1. ALGEBRA TILES
The University of Texas at Dallas

2. INTRODUCTION
Algebra tiles can be used to model algebraic expressions and operations with algebraic expressions.
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 1 1 x x 1
The University of Texas at Dallas

3. INTRODUCTION Each tile represents an area.x
Area of large square = x (x) = x2 x 1 x
Area of rectangle = 1 (x) = x 1 1
Area of small square = 1 (1) = 1
The University of Texas at Dallas

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

5. ALGEBRAIC EXPRESSIONS
To model 2x2, you need 2 large squares x2 x2
The University of Texas at Dallas

6. AGEBRAIC EXPRESSIONS
To model x2 + 3x, you need 1 large square and 3 rectangles x2 x x x
The University of Texas at Dallas

7. ALGEBRAIC EXPRESSIONS
How would you model 2x2 + x + 4?
ANSWER x2 x2 x 1 1 1 1
The University of Texas at Dallas

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

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

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

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

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

13. ALGEBRAIC EVALUATION If x = 2 3 (2) + 5 6 + 5 = 11 2) If x = 5
The algebraic expression for the tiles is 3x + 5
If x = 2
3 (2) + 5
= 11
2) If x = 5
3 (5) + 5
= 20
3) If x = 10
3 (10) + 5
= 35
The University of Texas at Dallas

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

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

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

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

18. ALGEBRAIC ADDITION + = (x2 + 3) + (2x2 + x +2) = 3x2 + x + 5
Find the sum:
(x2 + 3) + (2x2 + x + 2)
ANSWER x2 x2 + x2 x2 = x2 x 1 1 1 x 1 1 1 1 1 x2 1 1
(x2 + 3) + (2x2 + x +2) = 3x2 + x + 5
The University of Texas at Dallas

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

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

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

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

23. ALGEBRAIC SUBTRACTION
Use algebra tiles to find the difference.
(3x2 + 4x + 5) – (x2 + 3x + 4)
3x2 + 4x + 5
ANSWER x2 x2 x2 x x x x 1 1 1 1 1
x2 + 3x + 4 - x2 x x x 1 1 1 1
(3x2 + 4x + 5) – (x2 + 3x + 4) = 2x2 + x + 1
The University of Texas at Dallas

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

25. ALGEBRAIC MULTIPLICATION
Find the product.
x (x + 2)
ANSWER
(x + 2) x x 1 1 x x2 x x
x (x + 2) = x2 + 2x
The University of Texas at Dallas

26. ALGEBRAIC MULTIPLICATION
Find the product.
2x (x + 3)
ANSWER
(x + 3) 2x x 1 1 1 x x2 x x x x x2 x x x
2x (x + 3) = 2x2 + 6x
The University of Texas at Dallas

27. ALGEBRAIC MULTIPLICATION
Find the product.
(x+2) (x + 4)
ANSWER
(x + 4)
x +2 x 1 1 1 1 x x2 x x x x x x 1 1 1 1 1 1 1 1 1 1
(x+2) (x + 4) = 2x2 + 6x + 8
The University of Texas at Dallas

28. ALGEBRAIC MULTIPLICATION
Find the product.
(x+3) (2x + 1)
ANSWER
(2x + 1)
x +3 x x 1 x x2 x2 x x x x x 1 1 x x 1 1 1 1
(x+3) (2x + 1) = 2x2 + 7x + 3
The University of Texas at Dallas

29. 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

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

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

32. ALGEBRAIC DIVISION Find the quotient: 6x2 + 3x 3x 3x (6x2 + 3x)
ANSWER 3x x x 1
(6x2 + 3x) x x2 x2 x x x2 x2 x
6x2 + 3x 3x
= 2x + 1 x x2 x2 x
The University of Texas at Dallas

33. ALGEBRAIC DIVISION Find the quotient: 2x2 + 6x + 4 x + 2 x+ 2
ANSWER
x+ 2 x x 1 1
(2x2 + 6x + 4) x x2 x2 x x x x x x 1 1 1 1 1 1
2x2 + 6x + 4
x + 2
= 2x + 2
The University of Texas at Dallas