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## Let’s Do Algebra Tiles

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**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 +1xor 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**4 (+3) + (+1) = • Combine like objects to get four positives (-2) + (-1) = • Combine like objects to get three negatives -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) =**• 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**+8 (+3) – (-5) = Can’t take away any negatives so add five zeroes; Take away five negatives; Get eight positives**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) = • Divide into two equal groups 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 + 3·2 = 3x + 6 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**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**= 3x + 5 2x + 4 + x + 1 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)**Take away the second quantity or change to addition of the opposite 2x2+ -2x + 3 + -3x2+ -3x+ +2 Make zeroes and combine like terms + (-2x + -3x) (2x2 + -3x2) + (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 x and two positives are the same as three positives add two negatives to both sides of the equation; makes zeroes one xis the same as one positive**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 negativesis the same as one x**Solving Equations**· -1 · -1 • One half is the same as one negative x • Take the opposite of both sides of the equation • One half of a negativeis 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 • 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 3x + 2 = 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 + 3) = x2 + (2x + 3x) + 6 = x2 + 5x + 6 (x + 2) Model the dimensions (factors) of the rectangle Fill in each section of the area model Combine like terms and write in standard form