In this lesson, students build on two previously learned methods for multiplying whole numbers—using an area diagram and an algorithm—to find products of decimals.

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. The first optional activity in the lesson allows students to revisit these ideas.

In the main activity, 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. A second optional activity gives students additional practice with this line of reasoning.

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).