2000 and Beyond

495 Pages·2001·3.36 MB·English

Algebra 1:
2000 and Beyond
Mathematics Institute
http://www.utdanacenter.org/ssi/projects/texteams
Dwight D. Eisenhower Professional Development Program, Title II, Part B
Texas Education Agency
Texas Statewide Systemic Initiative in Mathematics, Science, and Technology Education
Charles A. Dana Center, The University of Texas at Austin
Permission is given to any person, group, or organization to copy and distribute Texas
Teachers Empowered for Achievement in Mathematics and Science (TEXTEAMS)
materials for noncommercial educational purposes only, so long as the appropriate credit is
given. This permission is granted by The Charles A. Dana Center, a unit of the College of
Natural Sciences at The University of Texas at Austin.
Acknowledgements
The TEXTEAMS Algebra I: 2000 and Beyond institute was developed under the direction
and assistance of the following:
Academic Advisors/Reviewers
Paul Kennedy Texas Christian University
Anne Papakonstantinou Rice University
Writer
Pam Harris Consultant
Advisory Committee
Linda Antinone Fort Worth ISD
Kathy Birdwell New Braunfels ISD
Kathi Cook Dana Center, University of Texas at Austin
Eva Gates Consultant
Juan Manuel Gonzalez Laredo ISD
Susan Hull Dana Center, University of Texas at Austin
Paul Kennedy Southwest Texas State University
Laurie Mathis Dana Center, University of Texas at Austin
Diane McGowan Dana Center, University of Texas at Austin
Bonnie McNemar Consultant
Barbara Montalto Texas Education Agency
Anne Papakonstantinou Rice University
Cindy Schimek Katy ISD
Jane Silvey ESC VII
JoAnn Wheeler ESC IV
Susan Williams University of Houston
TEXTEAMS Algebra I: 2000 and Beyond Institute
Table of Contents
About TEXTEAMS Institutes......................................................................................vii
Institute Introduction.....................................................................................................viii
Institute Overview.............................................................................................................ix
Section Overviews............................................................................................................x
Materials List .......................................................................................................................x v
I. Foundations for Functions
1 Developing Mathematical Models
1.1 Variables and Functions .....................................................................................1
Activity 1: Examples of Dependent Relationships .....................................11
Activity 2: Independent and Dependent Variables.....................................12
Reflect and Apply ............................................................................................13
1.2 Valentine’s Day Idea ........................................................................................14
Activity 1: Valentine’s Day Idea .....................................................................24
Activity 2: Using Tables to Find the More Economical Offer.......................25
Activity 3: Using Graphs to Find the Better Offer.........................................27
Activity 4: New Rose Offers ..........................................................................29
Activity 5: Using Tables for New Rose Offers.............................................30
Activity 6: Using Graphs for New Rose Offers............................................31
Reflect and Apply ............................................................................................33
Student Activity: Investigate Recursively ...................................................36
2 Using Patterns to Identify Relationships
2.1 Identifying Patterns ...........................................................................................41
Activity 1: Painting Towers..............................................................................51
Activity 2: Building Chimneys ........................................................................54
Activity 3: Constructing Trucks........................................................................57
Activity 4: Generating Patterns.......................................................................60
Reflect and Apply ............................................................................................62
Student Activity: Perimeter of Rectangles ....................................................63
2.2 Identifying More Patterns.................................................................................67
Activity 1: Building Blocks...............................................................................72
Activity 2: Starting Staircases .........................................................................75
Activity 3: Too Many Triangles.......................................................................78
Reflect and Apply ............................................................................................81
3 Interpreting Graphs
3.1 Interpreting Distance versus Time Graphs.....................................................82
Activity 1: Walking Graphs .............................................................................87
Activity 2: Walking More Graphs...................................................................88
Reflect and Apply ............................................................................................90
Student Activity: Walk This Way ...................................................................91
3.2 Interpreting Velocity versus Time Graphs .....................................................97
Activity 1: Matching Velocity Graphs.......................................................... 103
Activity 2: Connecting Distance and Velocity Graphs .............................. 105
Reflect and Apply ......................................................................................... 107
TEXTEAMS Algebra I: 2000 and Beyond iii
Table of Contents
II. Linear Functions
1 Linear Functions
1.1 The Linear Parent Function............................................................................ 108
Activity 1: ACT Scores................................................................................ 115
Activity 2: Temperatures.............................................................................. 116
Activity 3: Symbolic ..................................................................................... 117
Reflect and Apply ......................................................................................... 118
Student Activity 1: Age Estimates.............................................................. 119
Student Activity 2: Sales Goals.................................................................. 128
1.2 The Y- Intercept............................................................................................. 132
Activity 1: The Birthday Gift......................................................................... 139
Activity 2: Spending Money....................................................................... 142
Activity 3: Money, Money, Money............................................................ 145
Reflect and Apply ......................................................................................... 146
Student Activity: Show Me the Money! .................................................... 148
1.3 Exploring Rates of Change.......................................................................... 153
Activity 1: Wandering Around ..................................................................... 158
Activity 2: Describe the Walk ...................................................................... 160
Reflect and Apply ......................................................................................... 162
Student Activity: What’s My Trend? .......................................................... 163
1.4 Finite Differences............................................................................................ 170
Activity 1: Rent Me!...................................................................................... 176
Activity 2: Guess My Function .................................................................... 177
Activity 3: Finite Differences......................................................................... 179
Reflect and Apply ......................................................................................... 180
Student Activity: Graphs and Tables ......................................................... 181
2 Interpreting Relationships Between Data Sets
2.1 Out for a Stretch ............................................................................................. 183
Activity 1: Stretch It....................................................................................... 190
Activity 2: Comparing Graphs .................................................................... 194
