Bringing the Associative Property of Multiplication to Life

By Cassandra Hatfield, RME Assessment Coordinator, and Megan Hancock, Graduate Research Assistant

The Institute of Education Science (IES) Practice Guide for Improving Mathematical Problem Solving in Grades 4 through 8 Recommendation five states that it is important to “help students recognize and articulate mathematical concepts and notation” (Woodward et al., 2012). One way to carry out this recommendation is to “ask students to explain each step used to solve a problem in a worked example” and “help students make sense of algebraic notation” (Woodward et al., 2012).

The Associative Property of Multiplication will illustrate this recommendation by going beyond a procedural skill and making connections conceptually that support the symbolic notation. Our goal is to give evidence that the Associative Property of Multiplication can be taught through multiple representations. Through our research we found that some representations are mathematically accurate, but may not provide students with a compelling reason to use this property.

When developing the concept of volume of rectangular prisms, decomposing the rectangular prism into layers allows students to make the connection with content they are already familiar with, arrays and area. This decomposition also exemplifies the Associative Property of Multiplication. Here are some examples of how the rectangular prism shown above can be decomposed in different ways.

 

• A: 2 × (6 × 4)
• B: (2 × 6) × 4
• C: Supports commutative property of multiplication too
• 2 × 6 × 4; 2 × 4 × 6; (2 × 4) × 6

By designing activities and lessons that support the decomposition of rectangular prisms into different layers, teachers can support students in making sense of the notation of Associative Property of Multiplication, A x (B x C) = (A x B) x C, and finding the volume of rectangular prisms. Explorations like this also support teachers in holding students accountable for understanding the notation because students can use the different models to support their explanation of their understanding.

Woodward, J., Beckmann, S., Driscoll, M., Franke, M., Herzig, P., Jitendra, A., Koedinger, K. R., & Ogbuehi, P. (2012). Improving mathematical problem solving in grades 4 through 8: A practice guide (NCEE 2012-4055). Washington, DC: National Center for Education Evaluation and Regional Assistance, Institute of Education Sciences, U.S. Department of Education. Retrieved from http:// ies.ed.gov/ncee/wwc/publications_reviews.aspx#pubsearch/.