Multiplying Multi-Digit Numbers

Multiplying multi-digit numbers means finding the product of two numbers that each have more than one digit, such as \(34 \times 26\) or \(123 \times 45\). Because there is more than one digit in each factor, you cannot just multiply two single numbers. Instead, you break the problem into smaller, easier pieces based on place value, multiply each piece, and then add the results together.

This skill builds directly on arrays and factors and on multiplying by 10, 100, and 1000, since every partial product in multi-digit multiplication is really just a number multiplied by a power of ten.

The key idea behind multiplying multi-digit numbers is expanded form. For example, \(34 = 30 + 4\) and \(26 = 20 + 6\). Once both factors are broken apart, you multiply every part of the first number by every part of the second number, then add all the pieces. This works because multiplication distributes over addition.

For \(34 \times 26\), this gives four smaller multiplication facts:

• \(30 \times 20 = 600\)
• \(30 \times 6 = 180\)
• \(4 \times 20 = 80\)
• \(4 \times 6 = 24\)

Adding these partial products: \(600 + 180 + 80 + 24 = 884\), so \(34 \times 26 = 884\).

An area model is a visual way to organize those four partial products as the area of a rectangle split into smaller rectangles. The whole rectangle has a length of 34 and a width of 26, and it is cut into pieces of length 30, 4 and width 20, 6.

Each small rectangle's area equals one of the partial products found earlier. Adding all four areas gives the total area of the big rectangle, which is the final product, 884.

The standard column algorithm is a shortcut for the same idea, written vertically. To multiply \(34 \times 26\):

1. Multiply 34 by the ones digit of 26, which is 6: \(34 \times 6 = 204\).
2. Multiply 34 by the tens digit of 26, which is 2 tens (20): \(34 \times 20 = 680\). Write this partial product one place to the left, since it represents tens.
3. Add the two partial products: \(204 + 680 = 884\).

Notice that 204 and 680 are exactly the same numbers as the partial products from the area model, just grouped differently: \(204 = 180 + 24\) and \(680 = 600 + 80\). The standard algorithm and the area model always produce the same partial products, only organized in a different order.

Multiply \(123 \times 45\).

Multiply by the ones digit first: \(123 \times 5 = 615\).

Multiply by the tens digit next: \(123 \times 40 = 4920\).

Add the partial products: \(615 + 4920 = 5535\).

So \(123 \times 45 = 5535\). Each partial product came from multiplying a whole number by 10, 40, or another multiple of ten, which is why comfort with multiples of ten makes this process much faster.

Before or after multiplying, it helps to estimate. Rounding \(34 \times 26\) to \(30 \times 30 = 900\) tells you the exact answer of 884 is reasonable. Rounding \(123 \times 45\) to \(120 \times 50 = 6000\) confirms that 5535 is a sensible size. If an exact answer is far from the estimate, it is a sign to check the partial products and the addition step again.

Multiplying multi-digit numbers is the same place-value thinking used in reverse when you learn dividing multi-digit numbers. Both skills rely on splitting numbers apart by place value, working with smaller pieces, and combining results carefully, so strong practice here supports success across the rest of grade 5 arithmetic.

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• Math 5