Grids are Area Models

Geri Lorway

Geri Lorway

James Tanton

James Tanton

What’s an AREA MODEL?

A rectangular array used to demonstrate a rich and interconnected web of multiplicative ideas.

The hundred grid is a powerful model for developing a richly diverse and complex network of concepts. Do not reduce it to a “hundred chart.” Instead develop it as the powerful modeling tool it can be.

Screen Shot 2013-11-11 at 4.21.26 PMIn this image I am using the hundred grid as an area model for 100 (10×10). By colouring in tens and ones, students can learn to visualize relationships between numbers to 100.

The green represents 76 and with practice students use this area model to “know” and “perform with fluency”  addition and subtraction facts to 100.

The cuisenaire rods provide a concrete basis for developing “area” models into multiplication and division “facts” and relationships.  The rods fit cm grid paper.

Gr 3 facts I know with Rods


Can you use my numbers to distribute

ABOVE I am using the hundred grid to model multiplication facts as they are related by distributing within a hundred. The orange represents 8 x 8. I knew that without counting. I also see that 8×8= (5+3)(5+3). I can also describe that distribution as (5×5)+ (5×3) +(3×5) + (3×3).

Can you explain the 40?

Geri with a kid looks good

Two Digit Multiplication2 digit standard

The visual is a rectangle. You have likely seen this model for 2 digit by 2 digit but did you know that with practice students can perform these multiplications in their head?  And even more important to later success build confidence with the distributive property…..

And before you worry about division being too hard wait till you see these…2 digit puzzles

Area Models Apply to Mathematical Topics in Elementary School

click map

Area Models for Number Sense Place Value Sense to 100 Place Value Sense to 1000 Place Value Sense to 10 000 To a million and more Decimal Fluency Area Models for Mulitiplicative Reasoning Basic Multiplication/Division Create the facts Practice the facts Multi Digit Practice Division Practice Fitting in Fractions

Picture of

Dr. James Tanton, Princeton Phd

AREA MODELS continue to impact rich and deep learning of Foundational Concepts from Junior through Senior high

Click here to link to sample lessons and materials that support Junior and Senior High math courses.


Click on the map to enlarge it.

AREA MODELS connect “multiplicative reasoning” to the learning of “Math Facts”. AREA MODELS make evident number properties  (commutative, associative & distributive) in ways that “just make sense” to students.

   AREA MODELS  tap into the visual-spatial reasoning power of the human brain, a power that we can no longer ignore in our teaching.

It is the power of GENIUS and the untapped potential in all of us. (Lorway, 2015)

In the study of numerical and arithmetical abilities, there is compelling evidence demonstrating that number and space representations are connected to one another. (Walsh, 2003; Fias & Fischer, 2005; Hubbard, Piazza, Pinel & Dehaene, 2005; de Hevia, Vallar & Girelli, 2008)

The relation between spatial ability and mathematics is so well established that it no longer makes sense to ask whether they are related.  (Mix & Cheng, 2012)

Area models develop, sustain and challenge spatial reasoning. There is clear and direct evidence that spatial reasoning is a BASIC SKILL that can impact achievement for ALL STUDENTS. We know it makes a difference to achievement. It is our professional responsibility to incorporate it into our approach to teaching. (Lorway, 2014)

My Area Models July

Read more about AREA MODELS HERE.

geri at board and area