Three-digit numbers are big, but you already know the secret to working with them: place value. Today you will add and subtract numbers up to 1,000, use drawings of base-ten blocks, regroup tens and hundreds, estimate answers, and explain your thinking in words.
1. Hundreds, tens and ones
Every three-digit number is built from hundreds, tens and ones. A flat shows 100, a rod shows 10, and a cube shows 1. The number 347 has 3 flats, 4 rods and 7 cubes:
\(347 = 300 + 40 + 7\)
Regrouping means trading blocks of equal value: 10 ones make 1 ten, 10 tens make 1 hundred, and the other way around. The total value does not change when you regroup.
We keep each place in its own column, so we always add or subtract ones with ones, tens with tens, and hundreds with hundreds.
2. Adding with no regrouping
When every column adds up to 9 or less, you just add column by column.
Find \(234 + 152\).
Ones: \(4 + 2 = 6\). Tens: \(3 + 5 = 8\). Hundreds: \(2 + 1 = 3\).
So \(234 + 152 = 386\).
3. Adding with regrouping
When a column adds up to 10 or more, trade 10 of that unit for 1 of the next bigger unit. Look at the blocks for \(258 + 167\):
- Add the ones. If you have 10 or more, regroup 10 ones as 1 ten.
- Add the tens, including any ten you just made. If you have 10 or more, regroup 10 tens as 1 hundred.
- Add the hundreds, including any hundred you just made.
- Write the sum and check that it makes sense.
Find \(258 + 167\).
Ones: \(8 + 7 = 15\) ones, which is 1 ten and 5 ones.
Tens: \(5 + 6 + 1 = 12\) tens, which is 1 hundred and 2 tens.
Hundreds: \(2 + 1 + 1 = 4\) hundreds.
So \(258 + 167 = 425\).
4. Subtracting with no regrouping
If every digit on top is at least as big as the digit below it, subtract column by column.
\(586 - 243\): ones \(6 - 3 = 3\), tens \(8 - 4 = 4\), hundreds \(5 - 2 = 3\). So \(586 - 243 = 343\).
5. Subtracting with regrouping
If a digit on top is smaller than the digit below it, you need to trade. Take 1 from the next bigger place and turn it into 10 of the smaller place.
Find \(432 - 175\).
Ones: 2 is smaller than 5. Trade 1 ten for 10 ones: now \(12 - 5 = 7\) ones. The tens place has 2 tens left.
Tens: 2 is smaller than 7. Trade 1 hundred for 10 tens: now \(12 - 7 = 5\) tens. The hundreds place has 3 hundreds left.
Hundreds: \(3 - 1 = 2\).
So \(432 - 175 = 257\).
You can also think of subtraction as a missing part. The whole is 432 and one part is 175:
When you trade a ten for 10 ones, the tens digit goes down by 1. Forgetting this is the most common mistake. In \(432 - 175\), do not write \(3 - 7\) in the tens place; write \(2 - 7\) first, and then trade again.
Zeros need extra care. To find \(500 - 286\), there are no tens to trade, so trade 1 hundred for 10 tens first, then 1 of those tens for 10 ones. You get 4 hundreds, 9 tens and 10 ones: \(10 - 6 = 4\), \(9 - 8 = 1\), \(4 - 2 = 2\). So \(500 - 286 = 214\).
6. Counting up on a number line
Another way to subtract is to start at the smaller number and jump up to the bigger one. Add up all the jumps. To find \(432 - 175\):
The jumps are \(25 + 200 + 32 = 257\), the same answer as before. Two different strategies that agree is a great way to be sure.
7. Estimating
An estimate is a smart guess that is close to the exact answer. Think of the nearest hundred for each number, then add or subtract.
Estimate \(394 + 208\).
\(394\) is close to \(400\) and \(208\) is close to \(200\). So the sum is about \(400 + 200 = 600\). The exact sum is 602, which is very close.
If your exact answer is far from your estimate, go back and check your work.
8. Properties and checking
- Order does not matter: \(245 + 163 = 163 + 245\).
- Grouping does not matter: \((275 + 25) + 138 = 275 + (25 + 138)\). Look for numbers that make a friendly hundred.
- Adding 0 changes nothing: \(318 + 0 = 318\).
- Addition undoes subtraction: if \(432 - 175 = 257\), then \(257 + 175 = 432\).
Subtraction does not work in any order: \(9 - 4\) is not the same as \(4 - 9\). Use the addition check on every subtraction problem.
On my planet we always ask, “Can I make a friendly hundred?” To add \(275 + 138 + 25\), I add \(275 + 25 = 300\) first, then \(300 + 138 = 438\). Easy!
9. Explaining your strategy in writing
Good mathematicians say what they did and why. Use these sentence starters: “First, I…”, “Then, I regrouped because…”, “I checked by…”, “My answer makes sense because…”.
Example: “To find \(700 - 389\), I took away 400 and got 300. But I took away 11 too many, so I added 11 back. The answer is 311. I checked: \(311 + 389 = 700\).”
Key takeaways
- Line up the places: add or subtract ones, then tens, then hundreds.
- Regroup 10 ones as 1 ten and 10 tens as 1 hundred when adding.
- When subtracting, trade 1 ten for 10 ones or 1 hundred for 10 tens, and lower the digit you traded from.
- Estimate with friendly hundreds to see if your answer is reasonable.
- Check subtraction by adding, and use friendly hundreds to add smartly.
- Explain what you did in words, step by step.
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