Long Multiplication (2-digit by 2-digit)
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Learning Objectives
- 1Multiply a 2-digit number by a 2-digit number using the standard long-multiplication algorithm
- 2Write and add the partial products in the correct place-value columns
- 3Use the area model (box method) to decompose factors and find the same product
- 4Estimate products by rounding each factor to the nearest ten to check reasonableness
Mini Lesson
Long multiplication breaks a 2-digit by 2-digit problem into two smaller multiplications, called partial products. You multiply by the ones digit, then by the tens digit, and add the results. The area model shows the same idea as a rectangle split into four boxes.
The Standard Algorithm
- Stack the two factors so the digits line up by place value.
- Multiply the top factor by the ones digit of the bottom factor — this is the first partial product.
- Place a 0 in the ones column, then multiply the top factor by the tens digit — this is the second partial product.
- Add the two partial products to get the final product.
- Example: 47 x 23. First, 47 x 3 = 141. Next, 47 x 20 = 940. Add: 141 + 940 = 1,081.
Why the Placeholder Zero?
- When you multiply by the tens digit, you are really multiplying by tens, not ones.
- The placeholder 0 in the ones column shifts the partial product one place to the left.
- 47 x 2 written as 94 means 94 tens, which is 940 — the 0 makes the value correct.
The Area Model (Box Method)
- Break each factor into tens and ones: 47 becomes 40 + 7, and 23 becomes 20 + 3.
- Draw a 2 by 2 grid and label the columns and rows with each part.
- Multiply each pair of parts to fill the four boxes: 40x20=800, 7x20=140, 40x3=120, 7x3=21.
- Add the four partial products: 800 + 140 + 120 + 21 = 1,081.
- You get the same answer as the standard algorithm because both methods multiply every part of one factor by every part of the other.
Regrouping Inside Each Partial Product
- When a one-digit times one-digit product is 10 or more, write the ones and carry the tens.
- In 47 x 3: 3x7=21 (write 1, carry 2). Then 3x4=12 plus the carried 2 = 14. So 47 x 3 = 141.
- Cross out or erase the carried digit before starting the next partial product so it does not get reused.
Estimating to Check
- Round each factor to the nearest ten before multiplying.
- 47 rounds to 50 and 23 rounds to 20, so 47 x 23 is about 50 x 20 = 1,000.
- Compare your exact answer to the estimate — if 1,081 is close to 1,000, the answer is reasonable.
Tip: write each carry small and cross it out before the next partial product so you do not add it twice. The placeholder 0 on the second line is what makes the tens-digit product worth ten times as much.
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