Why does this innovative method of subtraction from a third grader always work?

My daughter is in year $3$ and she is now working on subtraction up to $1000.$ She came up with a way of solving her simple sums that we (her parents) and her teachers can't understand.

Here is an example: $61-17$

Instead of borrowing, making it $50+11-17,$ and then doing what she was told in school $11-7=4,$ $50-10=40 \Longrightarrow 40+4=44,$ she does the following:

Units of the subtrahend minus units of the minuend $=7-1=6$
Then tens of the minuend minus tens of the subtrahend $=60-10=50$
Finally she subtracts the first result from the second $=50-6=44$

As it is against the first rule children learn in school regarding subtraction (subtrahend minus minuend, as they cannot invert the numbers in subtraction as they can in addition), how is it possible that this method always works? I have a medical background and am baffled with this…

Could someone explain it to me please? Her teachers are not keen on accepting this way when it comes to marking her exams.

Solution 1:

So she is doing \begin{align*} 61-17=(60+1)-(10+7)&=(60-10)-(7-1)\\ & = 50-6\\ & =44 \end{align*} She manage to have positive results on each power of ten group up to a multiplication by $\pm 1$ and sums at the end the pieces ; this is kind of smart :)

Conclusion : If she is comfortable with this system, let her do...

Solution 2:

Your daughter is probably creating cognitive dissonance in her teacher because of the $7−1$ part. "Doesn't she know she is supposed to subtract the $7$ from the $1$, and so then has to do a borrow, not the other way around?" the teacher is probably thinking.

But this is actually a very common way of doing things!
Quick: What is $5003 - 7$? If you are like me, your mind went right to "whatever $5000 - 4$ is", that is $5000 - (7-3)$

To expand upon Netchaiev's answer, using Uppercase letters for numbers $>= 10$ and lowercase for numbers $< 10$:

$$(A+b)-(C+d) = A+b-C-d = (A-C)+(b-d)$$

If $b>d$, this works out easy without borrowing. But if you have to borrow, then you do your daughter's (easier!) solution:

$$(A-C)-(d-b)$$

So it's actually a neat trick: If you don't have to borrow, you use the normal method, but if you have to borrow, you use your daughter's method.

This method can be extended to subtracting numbers with three or more digits! But then the bookkeeping could be troublesome. Consider:

$$523-147 = (500-100) - (47-23), \space check!$$

But this could cause trouble, and you might want to see if this still works OK in your daughter's head:

$$517-161 = (500-100)-(61-17) = (500-100) - ( (60-10) - (7-1) ) = 400-44$$