visual college algebra for middle school teachers curriculum materials for preservice middle school...
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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