# Question b2995

Mar 28, 2016

One is half of 2

#### Explanation:

$\textcolor{b l u e}{\text{Discussion about what a fraction is.}}$

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 $\left(\text{numerator")/("denominator}\right)$

Consider the fraction $\frac{3}{5}$. 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 $\frac{2}{2}$. Which is the equivalent of 1 whole plank. Glue them back together and you have a whole plank again.
'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

$\textcolor{m a \ge n t a}{\text{Example: suppose we had 3 is half of what?}}$

I am using the sign $\to$ this means there is a connection of some sort

Remember that half is $\frac{1}{2}$ so

$3 \to \frac{1}{2}$

But for all of something we need the count part of $\frac{1}{2}$ to become 2 so we end up with $\frac{2}{2}$

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 $\frac{2}{2} \text{ "=" "1/2+1/2" "=" } \frac{1 + 1}{2}$

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

If we have doubled the right hand side of $3 \to \frac{1}{2}$
Then we need to double the left as well

$3 + 3 \to \frac{1}{2} + \frac{1}{2}$

So$\text{ " 6->2/2" }$ as we now have $\frac{2}{2}$ we have all of the something

'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
$\textcolor{b l u e}{\text{Answering your question}}$

1 is $\frac{1}{2} \text{ of something }$

$1 \to \frac{1}{2}$

$1 + 1 \to \frac{1}{2} + \frac{1}{2} = \frac{1 + 1}{2}$

$\textcolor{b l u e}{2 \to \frac{2}{2} \to \text{All of the something}}$