A User’s Guide to Implementing the Area Model in the Mathematics Classroom

Created Feb. 7, 2024 by Amy Velchoff

A User’s Guide to Implementing the Area Model in the Mathematics Classroom

This guide will provide an overview of the Area Model, topics it can be used to illustrate and teach in a deeper, more conceptual manner, and the resources available on OER Commons to do so.

Key Topics

  • Base 10
  • Area and Perimeter
  • Distributive Property
  • Multiplication and Division of Whole Numbers, Fractions, Decimals, Binomials, Polynomials
  • Addition of Polynomials
  • Solving One- and Two- Step Equations

Introduction: Standards and Definitions

What is the Area Model? The Area Model can build off of students’ prior Base 10 knowledge, and connects the geometric notion of area as equivalent to length times width to the basic operations, such as multiplication of whole numbers, fractions, and decimals. The model asks students to engage in the decomposition of numbers (e.g. 12 = 10 + 2), and can help develop and reinforce an understanding of the Distributive Property. Other benefits of the model include the ability to extend it to multiplication of Binomials and Polynomials. In addition, connecting in students’ geometric understanding of perimeter allows for the model to be expanded to include addition of polynomials.

Read this article for an overview of the Area Model and how it compares to the traditional multiplication algorithm.

Standards The Common Core State Standards (in particular, CCSS.MATH.CONTENT.3.MD.C.5, Geometric measurement: understand concepts of area and relate area to multiplication and to addition) specifically call for students to Use area models to represent the distributive property in mathematical reasoning.

Unit 9: Using Area Models to Understand Polynomials is a great overview of these topics: http://www2.cuny.edu/wp-content/uploads/sites/4/page-assets/academics/academic-programs/model-programs/cuny-college-transition-programs/adult-literacy/cuny-hse-curriculum-framework/Section4CUNYHSEFrameworkMath-Revised-10-24-17.pdf

Key Ideas:

The unit contains five activities that help to build students’ understanding of how the Area Model relates to various topics. Handouts for the activities are on pgs 216-221.

  1. The first activity (pgs 203-204) has students use arrays to understand multiplication. In addition, it reinforces the idea of “multiplication as repeated addition” and the commutative property.
  2. The second activity (pgs 205-209) transitions from the Array model to the idea of Area. in addition, the activity reviews Factors and connections to Division. Finally, the lesson concludes by introducing the Distributive Property and the idea of decomposing numbers.
  3. The third activity (pg 210) goes in depth with the Distributive Property, and
  4. The fourth activity (pgs 210-211) moves to a more abstract version of the Area Model.
  5. Finally, the fifth activity (pgs 212-215) extends the model to the multiplication of polynomials.

Math Intros – Flipped Class

I recommend taking a flipped class approach to introducing the Area Model, which involves assigning the students to watch one or more (depending on your students’ needs) of the following videos before attending class:

  1. No matter what the topic is in math, https://www.khanacademy.org/ likely has something to get you started For example, this video is an awesome review of Area and Perimeter: https://www.khanacademy.org/math/geometry/hs-geo-foundations/hs-geo-area/v/perimeter-and-area-basics
  2. For an OER-approved introduction to the Area Model and the Distributive Property, I recommend the following video: http://www.oercommons.org/courses/corbin-intermediate-distributive-property-of-multiplication/view 

Math Demonstrations and Modeling

  1. If you don’t have access to concrete manipulatives, try some simulations at https://phet.colorado.edu/. If the link does not work, google PHET Interactive Simulations. From the home page, click “Play With Simulations.” Use the topic feature to look at possibilities or type a keyword into the search feature to provide a list of simulations.
  1. Area Model Simulation https://www.oercommons.org/courses/area-model-algebra/view
  2. Algebra Tiles https://www.oercommons.org/courses/solving-multi-step-equations - long link to algebra tiles
  • From the ALEX Lesson Plan page, scroll to the bottom.
  • Click the Algebra Tiles hyperlink.
  • Choose the McDougal Littell Math Course 3 textbook (middle school math; state shouldn’t matter).
  • Click the Animations hyperlink, then choose Chapter 3: Algebra Tiles. Note: This uses Algebra Tiles for solving one-step equations. For two-step equations, use Chapter 3: Solving Two-Step Equations.
  • As an additional resource, Chapter 1: Perimeter and Area might work for review.

Group Activities For Team Building and Problem Solving

This activity provides good group work norms, as well as good sample work pieces to discuss either in a whole class or small group setting:

  1. http://www.oercommons.org/courses/maximizing-area-gold-rush/view: This OER includes a very detailed guide along with overhead slides, student work samples and common misconceptions.
  2. Extension Activity: https://www.illustrativemathematics.org/content-standards/tasks/215; Tips for Implementation: After working with the Area Model and strengthening the connections between Algebraic expressions and their visual representations, students should be able to tackle the extension activity. You can also assign the activity to be worked on in partners or in groups.

Exercises

  1. https://www.oercommons.org/courses/multiplying-polynomials/view; Suggestions: Make sure to use either MS Word or an OpenSource word processing platform that will read rich text format to open the rtf files.

Review Activities/ Formative/ Summative Assessments

Here is an Activity for making connections with Base 10: https://www.oercommons.org/courses/ten-10s-make-100

This phet simulation can be used to help students whose math facts are weaker practice their multiplication or division math facts while connecting them to the concept of Area:

  1. http://www.oercommons.org/courses/arithmetic-workout/view - practice multiplication, division, factoring Note: Flash player is needed

Suggestions for implementation:

  • The simulation shows a multiplication fact and highlights an area corresponding to the resulting product. Make sure to point out that the factors correspond to the dimensions of the rectangle, and the product corresponds to the area.
  • The Division setting is not as intuitive; the simulation does not highlight an area, so it does not reinforce the idea of division as a situation where we know the area, but need to find the missing side length.
  • For the factor setting, students are asked to highlight a rectangle with the desired area; they are given flexibility to choose any “side lengths,” or factors of their rectangle that would give the resultant area.

Review: Module 4, Topic C

This task assesses students’ understanding of how the Area Model relates to fractions: https://www.oercommons.org/courses/5-nf-connecting-the-area-model-to-context 

Task 3 is a good formative assessment of students’ understanding of the Area Model, and their ability to write algebraic expressions to represent Area or Perimeter: https://www.oercommons.org/courseware/lesson/1514/overview 

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