SUPPORTING
AND STRENGTHENING
STANDARDS -BASED
MATHEMATICS
TEACHER
PREPARATION
Guidelines for
Mathematics and
Mathematics Education
Faculty
The Charles A. Dana Center
The University of Texas at Austin
About the Charles A. Dana Center
The Charles A. Dana Center at The University of Texas at Austin works to support education leaders and policymakers in strengthening Texas education. As a research unit of The University of Texas at Austin’s College
of Natural Sciences, the Dana Center maintains a special emphasis on mathematics and science education. We
develop and offer professional development and research-based mathematics and science resources for educators to use in helping all students achieve academic success.
About the Dana Center’s work with higher education
The Supporting and Strengthening Standards-Based Mathematics Teacher Preparation (S3MTP) project builds upon
the work of a statewide network of Texas faculty that began in 1994 with a previous Fund for the Improvement
of Postsecondary Education (FIPSE) project, which resulted in the Dana Center’s 1996 publication, Guidelines
for the Mathematical Preparation of Prospective Elementary Teachers. That project also led to the Dana Center’s
annual statewide October Preservice Conference, which serves mathematics and education faculty who have
a special interest in mathematics teacher preparation. This conference has led to a growing network of faculty
interested in the mathematical preparation of teachers. For many faculty, this conference is their once-a-year
opportunity to share mathematics content and teacher preparation ideas with other faculty from across the
state, to hear from other higher education mathematics leaders, and to receive updates from statewide education agency leaders. The conference also provides opportunities for the Dana Center to solicit input for its
higher education publications and initiatives.
S3MTP was developed from ideas generated by participants at one of these October Preservice conferences.
For more information, see the Dana Center’s higher education website, at www.utdanacenter.org/mathematics/
highered/.
About the development of this guidelines book
The development of Supporting and Strengthening Standards-Based Mathematics Teacher Preparation (S3MTP)
was supported by Grant Award #P116B011116 from the Fund for the Improvement of Postsecondary Education (FIPSE), a program of the Office of Postsecondary Education of the U.S. Department of Education. Any
opinions, findings, conclusions, or recommendations expressed in this material are those of the author(s) and
do not necessarily reflect the views of the U.S. Department of Education or The University of Texas at Austin.
One hundred percent of the total costs for this project were financed with federal funds. The federal funds for
this project amount to $515,185. This information is provided in compliance with Public Law 108-7 Sec. 507,
the Consolidated Appropriations Resolution, 2003, February 20, 2003.
For more information, visit the Dana Center website at www.utdanacenter.org.
The Charles A. Dana Center
The University of Texas at Austin
2901 North IH-35, Suite 2.200
Austin, Texas 78722-2384
First printing February 2004.
Copyright 2004 The University of Texas at Austin. All rights reserved.
Permission is given to any person, group, or organization to copy and distribute this publication for noncommercial educational purposes only, so long as appropriate credit is given. Duplication for profit is prohibited.
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.
ii
Acknowledgements
Authors
Lesa Beverly, The University of Texas at Tyler
Tommy Bryan, Baylor University
Kimberly Childs, Stephen F. Austin State University
James Epperson, The University of Texas at Arlington
Christopher Kribs Zaleta, The University of Texas at Arlington
Deborah Pace, Stephen F. Austin State University
Colin Starr, Willamette University, Salem Oregon
Editors
James Epperson, The University of Texas at Arlington
Deborah Pace, Stephen F. Austin State University
Kimberly Childs, Stephen F. Austin State University
Project directors
Project Director Uri Treisman, Professor of Mathematics and Director,
Charles A. Dana Center, The University of Texas at Austin
Co-Project Director Susan Hudson Hull, Director of Mathematics,
Charles A. Dana Center, The University of Texas at Austin
Co-Project Director James Epperson, Assistant Professor of Mathematics,
The University of Texas at Arlington
Co-Project Director Deborah Pace, Associate Professor of Mathematics and Statistics,
Stephen F. Austin State University
External Evaluator
Ester Smith, EGS Research and Consulting
***
With special thanks to…
We would especially like to thank the contributions of our many reviewers and the members of the
advisory and steering committees for the S3MTP project. Their feedback and guidance significantly
strengthened the quality of this work.
Note: For reviewers, steering committee members, and advisory committee members, the professional
affiliations listed were those at the time of their participation; in some cases the affiliation has since
changed.
iii
Initial Reviews
First Draft
Dick Stanley, University of California at Berkeley
Michael Starbird, The University of Texas at Austin
Dixie Ross, Round Rock High School, Round Rock, Texas
Public Draft
Tom Fox, The University of Houston—Clearlake
Jeff Lawler, Trinity University
Michelle Moravec, McLennan Community College
Jane Schielack, Texas A & M University
George Tintera, Texas A & M University—Corpus Christi
Betty Travis, The University of Texas at San Antonio
Contributed Reviews
Tom Butts, The University of Texas at Dallas
Xuhui Li, The University of Texas at Austin
Advisory Committee
The Advisory Committee built and supported a collaborative leadership network from different levels of the
educational system, from different types and sizes of educational institutions, and from different regions
across the state.
Jasper Adams, Stephen F. Austin State University
Stuart Anderson, Texas A&M University—Commerce
Jamie Whitehead Ashby, Texarkana College
Evelyn R. Brown, University of Houston Downtown
James Epperson, The University of Texas at Arlington
Amy Gaskins, Alliance for the Improvement of Mathematics Skills (AIMS) PreK–16,
Del Mar College, Corpus Christi.
Basia Hall, Houston ISD–East District
Susan Hudson Hull, Charles A. Dana Center, The University of Texas at Austin
Paul Kennedy, Colorado State University, Fort Collins, Colorado
Mark Klespis, Sam Houston State University, Huntsville
Lee Von Kuster, University of Texas—Pan American
Barbara Montalto, Texas Education Agency, Austin
Deborah Pace, Stephen F. Austin State University
Anne Papakonstantinou, Rice University, Houston
Pamela Powell, The University of Texas at Austin
Karen Rhynard, Texas A&M University—Commerce
Dixie Ross, Round Rock High School, Round Rock
Michael Starbird, The University of Texas at Austin
Mourat Tchoshanov, University of Texas at El Paso
Frances Thompson, Texas Woman’s University, Denton
Reginald Traylor, University of the Incarnate Word, San Antonio
Ann Webb, University of Texas at Tyler
Connie Yarema, Abilene Christian University
iv
Steering Committee
The Steering Committee provided connections between policy leaders and the designers of teacher education programs.
Uri Treisman, Professor of Mathematics and Director, Charles A. Dana Center,
The University of Texas at Austin
Jasper Adams, Professor and Chairman, Department of Mathematics and Statistics,
Stephen F. Austin State University
Lynn M. Burlbaw, Department of Teaching, Learning, and Culture,
Texas A&M University
Jean Miller, Interstate New Teacher Assessment and Support Consortium,
Council of Chief State School Officers
Patricia Porter, Texas State Board for Educator Certification
Janet Russell, Curriculum and Professional Development,
Texas Education Agency
Gloria White, Texas Higher Education Coordinating Board
Charles A. Dana Center Production Team
Rachel Jenkins, Senior Editor
Brenda Nelson, Proofreader
Phil Swann, Senior Designer
Rob Starkey, Freelance Designer
***
Some trademarked designations are used in this publication. Where we were aware of such a designation, the trademarked term has been printed with initial capitalization.
Cabri Geometry is a trademark of Cabrilog, for more information, see www.cabri.com/en. Cuisenaire
is a registered trademark of ETA/Cuisenaire; for more information, see www.etacuisenaire.com.
Fathom and Fathom Dynamic Statistics are trademarks of KCP Technologies; for more information,
see www.keypress.com/fathom. The Geometer’s Sketchpad is a registered trademark of Key Curriculum
Press; for more information, see www.keypress.com/sketchpad. Maple is a trademark of Waterloo
Maple Inc.; for more information, see www.mapleapps.com. Mathematica is a registered trademark of
Wolfram Research, Inc.; for more information, see www.wolfram.com. MATLAB is a registered trademark of The MathWorks, Inc.; for more infomration, see www.mathworks.com.
v
***
This publication reproduces the Texas State Board for Educator Certification’s nine Mathematics Standards,
and quotes extensively from the SBEC Mathematics Standards knowledge and skills statements for early childhood–4, 4–8, and 8–12. As of fall 2003, these documents could be downloaded in their entirety from the web
via the following URLs:
SBEC standards for EC–4 teacher certification level:
www.sbec.state.tx.us/SBECOnline/standtest/standards/ec4math.pdf;
SBEC standards for grades 4–8 teacher certification level: www.sbec.state.tx.us/SBECOnline/standtest/
standards/4-8math.pdf; and
SBEC standards for grades 8–12 teacher certification level: www.sbec.state.tx.us/SBECOnline/standtest/
standards/8-12math.pdf.
The first page of each of these documents lists the nine Mathematics Standards reproduced in this publication.
For more information about the State Board for Educator Certification, see www.sbec.state.tx.us/.
***
Extensive efforts have been made to ensure the accuracy of the information in this publication. The Charles A.
Dana Center and The University of Texas at Austin, as well as the author and editors, assume no liability for
any loss or damage resulting from the use of this book.
Every effort has been made to provide proper acknowledgement of original sources and to comply with copyright law. If cases are identified where this has not been done, please contact the Charles A. Dana Center to
correct any omissions.
Related Resources
Supporting and Strengthening Standards-Based Mathematics Teacher Preparation (S3MTP) builds on the
work begun with a previous Fund for the Improvement of Postsecondary Education (FIPSE) project, which
resulted in the Dana Center’s 1996 publication, Guidelines for the Mathematical Preparation of Prospective Elementary Teachers (available for download from the Dana Center’s website at www.utdanacenter.org/
ssi/docs/GuideMath97.pdf.) That project also led to the Dana Center’s annual statewide October Preservice Conference, which serves mathematics and education faculty who have a special interest in mathematics teacher preparation (for more information, see the Dana Center’s higher education website, at
www.utdanacenter.org/mathematics/highered/). In 2003, the Dana Center also published Advanced Mathematics Educational Support: Support, recommendations, and resources for facilitating collaboration
between higher education mathematics faculty and Texas public high schools. To obtain a copy of the
AMES document, and for information about other Dana Center publications, visit the Dana Center home
page at www.utdanacenter.org. A policy position statement related to this S3MTP book is available through
the Dana Center higher education website at www.utdanacenter.org/mathematics/highered/.
vi
Table of Contents
Foreword ................................................................................................................................................viii
Preface .................................................................................................................................................... ix
Introduction
............................................................................................................................................ 1
Chapter 1: Tasks Vertically Connected Across Teacher Certification Levels ................................................. 7
Section 1.1: Mathematical Processes: Exploring Positional Systems Through Divisibility Rules
Section 1.1.1: EC–Grade 4 Teacher Task: Exploring Positional Systems Through Divisibility Rules
Section 1.1.2: Grades 4–8 Teacher Task: Exploring Positional Systems Through Divisibility Rules
Section 1.1.3: Grades 8–12 Teacher Task: Exploring Positional Systems Through Divisibility Rules
Section 1.1.4: Another Path: Extensions of Exploring Positional Systems Through Divisibility Rules, for EC–12
Section 1.2: Patterns, Algebra, and Analysis: Exploring Infinite Processes
Section 1.2.1: EC–Grade 4 Teacher Task: Exploring Infinite Processes
Section 1.2.2: Grades 4–8 Teacher Task: Exploring Infinite Processes
Section 1.2.3: Grades 8–12 Teacher Task: Exploring Infinite Processes
Chapter 2: Early Childhood–Grade 4 Teacher Tasks......................................................................................................37
Section 2.1: EC–Grade 4 Teacher Task: Numeration Systems: An Even/Odd Algorithm in Base Five
Section 2.2: EC–Grade 4 Teacher Task: Patterns, Geometry and Algebra: Painting the Cube
Section 2.3: EC–Grade 4 Teacher Task: Rational Numbers, Area Models, and Fallacious Reasoning: Geoboard Eighths
Section 2.4: EC–Grade 4 Teacher Task: Probability: Assessing the Fairness of Games
Section 2.5: EC–Grade 4 Teacher Task: Number Theory: The Stamps Problem
Section 2.6: EC–Grade 4 Teacher Task: Geometry and Measurement: Tiling a Round Patio
Chapter 3: Grades 4–8 Teacher Tasks.................................................................................................................................59
Section 3.1: Grades 4–8 Teacher Task: Polynomial Functions: Modeling Area and Volume
Section 3.2: Grades 4–8 Teacher Task: Geometry and Measurement: Pythagorean Relationships
Section 3.3: Grades 4–8 Teacher Task: Measures of Central Tendency and Spread: Designing Data
Section 3.4: Grades 4–8 Teacher Task: The Distributive Property: Patterns in Powers
Section 3.5: Grades 4–8 Teacher Task: Geometry, Measurement, and Modeling: The Paper Stacking Problem
Section 3.6: Grades 4–8 Teacher Task: Probability and Statistics: The Spicy Gumball
Chapter 4: Grades 8–12 Teacher Tasks ....................................................................................................... 87
Section 4.1: Grades 8–12 Teacher Task: Geometry and Measurement: Rain Gauges
Section 4.2: Grades 8–12 Teacher Task: Number Concepts: Cantor Sets
Section 4.3: Grades 8–12 Teacher Task: Mathematical Processes: Using Geometric Models to Predict Convergence
Section 4.4: Grades 8–12 Teacher Task: Probability and Statistics: Tests of Significance
Section 4.5: Grades 8–12 Teacher Task: History of Mathematics: The Life and Contributions of Pierre de Fermat
Section 4.6: Grades 8–12 Teacher Task: Geometry and Calculus Concepts: Using the Monte Carlo Method to Estimate The
Area Under a Curve
References ............................................................................................................................................................................ 109
vii
Foreword
Supporting and Strengthening Standards-based Mathematics Teacher Preparation captures the experience
and practice wisdom of Texas higher education faculty members who have committed themselves to
supporting the mathematics teachers in their communities. They share their wealth of knowledge
about approaches to preparing mathematically sophisticated teachers while incorporating various
styles of pedagogy. Their commitment to improving the education of all our children rings through.
This book was written for mathematics faculty members in two-and four-year colleges and
universities who seek to join the growing community of their peers committed to providing
prospective teachers with rich mathematical experiences to meet the growing demands on our
nation’s mathematics teachers. This set of tasks and guidelines pays special attention to important
structural issues, such as the challenges that institutions face when aligning programs to meet
standards-based teacher certification exams.
In the interests of strengthening Texas education, the authors took valuable time away from their
own programs to compose this document. The hope is to create mechanisms for teacher preparation
that will provide every Texas child with the well-prepared mathematics teachers they need and
deserve. This resource takes an important step toward ensuring that the accident of where Texas
children go to school will not determine the quality of mathematics education they receive. This
is a commitment that speaks not only to the American creed but to the practical importance of
preparing all children for responsible citizenship in the twenty-first century.
Uri Treisman
Professor of Mathematics and
Director, Charles A. Dana Center
The University of Texas at Austin
January 2004
viii
Preface
POLICY CONTEXT—————————————
The Texas State Board for Educator Certification
oversees prekindergarten–12 educator preparation
in Texas. In 2003, SBEC oversaw 70 institutions of
higher education and 28 alternative certification
programs with approved teacher preparation
programs. SBEC responsibilities include reviewing
and monitoring the quality of teacher preparation
programs, as well as developing and administering
TExES (Texas Examinations of Educator
Standards, the state exam that teachers must pass
to become certified).
1
The SBEC mathematics standards for
early childhood–4, 4–8, and 8–12 teacher
certification may be downloaded from the web
at www.sbec.state.tx.us/SBECOnline/standtest/
standards/ec4math.pdf; www.sbec.state.tx.us/
SBECOnline/standtest/standards/4-8math.pdf;
and www.sbec.state.tx.us/SBECOnline/standtest/
standards/8-12math.pdf.
2
In May 2002, Texas changed its state policy framework for
teacher preparation from a two-levels teacher certification
system (certification for teachers of grades 1–8 and of secondary
school) to a three-level system: early childhood through grade
4, middle level (grades 4 through 8), and secondary (grades 8
through 12). At that time, the Texas State Board for Educator
Certification(SBEC)1 released nine mathematics standards in
content and pedagogy for the beginning teacher of mathematics;
each standard is accompanied by statements of what the teacher
should know (knowledge statements) and be able to do (skills
statements). These teacher certification standards and knowledge
and skills statements are correlated to the state’s curriculum
standards for what every student should know and be able to do
in key academic areas—the TEKS, or Texas Essential Knowledge
and Skills.
For each teacher certification standard, the knowledge statements apply to beginning teachers at all grade levels, and the
skills statements describe the applications of that knowledge specific to the three certification levels—early childhood through
grade 4 (EC–4), grades 4 through 8, and grades 8 through 12.2
Supporting and Strengthening Standards-Based Mathematics Teacher
Preparation: Guidelines for Mathematics and Mathematics Education Faculty provides appropriate sample standards-based
mathematical tasks designed to clarify the intent of SBEC’s
Beginning Mathematics Teacher Standards for early childhood through grade twelve. The tasks are suitable for all college
students—those preparing to be teachers and those who are not.
The Guidelines document is intended to offer a foundation and
a stimulus for mathematics and mathematics education faculty
to continue to create standards-based resources that support the
design and implementation of standards-based teacher preparation and certification programs. Developing this document was
one of the goals of the Charles A. Dana Center’s Supporting and
Strengthening Standards-Based Mathematics Teacher Preparation
(S3MTP) project.
ABOUT THE S3MTP PROJECT—————————
S3MTP is designed to support the improvement of mathematics teacher preparation in Texas. This 3-year project, funded in
part by the U.S. Department of Education,3 began in October
2001 and is based on the premise that Texas’s new standardsbased policy guiding teacher preparation requires fundamental
changes in what teachers know and are able to do. The goal of
the S3MTP project is to broaden and diversify the network of
faculty and other key stakeholders who take leadership roles in
strengthening the preparation of mathematics teachers. Project
objectives are to
• provide opportunities for faculty collaboration, professional recognition, and leadership, and
• develop resources for implementing standards-based
teacher preparation and certification.
Through this project, we intend to create a model process and
resources that can be adapted for use in other content areas by
institutions that prepare teachers in Texas and in other states.
The initial stage of the project involved activating two leadership teams:
• the Steering Committee, which provided connections
between policy leaders and the designers of teacher
education programs; and
• the Advisory Board, which built and supported a collaborative leadership network from different levels of the educational system, from different types and sizes of institutions,
and from different regions across the state.
In collaboration with these groups, a writing team of six university mathematics faculty developed a draft guidelines document
to support the standards-based mathematical preparation of
teachers. One of the co-project directors also served as a writer.
x – preface
S3MTP is funded in part by a grant from the
Fund for the Improvement of Postsecondary
Education to the Charles A. Dana Center at The
University of Texas at Austin.
3
The October Preservice Conference brings
together K–12 education practitioners with higher
education faculty interested in the preparation
of mathematics teachers, for the purpose of
strengthening the preservice preparation of
teachers of mathematics.
4
After an external review of the initial draft, a revised draft
underwent further review at the Dana Center’s 2002 Annual
October Preservice Conference.4 After feedback gathered at the
conference was implemented, the draft was posted to the Dana
Center website for comment from faculty statewide. In addition
to this public review, we solicited review from six additional external reviewers . The conclusion of the second year (2002–03)
of the S3MTP project involves the publication and dissemination of this document. In its third and final year (2003–04),
S3MTP will host faculty leadership-development retreats that
will address 1) strategies for implementing the guidelines and 2)
the development of policy recommendations on a faculty reward
system for preparing teachers.
NEED FOR THE S3MTP GUIDELINES——————
Texas’s new state policy framework for teacher preparation
moves from a credit-based to a standards-based system for
teacher preparation, and creates a high-stakes accountability system for colleges, universities, and alternative certification agencies that prepare teachers. These changes in the state’s teacher
preparation system allow for greater flexibility and creativity in
program design while demanding more consistency and higher
quality in teacher preparation. The state’s new policy thus
requires that faculty reexamine the mathematics preparation of
teachers across the state and increased collaboration between
colleges of education and mathematics departments in designing
high-quality programs that meet state and national standards.
While these S3MTP guidelines are not intended to outline a
complete curriculum for a course or program, they do provide
examples of standards-based tasks that may be incorporated
into mathematics courses in teacher-preparation programs. The
mathematics tested for teacher licensure is based upon the SBEC
standards and must be embedded in teacher preparation programs.
preface – xi
ma
he tic
M
ure at
Co
ested f
sT
nt
lle
Mathema
ti
K1
te
eaching
2T
Ce
cs
Licens
or
athematics C
M
on
ge
ntral to
Beginning in 2003, all preservice teachers in Texas must pass
the Texas Examinations of Educator Standards (TExES), which
assess their knowledge of content and pedagogy for their subject
area and grade range. These exams are aligned with the Texas
Essential Knowledge and Skills (TEKS), the state-mandated
standards for kindergarten through grade 12. Low passing rates
on the TExES for students from a given institution can lead to
that institution losing its authority to certify teachers.
S3MTP GUIDELINES DEVELOPMENT——————
Project writers began work on the guidelines by focusing on
matters of mathematics content, seeking to create mathematical
tasks that explicate and contextualize those specific mathematical skills for which the State Board for Educator Certification
standard, and associated knowledge and skills statements,
seemed the broadest or most open to interpretation. Because the
primary audience for the guidelines is higher education mathematics and mathematics education faculty, the writers limited
their initial work to designing tasks to help clarify the first six
SBEC mathematics standards, covering mathematical content,
processes, and perspectives.
xii – preface
The Texas State Board for Educator Certification
mathematics standards for early childhood–4,
4–8, and 8–12, with associated knowledge and
skills statements, may be downloaded from the
web via the following URLs: www.sbec.state.tx.us/
SBECOnline/standtest/standards/ec4math.pdf;
www.sbec.state.tx.us/SBECOnline/standtest/
standards/4-8math.pdf; and www.sbec.state.tx.us/
SBECOnline/standtest/standards/8-12math.pdf.
The first page of each of these three documents lists
the nine Mathematics Standards reproduced here.
5
State Board for Educator Certification
Mathematics Standards5
Standard I. Number Concepts:
The mathematics teacher understands and uses numbers, number systems
and their structure, operations and algorithms, quantitative reasoning,
and technology appropriate to teach the statewide curriculum (Texas
Essential Knowledge and Skills [TEKS]) in order to prepare students to use
mathematics.
Standard II. Patterns and Algebra:
The mathematics teacher understands and uses patterns, relations, functions,
algebraic reasoning, analysis, and technology appropriate to teach the
statewide curriculum (Texas Essential Knowledge and Skills [TEKS]) in order
to prepare students to use mathematics.
Standard III. Geometry and Measurement:
The mathematics teacher understands and uses geometry, spatial reasoning,
measurement concepts and principles, and technology appropriate to teach
the statewide curriculum (Texas Essential Knowledge and Skills [TEKS]) in
order to prepare students to use mathematics.
Standard IV. Probability and Statistics:
The mathematics teacher understands and uses probability and statistics,
their applications, and technology appropriate to teach the statewide
curriculum (Texas Essential Knowledge and Skills [TEKS]) in order to
prepare students to use mathematics.
Standard V. Mathematical Processes:
The mathematics teacher understands and uses mathematical processes
to reason mathematically, to solve mathematical problems, to make
mathematical connections within and outside of mathematics, and to
communicate mathematically.
Standard VI. Mathematical Perspectives:
The mathematics teacher understands the historical development of
mathematical ideas, the interrelationship between society and mathematics,
the structure of mathematics, and the evolving nature of mathematics and
mathematical knowledge.
Standard VII. Mathematical Learning and Instruction:
The mathematics teacher understands how children learn and develop
mathematical skills, procedures, and concepts, knows typical errors students
make, and uses this knowledge to plan, organize, and implement instruction;
to meet curriculum goals; and to teach all students to understand and use
mathematics.
Standard VIII. Mathematical Assessment:
The mathematics teacher understands assessment and uses a variety of formal
and informal assessment techniques appropriate to the learner on an ongoing
basis to monitor and guide instruction and to evaluate and report student
progress.
Standard IX. Professional Development:
The mathematics teacher understands mathematics teaching as a profession,
knows the value and rewards of being a reflective practitioner, and realizes
the importance of making a lifelong commitment to professional growth and
development.
preface – xiii
As guidelines development progressed, it became apparent that
each teacher certification level (early childhood–grade 4, grades
4–8, and grades 8–12) had distinct challenges. For example,
there are many more resources for faculty preparing prospective
mathematics teachers of grades EC–4 than for those teaching
prospective mathematics teachers of grades 4–8 or 8–12. The
challenge for the EC–4 writers, then, became choosing the best
adaptations of mathematical tasks to help faculty preparing prospective educators teach them the knowledge and skills necessary to meet the standards. Since the 4–8 mathematics teacher
certification level is relatively new, and since there are relatively
few published textbooks targeting this certification level, the
4–8 writers faced the challenge of creating tasks that clearly illustrate the difference in the mathematical preparation expected
for 4–8 teachers, in contrast to that of EC–4 or 8–12 teachers.
The 8–12 mathematics certification level seems transparent in
that most faculty will read the SBEC standards and be able to
match many of the knowledge and skills statements to courses
in the mathematics major. Thus, the challenge for the 8–12
writers was to create tasks that illustrate the in-depth experiences with mathematical processes and mathematical learning
that prospective 8–12 teachers need so that their preparation
meets the spirit of the standards. The most compelling outcome
the writers found from the 8–12 work is the need for capstone
courses for prospective secondary mathematics teachers so that
their preparation meets the SBEC standards. (For specifics, see
the Task Correlation Guide 4 at the end of this preface.) This
outcome is in alignment with the Conference Board of the
Mathematical Sciences 2001 recommendations for secondary
mathematics teacher preparation. That is, that core knowledge
for prospective high school teachers may best be gained through
a program of study that includes many of the requirements of a
standard mathematics major, but that also includes a capstone
course sequence in which fundamental ideas from high school
mathematics are examined from an advanced standpoint.6 The
structure and presentation of the mathematical tasks in the
S3MTP guidelines evolved with ongoing reviews and comments
from the advisory board, external reviewers, and other contributed comments.
ORGANIZATION OF THE GUIDELINES ——————
It is worth mentioning that while the document’s sample tasks
are intended to clarify the intent of SBEC’s Beginning Mathematics Teacher Standards for early childhood through grade
xiv – preface
6
Tucker, A., Fey, J., Schifter, D., & Sowder, J.
(2001). The Mathematical Education of Teachers.
CBMS Issues in Mathematics Education (11), p 123.
Providence, R.I.: The American Mathematical
Society, Mathematical Association of America.
twelve, they are NOT meant to provide all the resources and
content necessary to master each standard. The S3MTP Guidelines document is organized as follows.
• Introduction
The introduction further explains the purpose and
audience of the Guidelines, and clarifies what they are
intended to do and what they are NOT intended to do.
It also defines terminology and maps the page format
used throughout the guidelines.
A key strategy to foster collaboration between
mathematics teachers across grade levels is the
formation of K-12 (or even K-16) “vertical
teams.” The primary goal of the vertical team
strategy is to enhance all students’ achievement
by increasing communication and cooperation
among teachers about the mathematics program
at their schools across grade levels. Vertical teams
can facilitate the implementation of academic
changes and support structures necessary to make
high achievement in mathematics by all students
a reality. More information about vertical teaming
can be found in the Advanced Placement Program
Mathematics Vertical Teams Toolkit (The College
Board and The Charles A. Dana Center, 1998)
and Advanced Mathematics Educational Support
(Charles A. Dana Center, 2003).
7
• Chapter 1: Tasks Vertically Connected Across
Teacher Certification Levels
Chapter 1 gives two sequences of connected mathematical tasks that illustrate how a single mathematical idea
or concept may be explored at each certification level
and how to gauge appropriate student responses to the
tasks for each certification level. Another way of saying
this is that the tasks connect “vertically” across grade
levels.7
• Chapter 2: Early Childhood–Grade 4 Teacher Tasks
Chapter 2 provides mathematical tasks that illustrate
the content strands of number concepts, patterns, algebra, geometry, measurement, and probability. Each task
is linked to one or more SBEC Mathematics Standards
and related knowledge and skills statements, for EC–4.
• Chapter 3: Grades 4–8 Teacher Tasks
Chapter 3 provides mathematical tasks that illustrate
specific knowledge and skills statements from the SBEC
Mathematics Standards for Grades 4–8. Each task highlights the concepts that are most appropriate for 4–8
teachers.
• Chapter 4: Grades 8–12 Teacher Tasks
Chapter 4 provides mathematical tasks that emphasize
the mathematical processes that should be understood
by teachers of grades 8–12.
In contrast to Chapter 1’s focus on connected tasks—tasks that
connect vertically across all three certification levels—Chapters
2 through 4 focus on specific knowledge and skills statements
from the SBEC mathematics standards and do not necessarily build on tasks from the preceding certification levels. It is
assumed, however, that prospective teachers of grades 4–8 will
have had mathematical experiences similar to those of prospec-
preface – xv
tive EC–4 teachers and that prospective teachers of grades 8–12
will have been exposed to mathematical experiences similar to
those of prospective EC–8 teachers.
The following four tables provide a quick reference for determining where the S3MTP tasks may be used in a teacher preparation
program. Of necessity, we use descriptive titles for the courses.
Divisibility Rules
Tasks
Infinite Processes
Connected Tasks Across Certification Levels
Task Correlation Guide 1:
Tasks vertically connected across teacher certification levels
xvi – preface
Courses where task may be most appropriate
EC–Grade 4
Teacher Task
Course in Foundations of Arithmetic for preservice elementary teachers
Grades 4–8
Teacher Task
Course in Number and Operation that extends the ideas from an EC–4
Foundations of Arithmetic course
Grades 8–12
Teacher Task
Course in number theory or a capstone course for preservice 8–12 teachers
EC–Grade 4
Teacher Task
Course in Foundations of Arithmetic for preservice elementary teachers
Grades 4–8
Teacher Task
Course in Concepts of Calculus for preservice 4–8 teachers or number theory course for
preservice 4–8 teachers
Grades 8–12
Teacher Task
Calculus or a capstone course for preservice 8–12 teachers
In the left-hand column we list the S3MTP mathematical tasks in the order that they appear
in this document.
Task Correlation Guide 2:
Tasks for early childhood through grade 4 teacher certification level
S3MTP Tasks for EC–Grade 4 Teacher
Certification Level
Courses where task may be most appropriate
Numeration Systems: An Even/Odd Algorithm in
Base Five
Course in Foundations of Arithmetic for preservice elementary
teachers
Patterns, Geometry, and Algebra: Painting the
Cube
Course in Number and Operation that extends the ideas from
an EC–4 Foundations of Arithmetic course
Rational Numbers, Area Models, and Fallacious
Reasoning: Geoboard Eighths
Course in number theory or a capstone course for preservice
8–12 teachers
Probability: Assessing the Fairness of Games
Course in Foundations of Arithmetic for preservice elementary
teachers
Number Theory: The Stamps Problem
Course in Concepts of Calculus for preservice 4–8 teachers or
number theory course for preservice 4–8 teachers
Geometry and Measurement: Tiling a Round Patio
Calculus course or capstone course for preservice 8–12 teachers
Task Correlation Guide 3:
Tasks for grades 4–8 teacher certification level
S3MTP Tasks for Grades 4–8 Teacher
Certification Level
Courses where task may be most appropriate
Polynomial Functions: Modeling Area and
Volume
College Algebra, College Algebra for Preservice Teachers,
Precalculus, Calculus
Geometry and Measurement: Pythagorean
Relationships
Geometry, College Algebra, Precalculus, and/or a standard “proofs”
course
Measures of Central Tendency and Spread:
Designing Data
Probability and Statistics course for preservice elementary teachers,
Statistics
The Distributive Property: Patterns in Powers
Foundations of Arithmetic course for preservice elementary teachers,
problem solving for preservice elementary teachers
Geometry, Measurement, and Modeling: The
Paper Stacking Problem
Geometry and Measurement course for preservice teachers, College
Algebra, Precalculus
Probability and Statistics: The Spicy Gumball
Geometry and Measurement course for preservice teachers, Statistics
preface – xvii
Task Correlation Guide 4:
Tasks for grades 8–12 teacher certification level
8–12 Certification Level Tasks
Courses where task may be most appropriate
Geometry and Measurement: Rain Gauges
Foundations of Geometry, capstone course for
secondary teachers, College Algebra
Number Concepts: Cantor Sets
Number theory, capstone course for secondary
teachers
Mathematical Processes: Using Geometric Models to Predict
Convergence
Capstone course for secondary teachers, Calculus
II
Probability and Statistics: Tests of Significance
Statistics, capstone course for secondary teachers
History of Mathematics: The Life and Contributions of Pierre
de Fermat
History of Mathematics course, capstone course
for secondary teachers
Geometry and Calculus Concepts: Using the Monte Carlo
Method to Estimate the Area Under a Curve
Capstone course for secondary teachers, Calculus,
Statistics
xviii – preface
Introduction
If K–12 mathematics education in the U.S. deserves criticism (and it surely has received a lot
of criticism in the wake of the TIMSS reports),
then a share of the blame falls to those university mathematicians who should be playing an
important role in the preparation of teachers but
are not. It is easy to make the case that among
the most important students mathematicians
teach are future school teachers — students who
will each pass on the mathematics they have
learned to hundreds of other young people.1
—American Mathematical Society Task
Force on Excellence, 1999
Significant involvement by mathematicians in all phases of
teacher preparation is critical to improving the preparation of
mathematics teachers, and, in turn, the mathematical education
of students in kindergarten through grade 12. Mathematicians,
with their depth of content knowledge, are uniquely qualified to help all their students, including prospective teachers,
make important connections within mathematics and between
mathematics and other fields of study.2 For prospective teachers,
this includes making connections between the content of their
college courses and the content they will be expected to teach.
AUDIENCE AND PURPOSE ——————————
1
Ewing, J. (ed.) (1999). Towards excellence:
Leading a mathematics department in the 21st century.
Providence, RI: American Mathematical Society
Task Force on Excellence, pp. 24–25. Available
on American Mathematical Society website, at
www.ams.org/towardsexcellence/.
Thus, the primary target audience of the S3MTP Guidelines is
mathematics department faculty in two- and four-year colleges
and universities.
For an example of a report drawn from TIMSS
(Third International Mathematics and Science
Study), see U.S. Department of Education (1996).
Pursuing Excellence: A study of U.S. eighth-grade
mathematics and science teaching, learning, curriculum,
and achievement in international context. National
Center for Education Statistics, NCES 97-198,
Washington, DC: U.S. Government Printing
Office.
The purpose of the guidelines is to support and encourage mathematicians and mathematics educators in efforts to improve the
mathematical preparation of prospective teachers.
The guidelines are intended to:
• Highlight sample connections between college-level and
K–12-level mathematics content.
For more information, see the following
publication, and in particular, its preface: Tucker,
A., Fey, J., Schifter, D., & Sowder, J. (2001). The
Mathematical Education of Teachers. CBMS Issues
in Mathematics Education (11), Providence, R.I.,
The American Mathematical Society.
2
• Provide sample mathematical tasks that are good for all
students.
• Help clarify state-mandated standards for the certification of teachers, early childhood through grade 12.
• Acknowledge limitations of any listing of standards or
content topics—most problems do not fit neatly into a
single category.
• Offer seed ideas of mathematical tasks for faculty to
model and expand upon.
Beyond the preparation of the next generation
of teachers, it is likely that colleges and universities will be called upon to play a larger role in
the important business of improving mathematics education in the U.S. This will require more
mathematicians taking a role in the continuing
education of teachers and making a contribution
to the public discussion of what is taught and
how it is taught. For most departments, this is
a fertile area for making a contribution to the
university’s mission.
• Motivate faculty to design additional resources for dissemination.
—American Mathematical Society Task
Force on Excellence, 19993
• Provide examples of mathematical tasks that connect
“vertically” through grade levels.
The guidelines are NOT intended to:
• Outline a complete curriculum for a course or program.
• Attempt to describe all prior knowledge or experience
necessary for successful completion of the mathematical
tasks.
3
Ewing, J. (ed.) (1999). Towards excellence:
Leading a mathematics department in the 21st century.
Providence, RI: American Mathematical Society
Task Force on Excellence, pp. 24–25. Available
on American Mathematical Society website, at
www.ams.org/towardsexcellence/.
• Limit or prescribe course or program content.
• Suggest that teacher preparation programs should be
entirely determined by an external set of standards.
• Imply that the tasks have completely covered all the
standards.
The primary goals of the mathematics content portion of the
guidelines are to:
• Provide examples of tasks that clarify the intent of the
Beginning Mathematics Teacher Standards for early
childhood through grade twelve (EC–12) from the
Texas State Board of Educator Certification.
• Stimulate mathematicians and mathematics educators
to think deeply about appropriate programs and courses
to support the preparation of mathematics teachers.
• Motivate mathematicians and mathematics educators to
expand upon the ideas presented and disseminate them
statewide.
In addition, prospective teachers at all levels should have the opportunity to work through each sequence of tasks at all certification levels.
For example, prospective middle-level teachers should first
explore the EC–4-level tasks, and prospective secondary teach2 – introduction
Teachers need to understand the big ideas of
mathematics and be able to represent mathematics as a coherent and connected enterprise
.…4 This kind of knowledge is beyond what
most teachers experience in standard preservice
mathematics courses in the United States.
—National Council of Teachers of
Mathematics (2000). Principles and
Standards for School Mathematics5
4
The ellipses in this quotation designate where
we removed the parenthetical citations to the
following two sources, presented here in full citation
form: Ma, Liping. (1999). Knowing and teaching
elementary mathematics: teachers’ understanding
of fundamental mathematics in China and the
United States. Mahwah, N.J.: Lawrence Erlbaum
Associates. Schifter, Deborah. (1999). Reasoning
about Operations: Early Algebraic Thinking in
Grades K–6. In Developing Mathematical Reasoning
in Grades K–12. 1999 Yearbook of the National
Council of Teachers of Mathematics, edited by Lee
V. Stiff, pp. 62–81. Reston, Va.: National Council
of Teachers of Mathematics.
5
National Council of Teachers of Mathematics
(2000). Principles and Standards for School
Mathematics. Reston, VA: Author. p 17.
ers should do both the EC–4 and 4–8 level tasks as well as those
designed for the 8–12 level. Due to time restrictions, this type of
exploration might best be done in a capstone course.
TERMINOLOGY ——————————————
To avoid confusion, this document uses the following terminology:
• Students refers to all college students, unless otherwise
noted.
• Faculty refers to faculty at colleges (both two- and fouryear) and universities.
• Teachers refers to teachers in the early childhood–12
system.
• Grade-level designations—early childhood–grade 4
(EC–4), grades 4–8, and 8–12—refer to teacher certification levels, rather than to specific grades in K-12 schools.
For example while an EC–4 task in this book would
seem inappropriate for a fourth-grade student, that is not
its intended audience; rather it’s intended for a teacher at
that level.
• Task is used here to describe the process in which a
college student investigates the specified question—in
contrast to a question posed for which students should
“get an answer” and move on to the next one.
Each teacher task section is structured similarly. That is, the
mathematical task appears in a text box early in the main narrative of the section. Along the right margin of each section are
text boxes with gray shading; these boxes list the State Board
for Educator Certification standards and teacher knowledge and
skills statements that are most closely related to that section’s
mathematical task. The right margin also includes text boxes
that include—as relevant—extension ideas, assessment ideas,
historical connections, and notes on relevant mathematicsspecific technology. Also, throughout the narrative are indented
italicized statements that emphasize points ranging from mathematical connections that preservice teachers need to make as
they do the tasks, to general statements about the mathematical preparation needs of preservice teachers. The “Supporting
Discussion” subsection for each task provides solution strategies,
classroom implementation ideas, explications of mathematical
introduction – 3
notions embedded in the tasks, and other information relevant
to the task. “Supporting Discussion” is not intended to provide
complete solutions to the mathematical tasks or prescriptive
rubrics for implementing the tasks.
The tasks in this book can take anywhere from a class period
to a week to complete. We have not specified a time range for
each task, however, because the tasks are designed as seed ideas
that can be freely adapted both into existing course structures
and into new investigations. Finally, it is important to note that
the boxed mathematical tasks are intended as a focus for student
(preservice teacher) investigation. That is, while it is important that the student find an accurate result, the most valuable
outcome of exploring these tasks is the mathematical experience
and habits of mind cultivated through exploration, discovery,
and discussion of these tasks.
The following illustration maps the page format used throughout
much of the document.
4 – introduction
Illustration of page format
SECTION 1.2.2
GRADES 4–8 TEACHER TASK:
EXPLORING INFINITE PROCESSES

