Addition with Regrouping up to 100

When you add two-digit numbers, sometimes the ones column adds up to 10 or more. Since only one digit fits in the ones place, you have to trade, or "regroup," ten ones for one ten. That extra ten gets carried over and added into the tens column. This is why the process is often called addition with carrying.

For example, in \( 8 + 7 = 15 \), you cannot write "15" in the ones column of a bigger addition problem. Instead, you keep the 5 in the ones place and carry the 1 (which really means one ten) into the tens column.

Every two-digit number is made of tens and ones. In 48, the 4 stands for 4 tens (40) and the 8 stands for 8 ones. Regrouping only works because 10 ones are always equal to 1 ten. Before tackling regrouping, it helps to be comfortable with adding digits up to 100 without carrying, so you can clearly see the difference between the two situations.

Follow these steps for any two-digit addition problem up to 100:

1. Line up the numbers by place value, ones under ones and tens under tens.
2. Add the ones column first.
3. If the ones sum is 9 or less, write it directly under the ones column.
4. If the ones sum is 10 or more, write only the ones digit, and carry the tens digit (a 1) above the tens column.
5. Add the tens column, including any carried ten.
6. Write the final tens digit as the answer's tens place.

Add the ones column first: \( 8 + 7 = 15 \). Write the 5 in the ones place of the answer and carry the 1 (one ten) to the tens column. Then add the tens column: \( 4 + 3 + 1 = 8 \). The final answer is \( 48 + 37 = 85 \).

Add the ones: \( 6 + 9 = 15 \). Write 5 in the ones place and carry 1 ten. Add the tens: \( 5 + 2 + 1 = 8 \). So \( 56 + 29 = 85 \). Notice that even though the digits look different from the first example, the same carrying process applies whenever the ones column reaches 10 or more.

Regrouping is one of several tools for adding numbers up to 100. Once you're confident carrying with two numbers, you can apply the same idea to problems with adding with 3 numbers up to 100, where you may need to carry more than once. It's also worth comparing regrouping to other addition strategies, since some problems are easier to solve mentally without carrying at all.

Watch out for these frequent errors:

• Forgetting to add the carried ten into the tens column.
• Writing both digits of a two-digit ones sum in the ones place (for example, writing "15" instead of "5" with a carried 1).
• Carrying the ones digit instead of the tens digit from a sum like \( 9 + 8 = 17 \), where the 1 (not the 7) is what gets carried.
• Losing track of which column the carried ten belongs to when numbers are not lined up neatly.

Write problems in vertical column form so ones and tens stay aligned. Say the carrying step out loud as you work, such as "8 plus 7 is 15, write the 5, carry the 1." Practicing with a mix of problems that do and do not require regrouping helps you quickly recognize when the ones column passes 9 and a ten needs to move over.

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• Math 2