Title: Middle School Mathematics Teachers
1Middle School Mathematics Teachers
CircleInstitute for Mathematics EducationNov.
3, 2009
Using Algebra Blocks for Teaching Middle School
Mathematics Exploring a Variety of Topics
- Cynthia Anhalt
- canhalt_at_math.arizona.edu
2Algebra Blocks
3Identifying the pieces
4Algebra Expression Mat
-
5Using algebra blocks for combining similar terms
6Combining 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
7Using algebra blocks for multiplying binomials
8Consider (x y) (x y)
- What is the common solution that students choose
on the AIMS exam? - y X2 y2
- 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 dont 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?
9Use 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
10Use 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
11Create your own multiplication of binomials
12Using algebra blocks for teaching perimeter
13Determine the perimeter
P 4x 2y
P y 5 (y -1) P 2y 4
14Determine the perimeter
P 1 x (y-1) 1 (y-1) 3 1 x 2x
2y 4
15Determine 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
16Create your own shape for the class to find the
perimeter
17Algebra Equation Mat
- -
18Equations solve for x
2(x-3) -4
- -
19Equation solve for x
- -
2x -6 -4
6 6
2x 2
_ _ 2 2
x 1
20Equation solve for y
- -
5y -20 10
20 20
5y 30
_ _ 5 5
y 6
21Create your own equation for the class to solve
for x
- -
22Discussion
- 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?
23How 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?
24Thank you for sharing your evening with me.
- Cynthia Anhalt
- canhalt_at_math.arizona.edu
25How 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?
26Multiplication Models
What models would represent 2 x 4?
2 groups of 4 discrete objects
27Consider the area model for multiplication of
whole numbers
4
3
28Base 10 Blocks
How can we convert these sets into an area model?
29Area model using Base 10 Blocks
3 x 12 36
30Consider the area model to make the connection of
arithmetic to algebraic thinking
100 20 30 6
Using base-10 blocks, we have
31Show the area model of 14 x 23 with base 10
blocks
(104)(203) 200803012 322
32Discussion
- 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?
33Thank you for sharing you evening with me.
- Cynthia Anhalt
- canhalt_at_math.arizona.edu