Let’s Do Algebra Tiles Roland O’Daniel, The Collaborative for Teaching and Learning October, 2009 Adapted from David McReynolds, AIMS PreK-16 Project and Noel Villarreal, South Texas Rural Systemic Initiative

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.

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.

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.

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.

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.

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

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.

Addition of Integers (+3) + (+1) = 4 Combine like objects to get four positives (-2) + (-1) = Combine like objects to get three negatives 4 -3

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) = -1 Make three zeroes Final answer one negative After students have seen many examples of addition, have them formulate rules. -1

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

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

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 -5

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

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

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

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

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.

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.

Division of Integers (+6)/(+2) = Divide into two equal groups 3 in each group (-8)÷(+2) = 3 -4

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

Division of Integers (-12)/(-3) = After students have seen many examples, have them formulate rules. +4

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

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)

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 + 3·2 = 3x + 6

Modeling Polynomials Algebra tiles can be used to model expressions. Model the simplification of expressions. Add, subtract, multiply, divide, or factor polynomials.

Modeling Polynomials 2x2 4x 3 or +3

More Polynomials Represent each of the given expressions with algebra tiles. Draw a pictorial diagram of the process. Write the symbolic expression. x + 4

More Polynomials 2x + 3 4x – 2

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

More Polynomials 2x + 4 + x + 1 = 3x + 5 Combine like terms to get three x’s and five positives

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) 2x2 + -2x + 3 + -3x2 + -3x + +2 Take away the second quantity or change to addition of the opposite 2x2 + -2x + 3 + -3x2 + -3x + +2 Make zeroes and combine like terms (2x2 + -3x2) + (-2x + -3x) + (3+ 2) Simplify and write polynomial in standard form -x2 – 5x + 5

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

Substitution 3 + 2x = 3 + 2(4) = 3 + 8 = 11 let x = 4

Substitution 4x + 2y = let x = -2 & y = 3 4(-2) + 2(3) = -8 + 6 = -2

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).

Solving Equations x + 2 = 3 -2 -2 x = 1 -2 -2 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

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

Solving Equations · -1 · -1 One half is the same as one negative x · -1 · -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

Solving Equations • 3 • 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

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

Solving Equations 3(x – 1) + 5 = 2x – 2 3x – 3 + 5 = 2x – 2 – 2 or + -2 3x = 2x – 4 -2x -2x x = -4 “x is the same as four negatives”

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

Multiplication using “Area Model” (12)(13) = (10+2)(10+3) = 100 + 30 + 20 + 6 = 156 12 x 13 36 +120 156 10 x 10 = 102 = 100 10 x 3 = 30 10 x 2 = 20 10 x 2 = 20 2 x 3 = 6

Multiplying Polynomials (x + 2) (x + 3) = x2 + (2x + 3x) + 6 = x2 + 5x + 6 Model the dimensions (factors) of the rectangle Fill in each section of the area model Combine like terms and write in standard form

Multiplying Polynomials (x – 1)(x +4) Model the dimensions of the rectangle Fill in each section of the area model Make Zeroes or combine like terms and simplify x2 + 4x – 1x – 4 = x2 + 3x – 4

Multiplying Polynomials (x + 2)(x – 3) (x – 2)(x – 3)

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.

Factoring Polynomials 3x + 3 2x – 6 (x + 1) = 3 · = 2 · (x – 3) Note the two is positive, because the green is positive, this decision takes time to develop

Factoring Polynomials Model the expression to be factored x2 + 6x + 8 = (x + 2)(x +4) Create a rectangle from the dividend Determine & model the dimensions of the rectangle. It takes trial and error to be able to create the rectangles, observation & analysis will let students make the connections between the factors of the constant and how to divide the x term. The factored form of the expression is …

Factoring Polynomials Model the expression to be factored x2 – 5x + 6 = (x – 2)(x – 3) Create a rectangle from the dividend Determine & model the dimensions of the rectangle. Note the factors have negative constants and the expression has a positive constant, students will need to be able to explain how they make the decision to make the constants negative. The factored form of the expression is …

Factoring Polynomials x2 – x – 6 = (x + 2)(x – 3) Model the expression to be factored Create a rectangle from the dividend In this case we need to add zeroes in order to complete the rectangle. 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 …

Factoring Polynomials x2 + x – 6 x2 – 1 x2 – 4 2x2 – 3x – 2 2x2 + 3x – 3 -2x2 + x + 6

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.

Dividing Polynomials x2 + 7x +6 = (x + 6) x + 1 Model the problem, dividend and divisor = (x + 6) Create a rectangle from the dividend with the width of x + 1 Determine and model the length of the rectangle `````

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

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

x2 + 4x +7 x2 + 4x +7 (x + 2) • (x + 2) = (x + 2)2 + 3 Model the expression Form a square with the terms of the expression (as close as possible) x2 + 4x +7 = (x + 2)2 + 3 Determine and model the dimensions of the square Rewrite in perfect square form including the “extra values” to complete the square.

x2 + 6x +7 x2 + 6x +7 (x + 3) • (x + 3) = (x + 3)2 + -2 Model the expression Form a square with the terms of the expression (as close as possible) x2 + 6x +7 = (x + 3)2 + -2 Add 2 zeroes to complete the square Determine and model the dimensions of the square Rewrite in perfect square form including the “extra values” to complete the square.

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

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’Daniel Collaborative for Teaching and Learning Partnership for Reform Initiatives in Science and Mathematics