i TEXTEAMS Algebra II/Precalculus Institute Acknowledgements The Algebra II

237 Pages·2004·17.03 MB·English

TEXTEAMS Algebra II/Precalculus Institute
Acknowledgements
The Algebra II/ Precalculus Institute was developed under the direction and assistance of
the following:
Writers:
Pam Harris Consultant
Susan May Consultant
Diane McGowan Charles A. Dana Center
.
Writers who contributed material that was used from the previous Algebra
II/Precalculus Institute:
Pam Chandler, Consultant; Susan Cinque, Fort Bend ISD; Maggie Dement, Consultant;
Tommy Eads, North Lamar ISD; Lois Moseley, Consultant; Richard Parr, Rice
University; Rachel Pinkston, Alief ISD; Diane Reed, Ysleta ISD; Rozanne Rubin, Alief
ISD; GT Springer, Texas Instruments; Susan Williams, University of Houston
TEXTEAMS Part 1: Algebra II and Precalculus Institute i
Advisory Team:
Beverly Anderson Region XVII: Educational Service
Cecilia Avendano Brownsville ISD
Tommy Bryan Baylor University
Nick Carter Independent Consultant
Oscar Chavarria Pasadena ISD
Libby Chaskin Northside ISD
Kathi Cook Charles A. Dana Center
Beth Glassman Leander ISD
Gaye Glenn Region II Educational Service Center
Basia Hall Houston ISD
Pam Harris Consultant
Susan Hull Charles A. Dana Center
Hee Joon Kim Charles A. Dana Center
Laurie Mathis Charles A. Dana Center
Susan May Consultant
Diane McGowan Charles A. Dana Center
Paul Mlakar Region IV Educational Service Center
Barbara Montalto Texas Education Agency
Richard Parr Rice University
Erika Pierce Charles A. Dana Center
Tim Pope Hays ISD
Diane Reed Ysleta ISD
Rozanne Rubin Alief ISD
Cindy Schimek Katy ISD
Susan Thomas Alamo Heights ISD
Dick Stanley University of California at Berkeley
Susan Williams University of Houston
Jeanne Womack Region I Educational Service Center
Beverly Weldon Region X Educational Service Center
Ward Roberts Witchita Falls: ISD
TEXTEAMS Part 1: Algebra II and Precalculus Institute ii
Table of Contents
Part I: Algebra II/Precalculus
Foundations of Functions
I. Foundations of Functions
Introduction: Ball Drop ………………………………………………………….1
1. Foundations
1.1 Transformations ………………………………………………….. 6
Activity 1: Explore Transformations………………………. 17
Activity 2: Domains ………………………………………… 22
Activity 3: Transformations with Technology ……………..….…. 23
Activity 4: Summarize Your Findings …………………….. 29
Reflect and Apply …………………………………………… 30
Student Activity 1: Move the Monster Algebra II …………….…. 31
Student Activity 2: Move the Monster Precalculus …………. 35
Student Activity 3: Combinations of Transformations: ……. 40
Student Activity 4: Transformations on Generic Graphs... …. 44
Student Activity 5: Transformation Practice Algebra II ……. 48
Student Activity 6: Transformation Practice Precalculus .. ….…. 54
1.2 What Is the Difference? ………………………………………… 64
2.Applying Transformations
2.1 Applications ………………………………………………………. 77
Activity 1:Good Viewing Windows ………………………… 85
Activity 2:Friendly Viewing Windows ……………………... 90
Reflect and Apply …………………………………………… 91
Student Activity: Matching Parent Functions …………….….…. 92
2.2 Airlines ……………………………………………………………. 102
Activity 1:Trend Lines ………………………………………. 105
Activity 2:Another Trend Line ……………………………… 106
Reflect and Apply …………………………………………… 107
3. Making Connections
3.1 Inverses …………………………………………………………… 108
Activity 1:Parametric Equations …………………………… 113
Activity 2:Restricted Domains ………………………………….….114
Reflect and Apply …………………………………………… 115
Student Activity: MIRA Reflections ………………………. 116
TEXTEAMS Part 1: Algebra II and Precalculus Institute iii
3.2 Stays the Same ………………………………………………….. 128
Activity: Solving Equations Step by Step Graphically …... ….…. 133
Reflect and Apply……………………………………………. 134
II. Exponential and Logarithmic Functions
Introduction: Linear and Exponential Models ……………………………….. 135
1.Exponential Growth and Decay
1.1 Bacteria Growth...………………………………………………… 142
Activity 1:E .Coli Growth Rate …………………………..… 150
Activity 2:The Flu Epidemic: A Logistic Growth Model ……. 153
Reflect and Apply …………………………………………… 156
Student Activity 1: Recursion and Bacteria Cell Division ….…. 157
Student Activity 2: Space Debris in Earth ’s Orbit …………. 166
1.2 Exponential Decay ……………………………………………… 171
Activity: Cooling Down……………………………………… 175
Reflect and Apply
2. Logarithmic Functions
2.1 The Energy of Earthquakes ………………………………………. 180
Activity 1:The Richter Scale ……………………………….… 187
Activity 2:Graphing the Magnitude………………………….… 189
