Visual College Algebra for Middle School Teachers

Curriculum Materials for Preservice Middle School Mathematics Teachers

Laurie BurtonWestern Oregon University

www.wou.edu/~burtonl

Course Content

Integer operations (to learn how to use the models) through basic work with polynomials.

Emphasis on connecting verbal, visual and symbolic ideas and techniques

Connecting all ideas to clear graphing (paper and pencil and graphing calculator).

Course Pedagogy

Hands On & Active

Discovery Based

Visual Patterns & Visual Models

Easy ways to make “abstract” concepts concrete

Students in this course often say “I really understand algebra for the first time”

Materials: Black and Red Tiles / Algebra Pieces

1

1

Black tiles: 1 1 or -1 -1Positive

Red tiles: -1 1 or 1 -1Negative

These are used initially for integer operations and throughout the text to model integers

-1

1

MaterialsBlack and Red Tiles / Algebra Pieces

Black n-strip (n)1 n or -1 -n

Black and red n-strips are used in the beginning of the course to model expressions and equations when looking at patterns indexed by whole numbers

This edge piece is n units long

This edge piece is 1 unit long

Red n-strip (-n) 1 -n or 1 -n

This edge piece is -n units long

This edge piece is 1 unit long

Black Strip Example

Words: 2 tiles from the left and right of the bottom row and the figure number of tiles in the middle column.

Symbols: T = 2 + n

MaterialsBlack and Red Tiles / Algebra Pieces

x-strip (x)1 x or -1 -x

This edge piece is x units long

xThis edge piece is 1 unit long

White x and opposite x strips are used when the domain is extended to integers and then to real numbers. x and –x can be both positive and negative

This edge piece is x units long

oooooo

ooo

-xooo

This edge piece is -1 unit long

opposite x-strip (-x)1 -x or -1 x

MaterialsBlack and Red Tiles / Algebra Pieces

x-square (x2) and –x-square (-x2)

Example: Solving Equationsx2 – x - 6 = 0

ooo

-x

ooo

Students form pieces into a rectangle and then measure the edges.

They must “add zero pairs” of white strips to achieve this.

x

-x

ooo

ooo

x

ooo

-x

ooo

ooo

-x

ooo

Example: Solving Equationsx2 – x - 6 = 0

x - 3 = 0; x = 3x + 2 = 0; x = -2

0-3

2

x

x

HistoryVisual algebra ideas designed for middle, beginning high students (Math Learning Center: www.mathlearningcenter.org)

2001 – 2005: Used modified version of MLC materials in my “College Algebra for Teachers” course.

Course populated by preservice elementary and middle school teachers.

Also used advanced version for a graduate course for inservice middle school teachers.

Creating “Visual Algebra for College Students”

(Sabbatical project)

Redesigned, reorganized and extended original MLC materials so that other college instructors could use them in any college classroom.

Text now uses mathematically mature and appropriate language for adult learners.

Creating “Visual Algebra for College Students”

Student text materials designed to reach grade 5 – grade 10 teachers.

Topics are now in typical college presentation; emphasis on individual ideas building toward a more in depth and complete picture.

Chapter 1Integers and Integer Operations

1.1: Modeling Integers (Black and Red Tiles)1.2: Adding Integers (Black and Red Tiles)1.3: Subtracting Integers (Black and Red Tiles)1.4: Arrays with Black and Red Tiles1.5: Multiplying Integers (Black and Red Tiles)1.6: Dividing Integers (Black and Red Tiles)

Chapter 2Linear Expressions, Equations

and Graphs2.1: Introduction to Toothpick Figure Sequences2.2: Alternating Toothpick Figure Sequences2.3: Introduction to Tile Figure Sequences2.4: Tile Figures and Algebraic Equations2.5: Linear Expressions and Equations2.6: Extended Sequences and Linear Functions

Chapter 3Real Numbers and Quadratic

Functions3.1: Graphing with Real Numbers3.2: Introduction to Quadratic Functions3.3: Algebra Pieces and Quadratic Functions3.4: Completing the Square, the Quadratic Formula and Quadratic Graphs3.5: Inequalities

Chapter 4Polynomials

4.1: Introduction to Higher Degree Polynomials4.2: Special Polynomials Factors, FOIL and Polynomial Division4.3: Introduction to Complex Numbers4.4: Working with Complex Numbers and Polynomial Roots

Student MaterialsEach Unit (Section)

Hands on, discovery based, activity set ideal for a class set up in small groups.

Each set includes:Purpose of setMaterials ListIntroduction (definitions, examples,

background)Sketching Notes (as appropriate)Technology Notes (as appropriate)

Student MaterialsEach Unit (Section)

Follow up homework assignmentElectronic materials include algebra piece images, graph paper, etc.

Each set is in a worksheet format (students write in text)

Classroom UseEach section covered in 2 to 4 hours (our classes meet 2 hours per day).

One possible pedagogical approach:Start with lead in discussion / covering

and extending introductory materials in activity set

Students work on activity set in groupsStudents report out on group workFinish with overall follow up discussion

Student MaterialsEach Chapter

Vocabulary and Review topics list

Practice Exam

Student MaterialsEnd of Text

Selected Answers to Activity Set Activities

Complete solutions for end of Chapter Practice Exams

Instructor MaterialsEach Unit (Section)

Instructor Notes including:

Unit Topic Significance & Purpose

Instructor Materials List

Lead In Discussion Ideas

Student Work (notes, pitfalls)

Follow Up Class Discussion Ideas and Worked Out Examples

Overheads for Discussions

Instructor MaterialsEach Unit (Section)

Brief comments and recommendations

Notes on student resources such as student electronic resources

Notes on correlation between section and exam questions (for quizzes, etc.)

Homework Notes which include:

Instructor MaterialsEach Chapter

Editable Chapter Exams

Complete Solutions for Instructor Chapter Exams

Instructor MaterialsOverall

Complete Answers to Activity Set Activities

Complete Answers to Homework Questions

Program Use

Ideal for follow up course to foundational series for elementary and middle school teachers.

Excellent preparation course for Calculus for Middle School Teachers course

Can be used for graduate inservice teachers

Class Testing

Burton—Fall 2007

Kruczek—Spring 2008

Dissemination

The Math Learning Center will include in their catalog; planned for 2009

orPossible work with mainstream publishing company.

I will happily share materials for your classrooms.

Visual Algebra for College Students

Laurie BurtonMathematics Department

Western Oregon University

www.wou.edu/~burtonl

burtonl@wou.edu