B. Number · Operations: Adding and Subtracting to 1000, Multiplying and DividingSubtracting within 1000: From Blocks to the Column Method
B2.4 B2.5About 40 minutes
Learning goals- I can subtract 3-digit numbers with base-ten blocks or drawings.
- I can trade 1 ten for 10 ones, or 1 hundred for 10 tens, when I need to.
- I can use the column method (algorithm) to subtract, even with zeros.
- I can check a subtraction answer by adding.
Words to know
difference
The answer when you subtract.
regroup (trade)
In subtraction, break 1 ten into 10 ones, or 1 hundred into 10 tens, so you have enough to take away.
algorithm
A set of steps that always works, done in the same order every time.
count up
Start at the smaller number and jump up to the bigger number. The jumps add up to the difference.
check
Make sure an answer is right, for example by adding the answer to the number you took away.
Remember from Grade 2
In Grade 2 you subtracted numbers like 71 − 35. When there were not enough ones, you traded 1 ten for 10 ones. In Grade 3 we subtract numbers up to 1000. Sometimes we also need to trade 1 hundred for 10 tens.
Let's start with a story
A school has 352 library books. The Grade 3 class borrows 127 of them. How many are still on the shelves?
We need 352 − 127. Let's build 352 with blocks.
We need to take away 7 ones, but there are only 2 ones. Not enough!
Trade 1 ten for 10 ones. Now we have 3 hundreds, 4 tens and 12 ones. It is still 352, just shown a different way.
Now take away 127 (1 hundred, 2 tens, 7 ones):
- Ones: 12 − 7 = 5
- Tens: 4 − 2 = 2
- Hundreds: 3 − 1 = 2
So 352 − 127 = 225. There are 225 books on the shelves.
The column method (an algorithm)
The column method writes the same steps in a short way.
- Line up the digits: ones under ones, tens under tens, hundreds under hundreds. The bigger number goes on top.
- Start with the ones. If the top digit is too small, trade 1 ten from the tens column: cross out the tens digit and write one less above it. The ones get 10 more.
- Subtract the tens. If the top digit is too small, trade 1 hundred for 10 tens.
- Subtract the hundreds.
- Check by adding.
Worked exampleExample 1: One trade
- Ones: 2 − 7? Not enough. Trade 1 ten: the 5 tens become 4 tens, and the 2 ones become 12 ones.
- Ones: 12 − 7 = 5.
- Tens: 4 − 2 = 2.
- Hundreds: 3 − 1 = 2.
- Check by adding: 225 + 127 = 352. ✓
Worked exampleExample 2: Two trades
- Ones: 4 − 8? Not enough. Trade 1 ten: tens 3 → 2, ones 4 → 14. Then 14 − 8 = 6.
- Tens: 2 − 7? Not enough. Trade 1 hundred: hundreds 5 → 4, tens 2 → 12. Then 12 − 7 = 5.
- Hundreds: 4 − 2 = 2.
- Check: 256 + 278 = 534. ✓
Column method for 534 − 278 with two trades.Subtracting from a number with zeros
What about 400 − 163? There are no tens to trade for ones! So we work in two steps:
- Trade 1 hundred for 10 tens. Now: 3 hundreds, 10 tens, 0 ones.
- Trade 1 ten for 10 ones. Now: 3 hundreds, 9 tens, 10 ones.
It is still 400: 300 + 90 + 10 = 400.
Worked exampleExample 3: Subtract across zeros
- Trade so that 400 = 3 hundreds, 9 tens, 10 ones.
- Ones: 10 − 3 = 7.
- Tens: 9 − 6 = 3.
- Hundreds: 3 − 1 = 2.
- Check: 237 + 163 = 400. ✓
Column method for 400 − 163. The 4 hundreds became 3 hundreds, 9 tens and 10 ones.Another way: count up
The column method always works, but sometimes counting up is faster, especially with zeros. For 400 − 163, start at 163 and count up to 400:
- 163 + 7 = 170
- 170 + 30 = 200
- 200 + 200 = 400
Jumps: 7 + 30 + 200 = 237. Same answer!
A very common mistake
Some people take the smaller digit from the bigger digit in every column, even when the bigger digit is on the bottom. For 352 − 127 they write 2 − 7 as "7 − 2 = 5" and get 235. That is wrong!
Always subtract bottom from top. If the top digit is too small, trade. Then check by adding: 235 + 127 = 362, not 352, so 235 cannot be right.
Just for funWhy do subtraction questions with zeros, like 400 − 163, always look nervous?
They know they will have to trade twice before they can start!
Hands-on activityShop and make change with play money
You need: play money or paper bills you make (write $100, $10 and $1 on paper rectangles), and a few toys or books for a pretend store.
