Presentation on theme: "Let’s Do Algebra Tiles Roland O’Daniel, The Collaborative for Teaching and Learning October, 2009 Adapted from David McReynolds, AIMS PreK-16 Project."— Presentation transcript:
1Let’s Do Algebra TilesRoland O’Daniel, The Collaborative for Teaching and Learning October, 2009Adapted from David McReynolds, AIMS PreK-16 Projectand Noel Villarreal, South Texas Rural Systemic Initiative
2Algebra TilesManipulatives 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.
3Algebra TilesSupport 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.
4Algebra TilesAlgebra 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.
5Algebra TilesAlgebra 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.
6Algebra TilesLet 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.
8Zero PairsCalled 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
9Addition 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 performedThe following problems are animated to represent how these problems can be solved.
10Addition of Integers (+3) + (+1) = 4 Combine like objects to get four positives(-2) + (-1) =Combine like objects to get three negatives4-3
11Addition of Integers (+3) + (-1) = Make zeroes to get two positives Important to have students perform all three representations, to create deeper understandingQuality not quantity in these interactions+2
12(+3) + (-4) = -1 Make three zeroes Final answer one negative After students have seen many examples of addition, have them formulate rules.-1
13Subtraction 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 processWrite a description of the actions taken
14Subtraction of Integers (+5) – (+2) =Take away two positivesTo get three positives(-4) – (-3) =Take away three negativesFinal answer one negative+3-1
15Subtracting Integers (+3) – (-5) = Can’t take away any negatives so add five zeroes;Take away five negatives;Get eight positives+8
16Subtracting Integers(-4) – (+1)= Start with 4 negatives No positives to take away; Add one zero; Take away one positive To get five negatives-5
17Subtracting Integers (+3) – (-3)= After students have seen many examples, have them formulate rules for integer subtraction.(+3) – (-3) is the same as to get 6
18Multiplication 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 processWrite a description of the actions performed
19Multiplication of Integers The counter indicates how many rows to make. It has this meaning if it is positive.(+2)(+3) =(+3)(-4) =+6Two groups of three positives-12Three groups of four negatives
20Multiplication 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) =-6Two groups of three‘Opposite of’ to get sixnegatives+3‘Opposite of’ three groups of negative oneTo get three positives
21Multiplication of Integers After students have seen many examples, have them formulate rules for integer multiplication.Have students practice applying rules abstractly with larger integers.
22Division of IntegersLike 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.
23Division of Integers (+6)/(+2) = Divide into two equal groups 3 in each group(-8)÷(+2) =3-4
24Division of IntegersA negative divisor will mean “take the opposite of” (flip-over)(+10)/(-2) =Divide into two equal groupsFind the opposite ofTo get five negatives in each group-5
25Division of Integers (-12)/(-3) = After students have seen many examples, have them formulate rules.+4
26Polynomials“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
27Distributive 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)
28Distributive Property 3(x + 2)=Three groups of the quantity x and 2Three Groups of x to get three x’sThree groups of 2 to get 63·x +3·2 =3x + 6
34More PolynomialsUse 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 x + 2
35More Polynomials 2x + 4 + x + 1 = 3x + 5 Combine like terms to get three x’s and five positives
36More 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)
37(2x2 – 2x + 3) – (3x2 + 3x – 2) 2x2 + -2x + 3 + -3x2 + -3x + +2 Take away the second quantity or changeto addition of the opposite2x2 + -2x x2 + -3x + +2Make zeroes and combine like terms(2x2 + -3x2)+ (-2x + -3x)+ (3+ 2)Simplify and write polynomial instandard form-x2 – 5x + 5
38SubstitutionAlgebra 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
40Substitution4x + 2y =let x = -2 & y = 34(-2)+ 2(3) =-8=-2
41Solving EquationsAlgebra 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).
