Visualizing algebra

Algebra TilesManipulative tool kit for solving linear

equationsMultiplying two linear equations to form a

quadraticFactoring quadratic equations into their

linear roots

Tool Kit5-inch square tiles = x2

5-in by 1-in rectangle = xUnit squares = 1

Green tiles = +Red tiles = –

Algebra tiles illustrateSolving linear equationsBuilding quadratic equations from linear

equationsFactoring quadratic equations into their

linear roots

x + 4 =Tiles needed

1 green x rectangle4 green unit tiles

=

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

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x + 4 = Remove pairs of red and green tiles

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x + 4 = Remove pairs of red and green tiles

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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 tileIf the tiles are the same color, use a green

tileIf the tiles are different colors, use a red tile

A positive times a positive is a positiveA positive times a negative is a negativeA negative times a negative is a positive

Place x+2 down the sidePlace 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

(x+2)(x+3)

Fill in with unit squares

(x+2)(x+3)

Count up partsx2+5x+6

(x+2)(x+3)

Place x-3 on topPlace x+2 on the side

(x+2)(x-3)

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 red units on the top and a green x on the side, use red 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

(x+2)(x-3)

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)

Remove pairs of green x’s and red x’s

(x+2)(x-3)

Count up partsx2-x-6

(x+2)(x-3)

Place x+2 down the sidePlace 3-x across the top

(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

(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

(x+2)(3-x)

Remove pairs of green and red x’s

(x+2)(3-x)

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

(x+2)(3-x)

FactoringDetermine factorization of constant term

x2 –x – 12

1

12

2

6

3

4

Pick and place a factorization of -12

x2-x-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 green x’s

x2-x-12

x2-x-12

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

Pick and place a factorization of -12

x2-x-12

Place red x’s

x2-x-12

Place green x’s

x2-x-12

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

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

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

Things to point out in FactoringThe coefficient of the second term in a quadratic is the

sum of the roots in the linear factorsThe 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 rootsIf the last term is positive and the second term is

positive, both roots are positiveIf the last term is positive and the second term is

negative, both roots are negativeIf 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