2-Digit Subtraction with Regrouping (Borrowing) - Second Grade Mathematics
In subtraction, you will often encounter a situation where the top digit in the ones column is smaller than the bottom digit, such as trying to subtract 7 ones from 2 ones in the problem 52 - 27. You cannot simply flip the numbers upside down and subtract 7 - 2, because that would completely change the problem! Instead, you must reach into the tens column, unbundle 1 ten into 10 loose ones, and combine them with your existing ones. This process is called regrouping, decomposing, or borrowing. In this chapter, you will master the regrouping algorithm step by step and discover how base-ten models make borrowing clear and natural.
The Golden Rule of Subtraction: More on the Floor? Go Next Door!
A classic mathematical rhyme helps you remember what to do when looking at the ones column:
The Subtraction Rhyme:
More on top? No need to stop! (Just subtract!)
More on the floor? Go next door, and get 10 more!
Numbers the same? Zero's the game!
Never Flip Digits Upside Down
Look at 63 - 28. In the ones column, you have 3 on top and 8 on the floor (bottom). Many learners are tempted to say: "8 - 3 = 5!" STOP! That is a major mathematical error. The problem asks you to take 8 away from 3. If you have 3 cookies, you cannot give a friend 8 cookies without opening a fresh box of 10. You must never flip the bottom digit to the top.
The Step-by-Step Borrowing Algorithm
Let us walk through the exact steps to solve 63 - 28.
Problem: 63 - 28
Step 1: Look at the ones column: 3 - 8.
There are not enough ones. We must regroup!
Step 2: "Go next door" to the tens column.
The top number 63 has 6 tens.
Take 1 ten away: cross out the 6 and write 5.
Step 3: Trade that 1 ten for 10 ones.
Add the 10 ones to the 3 ones you already had:
10 + 3 = 13 ones!
Cross out the 3 and write 13 above it.
5 13
6 3
- 2 8
---------
Step 4: Subtract the ones column now!
13 ones - 8 ones = 5 ones.
Write 5 in the ones place below the line.
Step 5: Subtract the tens column.
5 tens - 2 tens = 3 tens.
Write 3 in the tens place below the line.
5 13
6 3
- 2 8
---------
3 5
Final Difference: 35
Why 63 Is Still 63
Notice what happened: 6 tens and 3 ones was renamed as 5 tens and 13 ones. Is 50 + 13 still equal to 63? Yes! Exactly equal! You did not change the total value of your starting number; you simply unbundled one package of ten into ten loose units so you could subtract.
Base-Ten Model of Regrouping
Let us observe this exact unbundling using a base-ten sketch for 63 - 28.
Step 1: Draw the Minuend (63):
| | | | | | . . .
(6 tens) (3 ones)
Step 2: Unbundle 1 ten rod:
Cross out 1 ten rod and draw 10 dots in the ones section:
| | | | | [X] . . . + . . . . . . . . . .
(5 tens left) (Now 13 ones!)
Step 3: Cross out 28 (2 tens and 8 ones):
Cross out 2 ten lines: | | | [X][X]
Cross out 8 ones dots: . . . . . [X][X][X][X][X][X][X][X]
Step 4: Count what remains untouched:
Tens remaining: 3 lines = 30
Ones remaining: 5 dots = 5
Result: 30 + 5 = 35!
Seeing the Trade with Your Eyes
The drawing proves that regrouping is simply physical trading. The crossed-out ten rod became the 10 new dots. Once 13 dots were on the paper, taking away 8 dots was completely straightforward.
Practice Exercises
Exercise 1: Solve 74 - 38 using the standard vertical algorithm. Show the crossed-out digits and new numbers above the columns.
Exercise 2: A student solved 51 - 27 and wrote down an answer of 36. Look at the student's work: the student subtracted 7 - 1 = 6 in the ones place, and 5 - 2 = 3 in the tens place. Explain the student's error and provide the correct calculation.
Exercise 3: Solve 82 - 45. Explain how 82 was renamed into tens and ones before subtracting.
Exercise 4: A school orchard had 90 apples on a tree. A sudden windstorm blew 34 apples down to the ground. How many apples remained on the tree?
Solutions and Explanations
Solution to Exercise 1: In the ones column for 74 - 38, you cannot take 8 from 4. Go to the tens column: cross out 7 tens and write 6 tens. Give 10 ones to the 4 ones: cross out 4 and write 14. Now subtract ones: 14 - 8 = 6. Subtract tens: 6 - 3 = 3. The final difference is 36.
Solution to Exercise 2: The student made the classic error of flipping the digits in the ones column. Instead of taking 7 from 1, the student subtracted 1 from 7 to get 6. You cannot flip digits in subtraction! The correct way is to regroup: borrow 1 ten from 5 (leaving 4 tens) to make 11 ones. Then 11 - 7 = 4 ones, and 4 tens - 2 tens = 2 tens. The correct answer is 24.
Solution to Exercise 3: In 82 - 45, you cannot take 5 ones from 2 ones. Rename 82 by taking 1 ten from 8 tens (leaving 7 tens) and adding 10 ones to 2 ones (making 12 ones). So 82 is renamed as 7 tens and 12 ones. Then subtract: 12 - 5 = 7 ones, and 7 - 4 = 3 tens. The difference is 37.
Solution to Exercise 4: To find the remaining apples, calculate 90 - 34. In the ones place, you cannot take 4 from 0. Regroup 1 ten from 9 tens (leaving 8 tens) to make 10 ones. Subtract ones: 10 - 4 = 6. Subtract tens: 8 - 3 = 5. There were 56 apples remaining on the tree.