Addition (with regrouping, 2 digits)

Addition with regrouping (also called addition with carrying) is one of the first complicated math algorithms that most students learn.  

When learning addition with regrouping, it's helpful if students understand what they are doing ("Why am I carrying that 1?"), but it's not essential that they fully understand it in order to execute it. Whenever students learns an algorithm, they will remember it better and execute it more accurately if they understand it.  However, the reason we have algorithms is so that we don't have to reinvent the wheel everytime we want to do a math problem.  So, explain what's going on, but also show what to do.  Hopefully, students' understanding of what's going on grows as students become more comfortable with the process.  

So, what is going on when we do addition with regrouping?

Take the problem: $37+14$

We know that:

• 37 can be broken into 30 + 7, which is the same as 3 tens + 7 ones. 
• 14 can be broken into 10 + 4, which is the same as 1 ten + 4 ones.
• When we do math, we start with the lowest place value, so we add the ones: $4+7=11$
• 11 can be broken down into 10 + 1, which is 1 ten and 1 one.  We write down the ones place 1 (in the ones column) and we "carry" the tens place 1 over to the tens column.  
• Now, when we add the tens, we add: $3+1+1 \text{ (the carried ten) }=5$  There are 5 tens.
• 5 tens and 1 one is $51$.

Using the "adding with regrouping" algorithm tries to demonstrate the concepts above and make it easy to execute the movement of amounts from one place value to another:

First we would rewrite the problem as:

$$\begin{array}{r} &37\\+\!\!\!\!\!\!&14\\ \hline \end{array}$$

Then we would point out that the column the farthest to the right is the ones column, and the next column is the 10s column.

So, first we add the ones colum:

$$\begin{array}{r} &\overset{1}{3}\mathbf{7}\\+\!\!\!\!\!\!&1\mathbf{4}\\ \hline &\mathbf{1} \end{array}$$

We get 11, which is 1 + 10, so we put the 1 from the 11 in the ones column and we bring the 10 from the 11 up to the tens colum.

Then we add the items in the 10s column (including the carried 1):

$$\begin{array}{r} &\mathbf{\overset{1}{3}}7\\+\!\!\!\!\!\!&\mathbf{1}4\\ \hline &\mathbf{5}1 \end{array}$$

Overall, we want students to understand that we write numbers vertically and line them up, so that numbers with the same value are in the same column (ones with ones, tens with tens).

Then we want them to understand that each column holds only one digit.  When we have a number that spans two digits (11, 31, 56), then we carry the front number (in this case, the ten) to the next column.

From here, you just practice.  Make sure that they always remember to write out their carry numbers, and check for accuracy.  Even if students don't fully understand why they are doing what they are doing, as they get better at the skill and gain more math knowledge, everything that they are doing will start to make more sense.  By explaining the concept behind the algorithm, you give the students the tools that they need to do math through understanding, rather than just through rote memory (even if, in the early stages, they rely a lot on memory too!).