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

Question

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.

Answer 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...

Answer 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:

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

Answer 3

A very useful generalization to decimal notation is to allow arbitrary integers for digits, rather than restrict digits to $0,1,\ldots,9$.

The semantics where each place corresponds to a multiple of a power of ten and then they are summed still applies.

In this notation, the calculation can be seen as first subtracting digitwise:

$$ \begin{matrix} & \fbox{6} & \fbox{1} \\ - & \fbox{1} & \fbox{7} \\\hline & \fbox{5} & \fbox{-6} \end{matrix}$$

Then, you can convert this to the usual form by normalizing the digits. In this case, you add ten to the one's place and subtract one from the ten's place to get $\fbox{4}\, \fbox{4} $. Note this operation is the same thing as borrowing, but it's being done at the end of the calculation rather than the beginning.

Arguably, the usual approach to subtraction is using the same idea, just rearranged differently: rather than normalize at the end as your daughter does, it denormalizes the number first, rewriting the subtraction as

$$ \begin{matrix} & \fbox{5} & \fbox{11} \\ - & \fbox{1} & \fbox{7} \\\hline & \fbox{4} & \fbox{4} \end{matrix}$$

Answer 4

I highly recommend that you read chapter 1 of Liping Ma's Knowing and Teaching Elementary Mathematics – Teachers' Understanding of Fundamental Mathematics in China and the United States, which is about how subtraction is taught in the US and China. Here is a Chinese teacher's discussion of the problem $53-26$ from this chapter (p. 10):

Here is a Chinese teacher's discussion of how she teaches subtraction (p. 11-12):

We start with the problems of a two-digit number minus a one-digit number, such as $34-6$. I put the problem on the board and ask students to solve the problem on their own, either with bundles of sticks or other learning aids, or even with nothing, just thinking. After a few minutes, they finish. I have them report to the class what they did. They might report a variety of ways. One student might say "$34-6$, $4$ is not enough to subtract $6$. But I can take of $4$ first, get $30$. Then I still need to take $2$ off. Because $6=4+2$. I subtract $2$ from $30$ and get $28$. So, my way is $34-6=34-4-2=30-2=28$." Another student who worked with sticks might say, "When I saw that I did not have enough separate sticks, I broke one bundle. I got $10$ sticks and I put $6$ of them away. There were $4$ left. I put the $4$ sticks with the original $4$ sticks together and got $8$. I still have another two bundles of $10$s, putting the sticks left all together I had $28$." Some students, usually fewer than the first two kinds, might report, "The two ways they used are fine, but I have another way to solve the problem. We have learned how to compute $14-8$, $14-9$, why don't we use that knowledge. So, in my mind I computed the problem in a simple way. I regrouped $34$ into $20$ and $14$. Then I subtracted $6$ from $14$ and got $8$. Of course I did not forget the $20$, so I got $28$." I put all the ways students reported on the board and label them with numbers, the first way, the second way, etc. Then I invite students to compare: Which way do you think is the easiest? Which way do you think is most reasonable? Sometimes they don't agree with each other. Sometimes they don't agree that the standard way I am to teach is the easiest way. Especially for those who are not proficient and comfortable with problems of subtraction within $20$, such as $13-7$, $15-8$, etc., then tend to think that the standard way is more difficult.

There's more great stuff in this chapter, but this is a good sample. You can read it for the rest.

Answer 5

I think what your daughter has discovered is "backwards" numbers. She may have a different word for them, but I think of them as "backwards" because you subtract the numbers backwards. A "forwards" number is 7 - 1 = 6, while a "backwards" number is 1 - 7 = "backwards" 6, because you do the subtraction backwards (7 - 1). Adults, of course, call them "negative" numbers, but that's a silly name to a 3rd grader. I think this because she intuitively knows that when you add a backwards number, you really subtract it, which is what she is doing.

Now, teachers know that few kids in 3rd grade can grasp negative numbers, so they teach borrowing instead. Frankly, I suspect that a significant number of adults can't really grasp negative numbers. But either way works mathematically.

As for your daughter and her teachers, I can think of two suggestions that may or may not help:

• Suggest that she calls them "negative" numbers and writes them as -6, and maybe her teachers can comprehend what she is doing.
• At her level, mathematics is simply counting and tricks (when we get into fractions, it is counting, measuring, and tricks, and so forth). Try to get her to understand that the more tricks she knows, the easier it will be. She found a great trick for subtraction, that of negative, or backwards, numbers. There is another useful trick called borrowing. Sometimes, one will be easier, sometimes the other will be easier. Encourage her to learn both.

