3-Digit Subtraction: Decomposing Tens - Second Grade Mathematics
When subtracting three-digit numbers, you will frequently encounter a situation where the top ones digit is smaller than the bottom ones digit, such as trying to subtract 8 ones from 3 ones in 543 - 218. Because you cannot subtract a larger number from a smaller number in whole-number arithmetic, you must unbundle 1 ten from the tens column, convert it into 10 loose ones, and add them to your ones column. This is three-digit subtraction with regrouping from the tens place. In this chapter, you will master decomposing tens in three-digit numbers, follow clear vertical models, and verify every answer with addition.
The Need to Decompose Tens
Look closely at the ones column in 543 - 218:
Problem: 543 - 218
Hundreds Tens Ones
5 4 3
- 2 1 8
---------------------
The Inspection Step
Before touching your pencil, inspect the ones column: 3 on top, 8 on the bottom. Can you take 8 from 3? No! "More on the floor? Go next door and get 10 more!" You must go to the tens column of the top number (543), borrow 1 ten from the 4 tens, and give 10 ones to the 3 ones.
The Step-by-Step Algorithm
Let us walk through the complete calculation for 543 - 218.
3 13 <--- Decomposed Tens and Ones!
5 4 3
- 2 1 8
-----------------
3 2 5
Step 1: Regroup from Tens to Ones.
Take 1 ten from 4 tens: cross out 4 and write 3.
Add 10 ones to 3 ones: 10 + 3 = 13.
Cross out 3 and write 13.
Step 2: Subtract the Ones Column.
13 ones - 8 ones = 5 ones.
Write 5 in the ones place.
Step 3: Subtract the Tens Column.
Remember to use the new tens digit: 3 tens!
3 tens - 1 ten = 2 tens.
Write 2 in the tens place.
Step 4: Subtract the Hundreds Column.
5 hundreds - 2 hundreds = 3 hundreds.
Write 3 in the hundreds place.
Final Difference: 325!
Why 543 Has Not Changed Value
Notice that 5 hundreds, 4 tens, and 3 ones was renamed as 5 hundreds, 3 tens, and 13 ones. 500 + 30 + 13 = 543. The starting quantity is completely preserved; we simply opened a package of ten so we had enough ones to subtract 8.
Base-Ten Sketch of Decomposing a Ten
Visualizing the trade with base-ten symbols makes the algorithm concrete.
Draw 543: [ ] [ ] [ ] [ ] [ ] | | | | . . .
Action: Unbundle 1 Ten Line |:
Cross out 1 line | and draw 10 dots:
[ ] [ ] [ ] [ ] [ ] | | | [X] . . . + . . . . . . . . . .
(3 lines) (13 dots)
Subtract 218 (2 hundreds, 1 ten, 8 ones):
Cross out 8 dots: 13 - 8 = 5 dots remain.
Cross out 1 line: 3 - 1 = 2 lines remain.
Cross out 2 flats: 5 - 2 = 3 flats remain.
Result: 3 flats (300) + 2 lines (20) + 5 dots (5) = 325!
Practice Exercises
Exercise 1: Solve 652 - 327 vertically. Show the crossed-out digits and new values above the tens and ones columns.
Exercise 2: Solve 874 - 439. State how the top number was renamed before subtracting the ones column.
Exercise 3: A student solved 761 - 245 and wrote 524, subtracting 5 - 1 = 4 in the ones place. Explain the student's error and provide the correct difference.
Exercise 4: Set up and solve the addition check for your answer to Exercise 1.
Solutions and Explanations
Solution to Exercise 1: In 652 - 327, you cannot take 7 from 2. Regroup 1 ten from 5 tens (leaving 4 tens) to make 12 ones. Subtract ones: 12 - 7 = 5. Subtract tens: 4 - 2 = 2. Subtract hundreds: 6 - 3 = 3. The difference is 325.
Solution to Exercise 2: In 874 - 439, 4 ones cannot subtract 9 ones. Rename 874 by unbundling 1 ten from 7 tens (leaving 6 tens) to make 14 ones. The number 874 is renamed as 8 hundreds, 6 tens, and 14 ones. Subtracting gives: 14 - 9 = 5 ones, 6 - 3 = 3 tens, and 8 - 4 = 4 hundreds. Difference is 435.
Solution to Exercise 3: The student flipped the digits in the ones column, subtracting the smaller top number (1) from the larger bottom number (5). In subtraction, you must take from the top! The student needed to borrow 1 ten from 6 tens (leaving 5 tens) to make 11 ones. Then 11 - 5 = 6 ones, 5 - 4 = 1 ten, and 7 - 2 = 5 hundreds. The correct answer is 516.
Solution to Exercise 4: To check 652 - 327 = 325, add difference (325) + subtrahend (327). Ones: 5 + 7 = 12 (write 2, regroup 1 ten). Tens: 1 (attic) + 2 + 2 = 5 tens. Hundreds: 3 + 3 = 6 hundreds. The check sum is 652, which matches the original minuend.