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**REL HYBIRD ALGEBRA RESEARCH PROJECT**

Let’s Do Algebra Tiles REL HYBIRD ALGEBRA RESEARCH PROJECT Adapted from David McReynolds, AIMS PreK-16 Project and 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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**Subtraction of Integers**

(+5) – (+2) = Take away two positives To get three positives (-4) – (-3) = Take away three negatives To get one negative +3 -1

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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 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) = +6 Two groups of three positives +12 Three groups of four negatives

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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) = -6 Two groups of three Opposite of To get six negatives = +3 Opposite of three groups of negative one To get three positives

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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) =

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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 x + 2

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**More Polynomials 2x + 4 + x + 1 = 3x + 5**

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

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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 = 11 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 -2 -2 x = 1**

x = 1 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

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**Solving Equations -5 = 2x ÷2 ÷2 2½ = x**

÷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 · -1 · -1 One half is the same as one negative x**

· · -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

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Solving Equations • • 3 x = -6 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

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**Solving Equations 2 x + 3 = x – 5 - x - x x + 3 = -5 + -3 + - 3 x = -8**

x = -8 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

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**Solving Equations 3(x – 1) + 5 = 2x – 2 3x – 3 + 5 = 2x – 2**

– 2 or 3x = 2x – 4 -2x x 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) = = 156 10 x 3 = 30 10 x 10 = 102 = 100 10 x 2 = 20 10 x 2 = 20 2 x 3 = 6

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**Multiplying Polynomials**

(x + 2)(x + 3) Fill in each section of the area model Combine like terms x2 + 2x + 3x + 6 = x2 + 5x + 6

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**Multiplying Polynomials**

(x – 1)(x +4) Fill in each section of the area model Make Zeroes or combine like terms and simplify x2 + 4x – 1x – 4 = x2 + 3x – 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 (x + 1) = 3 · = 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**

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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 National Library of Virtual Manipulatives

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**Resources David McReynolds AIMS PreK-16 Project Noel Villarreal**

South Texas Rural Systemic Initiative Jo Ann Mosier & Roland O’Daniel Collaborative for Teaching and Learning Partnership for Reform Initiatives in Science and Mathematics

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