Let’s Work With Algebra Tiles. 2 Algebra Tiles Algebra tiles can be used to model operations...

Let’s Work With Algebra Tiles

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Algebra Tiles

Algebra tiles can be used to model operations involving integers.

Let the small green square represent +1 and the small pink square represent -1.

The green and pink squares are additive inverses of each other.

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Algebra Tiles

Algebra tiles can be used to model operations involving variables.

Let the green rectangle represent +1x or x and the pink rectangle represent -1 x or -x .

The green and red rods are additive inverses of each other.

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Algebra Tiles

Let the green square represent x2. The pink square represents -x2.

As with integers, the green squares and the pink squares form a zero pair.

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Zero Pairs

Called zero pairs because they are additive inverses of each other.

When put together, they model zero.

Practice with Integers

Algebra tiles can be use to model adding, subtracting, multiplying, and dividing real numbers.

Remember, if you don’t have enough of something, you can add “zero pairs”

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Modeling Addition/Subtraction -2 – 3 Start with -2

Take away positive 3. But wait, I don’t have 3 so I must add zero pairs!

Now remove positive 3 You are left with -5

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Modeling Addition/Subtraction

-1 + 3 Start with -1

Now, add positive 3 Now, you must remove any “zero

pairs” You are left with 2.

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You try

-5 + 3 -1 + 4 4 – -2 2 + -3

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Modeling Multiplication

2 x -3 This means I need 2 rows of -3

Which is -6 This could also mean “the opposite of

3 rows of two”

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You try

1 x -4 3 x -3 -2 x 4 -2 x -2

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Modeling Division

6 2 The “6” is called the dividend. The “2” is called the divisor. Place the “2” outside, and then line up the 6 inside. You answer is what fits on top, which is 3. (called the quotient)

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You Try

4 / 2 8 / -4 -6 / 3 3 / -1

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Modeling Polynomials

Algebra tiles can be used to model expressions.

Model the simplification of expressions.

Add, subtract, multiply, divide, or factor polynomials.

To solve equations with polynomials.

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Modeling Polynomials

2x2

4x

3 or +3

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More Polynomials

2x + 3

4x – 2

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Adding Polynomials

Use algebra tiles to simplify each of the given expressions. Combine like terms. Look for zero pairs. Draw a diagram to represent the process.

Write the symbolic expression that represents each step.

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Adding Polynomials

(2x + 4) + (x + 1)

Combine like terms to get three x’s and five positive ones

= 3x + 5

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Adding Polynomials

(3x – 1) – (2x + 4)

Now remove 2x and remove 4. But WAIT, I don’t have 4 so I must add zero pairs.Now remove 2x and remove 4 And you are left with x - 5

You Try

(2x – 1) + (x + 2) (x + 3) + (x – 2) (2x – 1) – (x + 5) (3x + 5) + (x – 1) (4x – 3) – (3x – 2)

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Adding Polynomials

This process can be used with problems containing x2.

(2x2 + 5x – 3) + (-x2 + 2x + 5)

(2x2 – 2x + 3) – (3x2 + 3x – 2)

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Distributive Property

Multiplying a monomial to a polynomial 3(x – 2) = 3x - 6

Distributive Property

-2(x - 4)

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= -2x + 8

You try

4 (x + 2) 2 (x – 3) -2 (x + 1) -2 ( x – 1)

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Multiplying Polynomials

(x + 2)(x + 3)

x2 + 2x + 3x + 6 = x2 + 5x + 6

Fill in each section of the area model

Combine like terms

Multiplying Polynomials

(x – 1)(x +4)

= x2 + 3x – 4

Fill in each section of the area model

Make Zeroes or combine like termsand simplify

x2 + 4x – 1x – 4

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You Try

(x + 2)(x – 3)

(x – 2)(x – 3)

(x – 1) ( x + 4)

(x – 3) (x – 2)

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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.

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Dividing Polynomials

x2 + 7x +6

x + 1= (x + 6)

Dividing Polynomials

x2 – 5x + 6

x – 3 = (x – 2)

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You Try

x2 + 7x +6 x + 1 2x2 + 5x – 3 x + 3 x2 – x – 2 x – 2 x2 + x – 6 x + 3

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Factoring Polynomials

Factoring is the process of writing a polynomial as a product.

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.

Be prepared to use zero pairs to fill in the array.

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Factoring Polynomials

3x + 3

2x – 6

= 3 ·(x + 1)

= 2 ·(x – 3)

You Try

Factor 4x – 2 Factor 3x + 6 Factor Factor

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22x x

2 2x x

Factoring Polynomials

x2 + 6x + 8 = (x + 2)(x + 4)

Factoring Polynomials

x2 – 5x + 6 = (x – 2)(x – 3)

Remember: You must form a RECTANGLE out of the polynomial

Factoring Polynomials

x2 – x – 6 = (x + 2)(x – 3)

This time the polynomial doesn’t form a rectangle, so I have to add “zero pairs” in order to form a rectangle.

You Try

x2 + 3x + 2 x2 + 4x + 3 x2 + x – 6 x2 – 1 x2 – 4 2x2 – 3x – 2

Solving Equations

We can use algebra tiles to solve equations.

Whatever you do to one side of the equal sign, you have to do to the other to keep the equation “balanced”.

Solving Equations 3x + 4 = 2x – 1 First build each side of the equation

Now remove 2x from each side. Next, remove 4 from each side. But wait, I don’t

have 4 so I must add “zero pairs” Remove 4 from each side You are left with x = -5

=

Solving Equations 4x + 1 = 2x + 7 First, build each side of the equation

Next, remove 2x from each side. Remove 1 from each side. Now divide each side by 2. Your result is x = 3.

=

You Try

2x + 3 = x – 2 x – 4 = 2x + 1 3x + 1 = x – 5 8x – 2 = 6x + 4

Credits

Adapted by Marcia Kloempken, Weber High School from David McReynolds, AIMS PreK-16 Project