Reflect and Apply …………………………………………… 191
Student Activity 1: Exploring Logs ………………………… 192
Student Activity 2: Change of Base ……………………….… 196
Appendix: Bad Bad Bugs ………………………………….… 204
III. Power Functions and Polynomials
Introductory Activity: Spread of Aids …….….….….….….….….….….…. 209
I. Power Functions and Polynomials
1.1 Power Functions ………………………………………………… 214
Activity 1: Introduction to Power Functions ……………….… 223
Activity 2: Comparing Power Functions…………………… 224
Activity 3: Comparing Functions…………………………… 228
Reflect and Apply …………………………………………… 229
Student Activity 1: Truth or Consequences ……………….… 230
Student Activity 2: Classifying Functions ……………….…… 232
1.2 Compounding Interest …………………………………………… 236
Activity 1: Money Talk ……………………………………… 242
Activity 2: Compounding Continuously …………………… 244
Reflect and Apply …………………………………………… 246
TEXTEAMS Part 1: Algebra II and Precalculus Institute iv
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 illicit 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.
TEXTEAMS Part 1: Algebra II and Precalculus Institute v
About the Algebra II/Precalculus Institute
The Algebra II / Precalculus Institute is a professional development experience designed
to stretch and extend participants mathematical knowledge. The activities in this institute
are designed as a professional development for teachers, rather than simply as an
opportunity to go through classroom tasks. The topics address in this institute extend
well beyond what might typically be covered in the Algebra II or Precalculus classroom.
This institute begins by developing foundations for functions and exploring functions as a
unifying theme for this institute and the two additional institutes, Part 2: Algebra II and
Part 2: Precalculus. The three institutes include materials from the previous TEXTEAMS
Algebra II/Precalculus Institute.
As with other TEXTEAMS Institutes, this session does not represent a complete
professional development curriculum for Algebra II or Precalculus teachers. Additional
professional development resources for secondary teachers are available through
TEXTEAMS. Course specific TEXTEAMS institutes include Algebra I: 2000 and
Beyond, Practiced-Based Professional Development: Algebra I Assessments, Rethinking
Secondary Mathematics: Algebraic and Geometric Modeling, High School Geometry:
Supporting the TEKS and TAKS, Practiced-Based Professional Development: Geometry
Assessments, Rethinking Secondary Mathematics: Statistical Reasoning Across the TEKS,
Rethinking Secondary Mathematics: In-depth Secondary Mathematics (IDSM, and,
Practiced-Based Professional Development: Algebra II Assessments (in development).
TEXTEAMS Part 1: Algebra II and Precalculus Institute vi
References:
Connally, Hughes-Hallett, Gleason, et al. (2000)  Functions Modeling Change: A
Preparation for Calculus, New York, John Wiley & Sons, Inc.
Demana, Waits, Clemens. (1994).  Precalculus Mathematics: A Graphing Appproach,
Menlo Park, CA: Addison-Wesley.
Finney, Thomas, Demana, Waits (1994). Calculus: Graphical, Numerical, Algebraic,
Readin, MA: Addison-Wesley.
Forester, Paul, Calculus: Concepts and Applications (2000), Emeryville, California, Key
Curriculum Press,
Murdoch, Kamischke, Kamischke, (1998).  Advanced Algebra Through Data
Exploration, Berkekly, CA: Key Curriculum Press.
The University of Chicago School Mathematics Project, (1992). Precalculus and
Discrete Mathematics. Glenview, IL: Scott, Foresman.
TEXTEAMS Part 1: Algebra II and Precalculus Institute vii
Overview
Activity Overview Materials
I. Foundations of Functions
Introduction
Participants use data collection to Motion detector and
Ball Drop motivate the need for transformations of data collection device,
functions. overhead graphing
calculator, balls
1. Foundations
1.1 Introducing Participants transform a function Transparency pens, 1”
Transformations numerically and graphically and make grid paper, markers
generalizations.
1.2 What is the Participants will become aware of the Chart paper, markers,
Difference? TEKS for each course and clarify the copies of TEKS for
similarities and differences in the two Algebra II and
courses. Precalculus
2. Applying Transformations
2.1 Applications Participants use their knowledge of Graphing calculator,
transformations to determine appropriate set of activity cards for
viewing windows on graphing calculators for each group
transformed parent functions.
