Algebra tiles are the coolest little things. They are a wonderful hands-on tool that can be used to visually represent so many topics in algebra, from solving equations to multiplying binomials to factoring quadratic trinomials.

In this post, I want to focus on that last topic -- using algebra tiles to factor quadratic trinomials. Algebra tiles are a perfect way to introduce and practice this concept. They take a lot of the guesswork out of factoring, especially for trinomials that are not easily factored with other methods.

Below are 4 examples of how to use algebra tiles to factor, starting with a trinomial where A=1 (and the B and C values are both positive), all the way to a trinomial with A>1 (and negative B and/or C values).

Here is a look at the tiles in this post:

In my set of algebra tiles, the same-size tiles are double-sided with + on one side and - on the other. You can get a similar effect by printing this free printable set of algebra tiles on astrobrights paper (or glue 2 different colored pieces of paper together back-to-back before cutting).

Let's jump into the examples!

###
__Example 1__:

__Example 1__:

###
Factor **x**^{2} + 5x + 6

^{2}+ 5x + 6

Factoring this trinomial is a nice place to start because the A value is 1 and both the B and C values are positive. We'll be able to get a good feel for how the algebra tiles work with this example.

__Needed__:1 x

^{2}tile

5 rectangular x tiles

6 + tiles

Here we have all of the tiles we will need to factor this trinomial. Now it's just a matter of putting the puzzle together.

First, I tried (x + 4)(x + 1). This used all of the rectangular x tiles but not all of the + tiles. In order for the trinomial to be factored correctly, all tiles have to fit together to make a perfect rectangle.

Next I tried (x + 2)(x + 3), which allowed all 6 of the + tiles to fit.

*x*

^{2}+ 5x + 6 factors to (x + 2)(x + 3).###
__Example 2__:

__Example 2__:

###
Factor 2x^{2} + 3x + 1

Algebra tiles are great for factoring quadratic trinomials where the A value is not 1. I have always liked the AC method for factoring these trinomials, but even I will admit that factoring with algebra tiles in this instance is all around way better.

__Needed__:2 x

^{2}tiles

3 rectangular x tiles

1 + tile

Here we have all the tiles we need to factor this trinomial. With trinomials where the A, B and C values are all positive, we start and finish with the same number of tiles. (In a minute when we factor a trinomial where B and/or C are negative, the approach will be slightly different.)

Arranging the algebra tiles into (x + 1)(2x + 1) allowed all of the tiles to fit together into a nice, neat rectangle.

*2x*

^{2}+ 3x + 1 factors to (x + 1)(2x + 1).###
__Example 3__:

__Example 3__:

###
Factor x^{2} - x - 12

When we factor quadratic trinomials involving negatives, we may not start and end with the same number of tiles. We may have to add in some zero pairs as we go. In the case of x

^{2}- x - 12, our B value is -1.

__Needed (to start)__:1 x

^{2}tile

1 rectangular -x tile

12 - tiles

Since there was no way to make a rectangle that fit those 12 - tiles, we needed to add in some additional zero pairs (1 of each rectangular + and - x tile).

We can do this because:

-1 + 0 = -1 or

-2 + 1 = -1 or

-3 + 2 = -1 .... and so on...

By adding in equal amounts of rectangular + and - x tiles, we are not changing the trinomial's B value. It's still -1.

Here we have 3 rectangular -x tiles and 2 rectangular x tiles. Still not enough to fit those 12 - tiles, though the B value is still being represented as -1.

We got it! We needed 4 rectangular -x tiles and 3 rectangular x tiles to make all 12 - tiles fit. -4 + 3 - =1 so we did not change the trinomial's B value.

*x*

^{2}- x - 12 factors to (x - 4)(x + 3).###
__Example 4__:

__Example 4__:

###
Factor 2x^{2} + x - 3

Using Algebra tiles is so helpful when factoring quadratic trinomials where the A value is greater than 1 and B and/or C are negative. The tiles make the process so much more intuitive!

__Needed (to start)__:
2 x

^{2}tiles
1 rectangular x tile

3 - tiles

Right away it's obvious that we do not have enough pieces to make a nice, even rectangle. So we will need to start adding zero pairs to keep B at +1. Here I added a rectangular +x and a rectangular -x. This wasn't quite enough to fill it in.

I added one more zero pair of x tiles and our rectangle is complete!

*2x*

^{2}+ x - 3 factors to (x - 1)(2x + 3).I had gotten this set of algebra tiles secondhand online. If you don't have a set, you can print this free set of printable algebra tiles.

There are 2 versions of the algebra tiles in the pdf. The first page mirrors the dimensions of my plastic set. The algebra tiles on the second page are a little larger for kids who may have trouble with the smaller size.

There are also a bunch of online algebra tile sites. This one gives example problems to factor and has tiles to drag into a workspace.

I've been working with a friend on a new math game series called Voyage to the Treasure where students work together to beat the game board. This factoring trinomials game covers trinomials where A>1.

There are more Voyage to the Treasure games, a video and a more detailed set of directions in this post.

I added these algebra tile references into my algebra word wall so that students can make the connection between factoring and area. The vertex form model is cool because a quadratic in vertex form is missing a few pieces because of that k value at the end.

__Related posts:__Solving Equations Using Algebra Tiles

Cereal Factoring

Fun with Quadratics

Amazing!

ReplyDelete:)

Delete