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Using Algebra Blocks for Teaching Middle School Mathematics: Exploring a Variety of Topics Middle School Mathematics Teachers’ Circle Institute for Mathematics & Education Nov. 3, 2009 Cynthia Anhalt [email protected] Algebra Blocks Figure out the pieces. Identifying the pieces 1 (y •y) (1 • 1) y2 x x2 (x •1) (y •1) y x2 (x •y) xy (x •x) Algebra Expression Mat + - Using algebra blocks for combining similar terms Combining Similar Terms 5x + 3y + 4 - 2 + 4x - y x2 + 2y -2 + 5 + 2x2 - 3y 2xy - x + -3 + 3y2 - 2x2 + 5 + (-xy) + 2x 4x + 3x2 – 2xy + (-2x) – x2 Using algebra blocks for multiplying binomials Consider: (x + y) (x + y) What is the common solution that students choose on the AIMS exam? y = X2 + y 2 Why is y= x2 + 2xy + y2 not understood as the solution by so many students? “FOIL” is sometimes taught as a procedure without conceptual understanding. Do you agree or disagree? Why? Would “LOIF” or “OILF” work? Most 7-12th grade students don’t know. What is the underlying concept? …multiplication of two binomials When “FOIL” is taught as a procedure, most students have a difficult time multiplying (x + y + 3) (x + 2y). Why do you suppose? Use the area model of multiplication with the algebra blocks for multiplying the two binomials: (x + y) (x + y) x+y (x + y) (x + y) = x2 xy = x2 + xy + xy + y2 x+y = x2 + 2xy + y2 xy y2 Use the algebra blocks to show: (2x + y) (x + y + 4) = x x x2 x x2 y xy = 2x2 + 2xy + 8x + xy +y2 + 4y By combining similar terms: y xy xy y2 1 x x y 1 x x y 1 x x y 1 x x y = 2x2 + 3xy + 8x + y2 + 4y Create your own multiplication of binomials Using algebra blocks for teaching perimeter Determine the perimeter x2 xy P = 4x + 2y y P = y + 5 + (y -1) P = 2y + 4 Determine the perimeter y x P = 1 + x + (y-1) + 1 + (y-1) + 3 + 1 + x = 2x + 2y + 4 Determine the perimeter: x2 xy y x P = x + 1 + 1 + y + x + (y-x) + 1 + x + 1 + (x-1) + 1 + (y-1) + 1 + (y-x) + x + x P = 4x + 4 + 4y Create your own shape for the class to find the perimeter… Algebra Equation Mat + - + = - Equations: solve for x 2(x-3) = -4 + - + = - Equation: solve for x 2(x-3) = -4 + 2x -6 = -4 +6 +6 + x 2x _ =2 _ 2 2 x=1 x - = - Equation: solve for y + + 5(y-4) = 10 5y -20 = 10 +20 +20 5y = 30 _ _ 5 5 y = 6 - = - Create your own equation for the class to solve for x + - + = - Discussion… What do you suppose are the benefits of using the algebra blocks? What do you suppose are the drawbacks of using algebra blocks? Other comments? How do the algebra blocks interface with what NCTM advocates in teaching mathematics? Consider the NCTM process standards… Explain the potential of the NCTM process standards in teaching with algebra blocks. Connections Representation Communication Problem Solving Reasoning & Proof How does using the algebra blocks impact students who are ELL? Thank you for sharing your evening with me. Cynthia Anhalt [email protected] How can algebraic thinking begin in the elementary grades? How can the elementary mathematics curriculum can be taught with algebraic thinking as a goal? How can base-10 blocks be used to aid in teaching multiplication that leads to the distributive property? Multiplication Models FOCUS What models would represent 2 x 4? Area model Set model 2 by 4 covered area 2 groups of 4 discrete objects Array model Linear model 2 jumps of 4 2 rows by 4 columns of discrete objects Consider the area model for multiplication of whole numbers 3 x 4 = 12 4 3 Base 10 Blocks 3 x 12 How can we convert these sets into an area model? Area model using Base 10 Blocks 12 3 3 x 12 = 36 Consider the area model to make the connection of arithmetic to algebraic thinking: 12 x 13 = (10+ 2) (10+3) = 100 + 20 + 30 + 6 Using base-10 blocks, we have: 12 X 13 3x2 = 6 3x10 = 30 10x2 = 20 10x10=100 156 Four partial products Show the area model of 14 x 23 with base 10 blocks (10+4)(20+3) = 200+80+30+12 = 322 Discussion How do these ideas promote algebraic thinking in the elementary mathematics classrooms or in the middle grades? How can you incorporate a variety of models of mathematical representation into your teaching? Thank you for sharing you evening with me. Cynthia Anhalt [email protected]