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