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Converting Products to a Smaller Product and Sum

This lesson shows how to convert a product into a smaller product and sum by breaking one factor apart, an approach based on the distributive property. Using array models and step-by-step worked examples, students learn to turn a hard multiplication fact into two easier ones and add the results.

What It Means to Convert a Product to a Smaller Product and Sum

Some multiplication facts are hard to picture all at once, like \(7 \times 8\) or \(6 \times 9\). Converting a product to a smaller product and sum means taking one of the numbers being multiplied and splitting it into two smaller, friendlier numbers. Once the number is split, you multiply each part separately and then add the two answers together. The sum you get is exactly the same as the original product, but it is often much easier to work out.

For example, instead of trying to think about \(7 \times 8\) all in one step, you can split the 8 into \(5 + 3\). Now you only need two easier facts, \(7 \times 5\) and \(7 \times 3\), and then a simple addition.

Breaking Apart Numbers with an Array Model

An array, or area model, is a great way to see why this works. Picture a rectangle made of rows and columns of squares, where one side has 6 squares and the other has 8 squares. If you slice that rectangle into two smaller rectangles, one showing \(6 \times 5\) and the other showing \(6 \times 3\), the two pieces together cover exactly the same area as the original \(6 \times 8\) rectangle.

Splitting arrays like this is the same idea used when you first learned that repeated addition and multiplication give the same result, since each smaller rectangle is really a group of equal rows being added together.

Step-by-Step: How to Split a Product

To convert a product into a smaller product and sum, follow these steps.

Pick one of the two factors in the multiplication problem to split.
Break that factor into two numbers that add up to it, choosing numbers that are easy to multiply, such as 10, 5, or 2.
Multiply the other factor by each of the two new numbers.
Add the two smaller products together to get the final answer.

This is the same reasoning behind the distributive property, written as \( a \times (b + c) = a \times b + a \times c \). Splitting \(b\) (or \(a\)) into two pieces before multiplying always gives the same result as multiplying first and adding second.

Worked Examples

Example 1: Find \(9 \times 7\).

Split 9 into \(5 + 4\). Then \(9 \times 7 = (5 + 4) \times 7 = 5 \times 7 + 4 \times 7 = 35 + 28 = 63\).

Example 2: Find \(8 \times 6\).

Split 6 into \(5 + 1\). Then \(8 \times 6 = 8 \times 5 + 8 \times 1 = 40 + 8 = 48\).

Example 3: Find \(7 \times 7\).

Split one of the 7s into \(5 + 2\). Then \(7 \times 7 = 7 \times 5 + 7 \times 2 = 35 + 14 = 49\). This same splitting idea also helps when a number is multiplied by itself, which is worth remembering when you study square numbers.

Connecting to the Distributive Property

Converting a product to a smaller product and sum is simply another way of describing the distributive property in action. Instead of memorizing every large multiplication fact, you can rely on facts you already know well, like multiplying by 5, 2, or 10, and combine them to reach the larger fact. This is especially useful once numbers grow larger, such as when multiplying with digits up to 999, where breaking a big factor into hundreds, tens, and ones makes the whole problem manageable.

Practice Tips

When choosing how to split a number, look for a split that includes a 5 or a 10, since multiplying by those numbers is usually the fastest. Always check your work by adding the two smaller products and comparing the total to what you expect from estimation. With practice, splitting products becomes a quick mental shortcut rather than a written procedure.

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