In this lesson, students compare several different methods for computing products of decimals, including:

• By first expressing them as fractions with a power of 10 for the denominator.
• By reasoning about properties of base-ten numbers.
• By reasoning about the area of a rectangle.

Students see that they can multiply decimals by powers of 10 to get whole numbers, multiply the whole numbers, and then divide the result by the same powers of 10. For example, to multiply , students may compute and to have whole numbers 25 and 16, multiply these to get 400, and then divide 400 by 1,000 to invert the initial multiplication by 1,000.

In earlier grades, students learned to represent two factors as side lengths of a rectangle, as well as to decompose multi-digit factors by place value to facilitate multiplication. For instance, in an 18 by 23 rectangle, the 18-unit side can be decomposed into 10 and 8 units, and the 23-unit side can be expressed as 20 and 3, creating four sub-rectangles whose areas constitute four partial products. The sum of these partial products is the product of 18 and 23. Students also learned to organize these partial products in a vertical calculation, which enabled them to multiply whole numbers without relying on a diagram.

In this lesson, students extend both methods of reasoning to represent and find products such as . They see that decimal factors can likewise be decomposed by place value, the parts can be multiplied separately, and the results can be combined at the end.

In making connections—between multiplication of decimals and multiplication of whole numbers, between diagrams and calculations—students practice looking for and making use of structure (MP7).