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March 6, 2018
Using Visual Models

Interacting with visual models is a key strategy for developing students' understanding and fluency with multiplication. Moving to efficient and effective multiplicative strategies relies on understandings that are built through opportunities for students to construct their own models as well as interact with given visual models. 
Initially, models are used to build an understanding of a concept. As shown in the picture below, moving from an understanding of multiplication as equal groups to arrays to the area model helps students develop understanding of the commutative property and of the relationship between factors and dimensions.  Notice how each model is related to the next model, providing a progression that can be used instructionally to guide students toward more efficient and generalizable multiplication strategies. The ultimate goal is that students will have a mental model that helps them make sense of numbers and operations. The model provides an anchor for understanding and can be a tool to reconstruct an algorithm or concept when they cannot remember.
The open area model  is a model that should permeate instruction in grades 4 and 5 as it will be used for multiplication and division of whole numbers, fractions and decimals, as well as in algebra. Notice how the student in the example below uses an open area model, base 10 understanding and the distributive property to successfully solve the single-digit by double-digit multiplication problem. In order for students to solve multiplication problems with larger numbers, when seeing all the squares or items in an array isn't feasible or practical, they need to be flexibly solving multiplication problems using an open area model with smaller numbers.
Action Item
  • Where do you see visual models used in your curriculum materials to develop multiplicative reasoning and fluency?
  • Identify some OGAP questions that have students interact with visual models to help develop a concept (e.g. E2, PR25, RA13). Have students solve the problem. Use student solutions to further strengthen their understanding of the targeted concept.
  • Identify some OGAP questions that have the potential to elicit student use of visual models to solve the problem (e.g. E9, RA4, MCM1). Analyze the evidence in their work and decide where on the OGAP Multiplication Progression the solutions are and your next instructional step.
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Also, you can access past issues of OGAP Teacher Tips by visiting  www.ogapmath.com/ogap-teacher-tips-archive.