2-Digit Addition with Regrouping - Second Grade Mathematics

When adding two-digit numbers, what happens when the ones digits add up to 10, 12, 15, or more? In our base-ten number system, a single place value column can only hold a single digit from 0 to 9. When you collect 10 or more ones, you must bundle 10 ones together, trade them for 1 new ten, and send that ten into the tens column. This essential mathematical process is called regrouping, composing a ten, or trading. In this chapter, you will master the regrouping concept using base-ten block models and learn the standard vertical algorithm with complete confidence.

What Does Regrouping Really Mean?

Regrouping is not a magic trick; it is simply exchanging equal values of currency or blocks.

The Grand Trade:
10 Unit Cubes (Ones)  <====== EQUAL TO ======>  1 Ten Rod (Tens)
. . . . . . . . . .                             |

Why We Cannot Write Two Digits in One Column

Suppose you add 37 + 28. In the ones column, 7 + 8 = 15. You cannot write the entire number 15 in the ones column, because that would leave no room for the tens! The number 15 is made of 1 ten and 5 ones. We keep the 5 ones in the ones column and carry or regroup the 1 ten into the tens column where it belongs.

The Regrouping Action:
     1  <--- Regrouped Ten (Carried to the top)
     37
   + 28
   ----
     65

The Step-by-Step Regrouping Algorithm

Let us follow the step-by-step procedure to solve 48 + 36.

Problem: 48 + 36

Step 1: Align columns neatly.
        Tens   Ones
         4      8
      +  3      6
      -----------

Step 2: Add the ones column first.
        8 + 6 = 14 ones.
        Ask: Is 14 ten or greater? YES!
        14 ones = 1 ten and 4 ones.

Step 3: Record the regrouping.
        Write the 4 in the ones column below the line.
        Write a small 1 above the tens column (in the attic).

         1
         4      8
      +  3      6
      -----------
                4

Step 4: Add all the tens together, including the regrouped ten!
        1 (regrouped) + 4 + 3 = 8 tens.
        Write 8 in the tens column below the line.

         1
         4      8
      +  3      6
      -----------
         8      4

The Attic Ten

The small 1 written above the tens column is affectionately called the "attic ten." It serves as an unshakeable reminder that you created an extra ten from your ones. The most common error students make is forgetting to add the attic ten, so always point your pencil to it first when adding the tens column: 1 + 4 = 5, plus 3 = 8!

Base-Ten Model of Regrouping

Let us look at what physically happens with base-ten blocks during 48 + 36.

Addend 48:  | | | |        . . . . . . . .
Addend 36:  | | |          . . . . . .

Count Ones: There are 14 dots.
Action:     Circle 10 dots.
            Trade the circled 10 dots for 1 new line | in the tens group.
            There are 4 dots left outside the circle.

Count Tens: 4 original + 3 original + 1 new trade = 8 lines.
Total Blocks: 8 lines and 4 dots = 84!

Direct Proof of Equivalence

Notice that no blocks appeared out of thin air and no blocks disappeared. We had 14 ones, which we reorganized as 1 ten and 4 ones. Regrouping simply changes the grouping of the blocks so that each column obeys the base-ten rule of single digits.

Practice Exercises

Exercise 1: Solve 57 + 26 using the standard vertical algorithm. Show the regrouped ten in the attic and write down each step of your calculation.

Exercise 2: Solve 39 + 45. Explain how many ones were in the ones column before regrouping, and how they were separated into tens and ones.

Exercise 3: A student solved 68 + 25 and wrote down the answer 83. Identify the student's mistake, explain what caused it, and provide the correct sum.

Exercise 4: A toy shop sold 47 teddy bears on Saturday and 38 teddy bears on Sunday. How many teddy bears were sold in total over the weekend?

Solutions and Explanations

Solution to Exercise 1: Step 1: Align vertically. Step 2: Add ones: 7 + 6 = 13. Since 13 is ten or more, write 3 in the ones column and place a 1 in the attic above the tens column. Step 3: Add tens: 1 (attic) + 5 + 2 = 8 tens. Write 8 in the tens column. The final sum is 83.

Solution to Exercise 2: To solve 39 + 45, add the ones first: 9 + 5 = 14 ones. The 14 ones are separated into 1 ten and 4 ones. The 4 ones stay in the ones column, and the 1 ten is regrouped to the tens column. Adding tens: 1 + 3 + 4 = 8 tens. The final sum is 84.

Solution to Exercise 3: The student's mistake was forgetting to add the regrouped ten. In the ones column, 8 + 5 = 13; the student correctly wrote 3 in the ones place, but then simply added 6 + 2 = 8 in the tens place, ignoring the extra 1 ten. The correct tens sum is 1 + 6 + 2 = 9 tens. The correct answer is 93.

Solution to Exercise 4: To find the total teddy bears sold, add 47 + 38. Add ones: 7 + 8 = 15. Regroup 1 ten to the tens place and write 5 in the ones place. Add tens: 1 + 4 + 3 = 8 tens. In total, 85 teddy bears were sold over the weekend.