Modeling with Algebra Tiles Connie Schlimme Wake County Public School System cschlimme@wcpss.net

Link to Math Summit Program https://mathsummit2017.sched.com  

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 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 -1 x 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.

You may have an earlier version that look like this: These will work just as well. . . with a little alteration - - -

Zero Pairs Called zero pairs because they are additive inverses of each other. When put together, they model zero. Don’t say “cancel out” for zeroes! Instead say “they form a zero pair” or “they add up to zero.”

Addition of Integers Addition is “combining.” Combining involves the forming and removing 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.

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

Addition of Integers (+3) + (-1) = Make a zero pair, two positives remain. (+3) + (-4) = Make three zero pairs, one negative remains. Have students practice applying the concept with larger integers. Fade away the manipulatives. +2 -1

Subtraction of Integers Subtraction is “taking away.” If we do not have enough positive or negative tiles to take away, we represent the number in a different form by adding zero pairs. 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.

Subtraction of Integers (+5) – (+2) = Take away two positives, three positives remain (-4) – (-3) = Take away three negatives, one negative remains +3 -1

Subtracting Integers +8 -5 (+3) – (-5) = Add some zero pairs, take away five negatives, eight positives remain (-4) – (+1)= Add some zero pairs, take away one positive, five negatives remain +8 -5

Subtracting Integers (+3) – (-3)= After students have seen many examples, students will begin to see the pattern that subtracting gives the same result as adding the opposite. (+3) – (-3) has the same value as 3 + 3

Addition/Subtraction of Integers Try These: (+1) + (+ 3) (-3) + (-4) (+4) + (-6) (-7) – (-2)

Multiplication of Integers Multiplication is repeated addition – we are combining sets of tiles. The first factor tells us the number of sets needed. The second factor tells the size of the set (how many tiles) and the kind of numbers/tiles to use (positive or negative). If the first factor is negative, we interpret that as “the opposite of.”

Multiplication of Integers (+2)(+3) = (+3)(-4) = number of sets size and kind of set +6 Two sets of three positives +12 Three sets of four negatives

Multiplication of Integers If the first factor is negative it means “the opposite of.” (-2)(+3) (-3)(-1) = -6 The opposite of two sets of three positives = +3 The opposite of three sets of one negative

Multiplication of Integers Let’s Practice (+4)(+2) (+5)(-3) (-2)(+2) (-3)(-3)

Multiplication of Integers After students have seen many examples, students will begin to see the pattern for integer multiplication. Have students practice applying the concept with larger integers. Fade away the manipulatives.

Division of Integers Division is putting things into sets of equal size. We start with a total number of tiles – either positive or negative. The divisor tells us how many sets to make. The answer is how many tiles are in each set. If the divisor is negative, we interpret that as “the opposite of.”

Division of Integers 3 -4 (+6)/(+2) = (-8)/(+2) = Put the tiles into two equal groups of 3 positives (-8)/(+2) = Put the tiles into two equal groups of 4 negatives 3 -4

Division of Integers -5 A negative divisor means “take the opposite of” (flip-over) (+10)/(-2) = Divide ten tiles into two equal groups of +5 Find the opposite of +5 -5

Division of Integers +4 (-12)/(-3) = Divide 12 negatives into 3 equal groups of -4. Find the opposite of -4. +4

Division of Integers It’s Your Turn! (+8)/(+4) (+12)/(-3) (-12)/(+6) (-10)/(-5)

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.

Solving Equations -2 -2 x = 1 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 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 x = -6 One third of an x is the same as two • 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 - x - x x + 3 = -5 + -3 + - 3 x = -8 2 x + 3 = x – 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”

Evaluation Survey - Thank You for Your Feedback. http://tinyurl

Polynomials and Algebra Tiles

Link to Math Summit Program https://mathsummit2017.sched.com  

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

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

Zero Pairs Called zero pairs because they are additive inverses of each other. When put together, they model zero. Don’t say “cancel out” for zeroes! Instead say “they form a zero pair” or “they add up to zero.” + - + -

Find the Additive Inverse Model and find the additive inverse for: 5x + 1 3x2 + 1 -2 + 4x -4x2 - 7

Simplifying Polynomials: Distributive Property Use the same concept that was applied with multiplication of integers: think of the first factor as the number of sets. Consider the following: 3(x + 2) We need three sets of x + 2

Distributive Property 3(x + 2)= We have three x’s 3x and three groups of +2 +6 3·x + 3·2 = 3x + 6

Distributive Property Model each of the following: 2(x + 2) 4(x + 3) 3(2x – 3) 4. -3(3x -2)

Adding and Subtracting 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.

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)

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.

Solving Equations -2 -2 x = 1 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 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 x = -6 One third of an x is the same as two • 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 - x - x x + 3 = -5 + -3 + - 3 x = -8 2 x + 3 = x – 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”

Multiplying Monomials (2x)(3) Make 3 sets of two x’s

Multiplying Polynomials by Monomials x(x – 1) Fill in each section of the area model x2 – 1x

Multiplying Polynomials by Monomials x(2x + 3) = 2x2 + 3x Fill in each section of the area model

Dividing Monomials 4x/2x = 2x Find Zero Pairs and simplify

Multiplication Multiplication using “area model” (12)(13) Think of it as (10+2)(10+3) Multiplication using the array method allows students to see all four sub-products.

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

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

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

Multiplying Polynomials (x + 2)(x – 3) (x – 2)(x – 3) (x + 4)(x – 2)

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 (x + 1) = 3 · (x – 3) 2x - 6 = 2 * Note the two are positive, this needs to be developed

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

Factoring Polynomials x2 – 5x + 6 = (x – 2)(x – 3)

Factoring Polynomials x2 – x – 6 = (x + 2)(x – 3)

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

Evaluation Survey - Thank You for Your Feedback. http://tinyurl

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 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