# Subtraction (without regrouping, 2 digits)

Subtraction is almost always a little more complicated than addition for students. But, it's important, when teaching addition and subtraction to emphasize the relationship between the two process.  Subtraction can be tricky -- but it's not more difficult than addition.  And the rules are much the same.

To that end, make sure a student is comfortable with multi-digit addition (including setting a problem up vertically) before you start subtraction.  Then you can refer to the addition processes as you teach subtraction.

A student who has learned multi-digit addition should already be familiar with place value and how place value relates to setting a subtraction problem up vertically.  Remind the students that the ones digit goes above the ones digit, and the tens digit goes above the tens digit.

With subtraction you have the added complication of the fact that there is no commutative property of subtraction: unlike in addition, it does matter which number comes first in a subraction problem (or which number is written on top when the problem is written vertically).  Talk about this conceptually.  Do not tell a student that the smaller number always goes on the bottom.  That's one of those rules that, while true in the lower grades, becomes completely wrong when the student starts working with negative numbers.  We try very hard not to give students rules that don't always hold.  So, here's the rule that always holds:

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.

Take the problem: $89-4$

We know that:

• In this problem, we're starting with 89.  89 goes on top.
• We're taking away 4.  4 goes on the bottom.
• 89 is 8 tens and 9 ones.
• 4 is 4 ones.
• The 9 and the 4 are lined up when the problem is written vertically.

$$\begin{array}{r} &89\\-\!\!\!\!\!\!&4\\ \hline \end{array}$$

• As with addition, it can be helpful to write the numbers in columns to keep them organized.

$$\underline{\begin{vmatrix}8\\- \end{vmatrix} \begin{vmatrix} 9\\4 \end{vmatrix}}$$

Once students get the hang of writing in clear columns, you should be able to drop the vertical lines.

With the problem written vertically, the process is the same as doing two subtraction facts:

$$\begin{array}{r} &89\\-\!\!\!\!\!\!&4\\ \hline \end{array}$$

• Subtract the ones digit on the bottom from the ones digit on the top.

$$\begin{array}{r} &89\\-\!\!\!\!\!\!&4\\ \hline &5\end{array}$$

• Subtract the tens digit from the bottom from the tens digit on the top (in this case, there's nothing there, so $8-0=8$)

$$\begin{array}{r} &89\\-\!\!\!\!\!\!&4\\ \hline &85\end{array}$$

After they get comfortable adding a 2 digit number with a 1 digit number, it's time to transition students into subtracting two 2 digit numbers.  The principles are the same:

• Write the problem vertically.
• Make sure the starting number is on the top and the number that is being taken away is on the botom.
• Line up the ones digit with the ones digit and line up the tens digit with the tens digit (the lining up habits will last them a lifetime!).
• Subtract the ones column first.
• Then subtract the tens column.

Try a problem like like: $92-31=$

• Students should first write the problem vertically -- ideally explaining where the ones digits are and where the tens digits are:

$$\begin{array}{r} &92\\-\!\!\!\!&31\\ \hline \end{array}$$

• They subtract the smallest place value first, in this case the ones. The bottom number subtracted from the top number.

$$\begin{array}{r} &9\mathbf{2}\\-\!\!\!\!&3\mathbf{1}\\ \hline &\mathbf{1}\end{array}$$

• Then they move up the place values, in this case to the tens -- where they finish the problem!

$$\begin{array}{r} &\mathbf{9}2\\-\!\!\!\!&\mathbf{3}1\\ \hline &\mathbf{6}1\end{array}$$

Subtracting multi-digit numbers without regrouping is really an intermediary step.  Students should feel comfortable with this process (and especially the process of writing problems vertically) before they move on to Subtraction with regrouping.  But, this step is not an end in and of itself and once students get the hang of it, move on to Subtraction with regrouping (and remember to make sure that, after students learn to regroup, they can still recognize and execute a problem that does not require regrouping!  Borrowing when they don't have to is a very common mistake students make when first learning to subtract multi-digit numbers).