SBEC standard along with the
teacher knowledge and skills
statements most related to
the mathematical task. Note,
therefore, that all the items in a
numbered list are not included;
this list, for example, skips from
2.9s to 2.14s.
Statements indented
for emphasis.
Prospective middle level teachers will be expected to relate
middle school mathematics to the concept of limit as a conceptual
foundation of calculus. Hence, it is essential that they develop
their own conceptual understanding of limits.
Students preparing to teach at the middle school level should
be expected to justify their mathematical thinking in somewhat
more sophisticated ways. The questions posed below build on
the Shaded Rectangles extension in EC–4, above, but lead students to organize their thinking in a different way.
Shaded Rectangles Revisited: Suppose that n is a
positive number less than 1.
STATE BOARD FOR EDUCATOR
CERTIFICATION MATHEMATICS
STANDARD II: PATTERNS AND ALGEBRA
Grades 4–8 Teacher Skills
The beginning teacher of mathematics is
able to:
2.8s apply all skills specified for teachers in
grades EC–4, using content and contexts
appropriate for grades 4–8;
2.9s make, test, validate, and use conjectures about patterns and relationships in
data presented in tables, sequences, or
graphs;
• Argue that 0.999999… must be greater than n.
Statement of the
problem or task.
• What rational number is equal to the repeating
decimal 0.999999…? Justify your thinking.
Supporting Discussion
Middle level students are often taught an algorithm for converting
numbers from decimal to fraction form, but they rarely develop a
rich conceptual understanding of the underlying ideas.
This problem encourages students to further develop their
understanding of the concept of limit by making connections
between algebraic and geometric ideas. Posed this way, the question focuses on different representations for rational numbers
using the concept of limit. As students justify their thinking,
they should be encouraged to argue conceptually.
Most students are exposed to geometric series at some point in
their undergraduate education, but only a small number have the
opportunity to investigate and explain the derivation of geometric
series.
The following sequence of questions, adapted from the Dana
Center’s TEXTEAMS In-depth secondary mathematics institute,8
allows students to discover the formula for the sum of a geometric series, using a geometric construction that is accessible to a
student in college algebra.
Assessment ideas, historical
connections, notes on technology use, or problem extension
ideas.
Supporting discussion for
the problem or task.
Assessment
The goal of these questions is to have students do more than algorithmic manipulation. As students argue, they are building a
type of informal proof or justification. For
instance, they may observe that 0.999… is
certainly not bigger than 1, but it is larger
than any other number smaller than 1.
Therefore, they conclude that 0.999…
8
Rethinking Secondary Mathematics: In-depth
secondary mathematics is one of dozens of institutes
provided by TEXTEAMS (Texas Teachers
Empowered for Achievement in Mathematics
and Science, www.utdanacenter.org/texteams),
a Dana Center–managed statewide teacher
professional development program. TEXTEAMS
is a comprehensive system of professional
development for K–12 mathematics and science
teachers, delivered through a statewide network of
trainers. The program’s institutes provide a core set
of professional development materials and skills and
help teachers develop a common understanding of
important mathematics and science content and
the state’s curriculum standards (the Texas Essential
Knowledge and Skills).
6 – introduction
Chapter 1
TASKS VERTICALLY CONNECTED ACROSS
TEACHER CERTIFICATION LEVELS ————————
1
See for example, Ball, D.L. (1991). Research
on teaching mathematics: Making subject matter
knowledge part of the equation. In J. Brophy (Ed.),
Advances in research on teaching (Vol. 2, pp. 1-48).
Greenwich, CT: JAI Press; and Ma, L. (1999)
Knowing and teaching elementary mathematics:
Teachers’ understanding of mathematics in China and
the United States. Mahwah, NJ: Lawrence Erlbaum
Associates.
2
This statement reflects the position put forth
in the Conference Board of the Mathematical
Sciences monograph on the Mathematical
Education of Teachers: Tucker, A., Fey, J., Schifter,
D., & Sowder, J. (2001). The Mathematical
Education of Teachers. CBMS Issues in Mathematics
Education (11), p. 25.
Current research underscores the importance of strong content
preparation for all prospective mathematics teachers and emphasizes the need for those teachers to have a deep understanding of
the mathematics they will be expected to teach.1 Secondary-level teachers need the equivalent of a major in mathematics, but
their preparation should include significant learning experiences
that focus on making deep connections between the mathematics in their college content courses and the mathematics
taught in schools. The mathematical preparation of middle-level
teachers should be different from—not simply less than—that of
secondary teachers.2 And because the role of elementary teachers is so important in building the foundations of mathematical
reasoning in children, their mathematics preparation is critically
important, as is their understanding of the important foundations they are building for their students.3

