ALGEBRA TILES The University of Texas at Dallas

INTRODUCTION Ø Algebra tiles can be used to model algebraic expressions and operations with algebraic expressions. Ø There are three types of tiles: 1. Large square with x as its length and width. 2. Rectangle with x and 1 as its length and its width 3. Small square with 1 as its length and width. x x 1 1 The University of Texas at Dallas

INTRODUCTION Ø Each tile represents an area. x Area of large square = x (x) = x 2 x 1 x Area of rectangle = 1 (x) = x 1 1 Area of small square = 1 (1) = 1 The University of Texas at Dallas

INTRODUCTION Ø Does an x-tile need to be any particular length? Ø Does its length matter? Ø x-tile does not have to be a particular FOR TUTOR length because it. NOTE represents a variable. Its length does not matter. The University of Texas at Dallas

ALGEBRAIC EXPRESSIONS Ø To x 2 model 2 x 2, you need 2 large squares x 2 The University of Texas at Dallas

AGEBRAIC EXPRESSIONS Ø To model x 2 + 3 x, you need 1 large square and 3 rectangles x 2 x x x The University of Texas at Dallas

ALGEBRAIC EXPRESSIONS ► How would you model 2 x 2 + x + 4? ANSWER x 2 x 1 1 The University of Texas at Dallas

ALGEBRAIC EXPRESSIONS ► What algebraic expression is modeled below? ANSWER 2 x + 3 The University of Texas at Dallas

ALGEBRAIC EVALUATION ► Suppose x is standing for 4. What is the value of 2 x +3 ? (2 x means 2 times x). ► 2 x +3 = 2 (4) +3 = 8 +3 = 11 The University of Texas at Dallas

ALGEBRAIC EVALUATION Ø What is the value of 2 x + 3 if x = 8? ANSWER Ø 2 (8) + 3 Ø 16 + 3 Ø 19 2 x + 3 =19 if x = 8 The University of Texas at Dallas

ALGEBRAIC EVALUATION ► Remember that a variable can represent many number. The University of Texas at Dallas

ALGEBRAIC EVALUATION ► Find the value of this expression using these values of x. X X X 1 x=2 ► x=5 ► x = 10 1 1 ► ANSWER The University of Texas at Dallas

ALGEBRAIC EVALUATION ► The algebraic expression for the tiles is 3 x + 5 1) If x = 2 2) 3 (2) + 5 3) 6 3) If x = 10 3 (10) + 5 30 + 5 = 35 +5= 11 4) 2) If x = 5 5) 3 (5) + 5 The University of Texas at Dallas

ALGEBRAIC OPERATIONS ► We can use algebra tiles to model adding, subtracting, multiplying, and dividing algebraic expressions The University of Texas at Dallas

ALGEBRAIC ADDITION ► To use algebra tiles to model 3 + (2 x + 4) represent each addend with tiles. 3 1 1 1 + 2 x + 4 + x = 2 x + 7 x 1 1 = x x 1 1 1 1 Combine the tiles The University of Texas at Dallas

ALGEBRAIC ADDITION ► Find the sum: ► 2 x + (5 x + 4) x x + x x x 1 1 = ANSWER x x x 1 1 2 x + (5 x + 4) = 7 x + 4 The University of Texas at Dallas

ALGEBRAIC ADDITION ► Find the sum: ► (x + 3) + (2 x + 4) ANSWER x 1 1 1 + x x 1 1 = x x x 1 1 1 1 (x + 3) + (2 x + 4) = 3 x + 7 The University of Texas at Dallas

ALGEBRAIC ADDITION ► Find the sum: ► (x 2 + 3) + (2 x 2 + x + 2) ANSWER x 2 x 2 1 1 1 + x 2 x 1 1 = x 2 x 1 1 1 (x 2 + 3) + (2 x 2 + x +2) = 3 x 2 + x + 5 The University of Texas at Dallas

ALGEBRAIC SUBTRACTION ► To use algebra tiles to model subtraction, represent each expression with tiles. Put the second expression under the first. ► (5 x + 4) – (2 x + 3) x x x 1 1 1 1 x Now remove the 5 x + 4 tiles which match in each The answer is the expression 2 x + 3 that is left. (5 x + 4) – (2 x + 3)= 3 x +1 The University of Texas at Dallas

