3-Digit Addition: Regrouping Tens - Second Grade Mathematics

As you delve deeper into three-digit addition, you will often find that combining the ones column produces a sum of 10 or greater. Just as in two-digit addition, our base-ten rules dictate that only a single digit can occupy the ones place. When the ones add up to 10 or more, you must bundle 10 ones together, trade them for 1 new ten, and record that regrouped ten in the tens column attic. In this chapter, you will master three-digit addition with regrouping from ones into tens, follow step-by-step vertical models, and learn how to keep multi-digit columns organized and error-free.

The Regrouping Principle in 3-Digit Numbers

Regrouping in three-digit numbers follows the exact same logic as in two-digit numbers, with the hundreds column simply standing by.

The Action in 348 + 237:

       Hundreds   Tens   Ones
                    1  <--- Attic Ten from Regrouping!
          3         4      8
       +  2         3      7
       ---------------------
          5         8      5

The Step-by-Step Walkthrough

Step 1: Add the ones column first: 8 ones + 7 ones = 15 ones. Ask: Is 15 ten or greater? Yes! 15 ones = 1 ten and 5 ones. Write 5 in the ones column below the line. Write a small 1 in the attic above the tens column.

Step 2: Add all the tens, including the attic ten: 1 (attic) + 4 tens + 3 tens = 8 tens. Write 8 in the tens column below the line.

Step 3: Add the hundreds: 3 hundreds + 2 hundreds = 5 hundreds. Write 5 in the hundreds column below the line. The grand sum is 585.

Base-Ten Block Model of the Trade

Let us observe the trade using base-ten sketches for 348 + 237.

Hundreds:   [ ] [ ] [ ]  +  [ ] [ ]                     = 5 Squares (500)
Tens:       | | | |      +  | | |                       = 7 Lines   ( 70)
Ones:       . . . . . . . .  +  . . . . . . .           = 15 Dots

Action in Ones:
Circle 10 dots:  [ . . . . . . . . . . ]  == Trade ==>  Draw 1 new line |
Leftover Dots:   5 dots remain.
New Tens Count:  7 original lines + 1 traded line = 8 lines (80).
Final Block Count: 5 Squares (500), 8 Lines (80), 5 Dots (5) = 585!

Nothing Lost, Nothing Created

The sketch confirms that 15 ones was physically transformed into 1 ten rod and 5 unit cubes. Adding that new ten rod to the 7 existing ten rods gave 8 ten rods, leaving 5 unit cubes.

Keeping Columns Neat with Grid Paper

When working with three-digit numbers, messy handwriting is the number one cause of arithmetic errors.

Using Columns to Prevent Drift:
| H | T | O |
|   | 1 |   |  <--- Attic
| 4 | 5 | 9 |
|+2 | 1 | 4 |
|---|---|---|
| 6 | 7 | 3 |

Guarding Against Column Drift

Notice how every digit stays strictly inside its own vertical lane. If the attic ten drifts into the hundreds lane, you would mistakenly add 1 to the hundreds instead of the tens. Keeping your columns straight guarantees accuracy.

Practice Exercises

Exercise 1: Solve 458 + 236 using the standard vertical algorithm. Show the regrouped ten in the attic and write out your steps.

Exercise 2: Solve 627 + 145. State the sum of the ones column before regrouping and explain how it was partitioned into tens and ones.

Exercise 3: A student added 349 + 215 and recorded 554, but forgot to add the attic ten to the tens column. What was the student's written sum, and what is the true correct sum?

Exercise 4: A school library received two boxes of books: the first box contained 248 books and the second box contained 136 books. How many books did the library receive in total?

Solutions and Explanations

Solution to Exercise 1: Align vertically: 458 over 236. Step 1: Add ones: 8 + 6 = 14. Write 4 in the ones place and write 1 in the attic above the tens place. Step 2: Add tens: 1 (attic) + 5 + 3 = 9. Write 9 in the tens place. Step 3: Add hundreds: 4 + 2 = 6. Write 6 in the hundreds place. The final sum is 694.

Solution to Exercise 2: To solve 627 + 145, add ones first: 7 + 5 = 12 ones. The 12 ones partition into 1 ten and 2 ones. The 2 ones stay in the ones column, and the 1 ten is regrouped to the tens column. Adding tens: 1 + 2 + 4 = 7 tens. Adding hundreds: 6 + 1 = 7 hundreds. The sum is 772.

Solution to Exercise 3: In 349 + 215, 9 + 5 = 14 ones (write 4, regroup 1 ten). If the student forgot the attic ten and simply added 4 + 1 = 5 tens, the student wrote 564? No, the student wrote 554. The true sum must include the regrouped ten: 1 + 4 + 1 = 6 tens. The true correct sum is 564.

Solution to Exercise 4: To find the total books, add 248 + 136. Add ones: 8 + 6 = 14 (write 4, regroup 1 ten). Add tens: 1 + 4 + 3 = 8 tens. Add hundreds: 2 + 1 = 3 hundreds. The library received 384 books in total.