Reflect and Apply ......................................................................................... 195
Student Activity 1: Have You Lost Your Marbles?.................................. 196
Student Activity 2: Unidentified Circular Objects (UCO’s)....................... 206
Student Activity 3: Going to Great Depths................................................ 215
Student Activity 4: Height versus Arm Span ............................................ 224
2.2 Linear Regression.......................................................................................... 232
Activity 1: Sum of Squares ......................................................................... 245
Activity 2: Lines of Best Fit.......................................................................... 246
Activity 3: The Correlation Coefficient ........................................................ 248
Reflect and Apply ......................................................................................... 250
3 Linear Equations and Inequalities
3.1 Solving Linear Equations .............................................................................. 251
Activity 1: Concrete Models........................................................................ 259
Activity 2: Using Concrete Models............................................................. 262
Reflect and Apply ......................................................................................... 264
3.2 Stays the Same............................................................................................. 265
Activity: Stays the Same............................................................................. 274
Reflect and Apply ......................................................................................... 278
3.3 Solving Linear Inequalities ............................................................................ 279
Activity 1: Linear Inequalities in One Variable............................................ 288
Activity 2: Linear Inequalities in Two Variables.......................................... 290
Reflect and Apply ......................................................................................... 294
TEXTEAMS Algebra I: 2000 and Beyond iv
Table of Contents
3.4 Systems of Linear Equations and Inequalities............................................ 295
Activity 1: Using a Table.............................................................................. 302
Activity 2: Solve the System Graphically.................................................. 303
Activity 3: Solve the System Symbolically............................................... 305
Reflect and Apply ......................................................................................... 306
Student Activity: Concrete Models and Systems of Linear Equations .. 307
III. Nonlinear Functions
1 Quadratic Functions
1.1 Quadratic Relationships................................................................................. 319
Activity 1: Building a Sandbox.................................................................... 329
Activity 2: Projectile Motion.......................................................................... 332
Reflect and Apply ......................................................................................... 335
1.2 Transformations.............................................................................................. 336
Activity 1: Investigating the Role of a ......................................................... 344
Activity 2: Investigating the Role of k.......................................................... 345
Activity 3: Investigating the Role of h ......................................................... 346
Activity 4: Transformations .......................................................................... 347
Reflect and Apply ......................................................................................... 350
1.3 Lines Do It Too .............................................................................................. 351
Activity 1: Exploring Slope ......................................................................... 361
Activity 2: Exploring Vertical Shifts ............................................................. 362
Activity 3: Exploring Horizontal Shifts......................................................... 363
Activity 4: A Different Perspective.............................................................. 365
Reflect and Apply ......................................................................................... 366
2 Quadratic Equations
2.1 Connections ................................................................................................... 367
Activity 1: Roots, Factors, x-intercepts, Solutions..................................... 374
Activity 2: Which Form?............................................................................... 377
Activity 3: Jump!........................................................................................... 378
Reflect and Apply ......................................................................................... 379
2.2 The Quadratic Formula................................................................................... 380
Activity 1: Programming the Quadratic Formula......................................... 384
Activity 2: Hang Time................................................................................... 385
Reflect and Apply ......................................................................................... 387
Student Activity: Investigate Completing the Square.............................. 388
3 Exponential Functions and Equations
3.1 Exponential Relationships............................................................................ 392
Activity 1: Paper Folding.............................................................................. 405
Activity 2: Measure with Paper ................................................................... 407
Activity 3: Regions ...................................................................................... 409
Activity 4: How Big is a Region?................................................................ 411
Reflect and Apply ......................................................................................... 413
Student Activity: Recursion Again............................................................... 414
3.2 Exponential Growth and Decay................................................................... 420
Activity 1: Exponential Growth.................................................................... 427
Activity 2: Exponential Decay..................................................................... 428
Reflect and Apply ......................................................................................... 429
Student Activity: On the Wall ...................................................................... 430
3.3 Exponential Models...................................................................................... 434
Activity 1: Population Growth...................................................................... 441
Activity 2: Cooling Down............................................................................. 443
Reflect and Apply ......................................................................................... 444
TEXTEAMS Algebra I: 2000 and Beyond v
4 Interpreting Relationships Between Data Sets
4.1 Bounce It!........................................................................................................ 445
Activity 1: Collect the Data........................................................................... 452
Activity 2: A Bounce..................................................................................... 453
Activity 3: Bounce Height versus Bounce Number.................................. 454
Activity 4: Bounce Height versus Drop Height ......................................... 455
Reflect and Apply ......................................................................................... 456
Student Activity 1: Pattern Blocks............................................................... 457
Student Activity 2: Throw Up!..................................................................... 463
Student Activity 3: Radioactive Decay....................................................... 467
Student Activity 4: Pendulum Decay.......................................................... 474
Calculator Programs...................................................................................................477
TEXTEAMS Algebra I: 2000 and Beyond vi
About TEXTEAMS Institutes
TEXTEAMS Philosophy
• Teachers at all levels benefit from extending their own mathematical knowledge
and understanding to include new content and new ways of conceptualizing the
content they already possess.
