Let’s Do Algebra Tiles

REL HYBIRD ALGEBRA RESEARCH PROJECTAdapted from David McReynolds, AIMS PreK-16 Projectand Noel Villarreal, South Texas Rural Systemic Initiative

November , 2007

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Algebra Tiles

Manipulatives used to enhance student understanding of concepts traditionally taught at symbolic level.

Provide access to symbol manipulation for students with weak number sense.

Provide geometric interpretation of symbol manipulation.

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Algebra Tiles

Support cooperative learning, improve discourse in classroom by giving students objects to think with and talk about.

When I listen, I hear. When I see, I remember. But when I do, I understand.

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Algebra Tiles

Algebra tiles can be used to model operations involving integers.

Let the small yellow square represent +1 and the small red square (the flip-side) represent -1.

The yellow and red 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 red rectangle (the flip-side) represent -1 x or -x .

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

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Algebra Tiles

Let the blue square represent x2. The red square (flip-side of blue) represents -x2.

As with integers, the red shapes and their corresponding flip-sides 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. Don’t use “cancel out” for zeroes use

zero pairs or add up to zero

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Addition of Integers

Addition can be viewed as “combining”. Combining involves the forming and

removing of all zero pairs. For each of the given examples, use

algebra tiles to model the addition. Draw pictorial diagrams which show the

modeling. Write the manipulation performed

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Addition of Integers

(+3) + (+1) = Combined like objects to get four

positives

(-2) + (-1) = Combined like objects to get three

negatives

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Addition of Integers

(+3) + (-1) = Make zeroes to get two positives

(+3) + (-4) = Make three zeroes to get one negative

After students have seen many examples of addition, have them formulate rules.

+2

-1

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Subtraction of Integers

Subtraction can be interpreted as “take-away.”

Subtraction can also be thought of as “adding the opposite.” (must be extensively scaffolded before students are asked to develop)

For each of the given examples, use algebra tiles to model the subtraction.

Draw pictorial diagrams which show the modeling process

Write a description of the actions taken

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(+5) – (+2) = Take away two positives To get three positives

(-4) – (-3) = Take away three negatives To get one negative

Subtraction of Integers

-1

+3

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Subtracting Integers

(+3) – (-5) =

Add five zeroes; Take away five negatives

To get eight positives

(-4) – (+1)=

Add one zero; Take away one positive

To get five negatives

+8

-5

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Subtracting Integers

(+3) – (-3)=

After students have seen many examples, have them formulate rules for integer subtraction.

(+3) – (-3) is the same as 3 + 3 to get 6

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Multiplication of Integers

Integer multiplication builds on whole number multiplication.

Use concept that the multiplier serves as the “counter” of sets needed.

For the given examples, use the algebra tiles to model the multiplication. Identify the multiplier or counter.

Draw pictorial diagrams which model the multiplication process

Write a description of the actions performed

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Multiplication of Integers

The counter indicates how many rows to make. It has this meaning if it is positive.

(+2)(+3) =

(+3)(-4) =

Two groups of three positives

Three groups of four negatives

+6

+12

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Multiplication of Integers

If the counter is negative it will mean “take the opposite of.”

Can indicate the motion “flip-over”, but be very careful using that terminology

(-2)(+3)

(-3)(-1)

•Two groups of three

•Opposite of three groups of negative one

•To get three positives

•Opposite of•To get six negatives

= -6

= +3

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Multiplication of Integers

After students have seen many examples, have them formulate rules for integer multiplication.

Have students practice applying rules abstractly with larger integers.

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Division of Integers

Like multiplication, division relies on the concept of a counter.

Divisor serves as counter since it indicates the number of rows to create.

For the given examples, use algebra tiles to model the division. Identify the divisor or counter. Draw pictorial diagrams which model the process.

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Division of Integers

(+6)/(+2) =

Divide into two equal groups

(-8)/(+2) =

Divide into two equal groups

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Division of Integers

A negative divisor will mean “take the opposite of.” (flip-over)

(+10)/(-2) =

Divide into two equal groups Find the opposite of To get five negatives

-5

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Division of Integers

(-12)/(-3) =

After students have seen many examples, have them formulate rules.

+4

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Polynomials

“Polynomials are unlike the other ‘numbers’ students learn how to add, subtract, multiply, and divide. They are not ‘counting’ numbers. Giving polynomials a concrete reference (tiles) makes them real.”