Good luck, and as others have said, if she can come up with this at 3rd grade, she has a bright future ahead of her.

Answer 6

Although the other answers are correct, I feel like they are overcomplicating how the calculation is done.

Due to how subtraction is taught in schools (at least in my school), it becomes really easy to subtract numbers away from a multiple of 10. For example, you learn really early on that 4 + 6 = 10, and therefore 10 - 6 = 4.

To better illustrate, imagine the question was 61 - 7.

• First, subtract a number so that the 61 becomes a multiple of 10. In this case, 1 (i.e. 61 - 1 = 60, which is a multiple of 10).
• Second, calculate how much of the second number (i.e. the 7) you have "left". This is what your daughter is calculating when she performs the 7 - 1 = 6 calculation.
• Finally, subtract 6 from 60, to give 54.

This is the way that I find most intuitive and is how I perform mental subtraction too. As stated by others, your daughter is very gifted to have derived this approach herself.

Answer 7

As someone who is still a student, I can relate and explain. This method is almost exactly the same as how I learned addition and subtraction.

It sounds like your daughter is taking apart the numbers and breaking them into smaller parts. This is something I do too. If you want to teach her how to do the work the "normal" way, don't. This will only confuse her. Also, tell her teachers that as long as she understands what she's doing and can explain it, they shouldn't mark her down. Not everyone learns the same way and nobody should be told they're doing something wrong when they're not. She might not have the vocabulary to explain it all yet, so be patient. You'll see some amazingly complex things become very simple in her mind.

Now the math: Your daughter is looking at two different numbers, and thinking "How on earth does one go into the other?" For her and millions of others, the simplest way is just to break it up. Now, your daughter clearly is intelligent because she reasoned her way through this (yay future mathematician!!!) when others can't. If you take it down to the basics, she's adding negatives.

She sees a 60 and 10 and takes the difference between them: 60-10=50

Now she sees a 7 and a 1 and takes the difference: 7-1=6

In her mind, she is just taking the difference, and knows that that number can be played with. She might not know what a negative is yet, but she understands them perfectly. For her that 6 is positive, but you can think about it like a negative 6. i.e. -7+1=-6. Because the 17 is being subtracted, you have the negative 7 from -17 compared to the positive 1 from +61.

Now she remembers that she flipped over (they became negatives, she just doesn't have the vocabulary) the 7 and the 1 to find 6. Now, she knows that she needs to flip it back over so that the 7 is being subtracted from the 1.

To flip it back, she subtracts the 6 from the 50 (remember it's like a negative 6) to get 44. In essence, this is all very simple with a good explanation.

61 - 17 = (60-10) + (-7+1) = (60-10) - (7-1) = 44

Answer 8

It looks like she is taking $61 - 17 = 44$ and partially changing it to $17 - 61 = \!^-44$, then flipping the sign. She does $60 - 10 = 50$ in normal order, and then $7 - 1 = 6$ in reverse order. Doing another final subtraction flips the $6$ around to the correct sign again, so $50 - 6 = 44$.

Subtraction is actually just addition of negative numbers. If you have a Texas Instruments scientific calculator there is a different button for negative than subtraction. $61 + \!^-17 = 44$, just like $\!^-17 + 61 = 44$. Here is what she is really doing: $60 + \!^-10 = 50$ and $\!^-7 + 1 = \!^-6$. Add the two results together and you get $50 + \!^-6 = 44$.

What I remember from school is that not understanding that subtraction is really addition of negative numbers leads to a lot of problems down the road in more advanced math classes where you get problems like a $3(a -- b) = 4$. Just treat everything like addition and let negative signs stay with their numbers and cancel each other out and everything is less confusing!

Answer 9

The most important aspect of this is not the mathematical method ... but the child i.e. your daughter. Suppose she picked up a book and started reading from the last page and every word from right to left, but at the end (which is now the first page he he!) she could explain the story to you, then the objective of reading and the understanding of the story - has been achieved. Let your daughter find her own way in a very complicated world with many different paths and obstacles on the way. There is almost invariably no 'right' way of doing or achieving anything in our world, merely a conventional way ...

