Question

Question #b2995

Answer 1

One is half of 2

Explanation:

`color(blue)("Discussion about what a fraction is.")`

A fraction has two parts, the top and bottom number. I want you to think of the top number as the count and the bottom number as an indicator of size. Proper names are `("numerator")/("denominator")`

Consider the fraction `3/5`. Strait away you can see that we have three of them (top number) but how do we think about the 'size'

The number of 5 means that we need 5 of them to make a whole of something. Suppose you have a plank of wood and split it into 2 equal lengths. We have a count of 2 but the size is also 2 so we have `2/2`. Which is the equivalent of 1 whole plank. Glue them back together and you have a whole plank again.
'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

`color(magenta)("Example: suppose we had 3 is half of what?")`

I am using the sign `->` this means there is a connection of some sort

Remember that half is `1/2` so

`3-> 1/2`

But for all of something we need the count part of `1/2` to become 2 so we end up with `2/2`

To change the count of 1 into a count of 2 we have 1+1

It is the same with fractions

To change the count of 1 (of size 2) we add another 1 (of size 2)

That is `2/2" "=" "1/2+1/2" "=" "(1+1)/2`

`color(red)("You must not do this "1/2+1/2=2/4`
`color(red)("You are adding counts of one size. The 4 means you have changed size")`
`color(red)("As long they are all the same size, adding counts does not change the size")`

If we have doubled the right hand side of `3->1/2`
Then we need to double the left as well

`3+3->1/2+1/2`

So`" " 6->2/2" "` as we now have `2/2` we have all of the something

'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
`color(blue)("Answering your question")`

1 is `1/2" of something "`

`1->1/2`

`1+1->1/2+1/2 = (1+1)/2`

`color(blue)(2-> 2/2 -> "All of the something")`