An algorithm is a systematic procedure for carrying out a computation. Algorithms are an important part of the study of mathematics in elementary school. The Principles and Standards for School Mathematics (NCTM, 2000) states that “students must become fluent in arithmetic computation” and that “standard algorithms for arithmetic computation are one means of achieving this fluency.” The standard algorithm for two-digit addition with regrouping is introduced in this chapter. The focus throughout this chapter should not only be on how to carry out the algorithm, but also on why the algorithm works.

Addition of two-digit numbers is developed sequentially, beginning by finding the sums of numbers that are multiples of ten. For example, in order to find the sum of 20 and 30, children may think of the basic addition fact 2 + 3 and then append the zero to the answer. The important point to emphasize with children is that they really are not adding 2 and 3, but rather 2 tens and 3 tens.

Next, children move on to adding two-digit numbers with sums no greater than two-digits. This means that the sum of the tens must be less than 10 tens. This avoids regrouping. That is, having to exchange, for example, 15 tens for one hundred and 5 tens.

The addition algorithm is based on our base-ten positional numeration system, and when adding two-digit numbers, place value must be acknowledged. Consider the example of 34 + 52. The first digit on the left is in the tens place and the second digit from the left is in the ones place. It should be explained that, written in expanded form, 34 means 3 tens 4 ones, or 30 + 4. So an addition such as 34 + 52 can be written in the following ways.

34
+ 52
86
30 + 4
+ 50 + 2
80 + 6 = 86

Adding 34 and 52 is equivalent to finding the sum of 3 tens (30) and 5 tens (50) and then adding to this result the sum of 4 ones and 2 ones. Of course, in standard practice the numbers are not written in expanded form before finding the sum. The process of finding the sum in expanded form can help students develop an understanding of why the algorithm works.

Teaching Model 11.4: Add Three Numbers