MAY 24, 2007

Algebra Tiles Explained: What Do They Mean Exactly?

BY CAMILA TORRES-BERTRAND

Many people believe that algebra tiles are new in mathematics instruction, however, algebra tiles have been used in classrooms since the mid 1980s. Since then, algebra tiles have gained popularity in teaching circles and are regularly presented in conjunction with traditional methods in most textbooks. Unlike a calculator or an answer key, which does the thinking for you, algebra tiles are a tool that can guide the learner towards a greater understanding of the concepts involved with a particular skill.

Studies show that over 80% of students are visual learners rather than auditory learners. Therefore, it makes sense that the majority of students benefit from a visual, concrete representation of abstract algebraic concepts. Algebra tiles do just this ?they allow students to match a concept to a tile versus trying to imagine everything in their minds. (For more information visit our conceptual math techniques page.)

Area: How can students use what they already know?

Algebra Tiles are created to allow students to view symbolic representations through concrete models while applying previously explored concepts of area.

By the time students are introduced to algebra, they should have already explored the concepts of area and perimeter. However, assuming as student has not solidified their understanding of area and perimeter, consider showing them this figure:

 figure 1a.

When asked “What is the difference between area and perimeter?? students unfamiliar with the concepts will give responses such as “The perimeter is the outside of the rectangle, the area is the stuff inside the rectangle.? This response or similar responses provide teaching opportunities to review area/perimeter concepts and smoothly transition to algebraic concepts.

Consider this diagram which depicts figure 1a recreated with units inside. (Note: the units are square because area measure space in two dimensions: lw.) The measurement of this rectangle is now 5 units (l) by 3 units (w).

 figure 1b.

Instead of simply giving them the formula of the perimeter of a rectangle P = 2l + 2w, I encouraged exploration so they could apply the definition to a variety of shapes in the future. To explore perimeter, students would simply count the sides that have a black border: 5 sides across the top, 3 sides down the right, 5 sides across the bottom and 3 sides up the left. After counting carefully, students will determine that perimeter of the figure is 16 units. Make sure that students know and understand why perimeter is a single dimensional unit ?length (l). The space it is measuring is only being measured in one direction. (For younger students, useful images and tools for measuring one-dimensional space is measuring distances using a string or toothpicks.)

As for area, it is a very different concept. This time, both length and width is being measured. (Perimeter only measured length.) Again, to encourage student thinking, I never gave them the formula for area (lw), rather I allowed the students to explore area for themselves. We ask them to count up the squares in figure 1b. and they quickly determine that there are 15 squares in the figure. If given different rectangles of varying dimensions and units, they will create the formula for themselves. Teachers may further coach the students to discover the concepts by asking them to write or journal about relationships or shortcuts they discovered within the lesson. When working with area, make sure students know and understand why area is a two-dimensional unit ?length (l) and width (w). The space area measures in being measured in two directions. Answers should be given in squared units (i.e. square feet, feet2 , square miles, miles2 , square meters, meters2 , etc.)

It has been a concern that these methodologies are time-consuming and tedious; however they show a tremendous opportunity for students to gain ownership of their own learning. Exploration can solidify this learning in a more meaningful way because they have discovered these relationships on their own. Furthermore, once students start to integrate these concepts volume (measurement in three dimensions) is innate. Teachers may further coach the students to discover the concepts by asking them to write or journal about relationships or shortcuts they discovered within the lesson. As for the formulas for perimeter and area, perhaps one of the most powerful discoveries for some students will ever discover is that multiplication is a shortcut for addition and addition is a shortcut for counting. It refutes the argument that math gets “harder?as you get progress. Rather, in this situation, multiplication (third/fourth grade) made a counting problem (kindergarten/first grade math) easier. Believe it or not, these shortcuts continue all the way into calculus and beyond!

The following is an introduction to the different types of algebra tiles in a typical algebra tile set.

The Ones

The “ones?piece is a small square. Typically, one side is yellow and the reverse is red. The dimensions of the square are one (1) unit by one (1) unit. Therefore using the area model:

POSITIVE ONE: The positive one tile is represented by a small, yellow square. The dimensions of the square are 1 unit by 1 unit so therefore the area model would show that the small, yellow square has an area of 1 square unit. In addition to reinforcing the ideas behind the area model of a four sided figure, the notion of when a number is “squared?it actually can be represented as the shape of a square.

NEGATIVE ONE: Students need to understand the concept of negative being the opposite. When using algebra tiles, a negative one is represented by the same tile as a positive one. However, the negative is denoted by the red color of the same small square piece. When the yellow square is flipped over to its opposite side, the color is red which represents negative.

The X s

The ?x ?piece is a rectangle. Typically one side is green and the reverse is red. The orientation of the piece in space, vertical or horizontal does not affect the significance of piece. The dimensions of the rectangle are one (1) unit by  x unit(s).

Therefore using the area model:

POSITIVE X: The positive tile is represented by a green rectangle. The dimensions of the rectangle are 1 unit by an unknown length of the same unit. Since the sides do not have the same length, a rectangle is formed instead of a square.

If one were to line up the one tiles against the long side of  x tile, if would not line up perfectly because the length is unknown. It is important for students to understand that there is no specific length associated with the unknown length side. That’s the point, it can be anything. However, if a one tile was lined up against the shorter side of the rectangle, it would line up perfectly because it shares the same dimensions ?one unit.

NEGATIVE  X: The negative  x is represented by the same tile, however just as for the ones tiles, the red color denotes the negative. When the green (positive) tile is flipped to its opposite side and displays the color red (negative) because the opposite of positive  x is negative  x.

The X2

The x2 piece is a large square. Typically one side is blue and the reverse is red. The dimensions of the square are x unit(s) by unit(s).

POSITIVE X2: The positive x2 tile is represented by a large blue square. The dimensions of the blue square are one unknown length by the same unknown length with similar units

If one were to line up ones tiles against a side of the x2 tile, it would not line up perfectly because the length is supposed to be unknown. It is important for students to understand that there is no specific length associated with the unknown length side. However, if long side of the green tile was lined up against either side of the blue tile, it would line up perfectly because it shares the same unknown length. In other words, the unknown length of the blue square is the same length as the unknown length of the long side of the green rectangle.

NEGATIVE X2: The negative x2 tile is represented by the same blue square, but just as in all the previous examples, the red color denotes the negative. When the blue (positive) tile is flipped, the tile becomes red (negative) because the opposite of positive x2 is negative x2

The colors used in our explanations of algebra tiles are most commonly used in textbooks and manipulative sets sold commercially. It is important to note that previous and older versions of algebra manipulatives denoted the negative side with the color black. The positive side was usually represented using a variety of bright colors like red, yellow, green and blue. We at www.aplusalgebra.com have used the most widely-used color scheme for the A+ Algebra TabletTM used to create our lessons.

Want your own set of tiles to practice? Download, print and cut out your set for FREE!
(Helpful Hint: For maximum learning, use paper that has a different color on each side.)

Want something a little more “high tech? The A+ Algebra TabletTM brings algebra manipulatives to the 21st Century by allowing users to use the tiles on a desktop computer, laptop computer, tablet PC or interactive whiteboard.