Lesson 7

Students continue to use area diagrams to find products of decimals, while also beginning to generalize the process. They revisit two methods used to find products in earlier grades: decomposing a rectangle into sub-rectangles and finding the sum of their areas, and using the multiplication algorithm.

Students have previously seen that, in a rectangular area diagram, the side lengths can be decomposed by place value. For instance, in an 18 by 23 rectangle, the 18-unit side can be decomposed into 10 and 8 units (tens and ones), and the 23-unit side can be expressed as 20 and 3 (also tens and ones), creating four sub-rectangles whose areas constitute four partial products. The sum of these partial products is the product of 18 and 23. Students extend the same reasoning to represent and find products such as \((1.8) \boldcdot (2.3)\). Then, students explore how these partial products correspond to the numbers in the multiplication algorithm.

Students connect multiplication of decimals to that of whole numbers (MP7), look for correspondences between geometric diagrams and arithmetic calculations, and use these connections to calculate products of various decimals. 

Some students might find it helpful to use graph paper to help them align the digits for vertical calculations. Consider having graph paper accessible for the last activity: Connecting Area Diagrams to Calculations with Decimals.