Multi-Digit Multiplication for 5th Grade: Standard Algorithm

Multi-digit multiplication is used when calculating totals from repeated groups—students per classroom, items per box, or cost per unit. In Grade 5, students multiply whole numbers with two or more digits using the standard algorithm (also called long multiplication). The algorithm builds on place value: we multiply the top number by each digit of the bottom number, shifting partial products to the left according to place value, then add the partial products. This skill helps students solve real-world problems involving large quantities.

The standard algorithm organizes multiplication into clear steps: multiply the top number by the ones digit of the bottom number, write that partial product; then multiply by the tens digit and write the second partial product shifted one place to the left (because we’re multiplying by tens); continue for each digit, then add all partial products. Understanding why we shift partial products—because multiplying by 30 is the same as multiplying by 3 and then by 10—reinforces place value.

DETAILED EXPLANATION

Steps for the standard algorithm (e.g., 347 × 52):

1. Write the larger number on top, the smaller below, aligned by place value (ones under ones, tens under tens).

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2. Multiply the top number by the ones digit of the bottom number. Write the partial product, aligned so its ones digit is under the ones digit of the multiplier.

3. Multiply the top number by the tens digit of the bottom number. Write this partial product shifted one place to the left (a zero or blank in the ones column).

4. Add the partial products to get the final product.

Example: 24 × 36. Partial product 1: 24 × 6 = 144. Partial product 2: 24 × 3 = 72, but 3 is in tens place, so 24 × 30 = 720. Add: 144 + 720 = 864.

WORKED EXAMPLES WITH STEP BY STEP SOLUTIONS

Example 1

A school has 24 classrooms with 36 students each. How many students in total?

Solutions:

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Step 1: We need to multiply 24 × 36. Write vertically:

× 36

Step 2: Multiply 24 by the ones digit (6): \(24 \times 6 = 144\). Write 144, aligned so the 4 is under the 6.

Step 3: Multiply 24 by the tens digit (3). Since 3 is in the tens place, we are really computing \(24 \times 30 = 720\). Write 72 shifted one place to the left (so the 2 is under the 3, or equivalently put a 0 in the ones place of the second partial product).

Step 4: Add the partial products: 144 + 720 = 864.

Step 5: There are 864 students in total.

Answer: 864 students

Example 2

Multiply 347 × 52

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Solutions:

Step 1: Write:

347

× 52

Step 2: Multiply 347 × 2 = 694. Write 694.

Step 3: Multiply 347 × 5 = 1,735. Since 5 is in the tens place, this represents 347 × 50 = 17,350. Write 1,735 shifted left (ones digit of 1,735 under the 5).

Step 4: Add: 694 + 17,350 = 18,044.

Answer: 18,044

Example 3

Each box holds 125 pencils. A store orders 48 boxes. How many pencils?

Solutions:

Step 1: Multiply 125 × 48.

Step 2: Partial product 1: \(125 \times 8 = 1{,}000\).

Step 3: Partial product 2: \(125 \times 4 = 500\), but 4 is tens, so \(125 \times 40 = 5{,}000\). Write 5,000 shifted left.

Step 4: Add: 1,000 + 5,000 = 6,000 pencils.

Answer: 6,000 pencils

Example 4

Multiply 156 × 23

Solutions:

Step 1: \(156 \times 3 = 468\).

Step 2: \(156 \times 2 = 312\); since 2 is tens, \(156 \times 20 = 3{,}120\).

Step 3: Add: 468 + 3,120 = 3,588.

Answer: 3,588