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Let’s Do Algebra Tiles

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Algebra Tiles Manipulatives used to enhance student understanding of subject 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 indigo square represent +1 and the small red square (the flip- side) represent -1. The indigo and red squares are additive inverses of each other.

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Zero Pairs Called zero pairs because they are additive inverses of each other. When put together, they cancel each other out to model 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.

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Addition of Integers (+3) + (1) = (6) + (+6) = 6 + 6 12 (-2) + (-1) = -2 + (-1) = -3 3 + 1 = 4

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Addition of Integers (-3) + (-1) = -3 + (-1) = -4 (-4) + (-5) = -4 +(-5) -9 (-6) + (-6) = -6+ (-6) = -12

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Addition of Integers (+3) + (-1) = 3 + (-1) = 2 (+4) + (-4) = 4+ (-4) = 0 (-2) + 5 = 3 After students have seen many examples of addition, have them formulate rules.

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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.

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Subtraction of Integers Subtraction can be interpreted as “take-away.” Subtraction can also be thought of as “adding the opposite.” For each of the given examples, use algebra tiles to model the subtraction. Draw pictorial diagrams which show the modeling process.

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Subtraction of Integers (+5) – (+2) = Remove or take away +2 of the tiles: (+1) – (+1) = (-4) – (-3) = (-7) – (-2) = So,….What’s left?

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Subtracting Integers (+3) – (-5) = +3 “remove” -5, but you do not have any negatives to remove. Add zero pairs!

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Subtracting Integers (+3) – (-5) = Now you can remove -5 And, your left with…… Now, try: (-4) – (+1) = Zero pair

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Subtracting Integers (-4) – (+1) = Add a zero pair. Now you can remove +1 And, your left with…… Now, try these: (+3) – (-3) = (-8) – (+3) =4 – (-7) =-6 – (+9) = After students have seen many examples, have them formulate rules for integer subtraction. Zero pair

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So, what are the rules for subtracting integers? Look at the second number to see if you can remove that many from what you have. If not, add zero pairs. Remove the tiles you need to remove. Count tiles left. Your end result will be to simply add the opposite of whatever the sign is.

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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.

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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) =

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Multiplication of Integers If the counter is negative it will mean “remove” __ sets of __ (-2)(+3) Remove 2 sets of +3. Nothing to remove so start with zeros.

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Multiplication of Integers If the counter is negative it will mean “remove” __ sets of __ (-2)(+3) “Remove 2 sets of +3.” Nothing to remove, so start with zeros. Now remove 2 sets of +3 Try: (-3)(-1)

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Multiplication of Integers With many examples, students may develop a rule If the counter is negative it will mean “take the opposite of.” (flip-over) (-2)(+3) Then flip Two sets of 3 (-3)(-1) Then flip

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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) = (-8)/(+2) =

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Division of Integers A negative divisor will mean “take the opposite of.” (flip-over) (+10)/(-2) =

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

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NYS Common Core Learning Standards 3.OA.5 5. Apply properties of operations as strategies to multiply and divide. Knowing that 8 x 5 = 40 and 8 x 2 = 16, one can find 8 x 7 as 8 x (5 + 2) = (8 x 5) + (8 x 2) = 40 + 16 = 56. (Distributive property.) 6.NS.4 Use the distributive property to express a sum of two whole numbers 1–100 with a common factor as a multiple of a sum of two whole numbers with no common factor. For example, express 36 + 8 as 4 (9 + 2). 6.EE.3 Apply the properties of operations to generate equivalent expressions. For example, apply the distributive property to the expression 3 (2 + x) to produce the equivalent expression 6 + 3x; apply the distributive property to the expression 24x + 18y to produce the equivalent expression 6 (4x + 3y); apply properties of operations to y + y + y to produce the equivalent expression 3y. Distributive Property

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NYS Common Core Learning Standards 7.NS.2a a. Understand that multiplication is extended from fractions to rational numbers by requiring that operations continue to satisfy the properties of operations, particularly the distributive property, leading to products such as (–1)(–1) = 1 and the rules for multiplying signed numbers. Interpret products of rational numbers by describing real-world contexts. 8.EE.7b Solve linear equations with rational number coefficients, including equations whose solutions require expanding expressions using the distributive property and collecting like terms. Distributive Property

