# Lesson 7

Using Diagrams to Represent Multiplication

### Lesson Narrative

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.

### Learning Goals

Teacher Facing

- Comprehend how the phrase “partial products” (in spoken and written language) refers to decomposing a multiplication problem.
- Coordinate area diagrams and vertical calculations that represent the same decimal multiplication problem.
- Use an area diagram to represent and justify (orally and in writing) how to find the product of two decimals.

### Student Facing

Let’s use area diagrams to find products.

### Required Materials

### Required Preparation

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.

### Learning Targets

### Student Facing

- I can use area diagrams and partial products to represent and find products of decimals.

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