ALGEBRAIC SUBTRACTION ► Use algebra tiles to find the difference. ► (8 x + 5) – (6 x + 1) ANSWER 8 x x - x x x x 1 1 +5 1 6 x + 1 1 (8 x + 5) – (6 x + 1) = 2 x + 4 The University of Texas at Dallas

ALGEBRAIC SUBTRACTION ► Find the difference. ► (6 x + 1) – (3 x) x x x 6 x ANSWER +1 x 1 - x x x 3 x (6 x + 1) – (3 x) = 3 x +1 The University of Texas at Dallas

ALGEBRAIC SUBTRACTION ► Find the difference. ► (5 x + 6) – (5 x) - x x x x x ANSWER 5 x + 6 1 1 1 5 x (5 x + 6) – (5 x) = 6 The University of Texas at Dallas

ALGEBRAIC SUBTRACTION ► Use algebra tiles to find the difference. ► (3 x 2 + 4 x + 5) – (x 2 + 3 x + 4) 3 x 2 + 4 x + 5 ANSWER x 2 - x 2 x x 1 1 1 x 2 + 3 x + 4 x 2 x x x 1 1 (3 x 2 + 4 x + 5) – (x 2 + 3 x + 4) = 2 x 2 + x + 1 The University of Texas at Dallas

ALGEBRAIC MULTIPLICATION ► To multiply using algebra tiles, lay the factors in a rectangular array. ► Ex: 2 (x + 3) 2 1 1 x x+3 11 1 x x 1 11 Fill in this space to form a rectangle. And the result is your answer 2 (x + 3) = 2 x + 6 The University of Texas at Dallas

ALGEBRAIC MULTIPLICATION ► Find the product. ► x (x + 2) ANSWER (x + 2) x x 2 11 xx x (x + 2) = x 2 + 2 x The University of Texas at Dallas

ALGEBRAIC MULTIPLICATION ► Find the product. ► 2 x (x + 3) ANSWER (x + 3) x 2 x 111 x x 2 xxx 2 x (x + 3) = 2 x 2 + 6 x The University of Texas at Dallas

ALGEBRAIC MULTIPLICATION ► Find the product. ► (x+2) (x + 4) ANSWER (x + 4) x x +2 1 1 x 2 xx x 1111 xxxx 1111 (x+2) (x + 4) = 2 x 2 + 6 x + 8 The University of Texas at Dallas

ALGEBRAIC MULTIPLICATION ► Find the product. ► (x+3) (2 x + 1) ANSWER x 2 xxx x 1 1 1 x 2 xxx x x (2 x + 1) x +3 1 x 1 1 1 (x+3) (2 x + 1) = 2 x 2 + 7 x + 3 The University of Texas at Dallas

ALGEBRAIC DIVISION ► To model division using algebra tiles, use a rectangular array like you did for multiplication except the dividend (the numerator) goes where your multiplication answer was The University of Texas at Dallas

ALGEBRAIC DIVISION ► Ex: 1 1 x 2 x Fill in this space to complete the array. This is your answer! 11 1 x x 1 11 4 x + 6 The University of Texas at Dallas

ALGEBRAIC DIVISION ► Find the quotient: x x 3 x ANSWER 1 xxx 1 1 1 xxx (9 x + 3) 1 1 1 The University of Texas at Dallas

ALGEBRAIC DIVISION ► Find 6 x 2 + 3 x 3 x the quotient: x 3 x x ANSWER 1 (6 x 2 + 3 x) x x 2 x 2 x x 2 6 x 2 + 3 x 3 x = 2 x + 1 The University of Texas at Dallas x

ALGEBRAIC DIVISION ► Find 2 x 2 + 6 x + 4 x+2 the quotient: x x+ 2 x ANSWER 11 (2 x 2 + 6 x + 4) xx x 2 xx xx x 1 1 x 2 11 11 2 x 2 + 6 x + 4 = 2 x + 2 x+2 The University of Texas at Dallas