The Four Operations — Mixed Review with Theory (Grades 3–4): 4-Digit Addition with Carrying, 4-Digit Subtraction with Borrowing, 3-Digit Multiplication & Basic Division

The standard algorithm, one column at a time

To add two multi-digit numbers, you line them up by place value (ones under ones, tens under tens, hundreds under hundreds, thousands under thousands). Then you add each column starting from the ones on the right.

When the column sum is 9 or less, write it under the line and move on.

When the column sum is 10 or more, you can only fit one digit under the line — the ones digit of the sum. You carry the tens digit of that sum to the top of the next column to the left, where it gets added along with the digits already there.

Why it works. Ten ones equal one ten. Ten tens equal one hundred. Ten hundreds equal one thousand. Carrying is nothing more than re-packing the total number correctly by place value.

Signal to a first grader: the little 1 that sits above the next column is not a magic mark — it’s a whole ten (or a whole hundred, etc.) that got moved into the correct column. Say “carry one ten” out loud the first few times.

Borrowing = re-packing 1 ten into 10 ones

To subtract, line the numbers up by place value, top to bottom. Start on the right at the ones column and subtract each column moving left.

When the top digit is larger than or equal to the bottom digit, subtract normally.

When the top digit is smaller than the bottom digit, you cannot subtract directly. Borrow 1 from the next column to the left: reduce that column by 1, and add 10 to the current column’s top digit. Now the top digit is at least 10, so subtraction is possible.

Why the total doesn’t change. One ten equals ten ones. When you take 1 from the tens column and add 10 to the ones column, the total value of the number stays exactly the same — you’ve just re-packed it.

The double-borrow trap. When the column you want to borrow from is a 0, you have to keep borrowing farther left until you find a non-zero column. Each 0 in between becomes a 9 — because you turned 1 hundred into 10 tens (leaving 9 tens), which then loans 1 ten to the ones. This is where grade 3 students get stuck most often. Slow down and do the borrow chain before subtracting.

Check with addition. If a − b = c, then b + c = a. In the example, 2,847 + 2,356 should equal 5,203. Try it — it does. That’s how you catch mistakes.

3-digit × 1-digit — the standard algorithm

Line the numbers up so the 1-digit number is on the bottom, directly under the ones place of the 3-digit number. Multiply the bottom digit by each top digit one at a time, moving right to left. When a partial product is 10 or more, carry the tens digit to the top of the next column — just like addition — then add it to the next multiplication.

Key rule: the carry gets added to the next partial product, not multiplied. Grade 3 students commonly make the mistake of multiplying the bottom digit by the carry (6 × 2 × 2 instead of 6 × 2 + 2). Say the correct routine out loud: “six times three is eighteen, plus the carry-two is twenty…”

3-digit × 2-digit — two partial products

For a 3-digit × 2-digit multiplication, do the same routine twice. First multiply by the ones digit of the bottom number and write that partial product on the first row. Then multiply by the tens digit of the bottom number, but shift that partial product one column to the left (because it’s really being multiplied by that digit × 10). Add the two partial products.

Why the shift left. The 3 in “34” is really 30 — it’s in the tens place. So multiplying by it gives a result 10× bigger than multiplying by 3 alone. Putting a 0 in the ones column of the second row is the same as multiplying by 10. This is the single most important idea in multi-digit multiplication.

Division splits a total into equal groups

Division answers the question: “How many equal groups can I make?” or “How many go in each group?”

Three ways to write the same division problem:

• 24 ÷ 6
• 24 / 6
• “6 goes into 24” (long-division bracket notation)

In every case, 24 is the dividend (the total), 6 is the divisor (how you split it), and the answer is the quotient.

Division is the inverse of multiplication

The fastest way to solve a basic division problem is to ask its multiplication question. 24 ÷ 6 = ? means “6 times what equals 24?” The answer is 4, because 6 × 4 = 24. That’s why memorizing multiplication tables gives you division facts for free — the two operations share fact families.

Special divisions to memorize

Check every division with multiplication

If a ÷ b = c, then b × c = a. That’s the check. Grade 3 students who get in the habit of multiplying back to check catch almost every mistake themselves.