# ## Regrouping With Subtraction

The standard algorithm for two-digit subtraction with regrouping is introduced in this chapter. Again, throughout the chapter, the focus is not only on how to carry out the algorithm, but also on why the algorithm works. An understanding of place value, basic subtraction facts, and subtraction as the inverse of addition, serve as the foundation for extending subtraction by using multiples of ten. For example, in order to find the difference 60 - 40, children may rely on a basic fact as they think: 6 tens − 4 tens = 2 tens, or 20.

Next, children move on to subtracting two-digit numbers that do not require regrouping.

The standard subtraction algorithm is based on our base-ten positional numeration system. When subtracting two-digit numbers, place value must be acknowledged. Consider the example of subtracting 21 from 74. The first digit is in the tens place. The second digit is in the ones place. In expanded form, 74 means 7 tens 4 ones, or 70 + 4. So, 74 − 21 can be written in the following ways.

74
− 21
53
70 + 4
− (20 + 1)
50 + 3 = 53

Subtraction With Regrouping
The subtraction algorithm consists of subtracting first the ones, then the tens, then the hundreds and so on, regrouping when necessary. The reason the subtraction algorithm works can be illustrated by using the expanded form of the numbers. Have children consider 52 − 38.

52
− 38

5 tens 2 ones
− (3 tens 8 ones)

4 tens 12 ones
− (3 tens 8 ones)
1 ten 4 ones = 14

When 52 and 38 are written in expanded form, it can be seen that there are not enough ones in the minuend to subtract the ones in the subtrahend. Therefore, one ten is removed from the tens and added to the ones to create 12 ones. It is now possible to subtract the ones, resulting in 4 ones, and then subtract the tens, resulting in 1 ten.

Examining these steps in expanded form makes it easier for children to understand the standard algorithm, but it is less efficient and requires more time and space. Therefore, the steps are recorded in the standard algorithm as illustrated below. It is worth noting that when many children add and subtract two-digit numbers, they will first devise a method of their own for finding the sum or difference. Children's problem-solving approaches should be encouraged and discussed. However, as illustrated above, it is important to move children in the direction of using standard algorithms to solve problems.

When a horizontal subtraction problem is presented to children, be sure to emphasize the need to correctly align the digits before subtracting. Children should be reminded that addition can be used to check subtraction.

Estimating Differences
When an exact answer is not needed, an estimate can be used. The most common method of estimating differences is to round each number to a specific place and then subtract the rounded numbers. In this chapter, children round numbers to the nearest ten with the help of a number line.

Estimate: 91 − 37

91
− 37

90 Round each number to the nearest ten.
− 40
50 Subtract the rounded numbers.

Mental math can often be used to complete estimates. At this grade level, however, explain that errors can be more easily identified if children write down their work when estimating answers.

Teaching Model 12.3: Regroup Tens