Let’s Do Algebra Tiles

Roland O’Daniel, The Collaborative for Teaching and Learning October, 2009

Adapted from David McReynolds, AIMS PreK-16 Projectand Noel Villarreal, South Texas Rural Systemic Initiative

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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 opposite or 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 -1x 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. Draw pictorial diagrams which show the

modeling. Write the manipulation performed The following problems are animated to

represent how these problems can be solved.

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Addition of Integers

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

positives

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

negatives

4

-3

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Addition of Integers

(+3) + (-1) =

Make zeroes to get two positives

Important to have students perform all three representations, to create deeper understanding

Quality not quantity in these interactions

+2

(+3) + (-4) =

Make three zeroes Final answer one negative

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

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-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)

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 Final answer one negative

Subtraction of Integers

-1

+3

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Subtracting Integers

(+3) – (-5) =

Can’t take away any negatives

so add five zeroes;

Take away five negatives;

Get eight positives

+8

Subtracting Integers

(-4) – (+1)=

Start with 4 negatives

No positives to take away;

Add one zero;

Take away one positive

To get five negatives

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-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 3 in each group

(-8)÷(+2) =

Divide into two equal groups

-4

3

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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 in each group

-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 the quantity x and 2 Three Groups of x to get three x’s Three groups of 2 to get 6

3·x + 3x + 63·2 =

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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)

(2x2 – 2x + 3) – (3x2 + 3x – 2)

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Take away the second quantity or change to addition of the opposite

Make zeroes and combine like terms

Simplify and write polynomial in standard form -x2 – 5x + 5

2x2 + -2x + 3 + -3x2 + -3x + +2

(2x2 + -3x2) + (-2x + -3x) + (3+ 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

Substitution

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4x + 2y = let x = -2 & y = 3

4(-2) + 2(3) =

-8 + 6 =

-2

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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. (the additon, subtraction, mulitplication, or division properties of equality.)

Variables can be isolated by using the Additive Inverse Property (zero pairs) and the Multiplicative Inverse Property (dividing out common factors).

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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 multiplying the two

binomials (10+2)(10+3) Multiplication using the array method

allows students to see all four sub-products.

The array method reinforces the area model and the foil method

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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

10 x 2 = 20

12x 13 36+120

156

2 x 3 = 6

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Multiplying Polynomials

(x + 2) = x2 + (2x + 3x) + 6

Fill in each section of the area model

Combine like terms and write in standard form

Model the dimensions (factors) of the rectangle

(x + 3) = x2 + 5x + 6

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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

Model the dimensions of the rectangle

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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 is positive, because the green is positive, this decision takes time to develop

Determine & model the dimensions of the rectangle.

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Factoring Polynomials

x2 + 6x + 8 = (x + 2)(x +4)Model the expression to be factored

Create a rectangle from the dividend

The factored form of the expression is …

It takes trial and error to beable to create the rectangles, observation & analysis will let students make the connections between the factors of the constantand how to divide the x term.

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Factoring Polynomials

x2 – 5x + 6 = (x – 2)(x – 3)Model the expression to be factored

Create a rectangle from the dividend

The factored form of the expression is …

Determine & model the dimensions of the rectangle.

Note the factors have negativeconstants and the expressionhas a positive constant, studentswill need to be able to explain how they make the decision to make theconstants negative.

In this case we need to add zeroes in order to complete the rectangle.

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Factoring Polynomials

x2 – x – 6 = (x + 2)(x – 3) Model the expression to be factored

Create a rectangle from the dividend

Determine & model the dimensions of the rectangle.Note that we know to have a negative value in the product the values being multiplied must be + and – .

The factored form of the expression is …

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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)Model the problem, dividendand divisor

Create a rectangle from the dividend with the width of x + 1

Determine and modelthe length of the rectangle

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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

Completing the Square

Algebra tiles can be used to complete the square when factoring quadratics

Use tiles and frame to represent problem. Expression should form a square array inside the form.

The square factors will form the dimensions

Be prepared to use zero pairs of constants to complete the square

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Rewrite in perfect squareform including the “extra values” to complete the square.

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Determine and model the dimensions of the square

x2 + 4x +7

Model the expression

Form a square with the terms of the expression(as close as possible)

(x + 2) • (x + 2)

= (x + 2)2 + 3x2 + 4x +7

Determine and model the dimensions of the square

x2 + 6x +7

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Model the expression

Form a square with the terms of the expression(as close as possible)

Add 2 zeroes to completethe square

(x + 3) • (x + 3)

Rewrite in perfect squareform including the “extra values” to complete the square.

= (x + 3)2 + -2x2 + 6x +7

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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

“Let’s Do Algebra Tiles” by David McReynolds, AIMS PreK-16 Project & Noel Villarreal, South Texas Rural Systemic Initiative

Jo Ann Mosier & Roland O’DanielCollaborative for Teaching and Learning

Partnership for Reform Initiatives in Science and Mathematics