Addition (without regrouping, 1+2 and 2+2 digits)

When we first start to teach students to add multi-digit numbers, it's important to teach them to write problems vertically and line up place values.  Setting addition problems up vertically, even when there is no regrouping or carrying, helps students prepare for the next step: regrouping! And, understanding that place values line up also sets the groundwork for more advanced skills (such as lining numbers up properly when numbers have decimals!).

Although worksheets for small kids often present addition problems "stacked" with one number above the other, it's important to also present students with horizontal problems, so that they can learn to set problems up vertically on their own. 

So, when presented with a problem that involves a two-digit number, start by talking about what the place values mean.

Take the problem: $42+6$

We know that:

• 42 can be broken into 40 + 2, which is the same as 4 tens + 2 ones. If students don't understand this, make a note to go back and do some place value work, although it's not critical that you stop what you're doing and go back before moving on.
• 6 is 6 ones.
• When we add, we put stack the numbers on top of each other and line up the place value.

$$\begin{array}{r} &42\\+\!\!\!\!\!\!&6\\ \hline \end{array}$$

• Sometimes, when you first start, it's helpful to write the numbers in a grid, to help students see that the columns are distinct.

$$\underline{\begin{vmatrix}4\\+ \end{vmatrix} \begin{vmatrix} 2\\6 \end{vmatrix}}$$

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

Once students have the problem written vertically, the addition process is clear:

• First they add the numbers in the ones column.

$$\begin{array}{r} &4\mathbf{2}\\+\!\!\!\!\!\!&\mathbf{6}\\ \hline &\mathbf{8} \end{array}$$

• Then, they add the numbers in the tens column (in this case, with the second number having no tens digit, so it's $4+0=4$).

$$\begin{array}{r} &\mathbf{4}2\\+\!\!\!\!\!\!&6\\ \hline &\mathbf{4}8 \end{array}$$

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

• Write the problem vertically.
• 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!).
• Add the ones first.
• Then add the tens.

Try a problem like : $56+21=$

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

$$\begin{array}{r} &56\\+\!\!\!\!&21\\ \hline \end{array}$$

• They add the smallest place value first, in this case the ones.

$$\begin{array}{r} &5\mathbf{6}\\+\!\!\!\!&2\mathbf{1}\\ \hline &7\mathbf{7}\end{array}$$

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

$$\begin{array}{r} &\mathbf{5}6\\+\!\!\!\!&\mathbf{2}1\\ \hline &\mathbf{7}7\end{array}$$

Adding 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 addition with regrouping.  But, this step is not an end in and of itself and once students get the hang of it it's time to move on to Addition 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!).