• Research in Mathematics Education Blog

    Wednesday, May 29, 2013

    It's Not a Window Pane... It's an Area Model

    By Cassandra Hatfield, RME Assessment Coordinator

    As students develop an understanding of multiplication, instruction often moves from an equal sets into an area model. In my experience as a math specialist, sometimes an area model was called a “window pane.” In this blog we will focus on the conceptual understanding of an area model and the need to shy away from calling it a “window pane.” 

    Just last month, I was visiting my nephew who was very proud that he had memorized several multiplication combinations and he asked me to quiz him. He was giving me answers at a rapid pace until I got to 8 • 7. His response, “I don’t know that one yet!” He had just told me what 8 • 5 was and what 8 • 2 was. However, my nephew was simply memorizing combinations and did not have a strategy to compose the combination with other combinations he knows. The area model is a powerful way to assist students in composing combinations. 

    Mult. Array             

    When students know and understand combinations through 10 • 10 then they can decompose numbers to find any other combination. 

    I’ve been in classrooms and seen anchor charts titled “Window Pane Strategy” with an example of finding the product of a 2 digit number by a 2 digit number. However, notice that prior to this area models did not look like window panes. The models shown are proportional. The units are squares, so in the problem 12 • 8 the factor 12 is longer than the factor 8. When teaching students to multiply 2 digit by 2 digit numbers before transitioning to an open area model (without the grid lines) it is extremely important for students to make the connection to their prior understanding by using a model with grid lines. Using an area model is not a procedure for finding an answer; it’s a conceptual understanding of the distributive property. 

     Mult. Array 2               Correct vs. Incorrect

    Even after moving to an open area model, while students do not need to measure to make the parts perfectly proportional, it is important that students still draw the open area models to show that each of the parts is a different area. The open area model can continue to be used to support students in multiplying larger numbers. 

    Open Array

    In another post next month, I’ll share how the area model connects to the standard algorithm and can be a powerful model to support student understanding of the standard algorithm.



    Open area model/open array and Partial product

    Partial Product

     Standard  algorithm.  Students should be able to understand and explain what is going on in each step. 

     Standard Algorithm