Area Arrangements

Have you ever wondered if there’s a better way to teach tricky arithmetic and algebra procedures to your students? Think about multiplying fractions and polynomials, long division, simplifying radicals, and completing the square—these are some challenging tasks facing learners in middle and high school math class. Let’s try an AREA MODEL for each of these situations to make connections, build understanding, and help our students be successful with these topics.

What is an Area Model?

In its simplest form, an area model is a rectangle whose area represents the product of two factors, the lengths of its sides. When students first learn to multiply numbers in elementary school, they may encounter square tiles or arrays of dots to represent the multiplication of two numbers. The arrangements below can model 2 x 3 = 6 since there are 2 rows of 3 dots (or tiles) each; they also show 3 x 2 = 6 since there are 3 columns of 2 dots/tiles each.

Array of 6 dots (left) and 6 square tiles (right)

Extending this idea to a rectangle with sides of 3 and 2 demonstrates the same product of 6. Dashed lines can be shown initially, to be removed later when students understand the more abstract idea of area.

Rectangles with sides 2 and 3 showing that area = 6

Area vocabulary: It seems that teachers of younger students prefer the terms LENGTH and WIDTH, and by middle school or Algebra 1, terms have shifted to BASE and HEIGHT. Use whatever terms that allow you and your students to clearly communicate your mathematical ideas.

Different versions of the rectangle area formula

Multiplying Numbers and Fractions

As their fluency with multiplication grows, learners can extend the area model to multi-digit numbers, decomposing a factor into convenient parts for multiplying (often by place value, but that isn’t essential).  The corresponding rectangles get subdivided into as many subsections as needed for the partial products being calculated. These examples show 2 x 13 = 26 (left) and 702 x 43 = 30,186 (right).

Rectangular representations of multiplication computations

Notice that there isn’t a requirement to size the boxes to scale.  We don’t need to represent every place value, and learners are free to choose other decompositions.  The area method is one of several effective strategies that students may use when calculating products; it becomes one of the tools in their “mathematical toolbox” to use when they find it helpful.

I have found two distinct advantages of the area model diagrams for students:

• The concrete visual representation helps reduce the abstraction of the traditional multiplication algorithm.
• The area subsections ensure that none of the partial products are overlooked.

The area model can also be used for multiplying fractions. Some students get stuck with multiplying fractions because it works differently than fraction addition and subtraction did—no common denominators needed!^Note1  The area model is a useful scaffold as students are building proficiency with this fraction operation. Here the shaded regions represent the fractions and of the whole sides, and the overlap is the product . 

Rectangle subdivided to demonstrate fraction multiplication

Multiplying Polynomials

A key concept when multiplying algebraic expressions is applying the Distributive Property; distributing is implied in the area diagrams shown above for multi-digit numbers, but it really becomes the star of the show for multiplying polynomials.

We teach Algebra 1 students to distribute a coefficient or other monomial over a polynomial, often drawing arrows, and this process works well enough for many learners.  But then students move to multiplying binomials, requiring the double distribution of the two terms from the first factor over the two terms of the second. Historically, this is called FOIL (first, outside, inside, last), but this acronym is troubling because it masks the repeated distribution and does not scale to handle polynomial factors with more terms.

Area model to the rescue! Simply arrange the terms of the factors along the sides of the rectangle and multiply for each smaller box. All of the partial products are accounted for in an efficient package, and the like terms are usually located along diagonals (if the original polynomials are ordered by decreasing degree).  Most importantly, this works for any number of terms.^Note2

These diagrams show the multiplications (3x – 7)(x + 2) = 3x² – x – 14 and (x – 5)(x² + 2x – 3) = x³ – 3x² – 13x + 15.

Area Method for multiplying polynomials

A nice benefit of the area model is that students can see how squaring a binomial such as (x + 3)² actually creates a square since the side “lengths” are the same. Read more about this scenario in the Completing the Square section below.

Division with an Area Model

Long division is a tricky algorithm that many students never master; instead, we will let the area model do the hard work! Label the rectangle area with the dividend value (the first number in the division) and one side with the divisor. When we find it, the other side of the rectangle will be the quotient.

This process is done in stages, and the prompting questions at each step are “what multiplier do I know?” and “what’s left?”.  This series of diagrams shows the division 304 8 = 30 + 8 = 38.

Dividing 304 by 8

Here we tackle the division 7785 15. Notice that the student doesn’t need to use the “best” partial quotient in order to be successful calculating the correct quotient, there’s more than one way to do it!^Note3

Dividing 7785 by 15

Remainders can even be accommodated by using any leftover amount as the numerator over the divisor for a remainder fraction.

How about polynomial division?  We use two similar questions: “what is the multiplier?” and “what’s needed to make the next term?” This series of diagrams shows the division of (x³ + 2x² – 5x – 6) (x + 1).

We start (left diagram below) by filling in the first term of the dividend, which is the desired area, and ask “what is the multiplier?” to create that term. Since x² x = x³, the x² is placed above that column as the “length” dimension. The column is filled in by distributing the x² multiplier to any other terms in their rows.

The next question (right diagram below) is “what’s needed to make the next term?” of 2x². Look at the red circled sections; we have an x² in the first column, so we need another x² in the top box of the second column. The multiplier is x, and again we distribute the multiplication down to the rest of the column.

Dividing polynomials with area model

The process continues: on the left below, look at the yellow circled sections. What is needed to make –5x if we have x already? On the right below, the –6x is filled in to make the desired sum, which makes –6 the multiplier at the top of that column. Distribute to complete the work, finding the quotient of x² + x – 6.

