Counting Up (The Shopkeeper's Method)

Calculate $63 - 47$ using the counting up strategy.

Start at the smaller number: Start at 47 = 47

Count up to a friendly number: $47 + 3 = 50$ = Added 3

Count up to the target: $50 + 13 = 63$ = Added 13

Add the jumps together: $3 + 13 = 16$ = $63 - 47 = 16$

Answer: $63 - 47 = 16$ (We counted up 3 to reach 50, then 13 more to reach 63)

Compensation (Round and Adjust)

Calculate $82 - 39$ using the compensation strategy.

Round the subtrahend to a friendly number: Round 39 up to 40 (added 1) = 39 becomes 40

Subtract the rounded number: $82 - 40 = 42$ = 42

Adjust for the rounding: We subtracted 1 too many, so add 1 back = $42 + 1 = 43$

Write the answer: $82 - 39 = 43$ = $82 - 39 = 43$

Answer: $82 - 39 = 43$ (Subtract 40, then add 1 back because 40 is 1 more than 39)

Breaking Apart (Place Value Method)

Calculate $75 - 32$ using the breaking apart strategy.

Break apart the number being subtracted: $32 = 30 + 2$ = Tens and ones separated

Subtract the tens first: $75 - 30 = 45$ = 45

Subtract the ones: $45 - 2 = 43$ = 43

Write the final answer: $75 - 32 = 43$ = $75 - 32 = 43$

Answer: $75 - 32 = 43$ (First subtract 30 to get 45, then subtract 2 to get 43)

Subtracting from 100

Calculate $100 - 63$ mentally.

Find the complement of the ones digit to 10: $10 - 3 = 7$ (ones digit) = 7

Find the complement of the tens digit to 9: $9 - 6 = 3$ (tens digit) = 3

Combine the digits: Tens digit: 3, Ones digit: 7 = 37

Verify: $63 + 37 = 100$ ✓ = $100 - 63 = 37$

Answer: $100 - 63 = 37$ (The shortcut: 9 minus tens digit, 10 minus ones digit)

Choosing the Best Strategy

Calculate $91 - 58$ using the most efficient method.

Analyze the numbers: 58 is close to 60, so compensation works well = Use compensation

Round 58 to 60: Round up by 2 = 58 becomes 60

Subtract the rounded number: $91 - 60 = 31$ = 31

Adjust: Add back the 2 we over-subtracted: $31 + 2 = 33$ = $91 - 58 = 33$

Answer: $91 - 58 = 33$ (Compensation was efficient because 58 is close to 60)

Mistake: Adjusting in the wrong direction after compensation

Why: When you round up the number you're subtracting, you subtract more than needed. Students sometimes subtract more instead of adding back.

Correct: If you round UP the subtrahend, you must ADD back. If you round DOWN, you must SUBTRACT more. Example: $82 - 39$ → $82 - 40 = 42$, then add 1: $42 + 1 = 43$

Mistake: Forgetting to complete all jumps when counting up

Why: Students sometimes stop counting before reaching the target number.

Correct: Always verify: start + all jumps = target. Example: $47 + 3 + 13 = 63$ ✓

Mistake: Mixing up which digit to subtract from 9 vs 10 when subtracting from 100

Why: The rule is: tens digit from 9, ones digit from 10. Students often reverse this.

Correct: For $100 - 63$: tens from 9 ($9 - 6 = 3$), ones from 10 ($10 - 3 = 7$) → 37

Making Change

Cashiers and customers both benefit from quick mental subtraction when making change.

If you pay with a 50 dollar bill for a 34 dollar purchase, count up: $34 + 6 = 40$, then $40 + 10 = 50$. Change is $6 + 10 = 16$ dollars.

Time Calculations

Figuring out how much time remains or has passed requires quick mental subtraction.

If a 90-minute movie started 35 minutes ago, count up: $35 + 5 = 40$, then $40 + 50 = 90$. Remaining time: $5 + 50 = 55$ minutes.

Sports Score Differences

Sports fans constantly calculate point differences mentally.

If Team A has 73 points and Team B has 58 points, the difference is $73 - 58$. Using compensation: $73 - 60 = 13$, then $13 + 2 = 15$ points.

Q: When should I use counting up vs compensation?

A: Use **counting up** when the numbers are close together (like $63 - 58$). Use **compensation** when the number you're subtracting ends in 8 or 9 (like $82 - 39$), making it easy to round to a ten.

Q: What if I need to borrow/regroup? Can I still do it mentally?

A: Yes! Compensation handles regrouping naturally. For $52 - 19$: round 19 to 20, calculate $52 - 20 = 32$, then add back 1 to get 33. No regrouping needed!

Q: How do I get faster at mental subtraction?

A: Practice with specific strategies first. Start with problems that clearly fit one strategy. As you get comfortable, you'll automatically choose the best method for each problem.