David A. Reid, Acadia University

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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)

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Distributive Property

3(x + 2)=

Three Groups of x to get three x’s Three groups of 2 to get 6

3·x + 3·2 = 3x + 6

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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.

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Modeling Polynomials

2x2

4x

3 or +3

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More Polynomials

Represent each of the given expressions with algebra tiles.

Draw a pictorial diagram of the process.

Write the symbolic expression.

x + 4

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More Polynomials

2x + 3

4x – 2

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More 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.

2x + 4 + x + 2

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More Polynomials

2x + 4 + x + 1

Combine like terms to get three x’s and five positives

= 3x + 5

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More Polynomials

3x – 1 – 2x + 4

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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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.

3 + 2x let x = 4

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Substitution

3 + 2x =

3 + 2(4) =

3 + 8 =

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let x = 4

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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.

Equivalent Equations are created if equivalent operations are performed on each side of the equation. (Which means to use the additon, subtraction, mulitplication, or division properties of equality.) What you do to one side of the equation you must do to the other side of the equation.

Variables can be isolated by using the Additive Inverse Property ( & zero pairs) and the Multiplicative Inverse Proerty ( & dividing out common factors). The goal is to isolate the variable.

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Solving Equations

x + 2 = 3

•x and two positives are the same as three positives

•add two negatives to both sides of the equation; makes zeroes

•one x is the same as one positive

-2 -2 x = 1

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Solving Equations

-5 = 2x

÷2 ÷2

2½ = x

•Two x’s are the same as five negatives

•Divide both sides into two equal partitions

•Two and a half negatives is the same as one x

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Solving Equations

x2

1

•One half is the same as one negative x

•Take the opposite of both sides of the equation

•One half of a negative is the same as one x

· -1 · -1

x2

1

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Solving Equations

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x

•One third of an x is the same as two negatives

•Multiply both sides by three (or make both sides three times larger)

•One x is the same as six negatives

• 3 • 3 x = -6

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Solving Equations

2 x + 3 = x – 5

•Two x’s and three positives are the same as one x and five negatives

•Take one x from both sides of the equation; simplify to get one x and three the same as five negatives

•Add three negatives to both sides; simplify to get x the same as eight negatives

- x - xx + 3 = -5 + -3 + - 3

x = -8

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Solving Equations

3(x – 1) + 5 = 2x – 2

3x – 3 + 5 = 2x – 2

3x + 2 = 2x – 2

– 2 or + -2

3x = 2x – 4

-2x -2x

x = -4

“x is the same as four negatives”

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Multiplication

Multiplication using “base ten blocks.”

(12)(13) Think of it as (10+2)(10+3) Multiplication using the array method

allows students to see all four sub-products.

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Multiplication using “Area Model”

(12)(13) = (10+2)(10+3) =

100 + 30 + 20 + 6 = 156

10 x 10 = 102 = 100

10 x 2 = 20

10 x 3 = 30

2 x 3 = 610 x 2 = 20

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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

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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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Multiplying Polynomials

(x + 2)(x – 3)

(x – 2)(x – 3)

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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.

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

Draw a picture.

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Factoring Polynomials

3x + 3

2x – 6

= 3 ·(x + 1)

= 2 ·(x – 3)

Note the two are positive, this needs to be developed

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Factoring Polynomials

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

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Factoring Polynomials

x2 – 5x + 6 = (x – 2)(x – 3)

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Factoring Polynomials

x2 – x – 6 = (x + 2)(x – 3)

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Factoring Polynomials

x2 + x – 6

x2 – 1

x2 – 4

2x2 – 3x – 2

2x2 + 3x – 3

-2x2 + x + 6

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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)

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Dividing Polynomials

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

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Conclusion

Algebra tiles can be made using the Ellison (die-cut) machine.

On-line reproducible can be found by doing a search for algebra tiles.

Virtual Algebra Tiles at HRW http://my.hrw.com/math06_07/nsmedia/tools/Algebra_Tiles/Algebra_Tiles.html

National Library of Virtual Manipulatives http://nlvm.usu.edu/en/nav/topic_t_2.html

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Resources

David McReynoldsAIMS PreK-16 Project Noel VillarrealSouth Texas Rural Systemic Initiative Jo Ann Mosier & Roland O’DanielCollaborative for Teaching and Learning Partnership for Reform Initiatives in Science

and Mathematics