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Middle School Mathematics Teachers’ Circle Institute for Mathematics & Education Nov. 3, 2009

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 canhalt@math.arizona.edu. Algebra Blocks. Figure out the pieces. ( y • y ). (1 • 1). 1.

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Middle School Mathematics Teachers’ Circle Institute for Mathematics & Education Nov. 3, 2009

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  1. Using Algebra Blocks for Teaching Middle School Mathematics: Exploring a Variety of Topics Middle School Mathematics Teachers’ CircleInstitute for Mathematics & EducationNov. 3, 2009 Cynthia Anhalt canhalt@math.arizona.edu

  2. Algebra Blocks Figure out the pieces.

  3. (y •y) (1 • 1) 1 y2 x2 (y •1) (x •1) (x •y) y x x2 xy (x •x) Identifying the pieces

  4. Algebra Expression Mat -

  5. Using algebra blocks for combining similar terms

  6. 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

  7. Using algebra blocks for multiplying binomials

  8. Consider: (x + y) (x + y) What is the common solution that students choose on the AIMS exam? y = X2 + y2 Why is y= x2 + 2xy + y2not 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?

  9. x + y y2 x2 Use the area model of multiplication with the algebra blocks for multiplying the two binomials: (x + y) (x + y) (x + y) (x + y) = = x2 + xy + xy + y2 = x2 + 2xy + y2 xy x + y xy

  10. 1 1 1 1 y2 y y y y y y x x x x x x x x x x x x2 x2 xy Use the algebra blocks to show: (2x + y) (x + y + 4) = = 2x2 + 2xy + 8x + xy +y2 + 4y By combining similar terms: = 2x2 + 3xy + 8x + y2 + 4y xy xy

  11. Create your own multiplication of binomials

  12. Using algebra blocks for teaching perimeter

  13. x2 xy y Determine the perimeter P = 4x + 2y P = y + 5 + (y -1) P = 2y + 4

  14. x y Determine the perimeter P = 1 + x + (y-1) + 1 + (y-1) + 3 + 1 + x = 2x + 2y + 4

  15. 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

  16. Create your own shape for the class to find the perimeter…

  17. Algebra Equation Mat =

  18. Equations: solve for x 2(x-3) = -4 =

  19. 2(x-3) = -4 x x Equation: solve for x 2x -6 = -4 +6 +6 2x = 2 _ _ 2 2 = x = 1

  20. Equation: solve for y 5(y-4) = 10 5y -20 = 10 +20 +20 5y = 30 = _ _ 5 5 y = 6

  21. Create your own equation for the class to solve for x =

  22. 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?

  23. 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?

  24. Thank you for sharing your evening with me. Cynthia Anhalt canhalt@math.arizona.edu

  25. 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?

  26. FOCUS Area model 2 by 4 covered area Array model Linear model 2 rows by 4 columns of discrete objects 2 jumps of 4 What models would represent 2 x 4? Multiplication Models 2 groups of 4 discrete objects Set model

  27. Consider the area model for multiplication of whole numbers 3 x 4 = 12 4 3

  28. Base 10 Blocks 3 x 12 How can we convert these sets into an area model?

  29. 12 3 Area model using Base 10 Blocks 3 x 12 = 36

  30. 12 X 13 3x2 = 6 3x10 = 30 10x2 = 20 10x10=100 156 Four partial products Consider the area model to make the connection of arithmetic to algebraic thinking: = 100 + 20 + 30 + 6 12 x 13 = (10+ 2) (10+3) Using base-10 blocks, we have:

  31. Show the area model of 14 x 23 with base 10 blocks (10+4)(20+3) = 200+80+30+12 = 322

  32. 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?

  33. Thank you for sharing you evening with me. Cynthia Anhalt canhalt@math.arizona.edu

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