Subtraction (with regrouping, 3+ digits)

Once students learn how to subtract with regrouping, they can apply those skills to numbers of any size!  For more on regrouping (how and why it works), see lesson on Subtraction (with Regrouping, 2-1 and 2-2 digits).

The first step in subtracting multi-digit numbers is setting the problem up vertically, making sure that the place values are lined up (ones to ones, tens to tens, etc).

From there, start with the ones column (or the column farthest to the right), subtract the bottom number from the top number.  If the top number is smaller than the bottom number, then regroup, borrowing a 10 from the next column.

Example: 

$811-234=$

Start by writing the problem vertically:

$$\begin{array}{r} &811\\-\!\!\!\!\!\!&234\\ \hline \end{array}$$

From there, you start with the ones colum: 

$$\begin{array}{r} &81\mathbf{1}\\-\!\!\!\!\!\!&23\mathbf{4}\\ \hline \end{array}$$

• 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 ($811$) is not smaller than the subtracted number ($234$), 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.

$$\begin{array}{r} &8\overset{0}{\bcancel{1}}\! \!^1\!1\\-\!\!\!\!\!\!&2\;3\;4\\ \hline &\end{array}$$
 

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

$$\begin{array}{r} &8\overset{0}{\bcancel{1}}\mathbf{\! \!^1\!1}\\-\!\!\!\!\!\!&2\;3\;\mathbf{4}\\ \hline &\mathbf{7}\end{array}$$

• Next, the tens column. But, you can't subtract 3 from 0 without going negative, so we need to borrow from the hundreds column.

$$\begin{array}{r} &\overset{\mathbf{7}}{\bcancel{8}}\! \!\overset{^1\!0}{\bcancel{1}}\!^1\!1\\-\!\!\!\!\!\!&2\;\mathbf{3}\;4\\ \hline &\mathbf{7}\;7\end{array}$$

• After we borrow a hundred from the 8 and add it to the 0 in the tens, column, we have $10-3=7$, so we put 7 in the answer row.
• Finally, we have to subtract the last column.  As long as the top number is larger than the bottom number, you should never have to borrow on the last digit.

$$\begin{array}{r} &\overset{\mathbf{7}}{\bcancel{8}}\! \!\overset{^1\!0}{\bcancel{1}}\!^1\!1\\-\!\!\!\!\!\!&\mathbf{2}\;3\;4\\ \hline &\mathbf{5}\;7\;7\end{array}$$

The most common issue that students have with subtraction with borrowing is when they come across zeros.  The rule is a rule that works throughout math: if you have nothing, you can't borrow! So, when students come to a zero, they need to borrow from the NEXT column, borrow to make the zero a 10 and THEN borrow from the 10.

Example: 

$900-467=$

Start by writing the problem vertically:

$$\begin{array}{r} &900\\-\!\!\!\!\!\!&467\\ \hline \end{array}$$

From there, you start with the ones colum: 

$$\begin{array}{r} &90\mathbf{0}\\-\!\!\!\!\!\!&46\mathbf{7}\\ \hline \end{array}$$

• The first subtraction problem we have to do is in the ones column: $0 - 7 = $, but we can't do that problem without going into negative numbers!  And, you can't borrow from the tens column because it's a zero!  So, you have to keep moving to the left until you find a non-zero column.  In this case, that's the 9 in the hundreds column.  Borrow from the 9, turn the tens digit into a 10, and then borrow from that.  Always borrow from the first non-zero number you reach, and always remember to borrow back to the column you're working in, writing carefully, so that you can read your numbers when you subtract from those columns. 

$$\begin{array}{r} &\overset{8}{\bcancel{9}}\! \!\overset{9}{\bcancel{^1\!0}}\!^1\!0\\-\!\!\!\!\!\!&4\;6\;7\\ \hline &\end{array}$$
 

• Now, the student can subtract 7 from 10 ($10-7=3$) and put the 3 in the ones column.

$$\begin{array}{r} &\overset{8}{\bcancel{9}}\!\!\overset{9}{\bcancel{^1\!0}}\!\mathbf{^1\!0}\\-\!\!\!\!\!\!&4\;6\;\mathbf{7}\\ \hline &\mathbf{3}\end{array}$$

• Next, the tens column. Pay attention to the numbers written while borrowing for the ones column.  Although the tens column used to be a 0, it's now a 9!

$$\begin{array}{r} &\overset{8}{\bcancel{9}}\!\!\mathbf{\overset{9}{\bcancel{^1\!0}}}\!^1\!0\\-\!\!\!\!\!\!&4\;\mathbf{6}\;7\\ \hline &\mathbf{3}\;3\end{array}$$

• Finally, do the last column.  Pay attention to what the number is now:

$$\begin{array}{r} &\mathbf{\overset{8}{\bcancel{9}}}\!\!\overset{9}{\bcancel{^1\!0}}\!^1\!0\\-\!\!\!\!\!\!&\,\mathbf{4}\;6\;7\\ \hline &\mathbf{4}\;3\;3\end{array}$$

Once students master the process of borrowing (and remember to deal with 0s) they can subtract numbers of any size! They just need to practice and write neatly!  Please push students to show their work when borrowing. There are too many places for careless mistakes here for them to try to hold the numbers in their heads.