Visual College Algebra for Middle School Teachers Curriculum Materials for Preservice Middle School Mathematics Teachers Laurie Burton Western Oregon University

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 Black tiles: 1 1 or Positive Red tiles: -1 1 or 1 -1 Negative These are used initially for integer operations and throughout the text to model integers

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

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

Materials Black and Red Tiles / Algebra Pieces x-strip (x) 1 x or -1 -x 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 opposite x-strip (-x) 1 -x or -1 x

Materials Black and Red Tiles / Algebra Pieces x-square (x 2 ) and –x-square (-x 2 )

Example: Solving Equations x 2 – x - 6 = 0 Students form pieces into a rectangle and then measure the edges. They must add zero pairs of white strips to achieve this.

Example: Solving Equations x 2 – x - 6 = 0 x - 3 = 0; x = 3 x + 2 = 0; x = -2

History Visual algebra ideas designed for middle, beginning high students (Math Learning Center: – 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 1 Integers 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 Tiles 1.5: Multiplying Integers (Black and Red Tiles) 1.6: Dividing Integers (Black and Red Tiles)

Chapter 2 Linear Expressions, Equations and Graphs 2.1: Introduction to Toothpick Figure Sequences 2.2: Alternating Toothpick Figure Sequences 2.3: Introduction to Tile Figure Sequences 2.4: Tile Figures and Algebraic Equations 2.5: Linear Expressions and Equations 2.6: Extended Sequences and Linear Functions

Chapter 3 Real Numbers and Quadratic Functions 3.1: Graphing with Real Numbers 3.2: Introduction to Quadratic Functions 3.3: Algebra Pieces and Quadratic Functions 3.4: Completing the Square, the Quadratic Formula and Quadratic Graphs 3.5: Inequalities

Chapter 4 Polynomials 4.1: Introduction to Higher Degree Polynomials 4.2: Special Polynomials Factors, FOIL and Polynomial Division 4.3: Introduction to Complex Numbers 4.4: Working with Complex Numbers and Polynomial Roots

Student Materials Each Unit (Section) Hands on, discovery based, activity set ideal for a class set up in small groups. Each set includes: Purpose of set Materials List Introduction (definitions, examples, background) Sketching Notes (as appropriate) Technology Notes (as appropriate)

Student Materials Each Unit (Section) Follow up homework assignment Electronic materials include algebra piece images, graph paper, etc. Each set is in a worksheet format (students write in text)

Classroom Use Each 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 groups Students report out on group work Finish with overall follow up discussion

Student Materials Each Chapter Vocabulary and Review topics list Practice Exam

Student Materials End of Text Selected Answers to Activity Set Activities Complete solutions for end of Chapter Practice Exams

Instructor Materials Each 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 Materials Each 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 Materials Each Chapter Editable Chapter Exams Complete Solutions for Instructor Chapter Exams

Instructor Materials Overall 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 BurtonFall 2007 KruczekSpring 2008

Dissemination The Math Learning Center will include in their catalog; planned for 2009 or Possible work with mainstream publishing company. I will happily share materials for your classrooms.

Visual Algebra for College Students Laurie Burton Mathematics Department Western Oregon University