3
See also Charles A. Dana Center, Texas Statewide
Systemic Initiative (1996). Guidelines for the
mathematical preparation of prospective elementary
teachers. Austin, TX: Charles A. Dana Center.
Available for download from the Dana Center’s
website, at www.utdanacenter.org/ssi/docs/
GuideMath97.pdf.
In addition to developing deep understanding of the
mathematics central to their own certification levels,
all prospective teachers should understand the
mathematical knowledge and skills expected of
early childhood–12 students.
Further, all teachers need to understand the mathematical
thinking that their students bring with them and the future
mathematical learning for which they are building foundations.
However, as most mathematics department faculty have little
direct experience with the current early childhood–12 school
mathematics curriculum, it can be helpful to see examples of
S3MTP • Chapter 1
different knowledge and skills expectations that might be attached to a single “big idea” in mathematics at the different
certification levels. Thus, this chapter includes two sequences of
vertically connected mathematical tasks—that is, tasks in which
a single idea is explored at varying levels of depth as it carries
through the teaching certification levels from early childhood
through high school. These tasks are only samples of those that
might be employed to teach or assess the understanding of the
mathematical idea; they are not intended to be exhaustive. It is
hoped that faculty will use their broad experience to extend and
adapt these example tasks to fit their own programs.
The first set of vertically connected tasks, “Mathematical
Processes: Exploring Positional Systems Through Divisibility
Rules,” focuses on number concepts, with an emphasis on developing the habits of mind associated with mathematical thinking.

The study of number concepts is central to early childhood–12
mathematics; hence, a deep understanding of our base ten number
system is critical for all prospective teachers.
Experience reveals, however, that many prospective teachers
have only algorithmic or formulaic proficiency with basic operations and their properties but often equate this with conceptual
mastery. The mathematics preparation of prospective teachers
should certainly include opportunities to investigate and justify
many of the standard base ten algorithms. One method of challenging their understanding in this area is to have them investigate standard base ten algorithms in the context of a different
base. In this first set of connected tasks, the introductory focus
at each certification level is a statement about a familiar “sum of
the digits” algorithm for testing for divisibility by three. At each
certification level, students are asked to examine and explain
concepts underlying this familiar algorithm. Also, we provide
explanations describing the varying degrees of justification and
proof that might be expected at each level.
Mathematical content strands run throughout the school
curriculum that build upon and extend mathematical ideas.
Prospective teachers should be aware of these mathematical
content strands, also known as “vertical” connections. Prospective teachers who encounter vertically connected mathematical
tasks in their preparation gain an insight into the varying levels
of depth required for mathematical conjecture, justification,
and proof at each certification level. Such preparation promotes
understanding of how sophisticated mathematical ideas can be
built from, depend upon, and connect to less developed math-
8 – Tasks vertically connected across teacher certification levels
S3MTP • Chapter 1
ematical notions from earlier grade levels. Vertically connected
mathematical tasks delineate the increasing refinement of
mathematical reasoning required of teachers for EC–4, or 4–8, or
8–12 students. For instance, an informal convincing argument
may suffice as a justification for a prospective EC–4 teacher, but
a 4–8 teacher would be required to demonstrate greater depth
of understanding and less informality, including correct use of
mathematical notation and broader capacity for generalization.
And a prospective secondary teacher would be expected to be
able to give both informal and formal proofs or justifications.
The second set of vertically connected tasks, “Patterns, Algebra, and Analysis: Exploring Infinite Processes,” is a guided
exploration of one of the most powerful ideas in the history of
mathematics: infinite processes. According to the State Board
for Educator Certification’s standards, prospective middle-level
(grades 4–8) teachers are expected to be able to relate the concept of limit to middle school mathematics; thus, they need opportunities to investigate concepts of calculus at varying levels
of formality. Although the specific questions in the “Patterns”
section vary, the focus is on using both geometric and algebraic
thinking to justify and reason about ideas of infinity.

Prospective teachers at all levels should have the
opportunity to work through each sequence of tasks
at all certification levels.
That is, prospective middle-level teachers should first explore
the Early Childhood–4-level tasks, then the middle-level tasks
(grades 4–8) and then 8–12; prospective secondary teachers
should do the Early Childhood–4 and 4–8-level tasks as well as
those designed for the 8–12 level. Due to the amount of time
needed to explore the tasks for all three certification levels, this
might best be done in a capstone course.
The tasks in this book can take anywhere from a class period to
a week to complete. We have not specified a time range for each
task, however, because the tasks are designed as seed ideas that
can be freely adapted both into existing course structures and
into new investigations. This chart (also included in the preface
to this book) suggests some possible courses for which these tasks
might be appropriate.
Tasks vertically connected across teacher certification levels – 9
S3MTP • Chapter 1
Divisibility Rules
Tasks
Infinite Processes
Connected Tasks Across Certification Levels
Task Correlation Guide 1
Tasks vertically connected across teacher certification levels
Courses where task may be most appropriate
EC–Grade 4
Teacher Task
Course in Foundations of Arithmetic for preservice elementary teachers
Grades 4–8
Teacher Task
Course in Number and Operation that extends the ideas from an EC–4
Foundations of Arithmetic course
Grades 8–12
Teacher Task
Course in number theory or a capstone course for preservice 8–12 teachers
EC–Grade 4
Teacher Task
Course in Foundations of Arithmetic for preservice elementary teachers
Grades 4–8
Teacher Task
Course in Concepts of Calculus for preservice 4–8 teachers or number theory course for
preservice 4–8 teachers
Grades 8–12
Teacher Task
Calculus or a capstone course for preservice 8–12 teachers
10 – Tasks vertically connected across teacher certification levels
S3MTP • Chapter 1
The State Board for Educator Certification
mathematics standards for early childhood–4,
4–8, and 8–12, with associated knowledge and
skills statements, may be referenced on the web
at www.sbec.state.tx.us/SBECOnline/ standtest/
standards/ec4math.pdf; www.sbec.state.tx.us/
SBECOnline/ standtest/standards/4-8math.pdf;
and www.sbec.state.tx.us/SBECOnline/ standtest
/standards/8-12math.pdf.
4
SECTION 1.1.
MATHEMATICAL PROCESSES:
EXPLORING POSITIONAL SYSTEMS
THROUGH DIVISIBILITY RULES
STATE BOARD FOR EDUCATOR CERTIFICATION
MATHEMATICS STANDARD V. MATHEMATICAL PROCESSES:4
The mathematics teacher understands and uses mathematical
processes to reason mathematically, to solve mathematical
problems, to make mathematical connections within and
outside of mathematics, and to communicate mathematically.
SBEC MATHEMATICS STANDARD V.
MATHEMATICAL PROCESSES
Teacher Knowledge
The beginning teacher of mathematics
knows and understands:
5.1k logical reasoning, justification, and
proof in relation to the structure of and
relationships within an axiomatic system;
5.2k the role of logical reasoning in
mathematics and age-appropriate
methods and uses of informal and formal
reasoning;
5.3k the process of identifying, posing,
exploring, and solving mathematical
problems in age-appropriate ways.
5.4k connections among mathematical
concepts, procedures, and equivalent
representations;
5.6k how to communicate mathematical
ideas and concepts in age-appropriate
oral, written, and visual forms; and
5.7k how to use age-appropriate mathematical manipulatives and drawings
and a wide range of technological tools
to develop and explore mathematical
concepts and ideas.

Since the study of number concepts occupies such a significant part of prekindergarten–12 school mathematics, this deep
understanding is critical for prospective teachers. Unfortunately,
because they have a great deal of experience operating in the
base ten number system, many college students have only algorithmic proficiency with this system, but often equate this with
conceptual understanding. Asking students to explain the reasoning behind “familiar” concepts and algorithms is one means
of assessing their depth of conceptual understanding; another is
to have students investigate those concepts in the context of a
different number base.

SBEC MATHEMATICS STANDARD I.
NUMBER CONCEPTS
Teacher Knowledge
The beginning teacher of mathematics
knows and understands:
1.1k the structure of number systems, the
development of a sense of quantity, and
the relationship between quantity and
symbolic representation;
1.2k the connections of operations, algorithms, and relations with their associated
concrete and visual representations;
1.3k the relationship among number
concepts, operations and algorithms, and
the properties of numbers, including ideas
of number theory;
1.5k how number concepts, operations,
and algorithms are developmental and
connected across grade levels.
All students should have a deep understanding of our base ten
number system.
Students need practice exploring mathematical questions and
representing their results systematically.
Students should have multiple opportunities to investigate interesting questions and then record the results in ways that can
reveal important patterns.

Students need experience manipulating numerical and algebraic
representations into useful forms and formulating explanations
and proofs in terms of such representations.
A concept that is very important to all students of mathematics, including those in grades K–12, is that of composition and
decomposition of numerical expressions. Thus, when investigating questions involving number concepts, prospective teachers should readily consider strategies involving expansions of
numbers using powers of ten.
Tasks vertically connected across teacher certification levels – 11
S3MTP • Chapter 1
STATE BOARD FOR EDUCATOR CERTIFICATION
MATHEMATICS STANDARD I. NUMBER CONCEPTS:
The mathematics teacher understands and uses numbers,
number systems and their structure, operations and algorithms, quantitative reasoning, and technology appropriate
to teach the statewide curriculum (Texas Essential Knowledge and Skills [TEKS]) in order to prepare students to use
mathematics.

By starting with familiar ideas, students are able to make important connections to previous knowledge and to deepen their
understanding of our number system.
For example, most students have been taught the “sum of the
digits” test for divisibility by three in base ten (that is, if the sum
of a whole number’s digits is divisible by three, then that number
is divisible by three), but few, if any, can give an explanation for
why it works. The series of questions in this example leads to a
deeper exploration of this familiar algorithm.

Students need opportunities to focus on the particular features of
the base ten system that make familiar algorithms work.
The mathematical tasks in this section progress from a concrete
or operational level to a more abstract one by asking students
first to justify the familiar algorithm, then to explore its effectiveness in other bases, and finally, to develop and prove a
conjecture that generalizes their conclusions.

It is critical that students be given time to experience doing
mathematics.
Students often see the finished product and have little understanding of the underlying processes so familiar to mathematicians—looking for patterns; making, testing, revising, and justifying conjectures; and communicating their results in a variety
of ways. Recall that the boxed task statements in the following
discussions are intended as a focus for student investigation.
That is, while it is important that the student find an accurate
result, the most valuable outcome of exploring these tasks is the
mathematical experience and habits of mind cultivated through
exploration, discovery, and discussion of these tasks.
12 – Tasks vertically connected across teacher certification levels
S3MTP • Chapter 1
SECTION 1.1.1
EC–GRADE 4 TEACHER TASK:
EXPLORING POSITIONAL SYSTEMS
THROUGH DIVISIBILITY RULES
STATE BOARD FOR EDUCATOR
CERTIFICATION MATHEMATICS STANDARD I.
NUMBER CONCEPTS—TEACHER SKILLS
Students in the elementary and middle grades are
often taught a simple test for determining whether a
whole number is divisible by three:
Grades EC–4
The beginning teacher of mathematics
is able to:
if the sum of a whole number’s digits (in its base-ten
representation) is divisible by three, then that number
is divisible by three.
1.1s compare and contrast numeration
systems;
1.2s analyze, explain, and model the
structure of numeration systems and, in
particular, the role of place value and
zero in the base ten system;
1.6s analyze and describe relationships
among number properties, operations,
and algorithms involving the four basic
operations with whole and rational
numbers;
1.10s describe ideas from number
theory (e.g., prime numbers, composite
numbers, greatest common factors) as
they apply to whole numbers, integers,
and rational numbers and use these
ideas in problem situations;
1.12s apply place value and other number properties to develop techniques
of mental mathematics and computational estimation.
• Justify and explain this algorithm.
Supporting Discussion
Before most students can seriously examine the reasoning behind divisibility rules in base ten, they need considerable work
with the decimal representation of natural numbers in terms
of powers of ten. They often need prompting to progress to the
expanded form of a natural number.

