine for whole numbers but no good when the numbers are fractions, decimals and negative.

Example with 2 negatives.  Let's say you owe £10 and will pay it off at £2 per week.  How many weeks before it is paid off.

weeks = -10 / -2 = 5

Bob

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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

mathdrop

Member

Registered: 2022-03-07

Posts: 75

Re: What is the best way to understand division

10 / 1/2 = 10/1 / 1/2 = 10/1 * 2/1 = 10*2 = 20 if i'm right.

in general, Frank Cardulla, a teacher at the niles north high school in a chicago suburb,
said: by dividing you make the denominator 1.

he made high school chemistry video lectures,
in which he also explained a lot of math
(i would recommend the first edition, where he was younger,
but i can only find the second advert removed.
unfortunately they are ttc videos, which are not free.

Bob

Administrator

Registered: 2010-06-20

Posts: 10,621

Re: What is the best way to understand division

I was taught that to divide by a fraction you invert it and multiply.  No explanation was offered but years later I worked it out for myself.

Let's say you want to do

Multiply the top and bottom of the outer fraction by d/c leaving the value unchanged

This is another example of using 'division as the inverse of multiplication'.

Bob

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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