Section 7.5 Going Deeper: Algebra Tiles
For many students, one of their struggles with mathematics is that all the concepts feel extremely abstract to them. This can make math particularly difficult for students that are more visual or kinesthetic in their learning. A number of educational manipulatives have been created to help students connect to mathematical thinking, but not all teachers are familiar with them or have encouraged their students to embrace them. We will be discussing a few of these as we work our way through the content to help bring deeper insights into the material.
Algebra tiles were introduced in this section to help visualize products, but they can be used more generally to help represent a range of algebraic concepts. We will start by describing the tiles themselves.
There are three basic shapes of tiles, and each one comes in two colors. The three shapes are a small square, a rectangle, and a large square. The dimensions of the different shapes correspond to each other, so that the narrow side of the rectangle matches with the small square and the long side of the rectangle matches with the large square. The actual relationship between the sides of the small square and large square are irrelevant. They just need visibly different in size. The small square is a unit tile, the rectangle is an \(x\) tile, and the large square is an \(x^2\) tile. The two colors represent either a positive or negative version of the various tiles.
Subsection 7.5.1 Representing and Simplifying Algebriac Expressions
Algebraic expressions can be represented by collections of tiles. Here are some examples:
Adding algebraic expressions together can be accomplished by combining two collections of tiles together. When combining tiles, there is an additional rule where a positive and negative version of the same shape will cancel each other out. When working with manipulatives, this is accomplished by simply setting the canceled tiles off to the side. For visualization purposes in this book, we will use a light gray outline to represent a pair of canceled tiles.
Subtracting is similar to addition, except that we first need to swap the colors of the tile. This corresponds to distributing the negative sign across the parentheses.
The distributive property can be thought of as having multiple bundles of the tiles in the parentheses. If the distributive property is paired with subtraction, we create multiple copies first, then subtract by swapping the colors.
Subsection 7.5.2 Solving Equations
The process of solving equations with algebra tiles focuses on the idea that the two quantities on either side of the equation must be the same. On each side, we can use any of the simplification steps from before. In addition to this, there are two other operations. We can add the same set of tiles to both sides, and we can divide the blocks on each side into an equal number of groups.
Here is an example of solving the equation \(x + 1 = -x + 5\text{:}\)
This process directly mirrors the algebraic steps involved in solving equations. You should be able to recognize that adding tiles is either addition or subtraction of algebraic expressions (depending on whether positive or negative tiles are added) and the grouping step is related to division.
Subsection 7.5.3 Other Uses of Algebra Tiles
These are not the only ideas that algebra tiles can represent. In this section, we briefly mentioned how algebra tiles can be used for representing multiplication. The idea here is to think of the tiles representing the length and width of a rectangle, and then additional tiles can be used to represent the result as an area. We'll later see how tiles can be used to help factor polynomials.
