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Visualizing algebra. Algebra Tiles Manipulative tool kit for solving linear equations Multiplying two linear equations to form a quadratic Factoring quadratic.

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Presentation on theme: "Visualizing algebra. Algebra Tiles Manipulative tool kit for solving linear equations Multiplying two linear equations to form a quadratic Factoring quadratic."— Presentation transcript:

1 Visualizing algebra

2 Algebra Tiles Manipulative tool kit for solving linear equations Multiplying two linear equations to form a quadratic Factoring quadratic equations into their linear roots

3 Tool Kit 5-inch square tiles = x 2 5-in by 1-in rectangle = x Unit squares = 1 Green tiles = + Red tiles = –

4 Algebra tiles illustrate Solving linear equations Building quadratic equations from linear equations Factoring quadratic equations into their linear roots

5 x + 4 = Tiles needed 1 green x rectangle 4 green unit tiles =

6 x + 4 = Place 4 red unit tiles on each side of the equation (What you do to one side, you have to do to the other side) =

7 x + 4 = Remove pairs of red and green tiles =

8 x + 4 = Remove pairs of red and green tiles =

9 x + 4 = Remove pairs of red and green tiles =

10 x + 4 = Remove pairs of red and green tiles =

11 x + 4 = Remove pairs of red and green tiles =

12 x = -4 =

13 How to choose a red or green tile If the tiles are the same color, use a green tile If the tiles are different colors, use a red tile A positive times a positive is a positive A positive times a negative is a negative A negative times a negative is a positive

14 Place x+2 down the side Place x+3 across the top (x+2)(x+3)

15 Place x 2 (x+2)(x+3)

16 Place 3 xs on the right (x+2)(x+3)

17 Place 2 xs on the bottom (x+2)(x+3)

18 Fill in with unit squares (x+2)(x+3)

19 Count up parts x 2 +5x+6 (x+2)(x+3)

20 Place x-3 on top Place x+2 on the side (x+2)(x-3)

21 We have a green x on the top and a green x on the side, use a green x 2 (x+2)(x-3)

22 We have red units on the top and a green x on the side, use red xs (x+2)(x-3)

23 We have a green x on top and green units down the side, use green xs (x+2)(x-3)

24 We have red units on the top and green units on the side, use red units (x+2)(x-3)

25 Remove pairs of green xs and red xs (x+2)(x-3)

26 Remove pairs of green xs and red xs (x+2)(x-3)

27 Remove pairs of green xs and red xs (x+2)(x-3)

28 Count up parts x 2 -x-6 (x+2)(x-3)

29 Place x+2 down the side Place 3-x across the top (x+2)(3-x)

30 We have a red x on the top and a green x on the side, use a red x 2 (x+2)(3-x)

31 We have green units on the top and a green x on the side, use green xs (x+2)(3-x)

32 We have a red x on the top and green units on the side, use red xs (x+2)(3-x)

33 We have green units on the top and green units on the side, use green units (x+2)(3-x)

34 Remove pairs of green and red xs (x+2)(3-x)

35 Remove pairs of green and red xs (x+2)(3-x)

36 Remove pairs of green and red xs (x+2)(3-x)

37 Count up parts -x 2 +x+6 (x+2)(3-x)

38 Factoring Determine factorization of constant term x 2 –x –

39 Pick and place a factorization of -12 x 2 -x-12

40 Red units mean we have a positive and a negative, so use red xs x 2 -x-12

41 Red units mean we have a positive and a negative, so use green xs x 2 -x-12

42 Check for –x by removing pairs of green and red xs

43 x 2 -x-12 Check for –x by removing pairs of green and red xs

44 Too many red xs left, try another factorization of 12 x 2 -x-12

45 Pick and place a factorization of -12 x 2 -x-12

46 Place red xs x 2 -x-12

47 Place green xs x 2 -x-12

48 Check for –x by removing pairs of green and red xs x 2 -x-12

49 Check for –x by removing pairs of green and red xs x 2 -x-12

50 Check for –x by removing pairs of green and red xs x 2 -x-12

51 Check for –x by removing pairs of green and red xs x 2 -x-12

52 –x checks out x 2 -x-12

53 x 2 -x-12=(x+3)(x-4)

54 Things to point out in Factoring The coefficient of the second term in a quadratic is the sum of the roots in the linear factors The last term in the quadratic is the product of the roots in the linear factors The signs of the coefficients tell you the signs of the roots If the last term is positive and the second term is positive, both roots are positive If the last term is positive and the second term is negative, both roots are negative If the last term is negative, one root is positive and one root is negative If the second term is positive, the positive root is larger If the second term is negative, the negative root is larger


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