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Use the same concept that was applied with multiplication of integers, think of the first factor as the counter. The same rules apply. 3(12) You can split this up into manageable parts: 3(10 + 2) Three is the counter, so we need three rows of (10 + 2)

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Distributive Property 3(10 + 2) 30 + 6 36 Now try these 2(16)4(14)2(19)

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20 + 12 32 4(14) = 4(10 + 4) = 40 +16 = 56 2(16) = 2(10 + 6)

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2(19) = 2(20 – 1) =40– 2 = 38

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Distributive Property This also works with a variable in the parentheses. 3(X + 2) 3x + 6 Now try these: 3(x + 4)2(x – 2)4(x – 3)

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Distributive Property 3(x + 4) 2(x – 2) 2x – 4 3x + 12 4(x – 3) 4x – 12

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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. Variables can be isolated by using zero pairs. Equations are unchanged if equivalent amounts are added to each side of the equation.

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Solving Equations Use the green rectangle as X and the red rectangle (flip-side) as –X (the opposite of X). X + 2 = 3

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Solving Addition Equations X + 2 = 3 Therefore: X = 1 Now try these: x + 3 = 8 5 + x = 12

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X + 3 = 8 Therefore: X = 5

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5 + x = 12 Therefore: X = 7

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9 = x + 4 Therefore: 5 = x Now try these: 8 = x + 6 14 = 10 + x

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8 = x + 6 Therefore: 2 = x

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14 = 10 + x Therefore: 4 = x

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Solving Subtraction Equations X – 2 = 3 Therefore: X = 5 Now try these: x – 3 = 8 5 – x = 12

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X – 3 = 8 Therefore: X = 11

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5 – x = 12 Then flip all to make the x positive Therefore: X = -7

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9 = x – 4 Therefore: 13 = x Now try these: 8 = x – 6 14 = 10 – x

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8 = x – 6 Therefore: 14 = x

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14 = 10 – x Then flip the sides to make the x positive Therefore: -4 = x

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Multiplication Equations 2X = 6 Then split the two sides into 2 even groups. Therefore: x = 3 Now, try these:4x = 8 3x = 15

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Multiplication Equations 4X = 8 Then split the two sides into 4 even groups. Therefore: x = 2

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Multiplication Equations 3x = 15 Then split the two sides into 3 even groups. Therefore: x = 5

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Multiplication Equations 4x = -16 Then split the two sides into 4 even groups. Therefore: x = -4 Now, try these:-10 = 2x-2x = -10

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Multiplication Equations -10 = 2x Then split the two sides into 2 even groups. Therefore: x = -5

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Multiplication Equations -10 = -2x Then split the two sides into 2 even groups. But to make the x’s positive, you have to flip both sides. Therefore: x = 5

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Division Equations = 6 We can’t split or cut the x into 2 parts here. So let’s manipulate the two sides to fit the whole x. It takes 2 groups of the 6 to make the whole x. So make enough of the number side to make the whole x—make another group of 6. Therefore: x = 12

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Division Equations = 9 We can’t split or cut the x into 3 parts here. So let’s manipulate the two sides to fit the whole x. It takes 3 groups of the 9 to make the whole x. So make enough of the number side to make the whole x—make two other groups of 9. Therefore: x = 27

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Division Equations = 3 We can’t split or cut the x into 5 parts here. So let’s manipulate the two sides to fit the whole x. It takes 5 groups of the 3 to make the whole x. So make enough of the number side to make the whole x—make 4 other groups of 3. Therefore: x = 15

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Division Equations 2 = We can’t split or cut the x into 6 parts here. So let’s manipulate the two sides to fit the whole x. It takes 6 groups of the 2 to make the whole x. So make enough of the number side to make the whole x—make 5 other groups of 2. Therefore: x = 12

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Division Equations -2 = We can’t split or cut the x into 6 parts here. So let’s manipulate the two sides to fit the whole x. It takes 6 groups of the -2 to make the whole x. So make enough of the number side to make the whole x—make 5 other groups of -2. Therefore: x = -12

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Division Equations = -3 We can’t split or cut the x into 5 parts here. So let’s manipulate the two sides to fit the whole x. It takes 5 groups of the -3 to make the whole x. So make enough of the number side to make the whole x—make 4 other groups of -3. Therefore: x = -15

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Division Equations = -9 We can’t split or cut the x into -3 parts here. So let’s manipulate the two sides to fit the whole x. It takes -3 groups of the -9 to make the whole x. So make enough of the number side to make the whole x—make two other groups of -9. Only this time, the x is negative, so we have to flip everything in order to make it positive.

