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Let’s Explore Algebra Tiles

Let’s Explore Algebra Tiles. Simplifying Polynomials, Distributive Property, Substitution, Solving Equations, Multiplying & Dividing Polynomials and Factoring. Modeling Polynomials. Modeling Polynomials. Algebra tiles can be used to model expressions.

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Let’s Explore Algebra Tiles

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  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.

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

  5. Modeling Polynomials 1) 2x + 4 • z 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 + 4 + x + 2 simplified: 3x + 6 2) -3x + 1 + 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

  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 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 3 - 2x let x = -4

  16. Distributive Property Using Algebra Tiles to demonstrate the Distributive Property

  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. Solving Equations 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 - 2 -2 x = 1

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

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

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

  27. 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 Does it matter which factor goes on top and which factor goes on the side? (x + 2)(x + 3) 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 = (x-2)(x-3) x2 – 5x + 6

  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. Advanced Polynomials Modeling polynomials with xy

  49. Modeling Polynomials w/ y • Let the blue square represent x2, • the green rectangle xy, and • the yellow square y2. • The red square (flip-side of blue) represents –x2, • the red rectangle (flip-side of green) –xy, • and the small red square (flip-side of yellow) –y2.

  50. Modeling Polynomials • Represent each of the following with algebra tiles, draw a pictorial diagram of the process, then write the symbolic expression. 2x2 4xy 3y2

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