• Professional development experiences, much like the school mathematics
curriculum itself, should focus on few activities in great depth.
• Professional development experiences must provide opportunities for teachers
to connect and apply what they have learned to their day-to-day teaching.
Features of TEXTEAMS Institute Materials
Multiple representations (verbal, concrete, pictorial, tabular, symbolic, graphical)
Mathematical ideas will be represented in many different formats. This helps both teachers
and students understand mathematical relationships in different ways.
Integration of manipulative materials and graphing technology
The emphasis of TEXTEAMS Institutes is on mathematics, not on learning about particular
manipulative materials or calculator keystrokes. However, such tools are used in various
ways throughout the institutes.
Rich Connections within and outside mathematics
Institutes focus on using important mathematical ideas to connect various mathematical
topics and on making connections to content areas and applications outside of mathematics.
Questioning strategies
A variety of questions are developed within each activity that help elicit deep levels of
mathematical understanding and proficiency.
Hands-on approach with “get-up-and-move” activities
Institutes are designed to balance intense thinking with hands-on experiences.
Math Notes and Reflect and Apply
A feature called Math Notes includes short discussions of mathematical concepts
accompanying the learning activities. Similarly, the Reflect and Apply feature is designed to
extend and apply participants’ understanding of the mathematical concepts.
The Charles A. Dana Center is approved by the State Board for Educator Certification as a registered Continuing
Professional Education (CPE) provider. Hours received in TEXTEAMS institutes may be applied toward the required training
for gifted and talented in the area of curriculum and instruction. Individual district/ campus acceptance of these hours for
gifted and talented certification is a local decision.
TEXTEAMS Algebra I: 2000 and Beyond vii
TEXTEAMS Algebra I: 2000 and Beyond Institute
Introduction
The Algebra I: 2000 and Beyond Institute is based on the groundbreaking work of
the 1996 TEXTEAMS Algebra I Institute. Both institutes assert that “Algebra for All”
is a realistic and attainable goal. To teach “Algebra for All” will require educators to
possess a deep understanding of mathematical content, pedagogy, and strategies
to meet the needs of diverse student populations.
This institute is not meant as a scope and sequence for the Algebra I course, nor is
it a set of student activities for use in a classroom without careful thought and
modification on the part of a knowledgeable teacher. The Algebra I: 2000 and
Beyond Institute is a rich, carefully designed professional development tool which is
intended to deepen teacher content knowledge. It is also intended to model the
importance of content depth by focusing on carefully selected activities that are few
in number and grounded in the mathematics necessary to support teacher and
student learning. The concepts and ideas explored within this institute are
connected to classroom instruction and key assessments.
In this institute, not all Algebra I topics can be addressed. It is important for
teachers to develop a deep and powerful understanding of the concepts and ideas
of algebra. This requires educators to understand the mathematics in a different
capacity from that of the student. Therefore, content within the institute is
approached from a more complex perspective and some topics are addressed at
a level that is deeper than would be typical for Algebra I students. Much of the
institute is built upon learning experiences that develop and promote the power of
using concrete experiences to introduce and build mathematical concepts.
Throughout the institute, multiple representations are utilized as a powerful strategy
to assist educators and students in making connections, understanding
mathematical concepts, and reasoning in meaningful and complex ways.
The institute assumes some prior participant knowledge. If teachers are unfamiliar
with the following, presenters may need to supplement with introductory materials.
• Representing, adding and subtracting polynomials with algebra tiles (area
model)
• Using algebra tiles to model monomial and binomial multiplication
• Modeling factoring trinomials with algebra tiles
• Graphing calculators
The Algebra I; 2000 and Beyond Institute draws on the work of the 1996
TEXTEAMS Algebra I Institute; Principles and Standards for School Mathematics,
NCTM, 2000; Discovering Algebra, Key Curriculum Press, 2000; and Dr. Paul
Kennedy, Southwest Texas State University.
TEXTEAMS Algebra I: 2000 and Beyond viii
TEXTEAMS Algebra I: 2000 and Beyond Institute
Institute Overview
I. Foundations for Functions
1 Developing Mathematical Models
2 Using Patterns to Identify Relationships
3 Interpreting Graphs
II. Linear Functions
1 Linear Functions
2 Interpreting Relationships Between Data Sets
3 Linear Equations and Inequalities
III. Nonlinear Functions
1 Quadratic Functions
2 Quadratic Equations
3 Exponential Functions and Equations
4 Interpreting Relationships Between Data Sets
TEXTEAMS Algebra I: 2000 and Beyond ix