Most students will need to investigate many examples and counterexamples before they begin to see patterns and can isolate the
underlying concepts that make the algorithm work in general.
When considering the test for divisibility by three, most students tend to apply it to several numbers to verify that the
“test” works—they must be reminded to look for the underlying
reasons it works. That is, they must be reminded that verifying
that an algorithm works does not prove or indicate that it always
works.
Since ideas of decomposition are so important in mathematics,
supporting classroom discussion or activities may include the
following:
5
Note: the 1-block or unit block is the name of
the smallest of the base ten blocks; the 10-block
(corresponding to 10 units) is referred to as a rod
or a long; and the 100-block (corresponding to 100
units) is called a flat.
• Think about other ways to write 100 (e.g., 1(102),
9(10)+9(1)+1, 99+1, etc.).
• Demonstrate various numbers with base ten blocks.5
Tasks vertically connected across teacher certification levels – 13
S3MTP • Chapter 1
32
32 = 3(10) + 2(1)
= 3(101) + 2(100)
147
147 = 1(100) + 4(10) + 7(1)
= 1(102) + 4(101) + 7(100)
• Using base ten blocks, illustrate the test you have
developed for divisibility by three for
243ten
EXTENSION IDEA
Explain the similar test for divisibility by 9
in base ten.
***
Since the test for divisibility by 9 involves
principles similar to those for divisibility by
3, this task provides more experience in base
ten and an opportunity to probe students’
understanding of the core ideas involved in a
sound justification.
14 – Tasks vertically connected across teacher certification levels
S3MTP • Chapter 1
The shading in the figure above helps illustrate that two blocks
were taken from the hundreds place and four from the tens
place. These six units are added to the three units in the ones
place. These nine units are precisely the sum of the digits.
• What about base eight? Is there a divisibility test for
three in base eight? How do you show a base eight number using blocks?
243eight
Notice that squares in base eight would always have one left
over (when divided by three) but a long in base eight would
always have two left over. What does that indicate about the
process of dividing in base eight compared to base ten?
Prospective teachers need fluency with a variety of approaches
to teaching mathematical concepts. Working with concrete
objects, such as the blocks above, helps them make connections
from concrete to pictorial to abstract representations. This fluency is directly transferable to their future classrooms. As students
expand a number (such as 243), explain the underlying physical
process, and justify each step in the decomposition and regrouping, they give meaning to the “sum of the digits” algorithm.
243 = 2(100) + 4(10) + 3(1)
= 2(99+1) + 4(9+1) + 3(1)
= 2(99) + 2(1) + 4(9) + 4(1) + 3(1)
= 2(99) + 4(9) + 2(1) + 4(1) + 3(1)
RELATED TASK
Other familiar concepts, such as “even” and
“odd” numbers, can be understood in new
depth through investigations in other bases.
For a related task, see the task Numeration
Systems: An Even/Odd Algorithm in Base
Five (in Chapter 2, Early Childhood–Grade
4 Teacher Tasks).
= 2(99) + 4(9) + (2+4+3)(1)
Depending upon their level of experience, students may provide
a variety of reasons to justify that the only part of the resulting
sum whose divisibility by three is uncertain is the term described
by “the sum of the digits in the original number.” Clearly, at
Tasks vertically connected across teacher certification levels – 15
S3MTP • Chapter 1
least some discussion of divisibility properties related to sums
and products should accompany the justifications.

Mathematics teachers should be able to provide convincing
arguments to justify mathematical ideas.
As students think about the process used and determine which
features are peculiar to the specific base number chosen, they
build the groundwork for presenting an informal (as opposed to
formal mathematical) argument that clearly explains the underlying mathematical processes leading to the predicted outcome.
Without using symbolic notation, students may simply explain
that any base ten natural number can be expanded using powers of ten and then regrouped into addends that involve factors always divisible by three and a remaining term. Then this
remaining term, which represents the sum of the digits of the
original number, determines whether or not the original number
is divisible by three.
Students should eventually be able to argue using notation, perhaps specifically for three- and four-digit numbers. Such
an argument for a three-digit number htu may begin with an
expansion such as:
htu = h(10)2 + t(10)1 + u(10)0 = h(9 + 1)2 + t(9 + 1)1 + u(1)
and then use the same reasoning as in the cases above.
Investigations such as these also provide students with the opportunity to think about what is required to build a convincing
mathematical argument. Prospective elementary teachers often
have little experience writing formal mathematical proofs; it is,
however, important that they be encouraged to think about and
articulate their intuitive ideas about how and why things work.
The next task builds upon this one and extends the investigation to divisibility rules for other bases.
16 – Tasks vertically connected across teacher certification levels
S3MTP • Chapter 1
SECTION 1.1.2
GRADES 4–8 TEACHER TASK:
EXPLORING POSITIONAL SYSTEMS THROUGH
DIVISIBILITY RULES
STATE BOARD FOR EDUCATOR
CERTIFICATION MATHEMATICS STANDARD I.
NUMBER CONCEPTS
Grades 4–8 Teacher Skills
The beginning teacher of mathematics
is able to:
1.13s apply all skills specified for teachers in grades EC–4, using content and
contexts appropriate for grades 4–8;
1.17s analyze and describe relationships
between number properties, operations, and algorithms for the four basic
operations involving integers, rational
numbers, and real numbers;
Students in teacher preparation programs for the middle grades
who have had mathematical experiences similar to those in the
previous exercises are prepared for this section. Students who have
not experienced the EC–4 level task or ones like it will first need time
to investigate more concrete examples.
The following activity builds on the previous task while requiring thinking that is deeper, broader, and somewhat more abstract. Once again, the questions posed constitute a launching
place for student exploration and discussion.
Students in the elementary and middle grades are
often taught a simple test for determining whether a
whole number is divisible by three:
1.19s explain and justify the traditional
algorithms for the four basic operations with integers, rational numbers,
and real numbers and analyze common
error patterns that may occur in their
application.
if the sum of a whole number’s digits (in its base-ten
representation) is divisible by three, then that number
is divisible by three.6
• Predict whether this test works for numbers written
in base five, six, and seven.
6
• Provide examples or counterexamples to support
your prediction.
Here we emphasize the “test” (the sum of a whole
number’s digits is divisible by three) as a sufficient
condition for determining whether a whole number
is divisible by three. However, note that the “test” is
both necessary and sufficient.
Supporting Discussion

Students with little experience doing this type of investigation will
need guidance in exploring specific examples in many different
bases, recording results in a systematic way, and looking for patterns to determine the bases in which this algorithm seems to work.
For example, in a table, students might record results by writing many numbers in several different bases and then looking
for patterns that emerge by identifying those bases in which
the divisibility-by-three test appears to work. Given previous
experience with base ten expansions, they may try writing the
numbers in expanded form in the other bases, comparing the
expansions, and in those bases where the test appears to work,
noting similarities in the process of regrouping into addends that
are divisible by three as well as examining the part that remains.
Tasks vertically connected across teacher certification levels – 17
S3MTP • Chapter 1

When investigating questions or areas of mathematics that are
new to them, mathematicians often employ the strategy of considering examples and non-examples before proceeding to more
abstract conjectures.
However, this practice of considering examples and counterexamples before considering abstract ideas, does not come naturally to most students and must be developed purposefully. For
example, students might consider the number 12, known to be
divisible by three. When this number is represented in base five,
its representation is 22five and the sum of the digits is not divisible by three. This is sufficient evidence to guarantee that the
base ten test certainly does not work in all cases in base five.
But this same number, when represented in base seven, becomes
15seven, and the sum of its digits is divisible by three. Whereas
this example is not sufficient to guarantee that the base ten
“divisibility by three” test works in base seven, it does indicate
that there may be some fundamental difference between the
bases and their relationship to three. Such examples also prompt
discussion about what is required to verify that a mathematical
statement is true, as compared to what is sufficient to claim that
it is not true.

Students should be expected to justify their thinking with a valid
argument, but the particular approach and level of formality
expected will vary depending on the students’ mathematical
background and experience.
Building on the base ten expansion, and the observation that
there is “one left over” in each position, may lead students to
conjecture that the base ten test and the expansion approach
also work in bases such as seven. For example, students may
observe that 243ten = 465seven can be expanded in powers of seven
in much the same way as before:
243ten = 465seven
= 4(7)2 + 6(7)1 + 5(1)
= 4(6+1)2 + 6(6+1)1 + 5(1)
Then they can compare this expression to the similar one in
base ten and determine which parts of the expression are—or
are not—always divisible by 3, and why this may lead to an
understanding of the essential underlying ideas necessary for a
justification. This provides a nice opportunity for a discussion of
the Binomial Theorem in a context that is not usually encountered by preservice teachers.
18 – Tasks vertically connected across teacher certification levels
STATE BOARD FOR EDUCATOR
CERTIFICATION MATHEMATICS STANDARD V.
MATHEMATICAL PROCESSES
Grades EC–12 Teacher Skills
The beginning teacher of mathematics
is able to:
5.2s apply principles of inductive
reasoning to make conjectures and
use deductive methods to evaluate the
validity of conjectures;
5.3s use formal and informal reasoning to explore, investigate, and justify
mathematical ideas;
5.6s provide convincing arguments or
proofs for mathematical theorems;
5.9s use physical and numerical models
to represent a given problem or mathematical procedure;
5.15s explore problems using verbal,
graphical, numerical, physical, and
algebraic representations;
5.19s facilitate discourse between
the teacher and students and among
students to explore, build, and refine
mathematical ideas;
5.20s use questioning strategies to identify, support, monitor, and challenge
students’ mathematical thinking;
5.21s translate mathematical statements among developmentally appropriate language, standard English,
mathematical language, and symbolic
mathematics;
5.22s provide students with opportunities to demonstrate their understanding
of mathematics in a variety of ways
using a variety of tools;
5.23s use visual media such as graphs,
tables, diagrams, and animations to
communicate mathematical information;
5.24s use the language of mathematics
as a precise means of expressing mathematical ideas.
S3MTP • Chapter 1
Another approach to this task involves familiar ideas connected to clock arithmetic and remainders—important ideas in
K–12 mathematics. Students who have had more extensive and
formal experience with modular arithmetic may observe that the
bases in which the test works all have a remainder of one when
divided by three or that these bases are congruent to 1 (mod 3).
In that case, they may argue informally that the original whole
number can simply be expanded into powers of 1; in that form,
the “sum of the digits” algorithm appears rather naturally.
Tasks vertically connected across teacher certification levels – 19
S3MTP • Chapter 1
SECTION 1.1.3
GRADES 8–12 TEACHER TASK:
EXPLORING POSITIONAL SYSTEMS
THROUGH DIVISIBILITY RULES
Regardless of the underlying approach used to address the previous question, the justification will probably be informal. The use
of mathematical notation and broader generalization marks one
bridge between the EC–4 and 4–8 levels. Using the patterns and
results of investigations at those levels, students are equipped for
the more formal conjecture and proof expected at the secondary
level.
Students in the elementary and middle grades are
often taught a simple test for determining whether a
whole number is divisible by three:
if the sum of a whole number’s digits (in its base-ten
representation) is divisible by three, then that number
is divisible by three.7
STATE BOARD FOR EDUCATOR CERTIFICATION MATHEMATICS STANDARD I. NUMBER
CONCEPTS
Grades 8–12 Teacher Skills
The beginning teacher of mathematics
is able to:
1.22s apply all skills specified for teachers in grades EC–8, using content and
contexts appropriate for grades 8–12;
1.28s investigate and apply fundamental number theory concepts and
principles (e.g., divisibility, Euclidean
algorithm, congruence classes, modular
arithmetic, the fundamental theorem
of arithmetic) in a variety of situations.
• Develop a conjecture about all number bases for
which a number expressed in that base is divisible
by three if and only if the sum of its digits is divisible by three.
• Prove your conjecture.
Supporting Discussion

It is assumed that students attempting to answer the question
posed will have had mathematical experiences similar to those
described for early childhood–4 and grades 4–8.
When investigating a new concept or area of mathematics, even
sophisticated mathematical thinkers return to the consideration
of specific examples and counterexamples before developing
abstract arguments. The apparent simplicity and elegance of
many textbook proofs often disguises the great effort required
to produce them, a fact well known to mathematicians but only
rarely to their students!
Having explored the “sum of the digits” divisibility test in
multiple bases, and having learned the underlying mathematical
concepts, students are prepared to conjecture that if a base, b,
has remainder 1 when divided by three, then a number expressed in base b is divisible by three if and only if the sum of its
20 – Tasks vertically connected across teacher certification levels
7
The “test” (the sum of digits is divisible by three)
is presented as a sufficient condition for determining
whether a number is divisible by three. Here,
students conjecture and prove that the “test” is both
necessary and sufficient.
S3MTP • Chapter 1
digits is divisible by three. A proof of that conjecture will likely
be a more general and formal justification using approaches similar to those provided earlier.

HISTORICAL NOTE
We are all familiar with a variety of
transactions, such as making purchases on
the Internet or doing online banking, that
require the secure transfer of information.
The use of public key codes to encode and
decode secret messages is a relatively new
technique that uses computer technology
and a theorem about modular arithmetic
that was proven over 350 years ago.
Students preparing to become secondary school mathematics
teachers commonly have much more experience with abstraction
and formal proofs, but rarely have made their own connections
between the content in their college courses and the content they
will be expected to teach or that they remember from their own
high school courses.
As indicated in the State Board for Educator Certification
mathematics standards, beginning teachers of secondary school
mathematics should be able to apply all the skills for teachers in
grades EC–8 using content and contexts appropriate for grades
8–12. It is helpful for prospective secondary teachers to think
carefully about how and where their students have learned the
foundations for the concepts they will teach. Taking a familiar
concept like divisibility and tracing it back, perhaps in courses
such as number theory, modern algebra, or a capstone sequence,
can help them help their students make powerful connections.

It is essential that students develop the habits of mind associated
with a mathematical thinker.
Only in this way will prospective teachers be able to encourage
their own students to develop similar habits—such as extending
observed patterns, asking their own questions, conjecturing and
testing solutions, and validating and generalizing their results.
There are many ways that ideas similar to those used in these
examples can be expanded to give students more experience.
The following discussion describes a collection of investigations
that could follow this first set of tasks.
Tasks vertically connected across teacher certification levels – 21
S3MTP • Chapter 1
SECTION 1.1.4
ANOTHER PATH: EXTENSIONS OF EXPLORING
POSITIONAL SYSTEMS THROUGH DIVISIBILITY
RULES, FOR EC–12
EARLY CHILDHOOD–GRADE 4 ————————
In this series of investigations, the task for prospective teachers
at the EC–4 level would be much the same as the divisibilityby-three test described above. After examining what makes the
test work in base ten, as well as the characteristics of other bases
in which the test appears to work, prospective teachers could be
guided to consider the characteristics of those bases in which the
test does not work. By focusing their attention on the counterexamples, students can discover algorithms that “work” in these
other bases.
GRADES 4–8 ———————————————
Below is a set of questions that guide such discovery
at the 4–8 level.
Grades 4–8 Task Extension
• Find and justify a test for divisibility by three that
applies for a number represented in base nine.
Make a conjecture about other bases for which this
test would apply. Justify your thinking.
• Find and justify a test for divisibility by three that
applies for a number represented in base eight.
Make a conjecture about other bases for which this
test would apply. Justify your thinking.
At the EC–4 level, students were given a familiar algorithm and
asked to explain why it worked. After exploring the questions
given here, using many specific examples in different bases and
the same expansion and decomposition techniques as used in
base ten, students can uncover tests that work for bases eight
and nine.
In the case of base nine, students may expand in powers of nine,
note that because every counting number power of nine is divisible by three, a number is divisible by three when the last digit is
divisible by three.
22 – Tasks vertically connected across teacher certification levels
S3MTP • Chapter 1

The process of actually finding a new algorithm, explaining why it
works, and generalizing to other bases requires a higher level of
mathematical sophistication. This is a valuable and attainable experience, provided students are given enough time and guidance.
Developing a conjecture for base eight will likely require more
time for student exploration. Students may first approach the
question by building tables and looking for patterns, before
returning to the process of writing the numbers using expanded
powers of eight. But if they try the same strategy as that used
with base seven, they will note that eight is not “one more than”
a multiple of three, so the base eight expansion does not result
in the same form as the expansion of powers of ten or seven. In
response to a series of probing questions, students may notice
that it was the “one more than a multiple of three” feature
that made the expansion so friendly, and they may look for an
alternate way of producing these “ones.” Given sufficient time
and guidance, they may eventually arrive at a strategy of writing
the powers of 8 as powers of (9 – 1) and arrive at an “alternating
sum of the digits” algorithm for base eight.

This process of finding and testing mathematical patterns
is valuable for students; they need practice in trying familiar
strategies, observing when they do not work in a context, and
then either trying something new or isolating the features of the
strategy that require modification.
Students at the 4–8 teacher certification level should observe
that it appears that every base “behaves like” base eight, nine,
or ten. At a very informal level, they may observe that the tests
they discover form a “complete” set of tests for divisibility by
three. In fact, when stated more formally, the question “what
is a complete set of tests for divisibility by three” is appropriate
for prospective secondary teachers; thus, it can serve as a bridge
between the two certification levels, with the secondary level
requiring a higher level of experience and degree of formality in
expression.
GRADES 4–8 TO 8–12 ———————————
Grades 4–8 to 8–12 Task Extension
Find a “complete” set of tests for divisibility by three.
That is, describe a set of divisibility-by-three tests
such that, regardless of the base of representation, one
of the tests will apply.
Tasks vertically connected across teacher certification levels – 23
S3MTP • Chapter 1
Students may find ideas of remainders a useful avenue. That is,
they may observe that as in base ten, if a number base has a remainder of one when divided by three, then the “sum of the digits” test works. If the base has a remainder of zero when divided
by three, then the original number is divisible by three if and
only if the last digit is divisible by three. Finally, if the base has
a remainder of two when divided by three, then an “alternating
sum of the digits” test works. Alternatively, any given base is
congruent to –1, 0, or 1 (mod 3), so one of these tests works.
Assuming students at the 8–12 certification level have progressed through all the tasks or investigations described above,
another level up in generalization may be pursued in the following extension.
GRADES 8–12
Grades 8–12 Task Extension
• Suppose that a number, n, is expressed in base
twenty-one. Determine all the numbers d so that
n, expressed in base twenty-one, is divisible by d
if and only if the sum of its digits is divisible by d.
Justify.
• Suppose that a number n is expressed in a base b.
Find a rule to determine all of the numbers d so
that n, expressed in base b, is divisible by d if and
only if the sum of its digits is divisible by d. Justify.
Again, students benefit from being expected to make their own
conjectures based upon a careful examination of many examples.
The key characteristic in making the divisibility-by-three test
work in bases ten and seven was the fact that the bases were
congruent to 1 (mod 3). Thus, a reasonable conjecture would
be that if twenty-one is congruent to 1 (mod d), then the “sum
of the digits” test will apply. This is true since, if twenty-one is
congruent to 1 (mod d), the expansion
�
������� ��������������������������
is congruent to the sum of its digits (mod d), so both are divisible by d or neither is. Thus, we need only determine for which
numbers, d, twenty-one is congruent to 1 (mod d). These numbers are 2, 4, 5, 10, and 20. If twenty-one is not congruent to 1
(mod d), it is not necessarily congruent to the sum of its digits.
Reasoning as above, we see that the “sum of the digits” test will
work if and only if b is congruent to 1 (mod d). That is, d must
be a factor of b – 1.
24 – Tasks vertically connected across teacher certification levels
S3MTP • Chapter 1
SECTION 1.2
PATTERNS, ALGEBRA, AND ANALYSIS:
EXPLORING INFINITE PROCESSES
SBEC MATHEMATICS STANDARD II.
PATTERNS AND ALGEBRA
STATE BOARD FOR EDUCATOR CERTIFICATION
MATHEMATICS STANDARD II. PATTERNS AND ALGEBRA:
The mathematics teacher understands and uses patterns,
relations, functions, algebraic reasoning, analysis, and
technology appropriate to teach the statewide curriculum
(Texas Essential Knowledge and Skills [TEKS]) in order to
prepare students to use mathematics.
Teacher Knowledge
The beginning teacher of mathematics
knows and understands:
2.1k how to use algebraic concepts
and reasoning to investigate patterns,
make generalizations, formulate mathematical models, make predictions, and
validate results;
SBEC MATHEMATICS STANDARD III. GEOMETRY AND
MEASUREMENT: The mathematics teacher understands and
uses geometry, spatial reasoning, measurement concepts
and principles, and technology appropriate to teach the
statewide curriculum (Texas Essential Knowledge and Skills
[TEKS]) in order to prepare students to use mathematics.
2.2k how to use properties, graphs, and
applications of relations and functions
to analyze, model, and solve problems;
2.3k the concept of and relationships
among variables, expressions, equations, inequalities, and systems in order
to analyze, model, and solve problems;
2.4k the connections among geometric,
graphic, numeric, and symbolic representations of functions and relations;
2.5k that patterns are sometimes
misleading;
2.6k that in many situations, a pattern
is only a trend and is accompanied by
random variation from the trend; and
2.7k how patterns, relations, functions,
algebraic reasoning, and analysis are
developmental and connected across
grade levels.

