Algebra tiles are a set of square and rectangular-shaped tiles used to represent numbers and variables. A typical set of algebra tiles consists of blocks that represent positive and negative values of $1,$ $x,$ and $x^{2} .$

These tiles can be used to model any expression involving $x,$ $x^{2},$ and numbers. For example, to represent the number $3,$ you might use three unit tiles. Similarly, to represent $2 x,$ you can use two $x -$ tiles. Now, consider this problem.

Suppose one person in your team has two $x^{2} -$ tiles, three $x -$ tiles, and one unit tile on his desk and another person has one $x^{2} -$ tile, five $x -$ tiles, and eight unit tiles on her desk. You decide to put all of the tiles together on one desk. What is the name for this new group of tiles?

Let's begin by writing down the number and types of tiles. In one person's pile, there are two $x^{2} -$ tiles, three $x -$ tiles, and one unit tile. Then, let's use appropriate tiles to make this group.

Next, we read that these tiles are combined with one $x^{2} -$ tile, five $x -$ tiles, and eight unit tiles. Let's add them onto the graph.

By adding these tiles, we can create an algebraic expression modeled by the tiles.

2 x^{2} + 3 x + 1 + x^{2} + 5 x + 8

Finally, we will combine like terms to simplify the above expression. When doing so, keep in mind that terms that have the same variables and exponents are like terms. Also, all constants are like terms.

2 x^{2} + 3 x + 1 + x^{2} + 5 x + 8

CommutativePropAdd

Commutative Property of Addition

2 x^{2} + x^{2} + 5 x + 3 x + 1 + 8

AddPar

Add parentheses

(2 x^{2} + x^{2}) + (5 x + 3 x) + (1 + 8)

AddTerms

Add terms

3 x^{2} + 8 x + 9

We can confirm that this result is correct by rearranging the new pile. To do so, we will group each type of tiles together.

In this rearranged pile there are three $x^{2} -$ tiles, which represent $3 x^{2} .$ There are also eight $x -$ tiles that represent $8 x$ and nine unit tiles that are equivalent to $9 .$

Together combined they give $3 x^{2} + 8 x + 9,$ which is our previous result. This way we can be sure that our answer is correct!

Exercise