Lesson 7: Using Diagrams to Represent Multiplication
About this lesson
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 subrectangles 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 18unit side can be decomposed into 10 and 8 units (tens and ones), and the 23unit side can be expressed as 20 and 3 (also tens and ones), creating four subrectangles 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 \((1.8) \cdot (2.3)\). 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.
Lesson overview
 7.1 Warmup: Estimate the Product (5 minutes)

7.2 Optional Activity: Connecting Area Diagrams to Calculations with Whole Numbers (20 minutes)
 There is a digital applet in this activity.

7.3 Activity: Connecting Area Diagrams to Calculations with Decimals (20 minutes)
 Includes "Are you Ready for More?" extension problem
 There is a digital applet in this activity.
 7.4 Optional Activity: Using the Partial Products Method (10 minutes)
 Lesson Synthesis
 7.5 Cooldown: Find the Product (5 minutes)
Learning goals:
 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.
Learning goals (student facing):
 Let’s use area diagrams to find products.
Learning targets (student facing):
 I can use area diagrams and partial products to represent and find products of decimals.
Required materials:
 Graph paper
 Rulers
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 and Calculations with Decimals.
Glossary:
 Access the complete Grade 6 glossary.
Standards
 This lesson builds on the standards: CCSS.4.NBT.B.5CCSS.5.NBT.B.7MS.4.NBT.5MS.5.NBT.7MO.4.NBT.A.6MO.5.NBT.A.6MO.5.NBT.A.7MO.5.NBT.A.8
IM 6–8 Math was originally developed by Open Up Resources and authored by Illustrative Mathematics, and is copyright 20172019 by Open Up Resources. It is licensed under the Creative Commons Attribution 4.0 International License (CC BY 4.0). OUR's 6–8 Math Curriculum is available at https://openupresources.org/mathcurriculum/.
Adaptations and updates to IM 6–8 Math are copyright 2019 by Illustrative Mathematics, and are licensed under the Creative Commons Attribution 4.0 International License (CC BY 4.0).
Adaptations to add additional English language learner supports are copyright 2019 by Open Up Resources, and are licensed under the Creative Commons Attribution 4.0 International License (CC BY 4.0).
The Illustrative Mathematics name and logo are not subject to the Creative Commons license and may not be used without the prior and express written consent of Illustrative Mathematics.
This site includes public domain images or openly licensed images that are copyrighted by their respective owners. Openly licensed images remain under the terms of their respective licenses.