Why does multiplying a number on a clock face by 10 and then halving, give the minutes? ${}{}$

Solution 1:

Each number on the clock face is worth five minutes. One good way to multiply by $5$ is to first multiply by $10$, and then divide by $2$. This works because $5=10\div 2$.

Solution 2:

The minute hand passes over $12$ numbers in $60$ minutes.

That is $5$ minutes for each number.

Note that multiplying by $5$ is the same as multiplying by $10$ and dividing by $2$

Thus $3$ translates into $15$ which is $3(10)/2$.

Similarly $6$ translates into $30$ which is $6(10)/2$.

Solution 3:

The minutes are reckoned as

(Minutes pointing numeral N)(number of minutes in one hour =60)/(maximum minutes digits available on dial= 12) $= 5 N $

The procedure your daughter gave has effect of multiplying pointed figure $N$ by $5$; .. so it works.