Answer 10

I think this method is awesome, it might even be easier than the classical method in some cases. Consider the following 'easy' subtraction, where the digit of the first number is bigger than the corresponding digit of the second number.

\begin{align}462-231&=(400+60+2)-(200+30+1)\\ &=(400-200)+(60-30)+(2-1)\\ &=200+30+1\\&=231\end{align}

Anyone would do this sum with little thinking. You would normally write it down without any steps inbetween. The method of your daughter extends this method to work with numbers that don't have this nice property. \begin{align}431-262&=(400-200)+(30-60)+(1-2)\\ &=(400-200)-(60-30)-(2-1)\\ &=200-30-1\\ &=170-1\\ &=169 \end{align} This could also be extended to larger numbers. To compare this to the usual method of borrowing tens I think that the regular method would be better if you have a pen and paper at hand and the numbers are relatively large, if you have no paper at hand this method might be easier.

Answer 11

(This is not meant to answer why it works formally, because this already been done, but rather a tentative to dwell in the girl's mind.)

I was trying to put me in your daughter's shoes and wondering what could be the mental process going through her mind.

Since she always places the greater number as the subtrahend and the smaller one as the minuend when computing the subtractions it is not reasonable to think that she discovered the concept of a negative number.

Instead of that, it seems that she intuitevely understood the following two things:

1. A two-digit number is just the sum of the number formed by the "tens digit" followed by $0$ with the "unit digit". (For example, $47 = 40 + 7$.)

2. The way we count actually orders the numbers, that is, she can tell that a number $n_1$ comes first than another number $n_2$ by counting from $1$ to the latter, $n_2$. For example, if she wants to know if $3$ comes first than $7$, she (mentally) counts $$ 1, 2, \color{red} 3, 4, 5, 6, \color{red} 7 $$ and so if she (mentally) said $3$ before $7$, then $3$ comes before $7$; or mathematically $3 < 7$. Also she understands that $0$ comes before every "counting number".

Furthermore she may have mastered the skill of computing subtractions with the non-negative integers less than $10$, namely $\{0,1,2,3,4,5,6,7,8,9\}$, when the subtrahend is greater than the minuend. (This because she has only been taught doing this way and also one-digit subtractions are teached first).

So what she found is a pattern to find an easier and equivalent difference of two-digits numbers to a harder difference of two-digits numbers through her grasp of one-digit numbers subtraction.

This is enough to explain her mental process. Let's take your example: $61-17$. This difference of two-digit numbers is harder to do than the one she found using her method. Using her method she obtains an equivalent and easier difference that is $50-6$, a difference of a two-digit number and a one-digit number.

The thing is that her method always reduce the a difference of two-digit numbers to a difference or a sum of a two-digit number and an one-digit number.

But how does she knows when its a sum or a subtraction? For that she compares the unit digit of the subtrahend with the unit digit of the minuend. If the former is greater or equal the latter, then it is a sum, else it is a subtraction. Thus your daughter's evaluation of $61-17$ goes something like this in her mind:

Since $1$ comes first than $7$, I subtract the result of $(7-1)$, namely $6$, from the number formed by the result of $(6-1)$, namely $5$, followed by a $0$, that makes $50$. Then I obtain $50 - 6$ which now I can evaluate. It is equal to $44$ and this is the result of $61-17$.

What is amazing, and you really should keep an eye on it, is that this method is applicable to the much more general scenario of integers subtraction, that is, the subtrahend and the minuend have an arbitrary number of digits, including a different number of each other, and the subtrahend may be less than the minuend. This means that your daughter can further develop her method to encompass all the possible cases of integers subtractions as the classes progress with the subtraction content. If this is the case then perhaps you are raising a future mathematician at home. :)

The following is what a systematized and more advanced version of her method could look like. Consider $147 - 76$.

First we make the two numbers have the same quantity of digits: $147 - 076$.

Now we respectively compare digit to digit and add to the total if the subtrahend digit is greater of equal to the minuend digit or subtract otherwise.

Since $7 \ge 6$ we add $7-6 = 1$ to the total.

Since $4 < 7$ we subtract the number formed by $7-4 = 3$ followed by one zero from the total.

Since $1 \ge 0$ we add the number formed by $1-0 = 1$ followed by two zeros to the total.

Therefore $147 - 76 = 100 - 30 + 1$. (Which is much more easier to compute.)

I have to say that it is very surprising that a third grader could find such a clever pattern. And she does this reasoning at an intuitive level, which only makes it more impressing!

May the gods of mathematics guide your daughter's path!

