Middle School Mathematics Teachers’ CircleMiddle School Mathematics Teachers’ CircleInstitute for Mathematics & EducationInstitute for Mathematics & Education

Nov. 3, 2009Nov. 3, 2009

Cynthia AnhaltCynthia Anhalt

canhalt@math.arizona.ecanhalt@math.arizona.edudu

Using Algebra Blocks for Teaching Middle School

Mathematics: Exploring a Variety of Topics

Algebra BlocksAlgebra Blocks

Figure out the pieces.Figure out the pieces.

Identifying the piecesIdentifying the pieces

x2

xy

x

y

y2

x2

1 (1 • 1)

(x •1)

(x •x)

(y •1)

(y •y)

(x •y)

Algebra Expression MatAlgebra Expression Mat++

-

Using algebra blocks Using algebra blocks for combining similar for combining similar termsterms

Combining Similar TermsCombining Similar Terms

5x + 3y + 4 - 2 + 4x - y5x + 3y + 4 - 2 + 4x - y

xx22 + 2y -2 + 5 + 2x + 2y -2 + 5 + 2x22 - 3y - 3y

2xy - x + -3 + 3y2xy - x + -3 + 3y2 2 - 2x- 2x2 2 + 5 + (-xy) + 2x+ 5 + (-xy) + 2x

4x + 3x4x + 3x22 – 2xy + (-2x) – x – 2xy + (-2x) – x22

Using algebra blocks Using algebra blocks for multiplying for multiplying

binomials binomials

Consider: (x + y) (x + y)Consider: (x + y) (x + y)What is the common solution that students choose on What is the common solution that students choose on the AIMS exam?the AIMS exam?

y = Xy = X22 + y + y22

Why is Why is y= xy= x22 + 2xy + y + 2xy + y22 not understood as the solution by not understood as the solution by so many students?so many students?

““FOIL” is sometimes taught as a procedure without FOIL” is sometimes taught as a procedure without conceptual understanding. Do you agree or disagree? conceptual understanding. Do you agree or disagree? Why?Why?

Would “LOIF” or “OILF” work? Most 7-12Would “LOIF” or “OILF” work? Most 7-12thth grade students grade students don’t know.don’t know.

What is the underlying concept?What is the underlying concept?……multiplication of two binomialsmultiplication of two binomials

When “FOIL” is taught as a procedure, most students have When “FOIL” is taught as a procedure, most students have a difficult time multiplying (x + y + 3) (x + 2y). Why do a difficult time multiplying (x + y + 3) (x + 2y). Why do you suppose?you suppose?

Use the area model of multiplication with the Use the area model of multiplication with the algebra blocks for multiplying the two binomials: algebra blocks for multiplying the two binomials: (x + y) (x + y)(x + y) (x + y)

x2 xy

xy

y2

(x + y) (x + y) =

= x2 + xy + xy + y2

= x2 + 2xy + y2

x +

y

x + y

Use the algebra blocks to show: Use the algebra blocks to show: (2x + y) (x + y + (2x + y) (x + y + 4) =4) =

x x yx

y

1

1

1

1

x2 x2

y2

xy

xy

x

x

x

x

x

x

x

x

y

y

y

y

= 2x2 + 2xy + 8x + xy +y2 + 4y

xy

By combining similar terms:

= 2x2 + 3xy + 8x + y2 + 4y

Create your own Create your own multiplication of multiplication of binomialsbinomials

Using algebra blocks Using algebra blocks for teaching perimeterfor teaching perimeter

Determine the perimeterDetermine the perimeter

x2xy

P = 4x + 2y

y P = y + 5 + (y -1)

P = 2y + 4

Determine the perimeterDetermine the perimeter

y

x

P = 1 + x + (y-1) + 1 + (y-1) + 3 + 1 + x = 2x + 2y + 4

Determine the perimeter:Determine the perimeter:

x2xy

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

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

Algebra Equation MatAlgebra Equation Mat++ ++

-- --=

Equations: solve for Equations: solve for xx

++ ++

-- --

2(x-3) = -42(x-3) = -4

=

2(x-3) = -42(x-3) = -4

++ ++

-- --

x

x

2x -6 = -4

2x = 2

+6 +6

_ _2 2

x = 1

Equation: solve for Equation: solve for xx

=

5(y-4) = 105(y-4) = 10

++ ++

-- --

5y -20 = 10

+20 +20

5y = 30_ _5 5

y = 6

Equation: solve for Equation: solve for yy

=

Create your own equation for Create your own equation for the class to solve for the class to solve for xx

++ ++

-- --=

What do you suppose are the What do you suppose are the benefits of using the algebra blocks?benefits of using the algebra blocks?

What do you suppose are the What do you suppose are the drawbacks of using algebra blocks?drawbacks of using algebra blocks?

Other comments?Other comments?

Discussion…Discussion…

How do the algebra blocks interface How do the algebra blocks interface with what NCTM advocates in with what NCTM advocates in teaching mathematics?teaching mathematics? Consider the NCTM process standards… Explain the Consider the NCTM process standards… Explain the

potential of the NCTM process standards in teaching potential of the NCTM process standards in teaching with algebra blocks.with algebra blocks.

ConnectionsConnections RepresentationRepresentation CommunicationCommunication Problem SolvingProblem Solving Reasoning & ProofReasoning & Proof

How does using the algebra blocks impact students How does using the algebra blocks impact students who are ELL?who are ELL?

Thank you for sharing Thank you for sharing your evening with me.your evening with me.

Cynthia AnhaltCynthia Anhalt

canhalt@math.arizona.educanhalt@math.arizona.edu

How can algebraic How can algebraic thinking begin in the thinking begin in the elementary grades?elementary grades?

How can the elementary mathematics How can the elementary mathematics curriculum can be taught with curriculum can be taught with algebraic thinking as a goal?algebraic thinking as a goal?

How can base-10 blocks be used to aid How can base-10 blocks be used to aid in teaching multiplication that leads to in teaching multiplication that leads to the distributive property?the distributive property?

Multiplication Multiplication ModelsModels

Set modelSet model

What models would represent 2 x 4?

2 jumps of 4

Linear Linear modelmodel

FOCUS

Array Array modelmodel

2 rows by 4 columns of discrete objects

2 groups of 4 discrete objects

Area Area modelmodel

2 by 4 covered area

Consider the area model for Consider the area model for multiplication of whole multiplication of whole numbers numbers

3 x 4 = 123 x 4 = 12

3

4

Base 10 BlocksBase 10 Blocks

3 x 123 x 12

How can we convert these sets into an area model?

Area model using Base 10 Area model using Base 10 BlocksBlocks

3

12

3 x 12 = 36

Consider the area model to make Consider the area model to make the connection of arithmetic to the connection of arithmetic to algebraic thinking:algebraic thinking:

12 x 13 = (10+ 2) (10+3)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= 100 + 20 + 30 + 6

Show the area model of Show the area model of 14 x 23 with base 10 blocks14 x 23 with base 10 blocks

(10+4)(20+3) = 200+80+30+12 = 322

DiscussionDiscussion

How do these ideas promote How do these ideas promote algebraic thinking in the elementary algebraic thinking in the elementary mathematics classrooms or in the mathematics classrooms or in the middle grades?middle grades?

How can you incorporate a variety How can you incorporate a variety of models of mathematical of models of mathematical representation into your teaching?representation into your teaching?

Thank you for sharing Thank you for sharing you evening with me.you evening with me.

Cynthia AnhaltCynthia Anhalt

canhalt@math.arizona.educanhalt@math.arizona.edu