- Put a price tag on each item, like $128 or $264.
- The shopper starts with $500: five $100 bills.
- Buy one item. Pay for it! You may need to trade a $100 bill for ten $10 bills, and a $10 bill for ten $1 bills. That is exactly what the column method does.
- Count what is left, and write the subtraction: 500 − 128 = 372.
- Check by adding: 372 + 128 = 500.
- Buy something else with what is left. Keep going until you can't afford anything.
Math talk: "Why did you need to trade before you could pay $128?"
Your turn
333. No trades: 6 − 3 = 3, 8 − 5 = 3, 5 − 2 = 3.
334. Trade 1 ten: 12 − 8 = 4. Tens: 6 − 3 = 3. Hundreds: 4 − 1 = 3.
264. Ones: 5 − 1 = 4. Tens: 4 − 8? Trade 1 hundred: 14 − 8 = 6. Hundreds: 5 − 3 = 2.
264. Ones: trade 1 ten, 13 − 9 = 4. Tens: 1 − 5? Trade 1 hundred, 11 − 5 = 6. Hundreds: 6 − 4 = 2. Check: 264 + 459 = 723.
252. Count up: 248 + 2 = 250, 250 + 250 = 500. 2 + 250 = 252. (Or use the column method: 500 = 4 hundreds, 9 tens, 10 ones.)
6Check by adding: is 815 − 367 = 448 right?
7Nina says 362 − 145 = 223. What did she do wrong? What is the right answer?
She did 5 − 2 in the ones instead of trading. Trade 1 ten: 12 − 5 = 7. Tens: 5 − 4 = 1. Hundreds: 3 − 1 = 2. The answer is 217. Check: 217 + 145 = 362.
More practice: warm-up
21 ten is the same as how many ones?
31 hundred is the same as how many tens?
922. 4 − 2 = 2, 7 − 5 = 2, and the 9 hundreds stay.
More practice: on your own
315. Trade 1 ten: 11 − 6 = 5. Tens: 3 − 2 = 1. Hundreds: 8 − 5 = 3. Check: 315 + 526 = 841.
446. 903 = 8 hundreds, 9 tens, 13 ones. 13 − 7 = 6, 9 − 5 = 4, 8 − 4 = 4. Check: 446 + 457 = 903.
625. Count up: 375 + 25 = 400, 400 + 600 = 1000. 25 + 600 = 625.
4A sugar bush made 612 bottles of maple syrup. It sold 285 bottles. How many are left?
327 bottles. Ones: trade, 12 − 5 = 7. Tens: 0 − 8? Trade a hundred, 10 − 8 = 2. Hundreds: 5 − 2 = 3. Check: 327 + 285 = 612.
5Explain why "bigger digit minus smaller digit" in each column gives the wrong answer for 430 − 218. What is the right answer?
That way gives 8 − 0 = 8, 3 − 1 = 2, 4 − 2 = 2, which is 228. But we must take 8 from 0, so we trade 1 ten: 10 − 8 = 2, then 2 − 1 = 1, then 4 − 2 = 2. The right answer is 212. Check: 212 + 218 = 430.
Challenge
1Find the missing digits: 7□4 − 36□ = 358
724 − 366 = 358. Check by adding: 358 + 366 = 724. So the first missing digit is 2 and the second is 6.
2Would you use the column method or mental math for 600 − 299? Solve it.
Mental math is faster: 600 − 300 = 300, then add back 1 because 299 is 1 less than 300. 301.
3Write a story problem for 520 − 275 and solve it.
Example: "A rink had 520 skaters. 275 went home. How many are still skating?" Answer: 245. Check: 245 + 275 = 520.
Quiz
3. How can you check that 640 − 275 = 365?
4. In 503 − 248, there are no tens to trade. What do you do?
6. Jaden says 421 − 167 = 346. What is true?
Big idea: Grade 3 students subtract within 1000 (B2.5) and learn the column algorithm by connecting it to base-ten blocks and other strategies (B2.4). The key idea is that trading (regrouping) does not change the number; it only shows it a different way (352 = 3 hundreds, 4 tens, 12 ones).
Common difficulties:
- Smaller-from-bigger in every column (352 − 127 = 235). This is the most common error. Ask: "Which number is on top? Do you have enough?"
- Zeros (400 − 163). Children often get stuck or write 0 − 3 = 3. Show 400 with play money: four $100 bills, trade one for ten $10 bills, then one $10 for ten $1 bills.
- Forgetting to cross out after trading, then using the old digit. Encourage neat crossing out.
- Not checking. Make "check by adding" a habit; it catches almost every mistake.
How to help: Let your child choose: column method or counting up. Both are correct. Counting up is often easier for questions like 1000 − 375 or 500 − 248.