Answer 12

This will always work when subtracting two-digit numbers. If we subtract two two-digit numbers in decimal notation $ab$ and $cd$ we get $$ab - cd$$ $$= 10*a + b - (10*c + d)$$ $$= (10*a - 10*c) - (d-b)$$

And that last line is exactly what your daughter is doing.

Answer 13

Your daughter is trying to use negative numbers, but not articulating it properly. If you explain to her that it really is, she should be fine. Emphasis on what I changed in your description:

Units of the minuend minus units of the subtrahend $=1-6= -6$

Then tens of the minuend minus tens of the subtrahend $=60-10=50$

Finally she adds, not subtracts the first result and the second $=50 + (-6)=44$

Now, she can do her math is this fairly intuitive way, and articulate in a way that completely passes muster with the teachers.

Good for her for independently coming up with (90% of) the concept of negative numbers, and using those to simplify problems!

Answer 14

Your daughter is clever. She's breaking the numbers into smaller and easily subtractable numbers: $$61-17= (50+11)-(10+7)$$

Rearranging the order:$$(50-10)+(11-7)=44$$

Answer 15

What a coincidence, I used this exact same method when I was in school, but only mentally because it would be marked with a big red cross if it ever came on paper.

Its a brilliant little technique based on a little manipulation. It works like this: 61 - 17 is basically how much you have to add from 17 to get to 61. This can be done in two ways:

The method we use

We basically first compute how much you need to add to 17 (the smaller number) to get the unit digits to be the same: 4, since 17 + 4 = 21. Then we find out the amount we now have to add, which is of course 40. Adding the two gives us 44. The first part of it is basically the carry: the amount you have to add to 17 to make the unit digit 1 is the amount you have to add to 7, the minuend, to make it 11, the smallest 2-digit number with a unit digit 1.

Consequently, the tens digit increases by one in the smaller number, and this is taken into account by subtracting the 1 from 6 or adding 1 to the 1 in 17.

The method your daughter uses

Imagine first adding 9 to your 61 and making it 70.
Then you add 3 to your 17 and make it 20.
Now (61 - 17) = (70 - 9) - (20 - 3) = (70 - 20) - (9 - 3) Now it turns out that this (9-3) is the same as the difference between the minuend and the subtraend, (7-1). This is always the case.

And that is how your daughter does it.

Answer 16

One really easy way to know that this will always work is to think of it as the addition of a positive and a negative number:

$61 + -17$

From there you can see what shes really doing:

$(61-1) + (-17+1)$

$60 + -16$

The equation is still balanced, therefore clearly still the same sum. You can continue along this line of reasoning:

$(60-10) + (-16+10)$

$50 + -6$

$50-6=44$

Answer 17

My way of explaining it:

\begin{align*} 61−17 &= (6 \cdot 10 + 1 \cdot 1) - (1 \cdot 10 + 7 \cdot 1) \\ &= (6 - 1) \cdot 10 + (1-7) \cdot 1 \\ &= (6 - 1) \cdot 10 - (7-1) \cdot 1 \\ &= 5 \cdot 10 - 6 \cdot 1 \\ &= 50 - 6 = 44 . \end{align*}

---

On the one hand, you could generalise: \begin{align*} ABCD \ldots FG - abcd \ldots fg &= (A \cdot 10^\alpha + B \cdot 10^\beta + \ldots + F \cdot 10 + G) - (a \cdot 10^\alpha + b \cdot 10^\beta + \ldots + f \cdot 10 + g) \\ &= (A - a) \cdot 10^\alpha + (B - b) \cdot 10^\beta + \ldots + (F - f) \cdot 10 + (G-g) , \end{align*} and then rearrange to adjust signs.

On the other hand, you can be flexible :-) : \begin{align*} 61−17 &= (50 + 5 + 6) - (10 + 6 + 1) = 50 - 10 + 5 + 6 - 6 - 1 = 40 + 5 - 1 = 44\\ 61−17 &= (17+3+40+1) - 17 = 3 + 40 + 1 = 44 \\ 61−17 &= (99-38) - (34-17) = 99-38-34+17 = 99-72+17 = 27+17=44 \\ \ldots& \end{align*}

Answer 18

People can't help what method they use in their head to solve a problem but they can try and find a way to get the right answer and as long as her method works, she might be unable to learn how to use the school's method in her head so I think it's better to leave her doing her own method. It works because 61 - 17 = (50 + 11) - 17 = 50 - (17 - 11) = 50 - 6 = 44. Good for her for figuring out her own method. I believe that I figured out on my own how to divide by 2 in decimal before I learned how to do long division.