Making Math Work Algebra Tiles Visualizing algebra

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

Tool Kit 5-inch square tiles = x2 5-in by 1-in rectangle = x Unit squares = 1 Green tiles = + Red tiles = –

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

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

(What you do to one side, you have to do to the other side) 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) =

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

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

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

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

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

x = -4 =

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

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

(x+2)(x+3) Place x2

(x+2)(x+3) Place 3 x’s on the right

(x+2)(x+3) Place 2 x’s on the bottom

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

(x+2)(x+3) Count up parts x2+5x+6

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

We have a green x on the top and a green x on the side, use a green x2 (x+2)(x-3) We have a green x on the top and a green x on the side, use a green x2

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

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

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

Remove pairs of green x’s and red x’s (x+2)(x-3) Remove pairs of green x’s and red x’s

Remove pairs of green x’s and red x’s (x+2)(x-3) Remove pairs of green x’s and red x’s

Remove pairs of green x’s and red x’s (x+2)(x-3) Remove pairs of green x’s and red x’s

(x+2)(x-3) Count up parts x2-x-6

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

We have a red x on the top and a green x on the side, use a red x2 (x+2)(3-x) We have a red x on the top and a green x on the side, use a red x2

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

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

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

Remove pairs of green and red x’s (x+2)(3-x) Remove pairs of green and red x’s

Remove pairs of green and red x’s (x+2)(3-x) Remove pairs of green and red x’s

Remove pairs of green and red x’s (x+2)(3-x) Remove pairs of green and red x’s

(x+2)(3-x) Count up parts -x2+x+6

Factoring Determine factorization of constant term x2 –x – 12 12 1 6 4 3

Pick and place a factorization of -12 x2-x-12 Pick and place a factorization of -12

Red units mean we have a positive and a negative, so use red x’s x2-x-12 Red units mean we have a positive and a negative, so use red x’s

Red units mean we have a positive and a negative, so use green x’s x2-x-12 Red units mean we have a positive and a negative, so use green x’s

Check for –x by removing pairs of green and red x’s x2-x-12 Check for –x by removing pairs of green and red x’s

Check for –x by removing pairs of green and red x’s x2-x-12 Check for –x by removing pairs of green and red x’s

Too many red x’s left, try another factorization of 12 x2-x-12 Too many red x’s left, try another factorization of 12

Pick and place a factorization of -12 x2-x-12 Pick and place a factorization of -12

x2-x-12 Place red x’s

x2-x-12 Place green x’s

Check for –x by removing pairs of green and red x’s x2-x-12 Check for –x by removing pairs of green and red x’s

Check for –x by removing pairs of green and red x’s x2-x-12 Check for –x by removing pairs of green and red x’s

Check for –x by removing pairs of green and red x’s x2-x-12 Check for –x by removing pairs of green and red x’s

Check for –x by removing pairs of green and red x’s x2-x-12 Check for –x by removing pairs of green and red x’s

x2-x-12 –x checks out

x2-x-12=(x+3)(x-4)

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