Multiplication of Polynomials – Area Model

This is my third post in the series 'What's Up with The Rectangle?'. In the first post, I talked about the grid/area model as a model for arranging and understanding information. In the second post, I briefly covered polynomials and rapid mental calculation. That was a huge leap for some folks, so I am going to back track a bit and cover multiplication of polynomials assuming you know nothing.

Before we examine multiplication of polynomials, we first need to know:

What's a Polynomial?

A polynomial is a math expression that contains many (poly = many) terms. A polynomial contains one or more of the following terms:


Constants are values that don't change.​

1, 2, 5, 10, 1/2, 1/3 . . .


Variables are the things that freak people out when it comes to algebra. Variables can change in value.​

 x, y, xy, xyz . . .


An exponent is what happens when a quantity is multiplied by itself. The number in the exponent tells you how many times it is multiplied by itself.

​x2 , 32, y4, but not x-2 

Negative and fractional exponents of a variable are not allowed.

These are polynomials: 

5x - 5

5x2  - 13x + 12

 -5y2 - (5/9 x)


These are not polynomials


Polynomial terms can be combined using addition and subtraction, but not multiplication or division. 

Addition vs Multiplication of Variables

What is a variable? A variable, in math, is usually represented by a letter. X, Y or some other letter. It represents something that is unknown to us. But just because it is unknown doesn't mean we can't count it. 

​When we add a variable, we are just gathering them all together and counting them. Just as we would numbers.  I've used yellow Cuisenaire Rods to represent the variable because, with Cuisenaire Rods, the rods don't have a fixed value - one can be anything.

Adding variables with Cuisenaire Rods

Adding variables with Cuisenaire rods.

y + y

adding variables with Cuisenaire Rods

y + y + y + y + y + y

When multiplying, we know that 3 x 4 is the same as 4 x 3.

Multiplying with Cuisenaire Rods

All we're saying is that there are 3 of the 4's or there are 4 of the 3's.

Multiplying Variables with Cuisenaire Rods

The process is still the same except we don't calculate the final total. We leave it as p x g or pg.

​If we multiply a variable by itself we get exponents, just as we would get if we were multiplying a number we can count. 

Squaring variables with Cuisenaire Rods.

We can count the green rods in several ways:

The slowest way is to add them: g + g + g + g + g + g = 6g

We can count them by multiplying, there are 6 greens therefore:  6 x g = 6g

We could also count them: g x g = g2.

Multiplication of Polynomials Using Rods

If we set up the statement (x + 4) times (X + 5) we will let the orange rod = x and the cream colored square = x2:

We have x + 5 across the top and x + 4 down the side. 

Multiplication of Polynomials Step 1

Here we've multiplied x  by x and the result was x2.

Multiplication of Polynomials Step 2

In the image on the left, we multiplied x by 5. On the right we multiplied x by 4. 

Multiplication of Polynomials Step 3
Multiplication of Polynomials Step 3

One Last Step Before We Are Finished

Multiplication of Polynomials Step 5

The last thing is to multiply 4 x 5 and that will provide the the last corner on the right.

When we set up the grid, or the rods, it helps us to understand the math behind the distributive property. Everything must be multiplied by everything else. The rods and the grid help us to make sure we have expanded the brackets without missing anything. If you have forgotten to multiply something, it becomes visually obvious that you've missed it. 

Multiplication of Polynomials on a Grid/Area Model

So let's go back to the grid and multiply those same two polynomials together x + 4 and x + 5, you might want to check my first post on the rectangle if this is confusing:

This is exactly the same thing we are doing with the rods, except we've laid out the information in a table. You can see that the rods match the information in the grid. Mortensen and Crewton Ramone's House of Math have very strong opinions about how you set up the rods. Gattegno allows the student to decide what makes most sense to them. I have no opinion on the matter and you should do what makes most sense to you.

If you go back to my last post you can see how laying out the rods in a grid pattern can help you understand polynomials. In my next post on the rectangle, I'll be covering polynomials and fancy counting.

In the meantime, if you want to learn more about using Cuisenaire Rods or other base-ten blocks like Mortensen or Math-U-See to help your children learn math, you can join me over on Facebook in our math group for base-ten block users. We're a mixed group of homeschoolers, tutors and teachers who are all learning to use the rods to teach math. Some of us have a good grasp of math, but most of us had a poor math education and need a lot of hand holding. There is no judgement, and no fear of looking dumb. 

Arithmophobia No More