Prospective teachers should have opportunities to explore infinite
processes in several contexts.
The concept of limit is one of the more subtle and powerful
ideas introduced in the history of mathematics. Developing
early childhood–12 students’ understanding of limits provides a
foundation for their obtaining a more complete comprehension
of the concept as they pursue more advanced mathematics. To
do this, all prospective teachers need experiences investigating
the concepts of calculus at various levels of formality.
The following series of problems involves the fundamental idea
of limit and makes connections between geometry, measurement, numeration, and algebra.
Tasks vertically connected across teacher certification levels – 25
S3MTP • Chapter 1
SECTION 1.2.1
EC–GRADE 4 TEACHER TASK:
EXPLORING INFINITE PROCESSES
Shaded Squares: Assume that each shaded square represents of the area of the larger square bordering two
of its adjacent sides and that the shading continues
indefinitely in the indicated manner.
• How much of the total area of the square is shaded?
Thoroughly justify your thinking.
SBEC MATHEMATICS STANDARD III.
GEOMETRY AND MEASUREMENT
Teacher Knowledge
The beginning teacher of mathematics
knows and understands:
3.3k connections among geometric ideas
and number concepts, measurement, probability and statistics, algebra, and analysis.
SBEC MATHEMATICS STANDARD II.
PATTERNS AND ALGEBRA
Grades EC–4 Teacher Skills
The beginning teacher of mathematics is
able to:
2.1s use inductive reasoning to identify,
extend, and create patterns using concrete
models, figures, numbers, and algebraic
expressions;
2.2s formulate implicit and explicit rules to
describe and construct sequences verbally,
numerically, graphically, and symbolically;
2.6s model and solve problems, including proportion problems, using concrete,
numeric, tabular, graphic, and algebraic
methods.
Supporting Discussion
Many students hesitate to engage in mathematics for mathematics’ sake. Attempts to put the mathematics into a context that
builds on their experiences can make a difference in their comfort level and their ability to analyze it. For example, the Shaded
Squares problem could be posed in the context of cutting a cake:
Each time Billy’s mother slices his birthday cake, she slices it into
fourths. She always keeps one slice for further cutting, sets aside
a slice for Billy, and gives away the other slices to guests. With
the remaining slice of cake, she divides it into fourths. Again, she
keeps one for further cutting, sets aside a slice for Billy, and gives
away the other slices to guests. She continues to do this until it is
not possible to subdivide it further. Approximately how much cake
does she set aside for Billy?
26 – Tasks vertically connected across teacher certification levels
S3MTP • Chapter 1
EXTENSION IDEA
Shaded Rectangles: Consider the sequence of
repeating decimals 0.9, 0.99, 0.999, 0.9999,
0.99999, … . Use graph paper to outline
a large square whose total area is 1 square
unit.
1) Subdivide each side into 10 sections of
equal length.
2) With a vertical line, separate the square
into two pieces, one of area 0.9 square
units, and shade the larger region.
This context provides a simple example of how to put an important but abstract idea into a real-world context. It is a fun
context that could be modeled in class with a real cake. It may
also help students think about a non-symbolic solution to the
problem. For example, when doing this exercise, some students
have “seen” the solution in an a-ha moment. They note that
at each stage of shading, one out of three congruent squares is
shaded. Thus they conclude that if the process could be continued indefinitely, the area of the shaded portion would be �� .
3) With a horizontal line, separate the
unshaded rectangle into two pieces, one
of area 0.09 square units, and shade the
larger region.
4) Continue subdividing the unshaded
rectangles in this manner, alternating
between vertical and horizontal lines,
and always shading the region with 90%
of the area, until further subdivisions are
too fine to do with a pencil.
a) Explain the relationship between the
total area shaded and the numbers in
the sequence above.
b) If you were not restricted by the thick
ness of a pencil point and could keep
subdividing forever, what would the
total shaded area eventually be? Justify
your thinking.
The question posed in Shaded Squares asks students to investigate conceptual connections between geometric representations
of fractional parts and algebraic ideas involved in evaluating
infinite series.

Students need opportunities to construct and extend patterns
in sequences and to make connections between geometric and
algebraic representations of these sequences.
As students approach this problem numerically, they typically
begin by using tables to record the cumulative fractional parts,
convert them to decimal form, and calculate the sum. They are
often troubled by the fact that they are unsure of “when to stop.”
Some recognize the sequence as geometric and try to recall a formula for the sum. Others will model this by cutting and pasting
pieces of paper and rearranging them in various ways as in the
example above.
A related set of questions and a possible extension to the Shaded
Squares task asks students to begin with the decimal representation for the infinite sequence
0.9, 0.99, 0.999, ...
and employ geometric ideas to motivate its convergence to 1.
Tasks vertically connected across teacher certification levels – 27
S3MTP • Chapter 1
SECTION 1.2.2
GRADES 4–8 TEACHER TASK:
EXPLORING INFINITE PROCESSES

Prospective middle level teachers will be expected to relate
middle school mathematics to the concept of limit as a conceptual
foundation of calculus. Hence, it is essential that they develop
their own conceptual understanding of limits.
Students preparing to teach at the middle school level should
be expected to justify their mathematical thinking in somewhat
more sophisticated ways. The questions posed below build on
the Shaded Rectangles extension in EC–4, above, but lead students to organize their thinking in a different way.
Shaded Rectangles Revisited: Suppose that n is a
positive number less than 1.
• Argue that 0.999999… must be greater than n.
STATE BOARD FOR EDUCATOR
CERTIFICATION MATHEMATICS
STANDARD II: PATTERNS AND ALGEBRA
Grades 4–8 Teacher Skills
The beginning teacher of mathematics
is able to:
2.8s apply all skills specified for teachers in grades EC–4, using content and
contexts appropriate for grades 4–8;
2.9s make, test, validate, and use
conjectures about patterns and relationships in data presented in tables,
sequences, or graphs;
2.14s relate the concept of limit as a
conceptual foundation of calculus to
middle school mathematics.
• What rational number is equal to the repeating
decimal 0.999999…? Justify your thinking.
Supporting Discussion

Middle level students are often taught an algorithm for converting
numbers from decimal to fraction form, but they rarely develop a
rich conceptual understanding of the underlying ideas.
This problem encourages students to further develop their
understanding of the concept of limit by making connections
between algebraic and geometric ideas. Posed this way, the question focuses on different representations for rational numbers
using the concept of limit. As students justify their thinking,
they should be encouraged to argue conceptually.

Most students are exposed to geometric series at some point in
their undergraduate education, but only a small number have the
opportunity to investigate and explain the derivation of geometric
series.
The following sequence of questions, adapted from the Dana
Center’s TEXTEAMS In-depth secondary mathematics institute,8
allows students to discover the formula for the sum of a geometric series, using a geometric construction that is accessible to a
student in college algebra.
28 – Tasks vertically connected across teacher certification levels
Assessment
The goal of these questions is to have students do more than algorithmic manipulation. As students argue, they are building a
type of informal proof or justification. For
instance, they may observe that 0.999… is
certainly not bigger than 1, but it is larger
than any other number smaller than 1.
Therefore, they conclude that 0.999…
must, in fact, be equal to 1.
8
Rethinking secondary mathematics: In-depth
secondary mathematics is one of dozens of institutes
provided by TEXTEAMS (Texas Teachers
Empowered for Achievement in Mathematics
and Science, www.utdanacenter.org/texteams),
a Dana Center–managed statewide teacher
professional development program. TEXTEAMS
is a comprehensive system of professional
development for K–12 mathematics and science
teachers, delivered through a statewide network of
trainers. The program’s institutes provide a core set
of professional development materials and skills and
help teachers develop a common understanding of
important mathematics and science content and
the state’s curriculum standards (the Texas Essential
Knowledge and Skills).
S3MTP • Chapter 1
Note: These questions are designed so that the students
can generate their own unique approaches. With
this in mind, the instructor should limit the
amount of guidance offered to the student.
A Geometric Look at the Geometric Series: The figure
below shows a sequence of three squares. Each square
has a side length that is 0.7 times the length of the
one to the left. Continue constructing squares to the
right in this same pattern; then justify your responses
to the following.
1) Show that an infinite number of squares takes up
only a finite amount of space.
2) Find the total length of the infinite sequence of
squares.
3) Discuss the relationship between the scale factor
relating adjacent squares and the slopes of the line
segments connecting the upper right corners of the
squares.
4) If the bases of the squares lie on the x-axis beginning at the origin, what is the relationship between
the x-intercept of the line that passes through the
upper right corners of the squares and the x-intercept of the line that passes through the upper left
corners of the squares?
5) Using the geometry of this diagram, derive a formula for the sum of an infinite geometric series.
Tasks vertically connected across teacher certification levels – 29
S3MTP • Chapter 1
Supporting Discussion

This series of questions leads students to link geometric and
algebraic ideas.
Exploring these questions will help students develop a deeper
understanding of some of the core content ideas of grades 4–8
mathematics. These include ideas of infinity, proportional thinking, geometry and measurement, triangle similarities, slope,
derivations of formulas, and assigning meaning to algebraic expressions and equations. In the context of this task, some related
questions include:
• How do you know that “connecting the upper right
corners” or “connecting the upper left corners” will
generate lines?
• How can an “infinite amount” of anything take up a
finite amount of space?
• Can you explain and demonstrate the impact a scale
factor will have on area and volume?
• What do equations of lines have to do with areas of
squares?
Generalizing results of this investigation provides a transition to
the 8–12 teacher-preparation level.
To show that the infinite number of squares takes up only a
finite amount of space and to find the total length of the infinite
sequence of squares, a student may use the sequence of squares
to construct a triangle such as the one shown with angle measure. Using similar triangles and letting b represent the length of
the base of the largest triangle, the student will find that
.
Thus � � �� . Since the total length of the infinite sequence of
squares is the sum of the length of the first square and length of
the base of the large triangle, the students see that
2
3
1 + 0.7 + (0.7) + (0.7) may be calculated by adding 1 to 73 to
get
.
The scale factor for this sequence of squares is 0.7. The students
30 – Tasks vertically connected across teacher certification levels
S3MTP • Chapter 1
should note that the slope of the line segments that connect the
� ������� � ����α� .
upper right corners of the squares is ���
���
���
�
Thus ��� � ��������α� .
From the discussion of the total length of the sequence of
squares, many students will see that if the left bottom corner of
the first square was placed at the origin on the x-axis, the line
that connects the upper right corners of the squares would have
x-intercept ��� . By calculating the equation of the line that
passes through the upper left corners of the squares, the students
should find that the two lines share the same x-intercept.
Deriving a formula for the sum of an infinite geometric series
requires the students to generalize the information that they
gained in the preceding exercises. Some may need to construct
the sequence of squares using p (where 0 < p < 1) as the scale
factor so that they can more easily visualize the situation. By
using the same strategies as before, the students should see that
2
3
the sum of any infinite series of the form 1 + p + p + p + . . . is
�
.
�����
Tasks vertically connected across teacher certification levels – 31
S3MTP • Chapter 1
SECTION 1.2.3
GRADES 8–12 TEACHER TASK:
EXPLORING INFINITE PROCESSES

Prospective secondary teachers need in-depth problem-solving
experiences that incorporate mathematical modeling of infinite
processes.
The following task builds on the understanding of infinite processes developed in the preceding EC–4 and 4–8 tasks. In this
secondary-teacher-level task, students engage in determining a
bisection process to approximate �� .
STATE BOARD FOR EDUCATOR
CERTIFICATION MATHEMATICS
STANDARD II: PATTERNS AND ALGEBRA
Grades 8–12 Teacher Skills
The beginning teacher of mathematics
is able to:
2.18s apply all skills specified for teachers in grades EC–8, using content and
contexts appropriate for grades 8–12;
2.19s use methods of recursion and
iteration to model and solve problems;
Bunny Pedigree: Your friend claims that her new pet
rabbit, Tercio, is �� Black Himalayan.
• Is this possible? Explain.
• How many generations of Tercio’s pedigree must be
known for your friend’s claim to be accurate to three
decimal places? Completely justify your answer.
2.20s analyze the properties of sequences and series and use them to solve
problems involving finite and infinite
processes; including problems related
to simple, compound, and continuous
interest rates, as well as annuities.
Supporting Discussion

Students often have trouble with the notion of limit in calculus,
especially the formal definition of a limit, because they have little
experience working with models upon which to build a conceptual understanding.
However, it is possible to develop a conceptual understanding of
limits and to compute some limits without the formal definition.
In fact, mathematicians talked about limits for centuries before
they were able to define the concept clearly.
This task aims to illuminate this as well as to provide other contexts where bisection methods may be used in approximations.
Whereas one cannot find a rabbit that is exactly �� of a recognized breed (such as the Black Himalayan), one could always
find a rabbit that was as close to �� Black Himalayan as desired.

A historical context for this task would be to consider trisecting
an angle with compass and straight edge with successive
bisections.
32 – Tasks vertically connected across teacher certification levels
HISTORICAL NOTE
Three special problems of antiquity are
the quadrature of the circle, the duplication of the cube, and the trisection
of a general angle. When Hippocrates
arrived in Athens (around 450 B.C.E.)
these problems were already engaging
the attention of mathematicians. These
problems remain landmarks in the
history of mathematics. They provide
a source of stimulation and fascination
for students, teachers, and scholars.
S3MTP • Chapter 1
Students are often intrigued by the fact that using college-level
abstract algebra, it can be shown to be impossible to trisect an
angle with compass and straight edge. However, making the
connection between the Bunny Pedigree task and approximating the trisection of an angle, using a compass and straight edge
provides groundwork for their developing a bisection process
for determining the number of iterations necessary for a desired
“closeness” to �� .
Taking successive bisections of an angle is a limiting process
with the trisection of the angle as a “limit.” In the pictorial mod�
el shown below, students see that each offspring inherits � of its
ancestors’ genetics. In addition, the choice of a circular sector in
the pictorial model allows a representation of the trisection of
the corresponding angle of the sector.
Tasks vertically connected across teacher certification levels – 33
S3MTP • Chapter 1
Note that by examining the arc length of the circular sectors in
this offspring model, students can see that the process of bisecting to get closer and closer to the trisection is, arithmetically,
equivalent to the process of getting �� in terms of successive �� s
�
(i.e. summing terms � for many n). Thus, the limiting process
can be reduced to using bisections for estimating �� of a unit
length as in the following task.
n
Bisecting to get : Start with an interval of unit length
and bisect it. Choose the subinterval in which lies
(in this case it is clearly [0, ]). Now, bisect that
interval and choose the sub-interval [0, ] or [ , ]
in which lies. Continue this process for three more
iterations. Record your data in a table as follows.
Interval
[0,1]
Interval length
1
(1)
Midpoint
0+
0+
[0, ]
[
,
]
[
,
]
[
,
]
[
,
]
+
(1)
Midpoint expressed as left endpoint of interval plus the interval length.
1) From your table, give a nondecreasing sequence approaching . List this sequence. How do you know
that this sequence approaches and not some
other number?
2) Examine the column labeled Midpoint and plot
these points. Find and describe the pattern. Now,
can you find a decreasing sequence which approaches ?
3) We could approach this question via base two
decimals. So, writing as a decimal in base 2, we
get “0.010101...” the infinitely repeating decimal.
Notice that “.01” in base two is , “.0001” in base
2 is , “.000001” in base 2 is , and so on.
34 – Tasks vertically connected across teacher certification levels
S3MTP • Chapter 1
Write the nondecreasing sequence you found in
step 1) in base two. Describe any patterns you notice. Each term in the sequence is a partial sum of a
geometric series. What is the sum of this series?
TECHNOLOGY NOTE
4) Which column(s) in your table give(s) information about the accuracy of your estimate? After 20
bisections, how close is your estimate, expressed in
the form �� , to �� ? How about after n steps or bisections?
Students can use dynamic geometry software to perform the compass and straight
edge constructions. The circular sector
model shown above was created using The
Geometer’s Sketchpad.9
n
5) Explain in detail how we could use a compass and
straight edge to get within 1/1000 of the trisection.
Hint: Begin by drawing an arbitrary angle, making
it larger than 90 degrees to give you enough room to
bisect successively. After four bisections, how close are
you to the trisection?
9
The Geometer’s Sketchpad is a trademark of Key
Curriculum Press; for more information on the
Sketchpad, see www.keypress.com/sketchpad.
Note:
This task assumes that students have been exposed to
sequences and some geometry (enough to know how
to bisect an angle with a straight edge and compass).
The task is suitable for students in capstone courses
for secondary teachers, a lab setting in calculus, or a
lab setting for college algebra or precalculus. Because students find developing and identifying the
sequences difficult, having them work in groups and
asking them probing questions makes best use of this
task.
Key to this problem is the use of successive approximation to get
better and better estimates for a quantity. Notice that ideally,
we want a scheme for creating successive approximations, that
better approximate the desired quantity at each step or stage. At
each stage, the interval which contains is chosen for bisection and the endpoints of the interval provide upper and lower
estimates of . Because all the intervals chosen for bisection
contain , it is reasonable to think that the successive approximations actually approach .

