ALGEBRA TILES Expressions and Equations In this presentation, the words “participant” and “facilitator” may also be understood as “teacher” and “student”, although this PowerPoint is not appropriate or intended for use with students. The facilitator, however will be modeling parallel practice. It will be helpful to have an overhead projector and the overhead tiles from the kit. When a problem is posed, participants will solve the problem and then the facilitator will ask for a volunteer to show a solution on the overhead. The solution on the slide is given last, as a validation of the answer rather than the source of the answer.

OBJECTIVES Provide a visual model to help students develop conceptual understanding for variables as unknown values. Build conceptual understanding of solving equations. Construct meaning for algebraic solutions of equations. The rectangle model provides a concrete example of abstractions, which allows students to understand the operations which follow. By now, students have accepted and become comfortable with the use of tiles. The only new concept is the appearance of the variable x, so they are not confused or challenged by the move into algebra.

ALGEBRAIC EXPRESSIONS The value of each piece is determined by its area. A new piece is created by developing a new dimension, “x”. for practical reasons, “x” is not a multiple of the dimension representing “1”. The value of this piece is x, because 1  x = x. For the variables in this set of algebra tiles, the positive pieces are green, and the negatives are red. Students want to “measure” the piece, and determine the value of x in terms of the known unit, 1. Continue to insist that we just don’t now what x is. As they work problems, they will become more comfortable with the changing values of x. (In mathematics, students are conditioned to seek unique solutions and they are not comfortable with the concept of a value that can change, however, in other areas, they use unknowns comfortably. Pronouns are unknowns, and can represent different things at different times. The concrete model allows them to manipulate something they understand, in order to become more familiar with the behaviors of abstractions.) x 1

Use tiles to express the following: EXAMPLES Use tiles to express the following: x + 2 3x 2x -1 -4x The co-effiecient of a variable tell us how many of that variable are used. A negative co-efficient means to use the opposite. Students need practice modeling algebraic expressions. Have them create the expressions as you walk around the room. Students should show and explain their answers. In an algebraic expression, the operation of subtraction is modeled by using negatives. By now, students probably understand the concept, but reinforce often.

COMBINING EXPRESSIONS 3x – 2x x + 4 + x – 3 2x + 3 – x + 1 = x = 2x + 1 Using algebra tiles allows students to strengthen their understanding of the processes involved in combining algebraic expressions. Both subtraction models (take away and adding the opposite) are used on this slide, as well as a zero pair involving variables. It would be good to ask a participant to model adding zeros on the overhead, to demonstrate that the outcome is not changed. When a visual model is being used, it is not necessary to explicitly teach combining like terms. Students holding tiles will not attempt to simplify 2x + 1 as 3x, because the pieces are so obviously different. As students model the answers, they should be explaining : “2x + 3 – x + 1 means take one x from the 2x , leaving x and then put 1 and 3 together, making 4. = x + 4

DISTRIBUTION 3x + 2 2x - 4 3(x + 2) - 4 (5x - 6) -(3x - 2) The distributive property is well modeled with the tiles. It allows students to see what the property really means, and how easily it is applied. What is three sets of (x + 2)?. It is 3 x’s (3x) and 3 2’s (6), so 3(x+2) = 3x + 6. The distributive property is a grouping operation. The co-efficient tells how many groups there are. In the second problem, the distribution of the subtraction sign is not shown explicitly. The model simply takes away the two negatives. Ask a participant to model the symbolic representation on the board, and compare the results. 2x - 4 Tiles make the concepts simple.

MODEL THE EXPRESSION 3x - 1 -6x + 2 -2(3x -1) Begin inside the parenthesis. 3x - 1 There are two sets, and their signs are opposite of the originals. This models nesting operations (order of operations) Begin inside the parentheses and work out. The terminology should reflect the visual model, but then transfer to the later abstractions. Even complicated sets of tiles can be modeled by using the attributes of the tiles. -6x + 2

= SOLVING EQUATIONS x = 2 MODEL 2x + 1 = 5 To solve, SUBTRACT 1 tile from each side. NEW EQUATION 2x = 4 Isolate x by DIVIDING into two groups = The two sets of tiles have equal value. If tiles are added or subtracted from one group, the same must be done to the other group, to maintain the relationship. By isolating one green tile, we can find the number of square tiles it represents. Walk participants through the process. Each algebraic step can be performed concretely, but the parallel algebraic steps have the same effect. x = 2

= SOLVING EQUATIONS -3 -3 2x + 3 = 7 2x = 4 x = 2 2x = 4 x = 2 Subtract 3 from each side of the equation -3 -3 = x = 2 2x = 4 x = 2 Divide by 2 on each side of the equation Again, it is important to remember that Algebra Tiles are a model and not a method. Using the tiles to routinely solve equations is silly and impractical. As the student works through the problems here, they should always be reminded of the algebra being modeled. What we are providing is a way for students to see and know what is happening, so that when we ask “what is the next step”, they have an innate understanding of the process. This is not a set of rules. When tiles are modeled and used, students become more confident and their understandings do not fade. The operations and the result are exactly the same.

SOLVE x – 4 = 5 = In each example, allow participants to attempt on their own. What might be a concern in this problem? The negatives cannot be “subtracted” using the take away model. Add the opposite to form zero pairs. The solutions should be shown on the overhead by volunteers. These answers are merely confirmation. x = 9

SOLVE 2(x - 3) = 10  2x - 6 = 10 = Another example, using the distributive property. 2x = 16 x = 8

ALGEBRA TILES Demonstrate the behavior of numbers so students can form INNATE conceptual understandings. Reaffirm arithmetic and algebraic concepts using a visual model Reassure and encourage struggling students, allowing them to develop number sense and confidence with signed numbers Students who use visual models to develop number sense will become more confident and perform better. The rectangular model used with Algebra Tiles is extremely useful for use with students who have missed the basic concepts with which we have been working today, HOWEVER the model can be extended for use in higher level algebraic topics, such as multiplying binomials, factoring polynomials and division of algebraic expressions. When teachers INTRODUCE concepts with these models, the conceptual understandings are built from day one, and students can avoid the confusion and frustration of incomplete or incorrect understandings.