How to explain for my daughter that $\frac {2}{3}$ is greater than $\frac {3}{5}$? I was really upset while I was trying to explain for my daughter that $\frac 23$ is greater than $\frac 35$ and she always claimed that $(3$ is greater than $2$ and $5$ is greater than $3)$ then $\frac 35$ must be greater than $\frac 23$.
At this stage she can't calculate the decimal so that she can't realize that $(\frac 23 = 0.66$ and $\frac 35 = 0.6).$
She is $8$ years old.

In a first step you can observe with the child that 
$$
  \frac{2}{3} = 1-\frac{1}{3}
\quad \text{ and } \quad 
  \frac{3}{5} = 1 - \frac{2}{5}.
$$
In words, you are looking at what is remaining of the pie when you have taken either one third or two fifths.
Then, you can show that cutting a third into two pieces gives two sixths, hence $\frac{1}{3}= \frac{2}{6}$ and then 
$$  
\frac{2}{3} = 1-\frac{2}{6}
\quad \text{ and } \quad 
  \frac{3}{5} = 1 - \frac{2}{5}.
$$
You can then conclude by the fact that sharing fairly between more people leads to smaller shares, thus $\frac{2}{6} < \frac{2}{5}$. Hence, the comlplement to two shares is larger:
$$
\frac{2}{3} = 1-\frac{2}{6}
 >
  \frac{3}{5} = 1 - \frac{2}{5}.
$$
