Addition Within 1,000 with Regrouping - Third Grade Mathematics

Adding three-digit numbers is like assembling a building floor by floor, where each place value column has its own room. When a room gets too crowded with 10 or more units, we must regroup those units into one unit of the next larger room to the left. In this chapter, you will master vertical addition within 1,000 with regrouping in the ones place, the tens place, and both places at once.

Understanding Regrouping with Base-Ten Models

Regrouping (sometimes called carrying or trading) happens whenever an addition in any place value produces 10 or more.

Recall the golden rule of place value:
- 10 Ones regroup into 1 Ten.

- 10 Tens regroup into 1 Hundred.

- 10 Hundreds regroup into 1 Thousand.


When you have 14 ones:
[.] [.] [.] [.] [.] [.] [.] [.] [.] [.] [.] [.] [.] [.]
<------------- 10 ones --------------> <--- 4 ones --->
              ||
       Trade for 1 Ten rod                Keep in ones
       [||||||||||]                      [.] [.] [.] [.]

You write down the 4 in the ones column and carry the 1 ten into the tens column above the other tens.

The Standard Addition Algorithm Step-by-Step

To add multi-digit numbers accurately, follow these four reliable steps:
1. Line up the numbers vertically by place value: ones over ones, tens over tens, hundreds over hundreds.

2. Add the ones column first. If the sum is 10 or greater, write the ones digit below and regroup the 1 ten above the tens column.

3. Add the tens column, including any regrouped ten. If the tens sum is 10 or greater (10 tens = 100), write the tens digit below and regroup 1 hundred above the hundreds column.

4. Add the hundreds column, including any regrouped hundred.


Worked Examples

Let us look closely at three different cases of regrouping.

Example 1: Regrouping Ones Only

Add 347 + 218.

    1           (regrouped 1 ten from 7 + 8 = 15)
    3 4 7
  + 2 1 8
  -------
    5 6 5

Step-by-step breakdown:
- Ones: 7 + 8 = 15 ones. 15 ones is 1 ten and 5 ones. Write 5 in the ones column, place 1 above the tens column.

- Tens: 1 (regrouped) + 4 + 1 = 6 tens. Write 6 in the tens column.

- Hundreds: 3 + 2 = 5 hundreds. Write 5 in the hundreds column.
Sum: 565.

Example 2: Regrouping Tens Only

Add 483 + 274.

  1             (regrouped 1 hundred from 8 + 7 = 15 tens)
  4 8 3
+ 2 7 4
-------
  7 5 7

Step-by-step breakdown:
- Ones: 3 + 4 = 7 ones. No regrouping needed because 7 is less than 10.

- Tens: 8 + 7 = 15 tens. 15 tens is 1 hundred and 5 tens. Write 5 in the tens column, place 1 above the hundreds column.

- Hundreds: 1 (regrouped) + 4 + 2 = 7 hundreds. Write 7 in the hundreds column.
Sum: 757.

Example 3: Regrouping Twice (Ones and Tens)

Add 568 + 275.

  1 1           (regrouped 1 hundred and 1 ten)
  5 6 8
+ 2 7 5
-------
  8 4 3

Step-by-step breakdown:
- Ones: 8 + 5 = 13 ones. Write 3, regroup 1 ten to the tens place.

- Tens: 1 + 6 + 7 = 14 tens. Write 4, regroup 1 hundred to the hundreds place.

- Hundreds: 1 + 5 + 2 = 8 hundreds. Write 8 in the hundreds place.
Sum: 843.

Checking Your Work with Estimation and Inverse Operations

A mathematician always checks their work to make sure their answer is reasonable.

Checking by Rounding

In Example 3, we added 568 + 275 = 843.
- Round 568 to the nearest hundred: 600.

- Round 275 to the nearest hundred: 300.

- Estimated sum: 600 + 300 = 900.
Since 843 is reasonably close to 900, our answer makes practical sense!

Checking by Subtraction

Because addition and subtraction are inverse operations, you can check your sum by subtracting one of the addends: 843 - 275 should equal 568. When you calculate it, it matches perfectly!

Chapter Practice Exercises

  1. Line up vertically and solve: a. 256 + 138 b. 419 + 325 c. 374 + 281 d. 586 + 179
  2. Solve: a. 648 + 257 b. 739 + 185 c. 496 + 348 d. 287 + 496
  3. An elementary school collected canned goods for a food drive. Third grade collected 385 cans and fourth grade collected 449 cans. How many cans did the two grades collect altogether?
  4. Identify the error in this problem: A student calculated 378 + 245 and wrote 513 because: 8 + 5 = 13 (wrote 3, forgot to regroup 1) 7 + 4 = 11 (wrote 1, forgot to regroup 1) 3 + 2 = 5 (wrote 5) What is the correct answer?

Solutions and Step-by-Step Answers

  1. Practice problems: a. 256 + 138: Ones: 6 + 8 = 14 (write 4, regroup 1). Tens: 1 + 5 + 3 = 9. Hundreds: 2 + 1 = 3. Sum = 394. b. 419 + 325: Ones: 9 + 5 = 14 (write 4, regroup 1). Tens: 1 + 1 + 2 = 4. Hundreds: 4 + 3 = 7. Sum = 744. c. 374 + 281: Ones: 4 + 1 = 5. Tens: 7 + 8 = 15 (write 5, regroup 1). Hundreds: 1 + 3 + 2 = 6. Sum = 655. d. 586 + 179: Ones: 6 + 9 = 15 (write 5, regroup 1). Tens: 1 + 8 + 7 = 16 (write 6, regroup 1). Hundreds: 1 + 5 + 1 = 7. Sum = 765.
  2. Two-digit and three-digit regrouping: a. 648 + 257: 8 + 7 = 15; 1 + 4 + 5 = 10 (write 0, regroup 1); 1 + 6 + 2 = 9. Sum = 905. b. 739 + 185: 9 + 5 = 14; 1 + 3 + 8 = 12 (write 2, regroup 1); 1 + 7 + 1 = 9. Sum = 924. c. 496 + 348: 6 + 8 = 14; 1 + 9 + 4 = 14; 1 + 4 + 3 = 8. Sum = 844. d. 287 + 496: 7 + 6 = 13; 1 + 8 + 9 = 18; 1 + 2 + 4 = 7. Sum = 783.
  3. Cans collected: 385 + 449. Ones: 5 + 9 = 14 (write 4, regroup 1 ten). Tens: 1 + 8 + 4 = 13 (write 3, regroup 1 hundred). Hundreds: 1 + 3 + 4 = 8. Total = 834 cans.
  4. The student forgot to add the regrouped 1 into both the tens and hundreds columns! Correct calculation: Ones: 8 + 5 = 13 (write 3, carry 1 ten). Tens: 1 + 7 + 4 = 12 (write 2, carry 1 hundred). Hundreds: 1 + 3 + 2 = 6. The correct sum is 623.