The level of detail in the Bisecting to get task helps students
see how the original task can be reduced to simpler tasks. This
device is often used in doing mathematics.
The idea of “closeness” surfaces in determining the accuracy of
our estimates or how “far off” we are from the desired quantity.
The successive approximations of “jump around” . At odd
steps, the midpoint is an overestimate and at even steps, it is an
underestimate; however, the maximum possible error decreases
with each successive step.
Tasks vertically connected across teacher certification levels – 35
S3MTP • Chapter 1
The first part of the problem involves creating a sequence that
has a numerical limit of . At the end of the problem (from a
different perspective), in trying to find the trisection of an arbitrary angle, the compass and straight edge help illustrate that the
sequence of trisections produces lines that approach the trisection of the angle. Students should be asked to think geometrically about how they would determine whether the bisection
line gives an over- or underestimate for the trisection.
Many applications employ bisection for determining solutions.
This task offers preservice teachers an opportunity to connect
limit processes, geometry, and a historical mathematical problem. The task also illustrates how conceptual development of
limits can be introduced at various levels (geometry, algebra)
before calculus and exposes many student misconceptions about
sequences, limits, and geometrical constructions.
36 – Tasks vertically connected across teacher certification levels
Chapter 2
EARLY CHILDHOOD–GRADE 4
TEACHER TASKS ———————————
Teachers of grades EC–4 need an understanding of fundamental
mathematics that includes—but goes beyond—computational
and algorithmic proficiency. Learning experiences that develop
the habits of mind associated with mathematical thinking equip
prospective EC–4 teachers with tools to model and support
mathematical thinking for their students. Thus, their preparation should include opportunities to explore the conceptual
underpinnings of mathematics that is central to the early grades,
with an emphasis on developing their own critical reasoning
skills.
The sample mathematical tasks in this chapter span the State
Board for Educator Certification content strands (mathematics
standards) of number concepts, patterns and algebra, geometry
and measurement, and probability and statistics. Each task
focuses on an idea central to elementary school mathematics;
this chapter outlines the types of mathematical experiences
that should be included in the preparation of elementary school
teachers. The targeted State Board for Educator Certification
knowledge and skills objectives say that teachers of grades EC–4
should be able to:
• Apply place value and other number properties to solve
problems involving mental mathematics and estimation;
• Identify and extend patterns, and, using inductive
reasoning, express these patterns in a variety of ways,
including symbolic language;
• Use concrete, numeric, tabular, graphic, and algebraic
methods to solve problems and communicate mathematical ideas;
S3MTP • Chapter 2
• Demonstrate an understanding of the concepts of
length, perimeter, area, and volume;
• Explore concepts of probability through data collection,
experiments, and simulations;
• Evaluate the reasonableness of a solution to a given
problem, and be able to recognize fallacious reasoning;
and
• Use formal and informal reasoning to investigate
mathematical ideas, providing convincing arguments to
support conclusions.
Chapter 2 includes six tasks:
• Numeration Systems: An Even/Odd Algorithm in Base
Five;
• Patterns, Geometry, and Algebra: Painting the Cube;
• Rational Numbers, Area Models, and Fallacious Reasoning: Geoboard Eighths;
• Probability: Assessing the Fairness of Games;
• Number Theory: The Stamps Problem; and
• Geometry and Measurement: Tiling a Round Patio.
Although each task has a particular content focus, we intend
that prospective teachers make important connections between
these content strands. The following chart (also included in the
preface to this book) suggests some possible courses for which
these tasks might be appropriate.
38 – Early childhood–grade 4 teacher tasks
S3MTP • Chapter 2
Task Correlation Guide 2:
Tasks for early childhood through grade 4 teacher certification level
S3MTP Tasks for EC–Grade 4 Teacher
Certification Level
Courses where task may be most appropriate
Numeration Systems: An Even/Odd Algorithm in
Base Five
Course in Foundations of Arithmetic for preservice elementary
teachers
Patterns, Geometry, and Algebra: Painting the
Cube
Course in Number and Operation that extends the ideas from
an EC–4 Foundations of Arithmetic course
Rational Numbers, Area Models, and Fallacious
Reasoning: Geoboard Eighths
Course in number theory or a capstone course for preservice
8–12 teachers
Probability: Assessing the Fairness of Games
Course in Foundations of Arithmetic for preservice elementary
teachers
Number Theory: The Stamps Problem
Course in Concepts of Calculus for preservice 4–8 teachers or
number theory course for preservice 4–8 teachers
Geometry and Measurement: Tiling a Round Patio
Calculus course or capstone course for preservice 8–12 teachers
Early childhood–grade 4 teacher tasks – 39
S3MTP • Chapter 2
SECTION 2.1
EC–GRADE 4 TEACHER TASK:
NUMERATION SYSTEMS: AN EVEN/
ODD ALGORITHM IN BASE FIVE
STATE BOARD FOR EDUCATOR
CERTIFICATION MATHEMATICS STANDARD I.
NUMBER CONCEPTS1
Grades EC–4 Teacher Skills
The beginning teacher of mathematics is able
to:
1.1s compare and contrast numeration systems;
Students in the elementary grades learn a simple rule
for determining whether a number written in base ten
is even or odd.
1) Develop an algorithm that will allow someone to
look at a number written as a base five numeral and
know whether the number is even or odd without
having to perform a division calculation and without converting the quantity to base ten and using
the base ten algorithm.
2) Describe the base five algorithm and justify why
it works.
Supporting Discussion
Before beginning this task, it is important that students take
time to think about “evenness” and “oddness.” Some leading
questions are:
• Can you represent an odd or even number using Cuisenaire Rods?2
• What do you notice about adding two odd numbers?
Adding two even numbers?
• Can you explain the pattern you see in the sum of two
odd numbers? Two even numbers?
This investigation involves exploring fundamental notions of
place value and numeration, while developing mental computation techniques, reasoning and proof skills, and communication
ability.

Students often need help structuring the problem-solving process
via questions that break the problem into smaller parts.
As posed, this task may be overwhelming for prospective EC–4
teachers. The following leading questions or hints can help
them structure the problem into smaller parts.
1.2s analyze, explain and model the structure of
numeration systems and, in particular, the role
of place value and zero in the base ten system;
1.6s analyze and describe relationships among
number properties, operations, and algorithms
involving the four basic operations with whole
and rational numbers;
1.10s describe ideas from number theory (e.g.,
prime numbers, composite numbers, greatest
common factors) as they apply to whole numbers, integers, and rational numbers and use
these ideas in problem situations;
1.12s apply place value and other number
properties to develop techniques of mental
mathematics and computational estimation.
SBEC MATHEMATICS STANDARD V.
MATHEMATICAL PROCESSES
Grades EC–4 Teacher Skills
The beginning teacher of mathematics is able
to:
5.3s use formal and informal reasoning to
explore, investigate, and justify mathematical
ideas;
5.6s provide convincing arguments or proofs
for mathematical theorems.
5.21s translate mathematical statements
among developmentally appropriate language,
standard English, mathematical language, and
symbolic mathematics;
5.24s use the language of mathematics as a precise means of expressing mathematical ideas.
The State Board for Educator Certification
mathematics standards for early childhood–4, 4–8,
and 8–12, with associated knowledge and skills
statements, may be referenced on the web at
1
www.sbec.state.tx.us/SBECOnline/standtest/
standards/ec4math.pdf;
www.sbec.state.tx.us/SBECOnline/standtest/
standards/4-8math.pdf; and
www.sbec.state.tx.us/SBECOnline/standtest/
standards/8-12math.pdf.
Cuisenaire Rods are manipulatives, originally
invented by Georges Cuisenaire, that can be used to
help K–6 students learn mathematics.
2
40 – Early childhood–grade 4 teacher tasks
S3MTP • Chapter 2
1) In thinking about “evenness” and “oddness,” what method do we typically use to determine whether a quantity
written in our base ten system is even or odd without
having to divide by two and check the remainder?
EXTENSION IDEA
To which other bases can the even/odd
algorithms for base ten and base five be
generalized? Describe how.
Note: This task probes for student understanding of the reasons why each of the two
algorithms works in its given base—two
being a factor of the base in the first case,
and a factor of one less than the base in the
second.
This question places the problem in familiar territory.
2) Does this base ten procedure work in base five? Why or
why not?
Students begin to try examples here, a helpful approach when
beginning to develop a new method.
3) Make a list of even numbers and a list of odd numbers
written in base five.
This prompting helps students with their initial exploration.
4) Try to find a new algorithm for determining whether
a quantity expressed in base five is even or odd which
does not involve performing a division calculation or
converting the quantity to base ten and using our base
ten algorithm. Look for a pattern in your list of base five
numbers. Experiment with different strategies. Once you
think you have found an algorithm that works, test the
algorithm a reasonable number of times.
ASSESSMENT
Grading student performance on a problem
such as the one described above can be a
daunting task; creating a rubric to use in
grading can make the task more manageable.
The following rubric has been used in grading a similar assignment. This rubric is organized around a 4-point grading scheme, with
each element carrying a 0.5 point value.
5) Clearly describe and demonstrate this algorithm.
• Demonstrates an understanding
of the task
6) Write a justification for your algorithm (i.e., prove that
it will work for all numbers expressed in base five). This
involves more than demonstrating ten, twenty, or even
100 cases where it works.
• List—correct through 3 digits
• Algorithm—correct
• Algorithm—clear
• Algorithm—demonstrated
• Justification—reasonable attempt
• Justification—complete
• Other—at discretion of the grader
Steps 5) and 6) separate the proof from the description of the algorithm. Both proof and description are difficult and complicated
for students, so it may help to distinguish them explicitly. The
issue surfaces here that examples do not constitute a proof; this
is often difficult for students to grasp.

Development of the algorithm typically follows the common
problem-solving process of exploring via examples, identifying
patterns, and justifying the conjecture.
Early childhood–grade 4 teacher tasks – 41
S3MTP • Chapter 2
Part of the goal of mathematics courses for prospective teachers
is to familiarize them with the “anatomy” of a problem. Multiple
experiences with problem solutions that follow the above process (a gradual tightening of focus) provide them with a robust
model from which to approach future problems.

Understanding that there are multiple correct algorithms for
certain problems provides an opportunity to consider whether
different approaches are equivalent.
There are several ways that students might formulate an algorithm for determining “evenness” and “oddness” in base five. Regardless of the nature of their algorithm, it is somewhat surprising to most students that to make a correct determination, they
must consider digits other than the units digit. Some will base
their determination on whether the sum of the digits is even or
odd. Others recognize that a mere consideration of the number
of odd digits is sufficient.

Justifications can take many forms, even reaching back to students’ first concrete experiences with representing numbers in
bases other than ten.
Given enough time, most students will discover an algorithm
that works in general; a justification for why their algorithm
works, however, may be much more elusive. An exemplary justification, given by a student named Monica, is grounded in the
use of base five manipulatives (unit block, long, flat, and large
cube) to represent the first four place values in base five.
Monica’s Proof—A “Sum of the Digits” Algorithm
Preliminary exploration—Consider the following base five representations:
continued on next page
42 – Early childhood–grade 4 teacher tasks
S3MTP • Chapter 2
Observe that the sum of the digits of a base five number simply
represents the number of manipulatives needed to represent the
given base five quantity.
Since each block (i.e., each place value) represents an odd quantity, summing the digits is synonymous with counting a group of
odd numbers whose sum is the original base five quantity.
It follows that an odd number of manipulatives represents an odd
quantity (i.e., an odd number of odd addends yields an odd sum)
and an even number of manipulatives represents an even quantity
(i.e., an even number of odd addends yields an even sum).
Of course, Monica’s proof is only one of many ways that
students could achieve an adequate, even exemplary, justification.
Early childhood–grade 4 teacher tasks – 43
S3MTP • Chapter 2
SECTION 2.2
EC–GRADE 4 TEACHER TASK:
PATTERNS, GEOMETRY AND ALGEBRA:
PAINTING THE CUBE
A large, cubical space station is to be constructed using smaller cubical modules. Every module wall that
becomes an exterior wall requires special shielding.
STATE BOARD FOR EDUCATOR
CERTIFICATION MATHEMATICS
STANDARD II. PATTERNS AND ALGEBRA
Grades EC–4 Teacher Skills
• If the space station is built using 3 modules in
each dimension, how many of the 27 modules will
require no shielding at all? How many will require
shielding on one side? Two sides? Three sides?
More?
The beginning teacher of mathematics is
able to:
• Using n modules on a side, how many modules
will require no shielding? How many will require
shielding on one side? Two sides? Three sides?
More?
2.2s formulate implicit and explicit rules to
describe and construct sequences verbally,
numerically, graphically, and symbolically;
Supporting Discussion
The above space station scenario grounds the following (somewhat more abstract) mathematical task in a possible real-world
application. Below, the large cube corresponds to the space
station, the smaller unit cubes correspond to the station’s cubical
modules, and the paint corresponds to the shielding required on
the space station’s exterior walls.
Take an n by n by n cube made up of smaller unit cubes,
and paint all the exposed faces.
When you decompose the large cube into the component
cubes, how many of the smaller cubes will have paint on 0
faces? 1 face? 2 faces? 3 faces? More?
Mathematically, these tasks address algebraic patterns and functions, the relation between algebra and geometry/measurement,
rates of change, and spatial and inductive
reasoning.
Students may need some structuring questions to facilitate their
investigation. For example, they may consider a 2 x 2 x 2 module space station, then a 3 x 3 x 3 module space station. As they
answer the associated questions, they can observe patterns by
gathering their data in tables.
44 – Early childhood–grade 4 teacher tasks
2.1s use inductive reasoning to identify,
extend, and create patterns using concrete
models, figures, numbers, and algebraic
expressions;
2.3s illustrate concepts of relations and
functions using concrete models, tables,
graphs, and symbolic expressions;
2.4s apply relations and functions to represent mathematical and real-world situations;
2.5s translate problem-solving situations
into expressions and equations involving
variables and unknowns;
2.6s model and solve problems, including proportion problems, using concrete,
numeric, tabular, graphic, and algebraic
methods.
SBEC STANDARD V.
MATHEMATICAL PROCESSES
Grades EC–4 Teacher Skills
The beginning teacher of mathematics is
able to:
5.3s use formal and informal reasoning to
explore, investigate, and justify mathematical ideas;
5.9s use physical and numerical models to
represent a given problem or mathematical
procedure;
5.15s explore problems using verbal,
graphical, numerical, physical, and algebraic
representations;
5.21s translate mathematical statements
among developmentally appropriate
language, standard English, mathematical
language, and symbolic mathematics.
S3MTP • Chapter 2
Students tend to approach this task by building concrete models
using manipulatives. As students consider larger space stations,
the increasing tedium of constructing larger models by hand
encourages them to move to a more abstract view of the
n x n x n module space station.

Solutions to this problem commonly follow:
• the learning trajectory from concrete to semiconcrete to
abstract through which all learners pass;
• the problem-solving process, familiar to mathematicians,
of trying examples, observing patterns and rules, and
developing justifications for them.
ASSESSMENT
A full solution should include:
• the data and formulas
• explanations of where each type of
module appears in the space station
• each component of the formulas
• a discussion of why no modules will be
shielded on more than three faces.
After building concrete models, gathering tables of data, and
identifying patterns, students move on to the development and
justification of formulas, and finally to making connections to
geometric ideas corresponding to each formula. It is important
to allow students sufficient time in each phase (especially the
first few) to develop and maintain connections among the
representations used. When required to justify their steps in the
process, students often develop detailed diagrams identifying
“types” of modules (cubes) by color, shading or separation.
EXTENSION IDEAS
• Consider “cubes” of dimension other
than three: first, consider an n x n square
made of unit squares. How many unit
squares have 0, 1, 2 or more edges
“showing” on the exterior of the large
square? Then consider a four-dimensional
hypercube made of unit hypercubes.
• Again consider an n x n x n cube made
of unit cubes. Find a way to paint the
unit cubes using n distinct colors, so that
for each color there is a way to assemble
the unit cubes into a large cube with only
that color showing.
These extensions generalize the problem
further; parts of them are considerably more
difficult than the original problem.

Solutions to this task bring up connections to the geometry of
polyhedra and traditional rectangular measurement formulas.
Three of the formulas developed in the typical student table,
below, correspond to the number of vertices (V), edges (E), and
faces (F) of a cube. Three of the formulas also correspond to the
standard formulas for length, area, and volume. Justifying the
formulas therefore tends to raise these issues as well:
Early childhood–grade 4 teacher tasks – 45
S3MTP • Chapter 2
• how to count V, E, and F, and
• why we square side length to get area, and cube side
length to get volume.
As seen in the student table below, the formulas also involve the
quantity n – 2. To explain the meaning of this expression in the
context of the given situation requires a close mapping between
algebraic terms and physical objects.
Modules
in each
dimension
of space
station
0 sides
1 side
2 sides
3
sides
Total
number
of
modules
2
0
0
0
8
8
3
1
6
12
8
27
4
8
24
24
8
64
5
27
54
36
8
125
n
�n������
��n������
���n�����
8
n3
Modules with exterior shielding on:
Here is an excellent opportunity to talk about conservation of
both volume and surface area.
• Volume—showing that the four partial formulas add to
n3
• Surface area—adding up the number of shielded (exposed) module faces accounted for to obtain �n� , the
total surface area for a space station of side length n.

The problem provides a context for a situated comparison of
polynomial models of different orders (including cubic).
The formulas representing vertices, edges, face interiors, and
the space station’s interior are, respectively, constant, linear,
quadratic, and cubic, giving students an opportunity to observe
how each type of function grows, both absolutely and relative to
the other functions. All four of these formulas require significant
discussion to ensure student comprehension.
46 – Early childhood–grade 4 teacher tasks
S3MTP • Chapter 2
SECTION 2.3
STATE BOARD FOR EDUCATOR
CERTIFICATION MATHEMATICS STANDARD I.
NUMBER CONCEPTS
Grades EC–4 Teacher Skills
The beginning teacher of mathematics is
able to:
1.3s demonstrate a sense of quantity and
number for whole numbers, integers, rational numbers, and real numbers.
SBEC Mathematics Standard III.
Geometry and Measurement
Grades EC–4 Teacher Skills
EC–GRADE 4 TEACHER TASK:
RATIONAL NUMBERS, AREA MODELS, AND
FALLACIOUS REASONING: GEOBOARD EIGHTHS
In one of her instructional videos, mathematics
educator Marilyn Burns3 asked her students to use
geoboards(each with a 5x5 array of pegs) and rubber
bands to come up with as many ways as they could
to demonstrate the fraction by partitioning the
geoboard into 8 regions of equal area. Some of the
students accomplished this task in predictable ways
(as shown in example 1 below), while other students
were more creative (see example 2). One pair of students came up with the third example (example 3).
The beginning teacher of mathematics is
able to:
3.2s develop, explain, and use formulas to
find length, perimeter, area, and volume of
basic geometrical figures.
SBEC Mathematics Standard V.
Mathematical Processes
Grades EC–4 Teacher Skills
The beginning teacher of mathematics is
able to:
5.3s use formal and informal reasoning to
explore, investigate, and justify mathematical ideas;
5.4s recognize examples of fallacious reasoning;
5.5s evaluate mathematical arguments and
proofs;
5.7s recognize that a mathematical problem
can be solved in a variety of ways, evaluate
the appropriateness of various strategies, and
select an appropriate strategy for a given
problem;
5.8s evaluate the reasonableness of a solution to a given problem.
Marilyn Burns is an author of children’s books
specializing in mathematics. Her books, workshops,
and videos are used throughout the country in
helping introduce the teaching of mathematics to
children.
3
Suppose, in trying to justify Example 3, a student
made the following claim:
“All the outside segments are equal. All the
inside segments are equal. If you straightened
out the outside two segments of the quadrilateral, it would look like the triangle. Therefore,
each of the quadrilaterals is equal to each of the
triangles.”
•
Give an analysis regarding the correctness and
validity of the students’ proposed models for
and the proposed justification for the
third example.
Early childhood–grade 4 teacher tasks – 47
S3MTP • Chapter 2
Supporting Discussion
ASSESSMENT
This problem involves the use of area as a means of visualizing
rational quantities, the relationship between perimeter and area
(particularly whether area stays invariant among different shapes
with the same perimeter), and the analysis of a mathematical
argument with fallacious reasoning. Most powerfully, it requires
students to identify the key concepts involved in the proposed
justification for Example 3 and to distinguish between the correct original construction and the incorrect justification suggested for it.
This task is appropriate either for an inclass group exploration or an out-of-class
extended assignment. As for all extended
assignments, a carefully designed rubric can
be quite helpful in reducing the complexity
of assessing responses.

