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

2. Algebra Blocks Figure out the pieces.

3. Identifying the pieces x2x2 xy x y y2y2 x2x2 1 (1 1) (x 1) (x x) (y 1) (y y) (x y)

4. Algebra Expression Mat + -

5. Using algebra blocks for combining similar terms

6. Combining Similar Terms 5x + 3y + 4 - 2 + 4x - y x 2 + 2y -2 + 5 + 2x 2 - 3y 2xy - x + -3 + 3y 2 - 2x 2 + 5 + (-xy) + 2x 4x + 3x 2 – 2xy + (-2x) – x 2

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 = X 2 + y 2 Why is y= x 2 + 2xy + y 2 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 7-12 th 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. Use the area model of multiplication with the algebra blocks for multiplying the two binomials: (x + y) (x + y) x2x2 xy y2y2 (x + y) (x + y) = = x 2 + xy + xy + y 2 = x 2 + 2xy + y 2 x + y

10. Use the algebra blocks to show: (2x + y) (x + y + 4) = xxy x y 1 1 1 1 x2x2 x2x2 y2y2 xy x x x x x x x x y y y y = 2x 2 + 2xy + 8x + xy +y 2 + 4y xy By combining similar terms: = 2x2 + 3xy + 8x + y2 + 4y

11. Create your own multiplication of binomials

12. Using algebra blocks for teaching perimeter

13. Determine the perimeter x2x2 xy P = 4x + 2y y P = y + 5 + (y -1) P = 2y + 4

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

15. Determine the perimeter: x2x2 xy x y 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. ++ -- x x 2x -6 = -4 2x = 2 +6 +6 _ 2 x = 1 Equation: solve for x =

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

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

22. What do you suppose are the benefits of using the algebra blocks? What do you suppose are the benefits of using the algebra blocks? What do you suppose are the drawbacks of using algebra blocks? What do you suppose are the drawbacks of using algebra blocks? Other comments? Other comments? Discussion…

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

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

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

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

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

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 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? 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