samedi 31 octobre 2009

What are fractions?

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Sometimes the part of math that some homeschoolers have difficulty with is when it comes to working with fractions.



Now, I think the difficulty comes from not understanding what fractions are. A simplified way is to say that fractions are like portions, like the 1/8 slices of a pizza. It would be better to say that that is a particular kind of fraction.

A "fraction" is a proportion. Each 1/8 slice of a pizza or any other 1/8 part or fraction is in proportion to the whole pizza (which existed as such before it was sliced) as one is to eight.

The funny thing about proportions is ...

that it does not really matter how many are involved:

Take ten pizzas, slice them to eighty slices, the slice is still one to eight compared to the pizza.

Which explains that: 10/80=1/8.

that it may just as easily by "how much" "how long" "how heavy" "how much worth" et c as how many:

Take a pizza sliced into eight slices, again. Each piece, from the moment the slice is made, even before all the eight slices are there, is and remains one eighth of the size of the whole pizza.

But add two pizza slices like the ones before - your neighbour ate only six slices. Each slice will be 1/10 of the number of slices. It is a less proportion, but it will still be same amount of pizza. But the total amount, of which it is now only 1/10, will be those two slices bigger.

That involves:

  • - each slice is still 1/8 the size of the pizza it was made from
  • - each slice is now 1/10 the number of slices (since those two were added)
  • - each slice is a less proportion (1/10 is less than 1/8) but same amount
  • - total amount is, as such, same proportion to itself (1/1) but bigger, because new total amount is 10/8 to old total amount
  • - that last proportion, 10/8, simplifies into 5/4 or 1/1 and 1/4, and the amount of pizza you have is the same as if you had had a whole pizza and a quarter slice

AND the fact that I have to apologise to my readers for making them hungry by thinking of pizza while having a math lesson. I'm getting a bit hungry myself, though I have just eaten!

Back from a meal?

Good.

Each time you eat one pizza slice, the total amount of remaining slices is less, each remaining slice remains as big as it was (until you go on and eat it) AND each slice is a bigger proportion of the remainder of pizza:

  • Before you eat, each slice is 1/10 of the total amount of pizza (but still 1/8 of the pizza it was made from).
  • You eat one slice, each slice is now 1/9 of the total amount of pizza (but still 1/8 of the pizza it was made from).
  • You eat another, each slice is now 1/8 both ways (as much of the remainder as it was originally to the whole pizza - which remains true even if you've eaten a slice from the old pizza, which is replaced by one from the new).
  • Eat three slices more, each slice is now 1/5 of the remainder (but still 1/8 of the pizza it was made from). Since the proportion has risen from 1/10 to 1/5=2/10, you might think it was twice as big? Of course not. Something wrong with the logic? No: for on the other hand, you only have 5 left instead of ten, which is 1/2; so each slice is as big as it was (if you need to check it out in maths: 1/2*2 = 1/2*2/1 = 2/2 = 1/1).
  • Another big bite, each slice is 1/4 of the remainder - so, as 4/8 = 1/2 the remainder equals half a pizza. (But each slice is still 1/8 of the pizza it was made from). And if I had a pay pal account, u might have been sending me money for half a pizza (which I could not use until another one sent me money for the other half), but I have not.

If on the other hand u r a parent and think I have helped your young ones over much of the year, well consider a somewhat bigger donation. (Donativo.) If you think it needs more, click on 7 artes label or look at other related subjects at my main index page.

Hans-Georg Lundahl
Paris V, Mouffetard
31 october 2009

8 commentaires:

Hans-Georg Lundahl a dit…

This means of course that a proportion, as a fraction, is not a number. Numbers answer the question "how many" not only involve it as in "how many in proportion to how many others" but directly answer it.

Two (2) is a number, whereas twice (2/1) is not a number but a proportion. However, since the number two is twice the one, one says that 2 is equivalent to 2/1.

Hans-Georg Lundahl a dit…

And a whole pizza is as much pizza as eight slices, but not as many slices as them, since the whole is not sliced. And the slices of a pizza are as much as the whole pizza was, but not as many pizzas, since each is not a round pizza, but a slice.

Anonyme a dit…

And (no, I won't log in right now) the great difficulty in math teaching nowadays comes from making operations (like -2=take away two) or qualities (like -2=two of a negative quality) or symbols which in themselves mean nothing (like 0) or things other than uncompared wholes, and therefore proportionals (like 1/8 or pi) the same as numbers.

Hans-Georg Lundahl a dit…

Yes, there is a hinch: what if the added two slices are from a bigger pizza?

Well then, in that case, each of the 10 slices is still 1/8 of the original pizza it was mde from, AND still 1/10 of the number of slices, BUT the ones from the smaller pizza are less than 1/10 of the amount of pizza and the ones from the bigger pizza are more than 1/10 of the amount of pizza.

Which illustrates once again, number (as number of slices) and quantity as for stuffing a stomach or for weighing on a balance is not the same thing.

Hans-Georg Lundahl a dit…

But of course, there are people who say:

a) all numbers are written with numerals

b) all fractions are written with numerals

c) therefore all fractions are numbers

which is as stupid as:

a) all cats are mammals
b) all dogs are mammals
c) therefore all dogs are cats

"written with numerals" or "mammals" are what is called the middle term - the one you find in both premisses and which ties the other terms together or separates them before the conclusion sums up their relation

Now, for a middle term to do its work, it must be univocal, at least to the extent that something can be clearly identified as being the middle term in one and same sense, one and same instance in both premisses. Like if one premiss says "all" or "none" about middle term, that includes all instances of the middle term in the other premiss too.

But a predicate is not said "all" of, that is reserved for subject. And a predicate of a negative sentence is of course one you say "none" of. But neither premiss is negative.

Therefore the middle term "mammals" and "written with numerals" may be applicable in quite different parts of theirs to the subject "cat" or "number" and the subject "dog" or "fraction". And the conclusion does not follow.

Why don't they teach logic in these schools?

Anonyme a dit…

One thing was implied but not stated; as numbers and sizes and shapes and things are absolute, proportions are relative, competitive:

three (3) is three whether you compare it to nothing or to something, whether you compare it to one (1) or to three million (3.000.000); but if you compare it to one, it is the proportion thrice (3/1), if you compare it to three million it is the proportion one millionth (1/1.000.000).

A pizza slice remains same shape, same size and same proportion to its origin, the as yet uncut pizza, BUT when two slices are added, it diminishes its part of total amount of pizza still available from 1/8 to only 1/10, if you then eat six slices, each remaining slice increases from 1/10 to 1/4.

How is that an increase? Well, remember that a slice 1/4 of a pizza is bigger than the next cut, 1/8 slice.

The one thing is: 1/4 of a whole pizza is twice 1/8 of a whole pizza.

The other thing is: 1/4 or for that matter 1/8 remains same proportion whatever the sizes or numbers involved (remember the 80 slices from 10 pizzas).

So: 1/4 of half a pizza (=1/8 of a whole) is same proportion, though only half the size of 1/4 of a whole pizza.

A proportion may thus increase or decrease at one end for two reasons: either because that end itself increases or decreases, OR because the other end does the reverse.

Proportions are competitive.

Anonyme a dit…

Which, ALSO, illustrates once again: number, size and prooportion are three different things. Not one and the same.

Hans-Georg Lundahl a dit…

Maths in English:

1 A quote from Aristotle 2 Two more than two make four, but why? 3 What are fractions?