Evaluating the validity of the various partitions shown in the three
examples provides students the opportunity to see a variety of
approaches and to construct mappings among them.
The key to evaluating the third example involves determining whether the unusually divided figure has “equal parts” (i.e.,
whether the triangles and the quadrilaterals have equal area).
Reaching this determination consistently generates a great deal
of variety in types of correct responses. The questions posed
in this problem almost always produce new lines of reasoning
from prospective EC–4 teachers. Prospective teachers will often
either:
• provide a rigorous demonstration of equal areas using
the Pythagorean Theorem to compute the length of
needed segments, or
• use a more informal “cut-and-paste” approach to show
that one geometric figure can be decomposed and recomposed to cover the other.
Although the unusual third model does indeed partition the
geoboard into eight regions of equal area, the argument
proposed as justification for it is problematic in several
fundamental ways.
• The student correctly observes that in Example 3, the
perimeters of the two shapes—the quadrilateral and the
triangle—are equal, but incorrectly assumes that figures
of equal perimeter must have equal area.
• There is also the assumption that transformations which
preserve perimeter also preserve area. The student’s
argument misuses the word “equal” by failing to specify
in what sense(s) the two shapes—the quadrilaterals and
triangles—are “equal.”

Identifying the flaw in the proposed justification requires an understanding of the relationship between perimeter and area that
in some students may be underdeveloped.
48 – Early childhood–grade 4 teacher tasks
A key component of a well-designed
response is clear communication and coherence of the justification of all “equal parts”
arguments. Careful attention must be given
to evidence of students’ misconceptions
regarding relationships between area and
perimeter.
S3MTP • Chapter 2
Few students at the EC–4 teacher certification level have had
the opportunity, let alone the obligation, to rigorously evaluate
mathematical arguments. The fallacious thinking in the proposed justification of Example 3 can be countered with a simple
investigation of perimeter versus area. Having students build
various rectangles with 100 feet of perimeter and record the
respective areas that emerge can be very revealing.
Early childhood–grade 4 teacher tasks – 49
S3MTP • Chapter 2
SECTION 2.4
EC–GRADE 4 TEACHER TASK:
PROBABILITY: ASSESSING THE FAIRNESS OF GAMES
A disagreement arose between two children playing
an even/odd sum game with a pair of dice. One child,
who scored a point for every odd sum rolled, argued
that the other child, who scored a point for every
even sum rolled, had an unfair advantage. “There are
six ways to roll an even sum: 2, 4, 6, 8, 10, or 12; I can
only score a point in five ways: 3, 5, 7, 9, or 11.”
1) Construct a mathematically sound argument, based
on probabilistic reasoning, that either attacks or
defends the fairness of the even/odd sum game
described above. Base your argument on a carefully
constructed definition of “fairness” in the context
of playing a game.
STATE BOARD FOR EDUCATOR
CERTIFICATION MATHEMATICS STANDARD
IV. PROBABILITY AND STATISTICS
Grades EC–4 Teacher Skills
The beginning teacher of mathematics is
able to:
4.1s investigate and answer questions by
collecting, organizing, and displaying data
from real-world situations;
4.5s use the concepts and principles of probability to describe the outcome of simple
and compound events;
4.6s explore concepts of probability through
data collection, experiments, and simulations.
2) Support your argument with both empirical results
and theoretical justifications.
SBEC MATHEMATICS STANDARD V.
MATHEMATICAL PROCESSES
Grades EC–4 Teacher Skills
Supporting Discussion
This task addresses fundamental notions associated with probability in the context of analyzing the fairness of a dice game. Big
ideas include considering all outcomes in a sample space, deciding whether or not outcomes are equally likely, and determining
probability from both an empirical and a theoretical perspective.
Before beginning the task, the class should discuss and define
terms such as experiment, trial, outcome, event, equally likely, and
sample space in the context of thinking about probability. There
should also be some discussion regarding the distinction between
theoretical and empirical means of determining probability. The
notion of fairness in terms of probability is also important working terminology for this exploration. Students generally need a
prompt to be very explicit in their probabilistic descriptions of
what it means for a game to be “fair.”

Most prospective EC–4 level teachers have had some experience
with probability, yet few have a good working definition for what
the probability of an outcome really represents—that is, the longrun relative frequency of that outcome.
50 – Early childhood–grade 4 teacher tasks
The beginning teacher of mathematics is
able to:
5.3s use formal and informal reasoning to
explore, investigate, and justify mathematical ideas.
5.4s recognize examples of fallacious reasoning.
RELATED TASK
For a task that addresses the
long-run relative frequency definition
of probability, see the 4–8 level task,
The Spicy Gumball.
S3MTP • Chapter 2
EXTENSION IDEA 1
On another occasion, the two children are
playing a similar even/odd game with a pair
of dice; however, in this game the number of
dots showing on the two dice are multiplied rather than added. Is this a fair game?
Again, support your conclusion using both
an empirical and a theoretical argument.
***
Through experimentation, students may be
surprised to find out that when multiplying the dots, an even product occurs much
more often than an odd product. With the
experience gained from the sum game, they
should be able to generate theoretical quantifications for these disparate probabilities;
however, they may still need some direction
in further exploring just why it should be
the case that the ratio of these probabilities
should be 3:1.
EXTENSION IDEA 2
Once again, two children are playing a
game; however, this time they are tossing
a thumbtack onto the floor, with one child
scoring a point if the tack lands “point up”
and the other scoring a point if the tack
lands “point down.” Is this game fair? Support your argument in whatever way you
deem necessary and appropriate.
Through empirical explorations, students are able to appreciate the often overlooked connections between probability and
statistics, particular in the context of teaching probability and
statistics in grades EC–4. Often, discussions will arise concerning how much data is “enough,” or what kind of simulation is
appropriate. Technological simulations (such as those available
on certain graphing calculators) expedite a time-consuming
process of data collection.
Through this exploration, students can become more fluent with
terminology associated with a probabilistic setting.
Ultimately, one of the students’ key observations should be that
the outcomes in the sample space of an experiment can often be
listed in multiple ways; however, it is not always the case that
the outcomes listed are all equally likely. That is, it is not always
the case that n outcomes in a sample space each has probability
1
n .
Discussions and diagrams such as the following are often helpful:
Consider the sample space {1, 2} and the following
diagrams:
***
For this exploration, students have no
choice but to investigate the associated
probabilities through empirical means. This
results in the important revelation that on
occasion probabilities can only be “known”
(i.e., approximated) as relative frequencies
based on actual experience.
• Are the events 1 and 2 equally likely to occur in each
situation?
Several levels of sophistication are portrayed by thinking about
the sample space in the context of this problem. Initially, students might think of two outcomes, even or odd, and naively
�
assume that the probability must be � for both. Empirical results
will support this conclusion and lead them to correctly declare
the game fair, but this conclusion is often based on superficial
reasoning. (The “even/odd sum” dice game is indeed fair, as
there is an ��
probability for the sum to be odd, and an ��
��
��
probability for it to be even.)
Early childhood–grade 4 teacher tasks – 51
S3MTP • Chapter 2
Extension idea 1, in which the numbers of dots on the dice are
multiplied, can prompt students to reconsider such superficial
reasoning. They might consider the eleven outcomes specified in
the problem from a theoretical perspective and discover
• that these outcomes are not all equally likely (e.g., a sum
of 7 is more likely than a sum of 2), and
• which outcomes are more likely than others (e.g., a sum
of 3 is more likely than a sum of 2).
Undoing the notion that a sum of 3, like a sum of 2, can only
occur in “one way” can be accomplished by constructing a chart
of possible outcomes that may occur when tossing a red and a
green die at the same time.
52 – Early childhood–grade 4 teacher tasks
S3MTP • Chapter 2
SECTION 2.5
EC–GRADE 4 TEACHER TASK:
NUMBER THEORY: THE STAMPS PROBLEM
STATE BOARD FOR EDUCATOR
CERTIFICATION MATHEMATICS STANDARD I.
NUMBER CONCEPTS
1) Suppose you have an unlimited supply of 4-cent
stamps and 9-cent stamps. What amounts of
postage can’t you make?
Grades EC–4 Teacher Skills
2) What amounts of postage can’t you make with
an unlimited supply of 9-cent stamps and 21-cent
stamps?
The beginning teacher of mathematics is
able to:
1.10s describe ideas from number theory
(e.g., prime numbers, composite numbers,
greatest common factors) as they apply
to whole numbers, integers, and rational
numbers and use these ideas in problem
situations.
SBEC MATHEMATICS STANDARD II.
PATTERNS AND ALGEBRA
Grades EC–4 Teacher Skills
The beginning teacher of mathematics is
able to:
2.1s use inductive reasoning to identify,
extend, and create patterns using concrete
models, figures, numbers, and algebraic
expressions;
2.2s formulate implicit and explicit rules to
describe and construct sequences verbally,
numerically, graphically, and symbolically;
2.6s model and solve problems, including proportion problems, using concrete,
numeric, tabular, graphic, and algebraic
methods.
3) In general, what amounts can and cannot be made
with an unlimited supply of a-cent stamps and
b-cent stamps?
Supporting Discussion
This problem, well known within elementary number theory,
involves concepts of prime, relatively prime, representation,
algebra, conjecture, and proof. It not only makes tangible the
meaning of relatively prime but challenges students to develop
efficient recording procedures, and to construct and validate
conjectures. As students are often likely to conjecture without validating, discussions involving conjecture and proof will
surface.

Structuring the problem with particular examples helps students
understand the nature of the general problem.
Although this problem is often stated quite succinctly in traditional number theory texts, a full understanding of the problem
can be elusive for prospective EC–4 teachers because of its
abstraction. Therefore, guiding questions such as 1) and 2) in
the task are important in helping the student understand and
prepare to respond to the general problem.

Organizing and recording data in a useful form is a critical first
step to solving this problem.
Here, students approach the problem by recording data and then
looking for patterns—a daunting task. The first hurdle is the organization of data. Students usually develop some kind of tabular
representation that builds one or more of the stamp denominations into its structure. They often use either:
Early childhood–grade 4 teacher tasks – 53
S3MTP • Chapter 2
• a table that lists all amounts of postage that can be made
(increasing by one stamp denomination across to the
right and by the other denomination in a column downward), or
• a table that lists all possible postage amounts (using
one of the denominations as the row length) and marks
those that can be made by the given stamps.
Of course, these tables are theoretically infinite, and students
must find an appropriate place to stop writing numbers and
to develop justifications for extending their results to all the
(larger) numbers not included in the table.

Solving the general problem requires abstract reasoning.
Once they have resolved the first two questions in the task,
students must identify the reason for the underlying difference
between the two answers (the GCF(a,b)).
The first question has a short answer (only 12 amounts can’t be
made), while the answer to the second question results in an
infinite list. In order to generalize their results in the third question, students must state results in terms of a, b and GCF(a,b).
Perhaps the most difficult result to generalize is the point past
which all multiples of GCF(a,b) can be made. Commonly,
students will conjecture that the point occurs before LCM(a,b).
With sufficient time and prompting they may discover that the
last multiple of GCF(a,b) that cannot be made is:
• ab – a – b (in the relatively prime case)
SOLUTION NOTE
1) The only amounts that cannot be made
with 4-cent stamps and 9-cent stamps are
1, 2, 3, 5, 6, 7, 10, 11, 14, 15, 19 and 23
cents.
2) With 9-cent and 21-cent stamps, you
cannot make 3, 6, 12, 15, 24 or 33 cents,
nor can you make any amount that is not
a multiple of 3[the GCF(9,21)].
3) In general, you can only make multiples
of the GCF(a,b), and the last such multiple that cannot be made is LCM(a,b)-a-b.
For the case where a and b are relatively
prime, the justification for the result can
follow very quickly from appropriate use
of the tabular representations developed
for the first two problems. For the type
of table which lists only “makable”
amounts, justifications use “chains” of
consecutive integers forming a diagonal
pattern in the table; for the type of table
which lists all amounts and marks the
“makable” amounts, justifications focus
on marking one number in each column
(if the row length is one of the stamp
denominations, then any number in the
same column as, and below, a marked
number, will also be marked).
ASSESSMENT
The third question of the task contains an
embedded assessment for the problem, in
that it brings out the ability to state conclusions in general terms. For a homework
assignment or on a test, many simple applications of this problem can be devised for
specific values of a and b.
• LCM(a,b) – a – b (in the general case)
Even then, students may find it difficult to validate their
conjecture(s).
While proof is an important part of doing mathematics, this
particular problem situation provides an important opportunity
to discuss what to do with a problem when you either cannot
find an exact expression for an answer, or cannot justify the formula you develop. It also provides an opportunity to talk about
the distinction between a proven answer and a conjecture, a
distinction few students will have explored by this point in their
education.
54 – Early childhood–grade 4 teacher tasks
EXTENSION IDEA
The Chicken Nuggets Problem. If chicken
nuggets are sold in boxes of 6, 9 and 20,
what is the largest number of chicken nuggets you can’t buy?
Here one must keep track of numbers that
are combinations of all three; doing so requires significant extensions of the methods
developed in the Stamps Problem.
S3MTP • Chapter 2
SECTION 2.6
EC–GRADE 4 TEACHER TASK:
GEOMETRY AND MEASUREMENT:
TILING A ROUND PATIO
STATE BOARD FOR EDUCATOR
CERTIFICATION MATHEMATICS STANDARD
III. GEOMETRY AND MEASUREMENT
Suppose we have a round patio 10 feet in diameter,
which we want to tile. The tiles we have are square,
and we want to cover as much of the patio as possible
without breaking tiles, overlapping tiles, or having
parts of tiles hang over the edge of the patio.
Grades EC–4 Teacher Skills
The beginning teacher of mathematics is
able to:
3.1s extend the understanding of shape in
terms of dimension, direction, orientation,
perspective, and relationship among these
concepts;
1) What is the least amount of area that can be left
untiled if we use tiles measuring 1 foot on a side?
Show your tiling arrangement.
3.6s identify attributes to be measured,
quantify the attributes by selecting and
using appropriate units, and communicate
information about the attributes using the
unit measure.
2) What is the least amount of area that can be left
untiled if we use tiles measuring foot on a side?
Show your tiling arrangement.
3) Repeat the question for tiles measuring foot on a
side and for tiles measuring foot on a side. What
generalization can you make about the area left
untiled as a function of tile side length?
SBEC MATHEMATICS STANDARD II.
PATTERNS AND ALGEBRA
Grades EC–4 Teacher Skills
The beginning teacher of mathematics is
able to:
2.1s use inductive reasoning to identify,
extend, and create patterns using concrete
models, figures, numbers, and algebraic
expressions;
2.3s illustrate concepts of relations and
functions using concrete models, tables,
graphs, and symbolic expressions;
2.4s apply relations and functions to represent mathematical and real-world situations.
SBEC MATHEMATICS STANDARD V.
MATHEMATICAL PROCESSES
Grades EC–4 Teacher Skills
The beginning teacher of mathematics is
able to:
5.9s use physical and numerical models to
represent a given problem or mathematical
procedure.
Supporting Discussion
This task explores fundamental notions of area in geometry and
measurement, connecting geometry to algebra. The search for
patterns and for generalizations about the untiled area when decreasing the side length of the square tiles provides an informal
look at the notion of limit.

All too often, prospective EC–4 teachers view area as the
number you calculate when replacing symbols in an appropriate
formula, rather than as a measure of the covering for a
two-dimensional region.
Students need experiences using and arranging objects (units)
that cover a given two-dimensional region in order to calculate
the area of that region. It is also helpful to have them approximate the area of an irregular shape, perhaps the “map” of a lake,
using a rectangle approximation method with rectangles that
vary in width from one map to another.
Early childhood–grade 4 teacher tasks – 55
S3MTP • Chapter 2
CONNECTION TO THE CLASSROOM
A popular activity involves having students
trace either their hand or foot, or a leaf
on a piece of square grid paper. Then their
teacher asks them first to estimate, then
measure, the area taken up by the shape.
There is an obvious inclination to find a
more efficient way to count than one by
one. Some students naturally invent their
own rectangle approximation method to
expedite the process.
From the lake “maps,” students begin to see not only how decreasing the width of the rectangles used decreases the difference
between the overestimate and the underestimate of the area but
also how the smaller rectangle width (and in fact each smaller
width) results in a more reliable approximation of area.
In completing this task, students are forced to engage in a covering process in the context of tiling a patio, and to encounter
the problem that the objects used will never cover the region
entirely.

One issue not readily apparent, that quickly arises in this problem,
is that the answer (the least amount of untiled area) depends on
the arrangement of tiles.
Discussion questions may include:
• Should we arrange the tiles on a square grid?
• Can we leave less area untiled by tiling in nonaligned
rows?
• Where should we start tiling?
Ultimately, students must talk about the dependence of untiled
area on tile size. It may be problematic for students to find that
direct relationship (between the amount of patio left untiled and
the tile size) and quantitatively identify it.

