## Adding Two-Digit Numbers

Beginning of Algorithms
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 standard algorithms for two-digit addition and subtraction without regrouping are introduced in this unit. The focus throughout this unit should be not only carrying out the algorithms correctly to find sums and differences, but also understanding why the algorithms work.

Addition of two-digit numbers is developed sequentially, beginning by finding the sums of numbers that are multiples of 10. For example, to add 20 plus 30, a child may think of the basic addition fact, 2 + 3 = 5, and then append a zero to the five to get the sum, 50. When children do this, the important point to emphasize is that they are not really adding 2 and 3, but rather 2 tens and 3 tens.

Next in the sequential development is the addition of two-digit numbers limited to addends that would have sums no greater than two-digits. This means that the sum of the ones must be less than 10 ones, and the sum of the tens must be less than 10 tens. This avoids the skill known as regrouping; that is, having to exchange, for example, 15 tens for one hundred and 5 tens.

As discussed earlier, the addition algorithm is based on our base-ten positional numeration system, and so when adding two-digit numbers, place value must be acknowledged. Consider the example of 34 + 52. The first digit from the left is in the tens place, and the second digit from the left is in the ones place. Thus, using the concept of decomposition and that of writing numbers in expanded form discussed earlier, 34 means 3 tens and 4 ones, or 30 + 4.

So, the addition 34 + 52 can be written in the following way:

 34+ 52 30 + 4 + 50 + 2 86 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.

In standard practice, numbers are not written in expanded form before finding the sum. The rule of simply adding the ones and then adding the digits in the tens place is followed, assuming regrouping is not necessary. This is how the standard addition algorithm is carried out. However, by showing children the process of finding the sum in expanded form and having them practice several problems in this form, children will develop an understanding as to why they can simply add the numbers in the place-value columns to find the sum. In addition, by focusing on the positional value of the digits in the numbers to be added, children will develop a strong conceptual foundation that will prepare them for the concept of regrouping when it is formally introduced in second grade.

It is worth noting that many children, when first confronted with the problem of adding two-digit numbers, will devise a variety of their own methods for finding the sum. Childrens' problem-solving approaches should be encouraged and discussed. In many instances the discussion and analysis of a child's approach to finding the sum may contribute to the class's understanding of the concept. However, it is also important to encourage children to appreciate the efficiency of standard algorithms.

Teaching Model 21.2: Add With Two-Digit Numbers