42Solving Equations x + 2 = 3 -2 -2 x = 1 x = 1x and two positives are the same as three positivesadd two negatives to both sides of the equation; makes zeroesone x is the same as one positive
43Solving Equations -5 = 2x ÷2 ÷2 2½ = x ÷2 ÷22½ = xTwo x’s are the same as five negativesDivide both sides into two equal partitionsTwo and a half negatives is the same as one x
44Solving Equations · -1 · -1 One half is the same as one negative x · · -1One half is the same as one negative xTake the opposite of both sides of the equationOne half of a negative is the same as one x
45Solving Equations• • 3x = -6One third of an x is the same as two negativesMultiply both sides by three (or make both sides three times larger)One x is the same as six negatives
46Solving Equations 2 x + 3 = x – 5 - x - x x + 3 = -5 + -3 + - 3 x = -8 x = -8Two x’s and three positives are the same as one x and five negativesTake one x from both sides of the equation; simplify to get one x and three the same as five negativesAdd three negatives to both sides; simplify to get x the same as eight negatives
47Solving Equations 3(x – 1) + 5 = 2x – 2 3x – 3 + 5 = 2x – 2 – 2 or3x = 2x – 4-2x xx = -4“x is the same as four negatives”
48Multiplication 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
49Multiplication using “Area Model” (12)(13) = (10+2)(10+3) == 15612x 1336+12015610 x 10 = 102 = 10010 x 3 = 3010 x 2 = 2010 x 2 = 202 x 3 = 6
50Multiplying Polynomials (x + 2)(x + 3)= x2 + (2x + 3x) + 6= x2 + 5x + 6Model the dimensions (factors) of the rectangleFill in each section of the area modelCombine like terms andwrite in standard form
51Multiplying Polynomials (x – 1)(x +4)Model the dimensions of the rectangleFill in each section of the area modelMake Zeroes orcombine like termsand simplifyx2+ 4x– 1x– 4= x2 + 3x – 4
53Factoring 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.
54Factoring Polynomials 3x + 32x – 6(x + 1)= 3 ·= 2 ·(x – 3)Note the two is positive, because the green is positive, this decision takes time to develop
55Factoring Polynomials Model the expression tobe factoredx2 + 6x + 8= (x + 2)(x +4)Create a rectangle fromthe dividendDetermine & model thedimensions of the rectangle.It takes trial and error to beable to create the rectangles,observation & analysis will letstudents make the connectionsbetween the factors of the constantand how to divide the x term.The factored form of the expression is …
56Factoring Polynomials Model the expression tobe factoredx2 – 5x + 6= (x – 2)(x – 3)Create a rectangle fromthe dividendDetermine & model thedimensions 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.The factored form of the expression is …
57Factoring Polynomials x2 – x – 6= (x + 2)(x – 3)Model the expression tobe factoredCreate a rectangle fromthe dividendIn this case we need to addzeroes in order to completethe rectangle.Determine & model thedimensions of the rectangle.Note that we know to have anegative value in the productthe values being multipliedmust be + and – .The factored form of the expression is …
59Dividing 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.
60Dividing Polynomials x2 + 7x +6 = (x + 6) x + 1 Model the problem, dividendand divisor= (x + 6)Create a rectangle fromthe dividend with the widthof x + 1Determine and modelthe length of the rectangle`````
61Dividing Polynomials x2 + 7x +6 x + 1 2x2 + 5x – 3 x + 3 x2 – x – 2
62Completing the SquareAlgebra tiles can be used to complete the square when factoring quadraticsUse tiles and frame to represent problem. Expression should form a square array inside the form.The square factors will form the dimensionsBe prepared to use zero pairs of constants to complete the square
63x2 + 4x +7 x2 + 4x +7 (x + 2) • (x + 2) = (x + 2)2 + 3 Model the expressionForm a square with theterms of the expression(as close as possible)x2 + 4x +7= (x + 2)2 + 3Determine and model thedimensions of the squareRewrite in perfect squareform including the “extravalues” to completethe square.
64x2 + 6x +7 x2 + 6x +7 (x + 3) • (x + 3) = (x + 3)2 + -2 Model the expressionForm a square with theterms of the expression(as close as possible)x2 + 6x +7= (x + 3)2 + -2Add 2 zeroes to completethe squareDetermine and model thedimensions of the squareRewrite in perfect squareform including the “extravalues” to completethe square.
65ConclusionAlgebra 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 HRWNational Library of Virtual Manipulatives
66Resources“Let’s Do Algebra Tiles” by David McReynolds, AIMS PreK-16 Project & Noel Villarreal, South Texas Rural Systemic InitiativeJo Ann Mosier & Roland O’DanielCollaborative for Teaching and LearningPartnership for Reform Initiatives in Science and Mathematics