Grade by Grade
B. Number · Operations: Adding and Subtracting to 1000, Multiplying and Dividing

Adding to 1000: From Blocks to the Column Method

B2.4 B2.5About 40 minutes
Learning goals
  • I can add 3-digit numbers with base-ten blocks or drawings.
  • I can add by splitting numbers into hundreds, tens and ones.
  • I can use the column method (algorithm) to add.
  • I can explain how trading blocks matches carrying in the column method.

Words to know

algorithm
A set of steps that always works, done in the same order every time. The column method is an algorithm.
sum
The answer when you add.
regroup (trade)
Swap 10 ones for 1 ten, or 10 tens for 1 hundred.
partial sums
The hundreds, tens and ones added separately before putting them together.
place value
The value of a digit depends on its place: hundreds, tens or ones.

Remember from Grade 2

In Grade 2 you added two-digit numbers like 58 + 25. When you had 10 or more ones, you traded 10 ones for 1 ten. In Grade 3 we add bigger numbers, up to 1000. Now we may also trade 10 tens for 1 hundred.

Let's start with blocks

Base-ten blocks show place value:

  • a big square, called a flat, is 1 hundred
  • a long stick, called a rod, is 1 ten
  • a small cube is 1 one

Let's add 245 + 137 with blocks.

245137
245 is 2 flats, 4 rods and 5 cubes. 137 is 1 flat, 3 rods and 7 cubes.

Push all the blocks together:

  • Flats: 2 + 1 = 3 hundreds
  • Rods: 4 + 3 = 7 tens
  • Cubes: 5 + 7 = 12 ones

We have 12 ones. That is more than 9! Trade 10 ones for 1 ten. Now we have 3 hundreds, 8 tens and 2 ones.

beforeafter
Before the trade: 3 flats, 7 rods, 12 cubes. After trading 10 cubes for 1 rod: 3 flats, 8 rods, 2 cubes. That is 382.

So 245 + 137 = 382.

Strategy: Partial sums

You can do the same thing with numbers instead of blocks. Add the hundreds, the tens and the ones separately. These are partial sums. Then add them together.

245 + 137

  • Hundreds: 200 + 100 = 300
  • Tens: 40 + 30 = 70
  • Ones: 5 + 7 = 12
  • Total: 300 + 70 + 12 = 382

The column method (an algorithm)

The column method is a short way to write exactly what the blocks do.

  1. Write the numbers in columns so the ones are under ones, tens under tens and hundreds under hundreds.
  2. Add the ones first. If there are 10 or more, write the ones digit and carry 1 ten to the top of the tens column.
  3. Add the tens, including any carried ten. If there are 10 or more tens, carry 1 hundred.
  4. Add the hundreds.
HTO2451137+382
Column method for 245 + 137. H = hundreds, T = tens, O = ones. The small 1 above the tens is the 10 ones we traded for 1 ten.
Worked example

Example 1: Add with the column method

Solve 245 + 137.

  1. Ones: 5 + 7 = 12. Write 2 in the ones place. Carry 1 ten to the tens column. (This is the trade: 10 ones became 1 ten.)
  2. Tens: 1 + 4 + 3 = 8. Write 8 in the tens place.
  3. Hundreds: 2 + 1 = 3. Write 3 in the hundreds place.

245 + 137 = 382. The blocks, partial sums and column method all give the same answer.

Why the little 1?

The carried 1 is not a one. It is 1 ten (10 ones that we traded). When you carry into the hundreds column, the 1 means 1 hundred (10 tens). The column method is just the blocks, written in a short way.

Worked example

Example 2: Two trades

Solve 468 + 275. Estimate first.

  1. Estimate: 500 + 300 = 800.
  2. Ones: 8 + 5 = 13. Write 3, carry 1 ten.
  3. Tens: 1 + 6 + 7 = 14 tens. Write 4, carry 1 hundred. (14 tens = 1 hundred and 4 tens.)
  4. Hundreds: 1 + 4 + 2 = 7.
  5. Answer: 743. Is it close to the estimate of 800? Yes, it is reasonable.

