The test of 9 it was done by my EGB math teacher when we did multiplications or divisions on the blackboard, with chalk, of the classroom. I don't remember her with precision and I certainly didn't understand how that could make sense. So it has been very fun to find a text by Carlos Fernández Martínez, math teacher at a high school in Asturias, which explains in detail what the meaning of the 9 test is, what it is for, how it is used and especially a complete demonstration of its validity.
As Carlos indicates if a multiplication or division does not pass the test of 9, it can be said that there are errors in the calculation although compliance with it does not ensure that the operation is well performed. The mechanism followed to perform the 9 test starts by drawing a cross on the board.
In the case of multiplication in the upper angle, the multiplying, in the lower angle the multiplier and at the left and right angles the result of product and the result of multiply the multiplier and the multiplier. Although in a way I try to explain here:
Multiplying is a number that is obtained by adding all the figures until there is only one left. In the same way it is done with the multiplier and with the result of the product. So in the right-hand corner only the multiplier and the multiplying multiply, a very basic multiplication (a single figure). The multiplication will be correct if the numbers of the left and right angles match.
For children it can be a good way to practice very simple sums and multiplications in addition to ensuring a happy face if they get that the result of the product is checked correctly with the test of 9.
In the case of division it is similar, in the upper and lower angles the divisor and the quotient are placed respectively. In the left angle the dividend and in the right the result of the divisor by the quotient plus the rest. And how the dividend is equal to the divisor by the quotient and to that figure the rest is added, you can check the division quickly.
I recommend reading Carlos's article and his practice with the kids. Also, for those who have more interest, Carlos has documented the demonstration in mathematical language simple but with enough dose of abstraction to demand an effort according to a high school student.
From Peques and Más, we thank Carlos for his wonderful work of analysis and dissemination and we encourage children to practice with this type of operations and abstraction exercises to claim their attention and approach mathematics with joy and interest.