2.2 Airlines Participants graph real data on a scatter plot Graphing calculator
and find trend lines. They use transformations
to explain aspects of the trend lines.
Participants find a real application for
horizontal shifts.
3. Making Connections
3.1 Inverses Participants use parametric equations to graph Graphing calculator
inverse relations and inverse functions and to
investigate domain and range.
3.2 Stays the Same Participants use graphing calculators to model Graphing calculator,
the solutions to equations graphically and 1” grid paper,
make connections between transformations of markers, string,
functions, inverse functions, and algebraic washers, timers, tape
solutions. measures or meter
sticks, compasses
TEXTEAMS Part 1: Algebra II and Precalculus Institute viii
I. Foundations Introductory Activity: Trainers' Notes
Ball Drop Introduction
Overview: In this introductory activity, we use data collection to motivate the need for
transformations of functions.
Objective: Algebra II TEKS
(b.1.B) In solving problems, the student collects data and records results,
organizes the data, makes scatter plots, fits the curves to the appropriate parent
function, interprets the results, and proceeds to model, predict, and make
decisions and critical judgments.
(c.1.A) The student identifies and sketches graphs of parent functions, including
1
… inverse (y= ) … functions.
x
(c.1.B) The student extends parent functions with parameters such as m in
y= mx and describes parameter changes on the graph of parent functions.

Precalculus TEKS
(c.2.A) The student is expected to apply basic transformations, including
a• f(x), f(x)+ d, f(x−c), f(b•x), f(x), f(x), to the parent functions.
(c.2.B) The student is expected to perform operations including compositions
on functions, find inverses, and describe these procedures and results verbally,
numerically, symbolically, and graphically.
Terms: Transformation
Materials: Motion detector and data collection device, overhead graphing calculator, ball
(Racket balls, golf balls, and basketballs work well.)
Procedures: Introduce this initial activity by explaining the focus of the Foundation
section. The intent of the Foundation section is to tie together commonalities
across many of the topics found in the Algebra II and Precalculus TEKS. This
institute builds on the functions-based approach from the Algebra I Institute.
Understanding transformations is an essential part of this approach. This
activity uses data to drive the need for transformation of functions.
Activity: Ball Drop
Hold up a ball and then let it bounce in front of whole group. Next, introduce
the motion detector that collects data about the distance from an object to the
motion detector.
Explain that the motion detector sends out an ultrasonic pulse. The pulse
bounces off the ball, and the motion detector records the distance from the ball
to the motion detector at that time. The calculator displays the data as a graph
with the distance measured in meters or feet and the time measured in
seconds.
TEXTEAMS Part 1: Algebra II and Precalculus Institute 1
DRAFT 3-29-03
I. Foundations Introductory Activity: Trainers' Notes
Hold up the motion detector and drop the ball under it.
1. Ask participants to individually sketch their guess for a graph of the
relationship between the distance of the ball from the motion detector over
time. Have them compare their graph in small groups and arrive at a
group consensus.
2. Collect distance data using the motion detector for about 4 seconds. (Do
not use a ball bounce program; simply use a distance program.) Repeat if
necessary for good data. Have participants sketch the graph of the
collected data.
Sample Data:
• How does your sketch compare with the graph from our collected
data? If maximum and minimum points are not mentioned, ask
questions about them.
• What do the peak points on the graph represent? The ball’s maximum
distance from the motion detector. This distance should be close to the
distance between the motion detector and the floor.
Trace to a point on the scatter plot; have a participant hold up the ball at the
distance away from the motion detector represented by the y-coordinate of the
traced point. (Do not choose a point halfway in between the floor and the
motion detector because participants may confuse distance from the motion
detector and distance from the floor.)
• Write a sentence using the two numbers from the ordered pair in the
context of the ball drop. For example, for the ordered pair (0.84, 2.7),
the sentence could be “At 0.84 seconds after the ball was dropped, the
distance between the ball and the motion detector was 2.7 units.” This
is important because it is easy to confuse the path of the ball with the
graph of the relationship of the distance from the motion detector over
time.
• Did the volunteer hold the ball at the correct distance from the motion
detector? Why or why not?
• Is this a picture of the ball’s path? No, it is a graph of the relationship
of distance from the motion detector over time.
Discuss with participants where the data from the motion detector is stored on
the calculator and how the calculator creates a scatter plot from the data. Save
the data in 2 named lists to use in Exercise 4.
TEXTEAMS Part 1: Algebra II and Precalculus Institute 2
DRAFT 3-29-03