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Two Step Equations 2X + 4 = 6 The first thing is to get the x’s by themselves. Do this by subtracting 4 from both sides of the equation. Then, get rid of zero pairs. Now, split the two sides into equal parts. Therefore: x = 1

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Two Step Equations Now, try this: 4x + 2 = 10 First, make your equation with algebra tiles Next, add/subtract #’s to get the x’s by themselves. And get rid of the zero pairs. After that, divide the sides into even groups of x’s & #’s Therefore: x = 2

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Two Step Equations How about this one: 3x – 2 = 13 First, make your equation with algebra tiles Next, add/subtract #’s to get the x’s by themselves. And get rid of the zero pairs. After that, divide the sides into even groups of x’s & #’s Therefore: x = 5

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Solving Equations- Variables on both sides 2X + 3 = X – 5 2X + 3 = X - 5-X X + 3 = - 5

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Solving Equations- Variables on both sides 2X + 3 = X – 5 X + 3 = - 5- 3 X= - 8

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Solving Equations Check by substituting X with - 8 2X + 3 = X – 5

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

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Modeling Polynomials Let the blue square represent x 2, the green rectangle xy, and the purple square y 2. The red square (flip-side of blue) represents –x 2, the red rectangle (flip-side of green) –xy, and the large red square (flip-side purple –y 2). As with integers, the red shapes and their corresponding flip-sides form a zero pair.

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Modeling Polynomials X x 2 y 2 y 1

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Modeling Polynomials xy 1

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Modeling Polynomials Represent each of the following with algebra tiles, draw a pictorial diagram of the process, then write the symbolic expression. 2x 2 4xy 3

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Modeling Polynomials 2x 2 4y 3

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Modeling Polynomials 3x 2 + 2y 2 -2xy -3x 2 –xy

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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 -3x + 1 + x + 3

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More Polynomials 2x + 4 + x + 2 -3x + 1 + x - 3

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More Polynomials 3x + 1 – (2x + 4) This process can be used with problems containing x 2. (2x 2 + 5x – 3) + (-x 2 + 2x + 5) (2x 2 – 2x + 3) – (3x 2 + 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 let x = 4 3 + 2x let x = -4 3 – 2x let x = 4

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

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Multiplying Polynomials (x + 2)(x + 3) x+3 =x 2 + 5x + 6 x + 2

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Multiplying Polynomials (x – 1)(x +4) x + 4 x - 1

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Multiplying Polynomials (x – 1)(x +4) x + 4 x - 1

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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

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Factoring Polynomials x + 13x+3= 3(x+1) 3

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Factoring Polynomials x 2 + 6x + 8 Need to make a rectangle

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Factoring Polynomials x 2 + 6x + 8

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Factoring Polynomials x 2 + 6x + 8 x + 4 x + 2

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Factoring Polynomials x 2 + 6x + 8

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Factoring Polynomials x 2 + 6x + 8

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Factoring Polynomials x 2 + 6x + 8

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Factoring Polynomials x 2 – 5x + 6

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Factoring Polynomials x 2 – 5x + 6 + - - - - - + + + +++

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Factoring Polynomials x 2 – 5x + 6 x - 3 + + + - - - - - + + + -- - - - X - 2 +++

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Factoring Polynomials x 2 – x – 6

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Factoring Polynomials x 2 – x – 6 Can add zeros to “fill in”

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Factoring Polynomials x 2 – x – 6 Think about signs

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Factoring Polynomials x 2 – x – 6

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Factoring Polynomials x 2 + x – 6 x 2 – 1 x 2 – 4 2x 2 – 3x – 2 2x 2 + 3x – 3 -2x 2 + 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 x 2 + 7x +6 x + 1 X+1

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Dividing Polynomials x 2 + 7x +6 x + 1

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Dividing Polynomials 2x 2 + 5x – 3 x + 3 x 2 – x – 2 x – 2 x 2 + x – 6 x + 3

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Conclusion “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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