Checking Subtraction with Addition - Second Grade Mathematics
One of the greatest feelings in mathematics is knowing with 100 percent certainty that your answer is correct before anyone else looks at your paper. Because addition and subtraction are inverse operations that undo each other, you can use addition as a built-in lie detector to test every subtraction problem you ever solve. If your subtraction is accurate, adding your answer back to the amount you took away will reconstruct your starting number with absolute perfection. In this chapter, you will master the reverse-check algorithm, explore why the check works through part-part-whole relationships, and learn how to catch and fix your own mistakes.
The Inverse Check Mechanism
Subtraction breaks a starting whole into two parts. Addition puts those two parts back together.
The Subtraction Action:
Whole (Minuend) - Part (Subtrahend) = Part (Difference)
63 - 28 = 35
The Addition Check:
Part (Difference) + Part (Subtrahend) = Whole (Minuend)
35 + 28 = 63!
Matches the top starting number? YES! The answer is verified!
The Three Check Steps
Step 1: Take your calculated difference (your answer). Step 2: Add the subtrahend (the middle number you took away). Step 3: Compare your sum to the minuend (the top starting number). If they match, your subtraction is guaranteed to be correct. If they do not match, an error occurred and you must recheck your work.
Setting Up the Side-by-Side Check
Mathematicians frequently write the subtraction problem on the left and the addition check on the right.
Side-by-Side Verification:
Subtraction Problem: Addition Check:
4 12 1 <--- Check Regrouping!
5 2 (Minuend) 3 7 (Difference)
- 1 5 (Subtrahend) + 1 5 (Subtrahend)
--------- ---------
3 7 (Difference) 5 2 (Matches Minuend!)
How the Check Catches Regrouping Errors
Suppose a student forgot to reduce the 5 tens to 4 tens during regrouping and wrote 47 instead of 37. When the student runs the addition check: 47 + 15 = 62. The student looks at the top number of the subtraction problem: it was 52, NOT 62! The check immediately catches the error. The student instantly knows something went wrong in the tens place and can correct it before turning in their work.
Understanding the Part-Part-Whole Logic
The addition check is not an arbitrary ritual; it reflects the deep structural logic of our number system.
Visualizing with a Bar Model:
+-------------------------------------------------------+
| WHOLE: 52 |
+---------------------------+---------------------------+
| PART: 15 | PART: 37 |
+---------------------------+---------------------------+
52 - 15 = 37 <==== Undoes ====> 15 + 37 = 52
Whole Equals Part Plus Part
In any subtraction situation, the minuend is the whole bar, and the subtrahend and difference are the two parts that fill it. When you remove a part of the bar, the remaining piece is the difference. Pushing the two pieces back together must recreate the original whole bar.
What to Do When the Check Fails
When your addition check does not equal the original minuend, follow this troubleshooting checklist:
Troubleshooting Checklist:
1. Did I add the check correctly? (Re-add your check first!)
2. In the subtraction, did I flip digits in the ones place? (e.g., 2 - 7 called 5)
3. Did I forget to slash the tens place and reduce it by 1?
4. Did I forget to add 10 to the ones place?
Celebrate Finding the Error
Never feel discouraged when a check fails! Catching an error with an addition check is a moment of triumph: it means your mathematical safety net worked perfectly, allowing you to fix a mistake before it caused any harm.
Practice Exercises
Exercise 1: Solve 84 - 36 vertically. Then set up and calculate the addition check to verify your answer.
Exercise 2: A student calculated 71 - 25 = 56. Run the addition check on the student's answer. Does the check match the original minuend? What was the student's mistake?
Exercise 3: Write the addition equation you would use to check each of the following subtraction problems: a) 65 - 19 = 46, and b) 90 - 43 = 47.
Exercise 4: Explain in your own words why adding the difference to the subtrahend must always equal the minuend when a subtraction problem is solved correctly.
Solutions and Explanations
Solution to Exercise 1: For 84 - 36, regroup 1 ten from 8 (leaving 7 tens) to make 14 ones. Subtract ones: 14 - 8 = 8? No, 14 - 6 = 8. Subtract tens: 7 - 3 = 4. Difference is 48. Now run the check: Add difference (48) + subtrahend (36). 8 + 6 = 14 (write 4, regroup 1 ten). Tens: 1 + 4 + 3 = 8. The check sum is 84, which matches the original minuend perfectly.
Solution to Exercise 2: To check 71 - 25 = 56, add the difference and subtrahend: 56 + 25. Ones: 6 + 5 = 11 (regroup 1). Tens: 1 + 5 + 2 = 8 tens. Check sum is 81. The original minuend was 71, not 81! The check failed. The student forgot to reduce 7 tens to 6 tens when borrowing. The correct difference is 46 (check: 46 + 25 = 71).
Solution to Exercise 3: For problem a, the addition check equation is 46 + 19 = 65. For problem b, the addition check equation is 47 + 43 = 90.
Solution to Exercise 4: In subtraction, you start with a complete whole quantity (the minuend) and separate it into two parts: the part you take away (the subtrahend) and the part that remains (the difference). Because no items were destroyed, combining the part you took away with the part that was left over must restore the exact original whole amount.