468 + 275 = 743

HTO46811275+743
Column method for 468 + 275 with two carries: 1 ten and 1 hundred.
Worked example

Example 3: All the way to 1000

Solve 356 + 644.

  1. Ones: 6 + 4 = 10. Write 0, carry 1 ten.
  2. Tens: 1 + 5 + 4 = 10 tens. Write 0, carry 1 hundred.
  3. Hundreds: 1 + 3 + 6 = 10 hundreds. 10 hundreds is 1 thousand. Write 10: the 1 goes in the thousands place and the 0 in the hundreds place.

356 + 644 = 1000

StrategyWhat you do for 245 + 137
Base-ten blocksCombine 3 flats, 7 rods and 12 cubes. Trade 10 cubes for 1 rod.
Partial sums300 + 70 + 12 = 382
Column methodAdd ones, carry 1 ten. Add tens. Add hundreds.
Three ways to add. They all work for the same reason: place value.
Just for fun

Why did the little 1 climb to the top of the tens column?

It got carried away!

Hands-on activity

Paper base-ten blocks

You need: grid paper (or paper with squares you draw), a pencil, and safety scissors. A grown-up should help with the cutting.

  1. Cut out 9 squares of 10 by 10 boxes (hundreds), 20 strips of 10 boxes (tens) and 20 single boxes (ones).
  2. A grown-up says two numbers, like 263 + 158.
  3. Build both numbers with your paper blocks.
  4. Push them together. If you have 10 or more ones, trade them for a ten strip. If you have 10 or more tens, trade them for a hundred square.
  5. Write the same question with the column method. Each time you trade, write a small carried 1.
  6. Check: do the blocks and the column method give the same answer? (263 + 158 = 421.)

Math talk: "Where did the little 1 come from? Show me with the blocks."

Your turn

1

312 + 145 = ?

2

238 + 154 = ?

3

471 + 362 = ?

4

589 + 236 = ?

5

405 + 395 = ?

6

In 238 + 154, what does the small carried 1 above the tens column mean?

7

Estimate, then add: 627 + 288.

More practice: warm-up

1

Add the blocks: 3 hundreds 5 tens 4 ones and 2 hundreds 1 ten 3 ones.

2

123 + 456 = ?

3

14 ones is the same as how many tens and ones?

4

12 tens is the same as how many hundreds and tens?

5

640 + 250 = ?

More practice: on your own

1

347 + 286 = ?

2

158 + 667 = ?

3

Ben added 264 + 178 and got 332. Find his mistake and fix it.

4

Use partial sums: 526 + 317.

5

A school collected 385 cans in October and 427 cans in November. How many cans in all?

Challenge

1

Find the missing digits: 4□7 + 25□ = 703

2

Without using the column method, explain why 299 + 299 is less than 600. Then find the exact answer.

3

Add three numbers: 125 + 250 + 375.

Check your understanding

Quiz

1.

What is 236 + 151?

2.

In 457 + 238, what do you do in the ones column?

3.

What is 365 + 248?

4.

10 tens is the same as:

5.

What is 500 + 499?

6.

Before you add with the column method, what must you do?

For parents and teachers

Big idea: In Grade 3, students add whole numbers up to a total of 1000 (B2.5) and begin to understand algorithms by connecting them to tools they already know, such as base-ten blocks and partial sums (B2.4). The goal is understanding why the column method works, not just following steps.

Common difficulties:

  • Not lining up digits (adding 45 under the hundreds of 312). Use grid paper or draw H, T, O columns.
  • Writing 12 in one column (245 + 137 = 3712). Remind your child that only one digit fits in each place; extra ones become a ten.
  • Forgetting the carried 1. Have your child say: "10 ones make 1 ten, so I carry 1 ten."
  • Starting on the left. Partial sums can start with hundreds, but the column method starts with the ones.

How to help: Build numbers with real objects: bundle 10 straws or popsicle sticks with an elastic to make a ten, and put 10 bundles together to make a hundred. Always ask your child to estimate first and compare.