Exploring Functions through the Use of Manipulatives

Cover	cover
Title Page	i
Copyright Page	ii
Dedication	iii

An Overview and Directions	v
Table of Contents	vii

Essential to Every Activity
0: Speaking, Hearing, and Writing about Functions and their Graphs	1
0-1: Describe Graphs: Technical and Creative Perspectives	1
0-2: The Identity Function is the Input Function	2
0-3a: Describe a Graph by Creating a Wordbank	2
0-3b: Supply a Wordbank to Assist in Describing the Graph of Function	3
0-4: Present a "Paper"	3
0-5: Play 20 Questions	4
0-6: "I'm Thinking of a Function"	5
4: Visualizing Functions and Graphs	6
4-1: Draw a Function in the Air	6
4-2: Doing Math "Exercises"	6
4-3: Take a Limit with Your Hands	7

Families of Functions: Translations, Reflections
11: Addition of Functions, Vertical Slides	8
11-1: Vertical Slides	8
12: Product of Functions, Reflection about the Horizontal	11
12-1: Reflections about a Horizontal Line	11
13: Other Compositions of Functions, Horizontal Slides	13
13-1: Horizontal Slides or Translations	13
14: More Compositions of Functions, Reflections about the Vertical	15
14-0: Point-Plotting	15
14-1: Discussing the Composition of a Function	15
14-2: Reflection about a Vertical Line	17

Solving Equations and Systems
15: Solving Systems, Making Statements	18
15-1: Using A Graph as a Boundary	18
15-2: Solving An Equation Graphically	19

An Introduction to Functions
1: Teacher-Lead Point-Plotting of a Function	23
1-1: Examine a Function Through Point-Plotting	23
2: Student Exploration Using Point-Plotting	25
2-1: Discovery by Pairs of Students	25
3: Create a Function Poster Library	28
3-1: Make a Poster Library	28

A Closer Look at Functions
5: The Slope of a Curve at a Point	31
5-1: Read Slope from the Slopemeter	31
5-2: Read the Slope Using a Tangent to a Curve	33
5-3: Computed Slope by the Slope Formula	34
5-4: Guessing the Slope of A Function Whose Slope is Computed by Taking the Derivative	34

6: Reflections about Y = X	36
6-1: Discuss the Differences, Predict the Graph	36
6-2: Simple and Mental Reflection about Y=X	37
6-3: Manipulatively Finding a Reflection about Y=X	37
6-4: Verification through Verbalization and Algebraic Coding	39
6-5: Bounding an Area	39
7: Inversely So: Finding the Potential Inverse Function	41
7-1: The Vertical Line Test	41
7-2: The Horizontal Line Test	42
7-3: Using Vertical and Horizontal Lines to Ascertain Domain and Range	42
7-4: Determining the Inverse Function	44
8: Considering the Slope of Inverse Functions	47
8-1: Estimating Slopes of Inverse Functions	47
8-2: Generalizing about the Slopes of Functions and Their Inverses	47
9: Function Values at Neighborhoods and Extremes	48
9-1: Function Values in Neighborhoods and at Extremes	48
9-2: Consider the Value of the Function in a Neighborhood about a Restricted Value	50
9-3: Consider Boring Cases	51
10: The Unit Circle	53
10-1: Measurement of the Trig Functions of One Angle	53
10-2: Generalizing about the Behavior of A Trig Function	54
10-3: Pythagorean Identities	54
The Unit Circle Overhead Model and Activity Sheet	56

More on Dilation
16: By a Constant Functions, Making Things Fat and Skinny	57
16-1: Comparing Dilations	58
16-2: A Project	58
17: More Addition of Functions: Polynomials	60
17-1: Addition of Functions	60
18: More Product of Functions: Rational, Trig Functions, and Envelope Functions	63
18-1: Product of Functions [Everyday Functions]	64
18-2: Predicting A Graph Based on the Behavior of Related Functions	68
18-3: Graphing Nonroutine functions	70

APPENDIX
Table of Contents	A-1
Function and Relation Library	A2
Lines: Vertical and Horizontal lines, Linear Functions	A2
Three Most Important Functions: Identity, Opposite, Reciprocal	A3
Family of Quadratics, Squaring & Square Root, Polynomials, Roots and Exponents	A4
Exponential or Power Functions, A Bit about e, the Exponential Function	A5
Lograthmic Functions, the Natual Log Function, Absolute Value Function Conics	A6
Conics	A7
Trigonometric Functions and the Unit Circle, Trigonometric Functions and the Unit Circle,	A8, A9
Mechanical Manipulatives: Demo Pack and Unit Circle Assembly and Masters	A10 - A14
Parent Function Masters: Shapes for 1/x, (1 - x²), (4 - x²), (9 - x²),	A15
halfplane, sine or cosine, not x, x, exp(x) or ln(x), x, ¦x¦, x², and x	A16 - A19
Additional Function Masters: Shapes for secant or cosecant, tangent or	A20
cotangent, .25x², .5x², 2x², 4x², 10x², .1x², x² and not x², y>x², and y < x²,	A21 -A24
not 2x², not 4x², not 10x², and not .1x², not .25x² and not .5x²	A25 - A26
Coordinate Planes: Coordinate Plane Paper, 2 Planes Paper, 4 Planes Paper	A27 - A29
About the Author	A30

Additional Planes	Planes