1st grade

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).

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.

Once students learn their additional facts, they should learn to add multiple numbers. 

There are many ways to add columns of numbers.  Of course, the strategies of counting up and counting on (holding a number, or sum, in your head while counting on another number) always work. 

However, once students know their math facts, they can work faster than that.  And, if students know the pairs that add up to nicely added-to numbers (like 10), they can work even faster. 

Example: 

$7+5+3=$

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

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!).