Subtraction (with regrouping, 2-1 digits & 2-2 digits)

Once students have mastered the subtraction facts and learned to stack and subtract multi-digit numbers, it's time to teach them about subtraction with regrouping (or subtraction with borrowing).  It's important that students have some understanding of why and how they are regrouping (they'll need another version of the concept later, when they start working with fractions), but this concept is a difficult one and it's ok to teach it in stages (and not have the students fully understand every part as they go along).

Remind students that there is no commutative property of subtraction: unlike in addition, it does matter which number comes first:

When you subtract, you take the second number AWAY from the first number.  So, 5-3 means that you take 3 away from 5.  5 goes on top, 3 goes on the bottom.  The number you start with goes on top, and the number you are taking away goes on the bottom.

Also remind students that, when they set up problems vertically, they line up the place values:

The ones digit of the subtracted number goes under the ones digit of the initial number, the tens digit of the subtracted number goes under the tens digit of the initial number, and so on. 

While you are setting up the problem, it's a good time to talk for a moment about what each column represents.  This sets up borrowing.

Take the problem: $81-34$

We know that:

• In this problem, we're starting with 81.  81 goes on top.
• We're taking away 34.  34 goes on the bottom.
• 81 is 8 tens and 1 ones.
• 34 is three tens and 4 ones.
• The 1 and the 4 are lined up when the problem is written vertically.

$$\begin{array}{r} &81\\-\!\!\!\!\!\!&34\\ \hline \end{array}$$

With the problem written vertically, the process is the same as doing two subtraction facts, except we have a problem:

$$\begin{array}{r} &8\mathbf{1}\\-\!\!\!\!\!\!&3\mathbf{4}\\ \hline \end{array}$$

• The first subtraction problem we have to do is in the ones column: $1 - 4 = $, but we can't do that problem without going into negative numbers!  Why don't we go into negative numbers?  Because, at the moment, we're just dealing with the ones column.  The entire initial number ($81$) is not smaller than the subtracted number ($34$), so we don't want to go negative, we just want to borrow from the tens column so that we can do the subtraction in the ones column.
• Think about 81 again: it's 8 tens and 1 one.  We can borrow one of the tens and give it to the 1 in the ones column.  We are, literally, going to regroup this number, by breaking a 10 off of the 80 and adding it to the ones column: 

$$\eqalign{81 &= 8 \text{ tens} \qquad \qquad &&+ 1 \text{ one}\\ &= 7 \text{ tens} + 1 \text{ ten} &&+ 1 \text{ one}\\ &= 7 \text{ tens} + \bbox[5px, border: 2px solid red]{1 \text{ ten}}\rightarrow &&+  1 \text{ one}\\&=7 \text{ tens} &&+ \bbox[5px, border: 2px solid red]{1 \text{ ten}} + 1 \text{ ones}\\ &= 7 \text{ tens} &&+ 11 \text { ones}}$$

When writing a vertical subtraction problem, we'd write the above process like this:

• We'd start by writing the problem vertically:

$$\begin{array}{r} &81\\-\!\!\!\!\!\!&34\\ \hline \end{array}$$

• Then, when we could not subtract the ones column without going into the negative numbers ($1-4=$ a negative number), we would break down the tens column (symbolized by crossing out the 8, turning it into a 7) and bring that extra 10 over to the ones column (symbolized by writing a little 1 in front of our ones digit):

$$\begin{array}{r} &\overset{7}{\bcancel{8}}\! \!^1\!1\\-\!\!\!\!\!\!&3\,4\\ \hline &\end{array}$$

• Now, the student can subtract 4 from 11 ($11-4=7$) and put the 7 in the ones column.

$$\begin{array}{r} &\overset{7}{\bcancel{8}}\mathbf{\! \!^1\!1}\\-\!\!\!\!\!\!&3\,\mathbf{4}\\ \hline &\mathbf{7}\end{array}$$

• Finally, the student can subtract 3 from 7 ($7-3=4$) and put the 4 in the tens column.

$$\begin{array}{r} &\overset{\mathbf{7}}{\bcancel{8}}\! \!^1\!1\\-\!\!\!\!\!\!&\mathbf{3}\,4\\ \hline &\mathbf{4}7\end{array}$$

As you can see, it's helpful for students to understand how and why they are regrouping or borrowing.  There are a lot of steps in these problems and it's hard to "just memorize them."  However, we teach subtraction with grouping to children as young as 7  -- and they are not always ready to fully understand place value and the base-10 system!  Explain what's going on and how it works.  Walk them through the process while explaining what's happening (e.g., "We can't subtract 9 from 6, so we're going to borrow from the tens place next to the 6 and turn it into a 16, now we can do $16-9=7$).

Some critical details to make sure students pay attention to:

• Write neatly.  You want to be able to read your new numbers.
• Always cross out the numbers you borrow from (at EdBoost, we often say that you can't actually write the little 1 until you have changed the number you're borrowing from).  It's easy to forget that you borrowed and just subtract from the original number.

Some very common errors to look out for:

• The most common error we see is that, when there's a 0 on top, students skip borrowing and just bring the non-zero number down:

$$\begin{array}{r} &7\mathbf{0}\\-\!\!\!\!\!\!&2\mathbf{8}\\ \hline &8 \end{array}$$

We need to remind students that $0-8=$ is a math problem... and the answer is NOT $8$.  They need to borrow:

$$\begin{array}{r} &\overset{6}{\bcancel{7}}\! \!^1\!0\\-\!\!\!\!\!\!&2\,8\\ \hline &42\end{array}$$

• The other common error that we see is that once students get good at borrowing, they borrow through an entire problem (sometimes several place values in).  Eventually, this results in them subtracting a number that has an extra 10 added to it but does need it, leading to a 2-digit answer for that colum, which leads to the student trying to carry... and a disaster.  Here's what it usually looks like:

$$\begin{array}{r} &\overset{5}{\bcancel{6}}\! \!^1\!7\\-\!\!\!\!\!\!&3\,2\\ \hline &\end{array}$$

If they don't catch themselves, they do $17-2=15$, then they try to put the 5 in the ones column and carry the 1.  If you see a problem that looks like the one below, you can bet that your students are borrowing when they don't need to. The good part is, once they get to this step, they are usually too confused to continue!

$$\begin{array}{r} &\overset{1}{\overset{5}{\bcancel{6}}}\! \!^1\!7\\-\!\!\!\!\!\!&3\,2\\ \hline &5\end{array}$$

So, a good basic rule to recognize why a disaster has happened is to let students know that if they start to want to carry in a subtraction problem, they have done something wrong.  There should be no carrying.

To prevent the disaster from going this far, we recommend that students only borrow one column at a time.  If they need to borrow from two columns in order to bring a value to the column they are trying to subtract (so, if the column to the left of where they are working is a 0), they can borrow from two columns, but that's the only exception.  Work column by column.  Always remember that you might not need to borrow, so don't do more work than you have to.  Don't borrow until you need to.