1. Let’s Explore Algebra Tiles
Simplifying Polynomials, Distributive Property, Substitution, Solving Equations, Multiplying & Dividing Polynomials and Factoring

2. Modeling Polynomials

3. Modeling Polynomials Algebra tiles can be used to
model expressions ;
aid in the simplification of expressions ; 6.EE.3; 6.EE.4

4. Modeling Polynomials =1
= x
= -1
= - x
= x2
= - x2

5. Modeling Polynomials
1) 2x + 4
2) -3x + 1

6. Modeling Polynomials
3) 2x2 – 5x -4

7. Simplifying Polynomials
Students need to use the same idea of zero pairs with variables

8. Simplifying Polynomials
1) 2x x + 2
simplified: 3x + 6
2) -3x x + 3
simplified: -2x + 4

9. More Polynomials try: 3) 3x + 1 – 2x - 4
This process can be used with problems containing x2.
(2x2 + 5x – 3) + (-x2 + 2x + 5)

10. More Polynomials How would you show/demonstrate:
1) (3x + 5) – (2x + 2)?
2 ) (2x2 – 2x + 3) – (3x2 + 3x – 2)?

11. Substitution Using Algebra Tiles for evaluating expressions
; 6.EE.1; 6.EE.2

12. Substitution Algebra tiles can be used to model substitution.
Represent original expression with tiles.
Then replace each rectangle with the appropriate tile value.
Combine like terms.
For example:
3 + 2x let x = 4

13. Substitution
3 + 2x let x = 4
Therefore when x=4,
3 + 2x = 11

14. Substitution Simplify Therefore when x=-4, 3 + 2x = -5
3 + 2x let x = -4
Simplify
Therefore when x=-4,
3 + 2x = -5

15. Substitution How would you show/ demonstrate? 3 - 2x let x = 4

16. Distributive Property
Using Algebra Tiles to demonstrate the Distributive Property
(numbers only); ; 6.EE.3

17. Distributive Property
Use the same concept that was applied with multiplication of integers, think of the first factor as the counter.
The same rules apply.
3(x+2)
Three is the counter, so we need three rows of (x+2).

18. Distributive Property
3(x + 2)
simplified 3x + 6

19. Distributive Property
3(x - 2)
simplified 3x - 6

20. Distributive Property
Try these:
3(x – 4)
-2(x + 2)
-3(x – 2)

21. Using Algebra Tiles to show the steps for solving equations

22. Solving Equations
Algebra tiles can be used to explain and justify the equation solving process. The development of the equation solving model is based on two ideas.
Equations are unchanged if equivalent amounts are added to each side of the equation.
Variables can be isolated by using zero pairs.

23. Equations are unchanged if equivalent amounts are added to each side of the equation.
x + 2 = 3
Show using symbols
x + 2 = 3
x = 1

24. Solving Equations
2x – 4 = 8
Show using symbols

25. Solving Equations
2x + 3 = x – 5
Show using symbols

26. Questions at this point? How can you use this in your classroom?
Algebra tiles
Questions at this point?
How can you use this in your classroom?

27. Using Algebra Tiles in higher level math courses
Advanced Polynomials
Using Algebra Tiles in higher level math courses

28. More Advanced Polynomials
Algebra tiles can also be used to:
Multiply polynomials,
Divide polynomials, or
Factor polynomials.

29. Multiplying Polynomials
(x + 2)(x + 3)
Does it matter which factor goes on top and which factor goes on the side?
x+3
x+2
(x + 2)(x + 3)=x2+5x+6

30. Multiplying Polynomials
(x + 2)(x + 3)
x+2
x+3
(x + 2)(x + 3)=x2+5x+6

31. Multiplying Polynomials
(x – 1)(x +4)
(x – 1)(x +4)=x2+3x-4

32. Multiplying Polynomials
Try:
(x + 2)(x – 3)
(x – 2)(x – 3)

33. Dividing Polynomials Algebra tiles can be used to divide polynomials.
Use tiles and frame to represent problem. Dividend should form array inside frame. Divisor will form one of the dimensions (one side) of the frame.
Be prepared to use zero pairs in the dividend.

34. Dividing Polynomials
x2 + 7x +6
x + 1
= x+6

35. Dividing Polynomials
x2 + 5x +6
x + 2

36. Dividing Polynomials
x2 + 5x +6
x + 2

37. Dividing Polynomials
x2 + 5x +6
x + 2

38. Dividing Polynomials
x2 + 5x +6
x + 2
= x+3

39. Dividing Polynomials
Try:
x2 - 5x +6
x - 2
= x-3

40. Dividing Polynomials
Try:
x2 - 5x -6
x + 1
= x-6

41. Factoring Polynomials
Algebra tiles can be used to factor polynomials. Use tiles and the frame to represent the problem.
Use the tiles to fill in the array so as to form a rectangle inside the frame.
3x + 3
2x – 6

42. Factoring Polynomials
x2 + 6x + 8
We need to make a rectangle that uses all of the Algebra tiles

43. Factoring Polynomials
x2 + 6x + 8
= (x+2)(x+4)

44. Factoring Polynomials
x2 – 5x + 6
= (x-2)(x-3)

45. Factoring Polynomials
x2 – x – 6 (harder)
= (x+2)(x-3)

46. Factoring Polynomials
x2 - 1 (even harder)
= (x+1)(x-1)

47. Factoring Polynomials
Try these:
x2 + x – 6
x2 – 4
2x2 – 3x – 2
2x2 + 3x – 3
-2x2 + x + 6

48. Questions???????