Dividing polynomials with area model

This procedure might seem detailed, but in my experience, it is easier for students to grasp than the long division or synthetic division algorithms, which can seem abstract and devoid of meaning. Since the area diagram represents the division, students have a visual framework for the process. It is easy to check your work as you go, simply by multiplying lengths to get the area in the box.

One thing to note is that you need to account for “zero terms” just as you would with long division or synthetic division. So to do (x³ – 1) (x – 1), think of the cubic as x³ + 0x² +0x – 1.

Sometimes this is called the “Reverse-Area Model” or “Backwards Multiplication,” which are terms I like because they refer to how division is the inverse operation of multiplication. Try out one of the Area Model polynomial divisions^Note4 yourself to see how you like it!

Simplifying Radicals

Irrational numbers like are harder to represent geometrically than integers, but it turns out that the area model sheds light on the situation^Note5.

Think about perfect square numbers such as 4. A square with area = 4 will have each side length = = 2.  A square with area = 5 has each side length = .

Squares with areas of 4 and 5

Let’s leverage this approach to simplify a radical like . Visualize a square with area 12, so each side has length . Subdivide it into a perfect square number of smaller squares; this parallels the process of searching for a perfect square factor. Each smaller square has area = 3 and side length . Since the overall side length is , that is the simplified radical.

Simplifying a radical by subdividing into 4 squares

Here’s another example: simplifying . This time we subdivide into 9 smaller squares, since 54 isn’t divisible by 4.

Simplifying a radical by subdividing into 9 squares

We choose the area decomposition specifically with a square number factor in mind.  To simplify , both 6•2 and 4•3 are correct factor pairs, but choosing is a more useful pair for simplifying.

Completing the Square

Probably the toughest algorithm I teach to Algebra students is the procedure for completing the square. There are several steps and it is difficult for students to remember their order and details.  Most students never realize that we are “completing” an actual square, which is why an area model is the ticket to success.

First, students experiment with multiplying a binomial by itself, and I guide them to draw square boxes rather than rectangles, since the sides are equal.  The example (x + 3)² is on left below, and the general case (A + B)² is on right below.

Squaring a binomial

Then we “level up” by using algebra tiles to build the square area.  Using physical tiles or virtual ones (like these from Mathigon Polypad) allows students to explore features of these arrangements.

To lay the groundwork for completing the square, students area guided to notice that there are always the same number of the X’s (rectangular bars) in each section of the square, and thus the total number of X’s is even. 

Using the algebra tiles to represent area helps students to complete the square for problems with either an even or odd coefficient B of the linear term in Ax² + Bx + C. Recall that there are two reasons to complete the square: to convert the quadratic function y = Ax² + Bx + C to vertex form y = A(x – H)² + K; or to solve the equation Ax² + Bx + C = 0 if factoring doesn’t work out.

These diagrams demonstrate completing the square on y = x² + 8x + 1.

Completing the Square

The algebraic steps must include adding 15 to both sides of the original equation. Then factor into a binomial squared, which is the square area represented as (side)². Finally rearrange into vertex form.

Mathigon even supports slicing an area in half, to model completing the square when the B coefficient is odd. This example begins with y = x² + 7x + 4. Notice that 8.25 squares are added: 5 full squares, 6 half squares, and 1 quarter square. Students progress from these introductory examples to the realization that the unit squares will total .

Completing the Square with odd Linear coefficient

Wrapping Up

This post has covered several AREAS of math in which using an AREA MODEL can help your students visualize some tough mathematical procedures. Adding in the geometric representation to the numerical or algebraic algorithm will deepen your students’ understanding and help make their learning stick!

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Notes & Resources

¹ In fact, you can also use common denominators for fraction division, so multiplication is the only fraction operation that doesn’t use them. Thanks to Graham Fletcher (@gfletchy) for this insight. Here is a good explanatory post.

² I used to call this the “Box Method” for multiplying polynomials (which kind of misses the connection to AREA) and introduced it to my students only when we had trinomials as factors. Now I show this technique for all kinds of polynomial multiplication.

³ Howie Hua (@howie_hua) explains the dividing technique in this video. More than one correct way to compute is always an advantage!

⁴ To practice more area model polynomial division, check out the examples and explanations in this blog post from Math Recreation, which has a few follow-up posts. Dan calls it the “Grid Method.” And Howie Hua shares this video on polynomial division.

⁵ I learned about this method to simplify radicals from Marc Garneau (@314piman). A good summary is found in this blog post from Chris Hunter. Thanks to Anna Blinstein (@ablinstein) who reminded me of this method after reading my “Shout Out For Squares!” post last month. Anna learned about it from Henri Picciotto’s (@hpicciotto) wonderful Geometry Labs (section 9).

#SoME3 This post is an entry in the third “Summer of Math Exposition,” an annual competition to foster the creation of excellent math content online. Organized by Grant Sanderson (@3blue1brown) and others, there were over 500 entries last year in either video and blog format. More information is at some.3b1b.co . WISH ME LUCK!

This post is longer than I anticipated; I would have split it into two more compact posts, except that I can only submit one entry to SoME3. Thanks for reading!!

Source/Disclaimer: AFTER I had written this post, I discovered James Tanton’s excellent and thorough online course “The Astounding Power of Area.” It is worth a look to see how James explains these and other math applications of area.

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Reflections and Tangents by Karen D. Campe is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License.
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