How to Use Algebra Tiles to Multiply Polynomials -- with pictures!

This post shows how to use algebra tiles to demonstrate polynomial multiplication. Two different examples are shown through photos of the algebra tiles of multiplying polynomials.

I set a pretty average goal for myself a couple months ago to blog about all the ways to use algebra tiles. But then of course life got in the way and here I am only on post #3-- how to use algebra tiles to multiply polynomials. I'll get there, but it may take more time than anticipated!


New to algebra tiles? Watch my algebra tiles tutorial video here.


My favorite use of algebra tiles is for factoring quadratics, especially where the A value is greater than 1. Using the tiles makes this process so much more concrete than any other method (in my opinion). Because it's my favorite, I jumped to blog about it first. But before we even get to factoring, we learn how to multiply the binomials, which is what this post is about.

But first! If you don't have a set of algebra tiles, here is a free set of printable paper algebra tiles. If you print on 2-sided card stock (like Astrobrights) it'll mimic the positive and negative sides of the plastic tiles.

Here we go!


Wondering how to use algebra tiles to multiply polynomials? Algebra tiles are perfect for making abstract concepts more concrete for our hands-on and visual learners. In this post are 2 examples with pictures for using algebra tiles to multiply polynomials.


Example 1:
Multiply (x + 3)(x - 4)

Algebra tiles are perfect for making abstract concepts more concrete for our hands-on and visual learners. In this post are 2 examples with pictures for using algebra tiles to multiply polynomials.

Algebra tiles come in 3 shapes: large square (for +/-x2), long rectangle (for +/-x) and small square (for the +/- constant values). 

The goal of using them to multiply polynomials is to build a rectangular area. This area will have side side lengths of the two binomials you are multiplying (the picture above shows this better than I can put into words). 

Algebra tiles are perfect for making abstract concepts more concrete for our hands-on and visual learners. In this post are 2 examples with pictures for using algebra tiles to multiply polynomials.

The large blue square is now there to show (x)(x) = x2Now, because (+)(-) = (-), we will stack 4 (-x) rectangles horizontally below our blue x2 to show (+x)(-4). 

Algebra tiles are perfect for making abstract concepts more concrete for our hands-on and visual learners. In this post are 2 examples with pictures for using algebra tiles to multiply polynomials.

Now we lay 3 (+x) rectangles vertically to show (+x)(+3).
Algebra tiles are perfect for making abstract concepts more concrete for our hands-on and visual learners. In this post are 2 examples with pictures for using algebra tiles to multiply polynomials.

Lastly, we just fill in the space to complete this rectangular-shaped puzzle. (+)(-) = (-), so we need 12 small (-) constant tiles. 



Example 2:
Multiply (2x - 1)(x - 1)

Algebra tiles are perfect for making abstract concepts more concrete for our hands-on and visual learners. In this post are 2 examples with pictures for using algebra tiles to multiply polynomials.

This is the exact same process as Example 1, except we'll use 2 big blue x2 squares to show (2x)(x). We first line up tiles on the sides to show 2x - 1 and x - 1.


Algebra tiles are perfect for making abstract concepts more concrete for our hands-on and visual learners. In this post are 2 examples with pictures for using algebra tiles to multiply polynomials.

Here we've filled in the space with 2 (x2) tiles. 

Algebra tiles are perfect for making abstract concepts more concrete for our hands-on and visual learners. In this post are 2 examples with pictures for using algebra tiles to multiply polynomials.

Now we place 2 rectangular (-x) tiles horizontally below the blue x2 tiles to show that we are multiplying the 2 green (x) tiles on top by the one small (-) tile on the left side. We put them side by side in this example because the goal is to make a rectangle with the tiles, and this is the way they fit.

Algebra tiles are perfect for making abstract concepts more concrete for our hands-on and visual learners. In this post are 2 examples with pictures for using algebra tiles to multiply polynomials.

And then fill in that skinny little column on the right with another rectangular (-x) tile.
Algebra tiles are perfect for making abstract concepts more concrete for our hands-on and visual learners. In this post are 2 examples with pictures for using algebra tiles to multiply polynomials.

Lastly, to complete the rectangular puzzle, we plop one (+) tile in that tiny bottom right corner to show (-)(-) = +.

And that is it! If you have a large set of algebra tiles, you can pretty much multiply any two binomials. What would an x3 tile look like?



Additional resources:

Here is puzzle #5 of a multiplying polynomials digital math escape room. It presents polynomial multiplication as the side lengths of rectangles, the wall lengths of blueprints and as straightforward polynomial multiplication problems. To meet the needs of students working online, I've made over 50 of these digital math escape rooms, all built in Google Forms to be super easy to send to students.  

Puzzle #5 from a Multiplying Polynomials Digital Math Escape Room
Multiplying polynomials digital math escape room


If you'd like to learn more about ways to use algebra tiles, I have put together an algebra tiles tutorial video that covers ways to use algebra tiles in middle school math.


Using Algebra Tiles in Middle School Math:



watch video



2 comments:

Unknown said...

I love this idea!!!

I think on example two you should have put a positive small square.
(2x+1)(x-1)=
2x^2-2x+1x-1=
2x^2-1x-1
Final answer: 2x^2-1x-1

ScaffoldedMath said...

Thank you! I need you in my corner before I push publish! I think it's all fixed now. I really appreciate you pointing that out!