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D.9 Multiply whole numbers by 2-digit numbers

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What is multiplication by 2-digit numbers?

Multiplying by 2-digit numbers is a mathematical operation where we find the product of a number and a number between 10 and 99. This builds on basic multiplication skills and helps us solve more complex problems.

Examples:
  • 24 × 15
  • 37 × 42
  • 56 × 83
Note

When multiplying by 2-digit numbers, we break the problem into smaller, easier steps using place value understanding.

The Standard Algorithm for 2-Digit Multiplication

The standard algorithm is a step-by-step method for multiplying multi-digit numbers. It uses partial products and then adds them together to find the total.

Example: 36 × 24
  1. Write the numbers vertically, lining up the digits:
    2.             36
          × 24
        ——————
  2. Multiply 36 by the ones digit (4): 36 × 4 = 144
  3. Multiply 36 by the tens digit (2): 36 × 2 = 72, then add a zero because the 2 represents 20 → 720
  4. Add the two partial products: 144 + 720 = 864
            36
          × 24
        ——————
           144   ← 36 × 4
          +720   ← 36 × 20
        ——————
           864
Note

When multiplying by the tens digit, always add one zero (or shift the number one place to the left) because it represents tens, not ones.

Understanding partial products

Partial products are the results we get when we multiply each digit of one number by each digit of the other number, considering their place values.

Example: 47 × 23
  • 47 × 3 = 141 (ones × ones)
  • 47 × 20 = 940 (ones × tens)
  • 141 + 940 = 1,081 (sum of partial products)
Note

Thinking about partial products helps you understand why the standard algorithm works and makes it easier to check your work.

Using the area model for multiplication

The area model is a visual method that breaks numbers into their place value components and shows multiplication as finding the area of a rectangle.

Example: 28 × 34
  • Break 28 into 20 + 8
  • Break 34 into 30 + 4
  • Multiply each part: 20×30=600, 20×4=80, 8×30=240, 8×4=32
  • Add all parts: 600 + 80 + 240 + 32 = 952
Note

The area model helps you see how place value works in multiplication and connects to the partial products method.

Checking your work with estimation

Estimation is a strategy to check if your answer is reasonable by using rounded numbers that are easier to multiply mentally.

Example: 63 × 47
  • Round 63 to 60 and 47 to 50
  • Multiply: 60 × 50 = 3,000
  • Actual product: 63 × 47 = 2,961
  • Since 2,961 is close to 3,000, our answer is reasonable
Note

Estimation helps catch major errors. If your answer is very different from your estimate, check your work again.

Real-world applications

Multiplying by 2-digit numbers helps solve many real-world problems involving quantities, measurements, and calculations.

Examples:
  • Calculating total cost: If 24 students each pay $15 for a field trip, the total is 24 × 15 = $360
  • Finding area: A garden that is 35 feet long and 12 feet wide has an area of 35 × 12 = 420 square feet
  • Determining distance: If a car travels 65 miles per hour for 14 hours, it covers 65 × 14 = 910 miles
Note

Always make sure your answer makes sense in the context of the problem. Include units when appropriate.

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