The idea of successively closer approximations to areas of irregular shapes is a notion fundamental to the Riemannian development of the definite integral, as well as to a true understanding of
measuring area.
Students are asked here to determine the errors in their successive approximations to the area of a circle. Of course, one goal is
for them to recognize that the errors are decreasing toward zero
56 – Early childhood–grade 4 teacher tasks
SOLUTION NOTES
1) A complete answer should include not
only a quantitative description of the
relationship between untiled area and tile
size, but also some attempt to justify why
a given tiling arrangement is optimal for
a given tile size.
2) More important than finding (and proving) the optimal tiling at each stage is
the relationship between tile size and
the untiled area. For the first question,
superimposing the circle on a square grid
with the center at a vertex will allow 60
tiles to fit within the circle (untiled area
25π– 60 square feet). Superimposing the
circle on a square grid with the center of
the circle at the center of a square will
allow 61 tiles.
S3MTP • Chapter 2
as the size of the tiles becomes smaller and smaller. However,
it is a nice connection to notice that the total covering of the
tiles is nearing 25π, the area of the circle, as the size of the tiles
becomes smaller and smaller. Observing the trend in the tiled
and the untiled areas as the tile size becomes smaller and smaller
is in fact the idea of limit.
Early childhood–grade 4 teacher tasks – 57
S3MTP • Chapter 2
58 – Early childhood–grade 4 teacher tasks
Chapter 3
GRADES 4–8 TEACHER TASKS ——————————
Teachers of grades 4–8 need a deep understanding of the mathematics central to the early grades (early childhood–4), but they
also need a broader preparation than does the teacher of elementary grades. They must develop mathematical sophistication
that allows them not only to support their students in building
mathematical understanding but also to teach the mathematics
that is central to the secondary curricula. Thus, the preparation
of prospective grades 4–8 teachers involves some overlap with
that for EC–4 teachers but broadens it to include an extended
focus on proportional reasoning, variables and relations, algebraic thinking, and critical reasoning.
There is a national need for mathematics courses that address
4–8-level teacher content knowledge in a manner consistent
with state and national recommendations and certification standards. For this reason, we have provided tasks that clarify several
key knowledge and skills objectives from the State Board for
Educator Certification mathematics standards. Those targeted
objectives say that teachers of grades 4–8 should be able to:
• Use connections between algebra and geometry to solve
a variety of problems, and provide appropriate justifications for the manipulation and equivalence of algebraic
expressions;
• Use functions and relations to model and solve problems
and communicate mathematical ideas;
• Use concrete, verbal, numerical, tabular, graphical, and
algebraic methods, as well as a variety of tools and appropriate technology, to solve problems and communicate ideas;
S3MTP • Chapter 3
• Use concepts of calculus to analyze properties of functions and to answer questions about areas, volumes, and
rates of change;
• Explore geometric relationships, developing and proving
conjectures;
• Investigate real-world problems by designing, conducting, and analyzing statistical experiments, then interpreting the results using concepts of probability and
appropriate statistical measures;
• Use knowledge about the history of mathematics to
support student understanding of important concepts;
and
• Provide convincing justification for mathematical
theorems.
It is assumed that before investigating the tasks described in this
chapter, students will have had experiences similar to those for
prospective teachers of grades EC–4 (such as those in chapters 1
and 2).
Chapter 3 includes six tasks:
• Polynomial Functions: Modeling Area and Volume;
• Geometry and Measurement: Pythagorean
Relationships;
• Measures of Central Tendency and Spread:
Designing Data;
• The Distributive Property: Patterns in Powers;
• Geometry, Measurement, and Modeling:
The Paper Stacking Problem; and
• Probability and Statistics: The Spicy Gumball.
Although each task has a particular content focus, we intend
that prospective teachers make important connections between
these content strands. This chart (also included in the preface to
this book) suggests some possible courses for which these tasks
might be appropriate.
60 – Grades 4–8 teacher tasks
S3MTP • Chapter 3
Task Correlation Guide 3:
Tasks for grades 4–8 teacher certification level
S3MTP Tasks for Grades 4–8 Teacher
Certification Level
Courses where task may be most appropriate
Polynomial Functions: Modeling Area and
Volume
College Algebra, College Algebra for Preservice Teachers,
Precalculus, Calculus
Geometry and Measurement: Pythagorean
Relationships
Geometry, College Algebra, Precalculus, and/or a standard “proofs”
course
Measures of Central Tendency and Spread:
Designing Data
Probability and Statistics course for preservice elementary teachers,
Statistics
The Distributive Property: Patterns in Powers
Foundations of Arithmetic course for preservice elementary teachers,
problem solving for preservice elementary teachers
Geometry, Measurement, and Modeling: The
Paper Stacking Problem
Geometry and Measurement course for preservice teachers, College
Algebra, Precalculus
Probability and Statistics: The Spicy Gumball
Geometry and Measurement course for preservice teachers, Statistics
Grades 4–8 teacher tasks – 61
S3MTP • Chapter 3
SECTION 3.1
GRADES 4–8 TEACHER TASK
POLYNOMIAL FUNCTIONS: MODELING AREA
AND VOLUME
Dog Pen Problem (area): Suppose you have 100 meters
of fencing to build a pen for your dog named Cooper.
You would like to build a rectangular pen that provides Cooper with the maximum living space. Make
a conjecture about the rectangular shape that will
provide the maximum living space and explain your
thinking. Then determine a function that models this
situation and use it to determine the optimal dimensions of the pen. Think about what values of the
domain yield a meaningful solution for the real-world
problem. Compare your solutions to your original
conjecture and explain any differences.
Candy Box Problem (volume): Suppose you are teaching and your class wants to purchase sacks to hold
Valentine candy so that they can distribute the candy
to younger children. You decide to turn this activity
into a volume lesson, so you change the design of the
“sack” to be a “box” instead. You instruct the class to
use sheets of construction paper measuring 8.5 inches
by 11 inches and to make square cuts in each corner
and fold and tape to form a box. The class wants to
know what size cut will yield the maximum volume
(we all like the most candy we can get!). Approach
this problem using a similar process (conjecture, determining a function/model, etc.) to that used in the
Dog Pen problem.
Supporting Discussion
STATE BOARD FOR EDUCATOR
CERTIFICATION STANDARD III.
1
GEOMETRY AND MEASUREMENT
Grades 4–8 Teacher Skills
The beginning teacher of mathematics is
able to:
3.14s relate geometry to algebra and trigonometry by using the Cartesian coordinate
system and use this relationship to solve
problems; and
3.15s use calculus concepts to answer questions about rates of change, areas, volumes,
and properties of functions and their graphs.
SBEC STANDARD II.
PATTERNS AND ALGEBRA
Grades 4–8 Teacher Skills
The beginning teacher of mathematics is
able to:
2.10s use linear and nonlinear functions and
relations, including polynomial, absolute
value, trigonometric, rational, radical,
exponential, logarithmic, and piecewise
functions, to model problems;
2.11s use a variety of representations and
methods (e.g., numerical methods, tables,
graphs, algebraic techniques) to solve linear
and nonlinear equations, inequalities, and
systems;
2.12s use transformations to illustrate
properties of functions and relations and to
solve problems;
2.13s give appropriate justification of the
manipulation of algebraic expressions, equations, and inequalities.
Pattern observation and recognition are fundamental to the
study of mathematics.

As we study functions, we often move away from the study of a
particular function to the study of patterns and properties peculiar
to all members of a given family of functions.
The Dog Pen and Candy Box investigation invites students to use
algebraic functions to model and solve problems created by real-
62 – Grades 4–8 teacher tasks
The State Board for Educator Certification
mathematics standards for early childhood–4,
4–8, and 8–12, with associated knowledge and
skills statements, may be referenced on the web
at www.sbec.state.tx.us/SBECOnline/standtest/
standards/ec4math.pdf; www.sbec.state.tx.us/
SBECOnline/standtest/standards/4-8math.pdf;
and www.sbec.state.tx.us/SBECOnline/standtest/
standards/8-12math.pdf.
1
S3MTP • Chapter 3
world situations. Here, prospective teachers first investigate the
quadratic family and then, the cubic family.

SBEC MATHEMATICS STANDARD V.
MATHEMATICAL PROCESSES.
Grades 4–8 Teacher Skills
The beginning teacher of mathematics is
able to:
5.7s recognize that a mathematical problem
can be solved in a variety of ways, evaluate
the appropriateness of various strategies, and
select an appropriate strategy for a given
problem;
5.15s explore problems using verbal,
graphical, numerical, physical, and algebraic
representations;
5.16s recognize and use multiple representations of a mathematical concept (e.g., a
point and its coordinates, the area of a circle
as a quadratic function in r, probability as a
ratio of two areas).
5.17s apply mathematical methods to analyze practical situations;
5.24s use the language of mathematics as a
precise means of expressing mathematical
ideas.
HISTORICAL NOTE
From the ancient Greeks we adopt the
visual representation for completing the
square using tiles. Today, we use algebra
tiles to build area models in which we
can demonstrate squares of binomials and
square trinomials. We can use such models
to complete the square and solve quadratic
equations.
Solving these problems should include not only a discussion of
modeling using algebraic functions, but also an application of
analytical and graphical approaches to find a solution. Prospective teachers may be encouraged to use techniques from calculus
as well.
As prospective teachers work through the Dog Pen problem,
they may realize that though they are restricted to a rectangular pen, a circular pen might be an optimal choice for the dog.
When restricted to a rectangular shape, however, they may conjecture that a square will give the greatest area. They may begin
by building tables and systematically considering rectangles of
different sizes, such as a 1 x 49 meter rectangle, a 10 x 40, etc.,
until they reach a 25 x 25 and get what seems to be a “largest”
area. Therefore, they may provide an argument such as this: “A
long skinny rectangle doesn’t have very much area, so the sides
shouldn’t be very different in length. In fact, maybe if the sides
are the same, we will get the most area.”
For prospective teachers who have not seen a problem like this
one before, it usually takes a while to find a function that represents the area of the pen—namely, if x represents the length of
one side, then the area is given by ������������� . This function representation provides a nice opportunity to discuss the
domain and range of the real-valued function as opposed to the
real-world situation. Most prospective teachers will arrive at this
function by drawing a figure. One very common error they tend
to make is failing to define their variables; another is to write
������������� for the area.
A nice review of completing the square is also meaningful here.
This skill is useful for those who do not have a calculus background and will assist them in finding the maximum function
value.

The algebra involved in completing the square is sometimes elusive and is a skill that is often forgotten due to lack of use.
When using graphing calculators, the usefulness of the trace
and zoom features emerge during class discussion. Although the
graphing calculator approach makes exploration of the graph
relatively easy, making the connection between “maximum
area” and “highest point on the graph of the area function” often
needs further discussion.
Grades 4–8 teacher tasks – 63
S3MTP • Chapter 3
Proposed solutions to the Dog Pen problem extension questions
will vary widely. Some students will try a circular pen because
that is a simple shape, but they may need a little direction in
computing the area knowing only the circumference. It may
occur to others to try a triangle, hexagon, or octagon; those who
try all three may be led to the idea of a circle.

The typical reaction to doubling the amount of fencing is to say,
“of course you will enclose twice as much area!” and investigate
no further. These questions provide students with an opportunity
to learn that what seems “obvious” may not be true.
Analyzing the dog pen with only three sides of fencing may seem
straightforward after having successfully explored the original
Dog Pen problem. However, rich discussions can arise if students
are asked to explain the significance of the shape of the new pen.

The Candy Box problem adds significantly to the difficulty level, as
it now requires the prospective teacher to model a three-dimensional situation in two dimensions.
Individual, group, and class participation on the Candy Box
problem might include:
DOG PEN PROBLEM EXTENSION
• Developing a table of values to record the size of the
square cut in the corner of the paper, as well as the
dimensions and volume of the box formed;
• Writing an algebraic expression for the volume, V, in
terms of the size of the cut, x;
• Using a graphing calculator to graph the equation and
trace along the curve to find the best cut to yield maximum volume;
• Determining why the curve is “smooth” rather than a
collection of isolated points like those generated in the
table of values;
• Determining which part of the graph is applicable to
this real-world problem;
• Explaining how the graph allows the consideration of
smaller cuts in order to better approximate the size of
the optimal cut;
• Using the table-building feature of a graphing calculator
to consider smaller cuts; and
64 – Grades 4–8 teacher tasks
Suppose you were not restricted to a rectangular pen.
• Would you be able to use the 100 meters
of fencing to enclose a larger area?
Consider pens in the form of a regular
hexagon and a regular octagon.
• If you doubled the amount of fencing,
could you enclose twice as much area?
• Suppose the rectangular plot runs along
a building so that you need to fence only
three sides using 100 meters of fencing.
What dimensions will now yield a maximum area?
ASSESSMENT
The Dog Pen and Candy Box problems
expose many common pitfalls for prospective teachers–incuding analytical, graphical,
and algebraic. Assessment might include
an analysis of whether they avoided these
pitfalls or to what extent they were able to
manage the problems with the mathematical tools they had available.
The Dog Pen Problem extension also requires
some creativity in examining nonrectangular pens and presents a nice review of
geometric ideas.
S3MTP • Chapter 3
• Applying calculus principles to determine maximum
volume.
Prospective teachers may simply graph and trace for the maximum function value within an acceptable domain and range.
They may also use concepts of calculus including derivatives
and/or critical points in locating relative maxima and minima,
thus providing an excellent opportunity to use techniques different from the algebraic ones used in the Dog Pen Problem.
Grades 4–8 teacher tasks – 65
S3MTP • Chapter 3
SECTION 3.2
GRADES 4–8 TEACHER TASK
GEOMETRY AND MEASUREMENT:
PYTHAGOREAN RELATIONSHIPS
Using dynamic geometry software, first construct a
right triangle. Then:
2
1) Construct equilateral triangles on each leg and on
the hypotenuse of the right triangle.
2) Use the measure feature of the software to calculate the areas of the constructed equilateral
triangles. Explore the relationships between these
areas and make a conjecture.
3) Using algebraic techniques, verify or construct a
proof for the conjecture.
4) Repeat the process described in steps 1–3 using
regular hexagons; that is, measure the areas of
regular hexagons constructed on each side of the
right triangle, make conjectures, and verify or
construct a proof of the conjectures.
5) Finally, repeat the process described in steps 1–3
using semicircles. The sides of the triangle will
serve as the diameters for the semicircles.
STATE BOARD FOR EDUCATOR
CERTIFICATION STANDARD III.
GEOMETRY AND MEASUREMENT
Grades 4–8 Teacher Skills
The beginning teacher of mathematics is
able to:
3.8s develop, justify, and perform geometric
constructions using compass, straight-edge,
and reflection devices and other appropriate
technology;
3.9s investigate and prove geometric relationships within the axiomatic structure of
Euclidean geometry;
3.10s analyze and solve problems involving
one-, two-, and three-dimensional objects
such as lines, angles, circles, triangles, polygons, cylinders, prisms, and spheres.
SBEC STANDARD V.
MATHEMATICAL PROCESSES
Grades 4–8 Teacher Skills
Supporting Discussion
This task involves the basic ideas of measurement in connection
with a right triangle. The students will need to use algebraic manipulations and proof techniques to verify whether any conjectured relationships hold.

It is important that prospective teachers of middle grade students
be fluent in the processes of mathematical inquiry—especially
when dealing with geometric properties and relationships. It is
also important that teachers experience discovery and proof for
themselves so they can better model the process of conjecture
and justification for their students.
Prospective teachers will first observe the difference between
“drawing” and “constructing” using the dynamic software. After
constructing the shapes on the legs and hypotenuse of the right
triangle, they tend to look for the area measurements and will
observe that the sum of the areas of the triangles constructed on
the legs of the right triangle is equal to the area of the triangle
66 – Grades 4–8 teacher tasks
The beginning teacher of mathematics is
able to:
5.6s provide convincing arguments or proofs
for mathematical theorems.
NOTE
Determining the difference between “drawing” and “constructing” is not a simple idea.
Experiences with dynamic geometry software can help prospective teachers deepen
their understanding.
They may begin by comparing use of the
“draw” tool to the “construct” option to
discover that constructing and drawing are
quite different, although the same design
can be produced with each.
See, for example, The Geometer’s Sketchpad
(for more information, see www.keypress.com/
sketchpad) or Cabri Geometry (for more
information, see www.cabri.com/en).
2
S3MTP • Chapter 3
constructed on the hypotenuse. They will notice the same to be
true for regular hexagons and semicircles. Showing this result
algebraically involves generalizing the areas of the shapes, based
on legs a and b and hypotenuse c of the original right triangle.

SBEC STANDARD VII.
MATHEMATICAL LEARNING AND
INSTRUCTION
Grades 4–8 Teacher Skills
The beginning teacher of mathematics is
able to:
7.8s use a variety of tools, including, but
not limited to, rulers, protractors, scales,
stopwatches, measuring containers, money,
calculators, and software, to strengthen
comprehension and understanding;
7.18s use mathematics labs, simulations,
open-ended investigations, research projects, and other activities when appropriate
to guide students’ learning;
7.19s apply appropriate technology to
promote mathematical learning.
Students often have great difficulty understanding the
essence of proof.
In this case, students often make the following errors.
• Students make conjectures based on observed measurements and then determine that the proof is complete.
• Lacking understanding that a geometric interpretation
of the Pythagorean Theorem is a statement about areas
of squares, they try to use the Pythagorean Theorem directly to justify their conjectures about the areas of other
regular shapes.
Solution strategies often result in constructions similar to the
following diagrams. In each figure below, triangle ABC is a right
triangle with right angle at C. In each figure, notice that the
sum of the areas of the figures constructed on the legs is equal to
the area of the figure constructed on the hypotenuse.
Grades 4–8 teacher tasks – 67
S3MTP • Chapter 3
Area ∆ BDC = 4.46 cm2
Area ∆ CEA = 11.08 cm2
Area ∆ AFB = 15.54 cm2
((Area ∆ BDC) + (Area ∆ CEA)) - (Area ∆ AFB) = 0.00 cm2
A
F
E
C
B
D
Figure 1
SBEC STANDARD VI.
MATHEMATICAL PERSPECTIVES
Grades 4–8 Teacher Skills
The beginning teacher of mathematics is
able to:
6.1s use key events and knowledge of
specific individuals throughout the history
of mathematics to illustrate age-appropriate
mathematical concepts;
6.4s use historic mathematical problems as a
tool for assessing the mathematical knowledge of a particular period or culture.
68 – Grades 4–8 teacher tasks
S3MTP • Chapter 3
In the figure below, the area measurements refer to the hexagons
constructed on the sides of triangle ABC.
Area BCGFED = 3.28 cm2
Area AHIJKC = 14.11 cm2
Area BPNMLA = 17.40 cm2
((Area BCGFED) + (Area AHIJKC)) - (Area BPNMLA) = 0.00 cm2
E
F
K
G
D
B
J
N
I
A
H
ASSESSMENT
It is advantageous to have students work
in pairs to construct the drawings and the
algebraic proof that supports the observed
results. A scoring rubric for this task might
include criteria such as:
P
C
Figure 2
L
M
• Appropriate use of geometric formulas
• Accuracy of constructions
• Proper examples (numerical results)
• Overall organization of results
• An individual assessment of the student’s
understanding in the form of a separate
construction and verification.
Grades 4–8 teacher tasks – 69
S3MTP • Chapter 3
In Figure 3, the labels A1, A2, and A3 refer to the semicircles
constructed on the sides of triangle ABC.
Area A1 = 12.44 cm2
Area A2 = 3.55 cm2
Area A3 = 15.99 cm2
((Area A1) + (Area A2)) - (Area A3) = 0.00 cm2
A3
B
A2
A
C
A1
Figure 3
HISTORICAL NOTE
Prospective teachers often report in the form of a conjecture.
“Here is my conjecture: If similar figures are constructed on the three sides of a right triangle, then the
sum of the areas on the legs is equal to the area on the
hypotenuse.”
However, the instructions call for a more complete explanation
in the form of a verification or proof (depending on the level of
the student).
Students typically begin proving their conjectures by expressing
the areas of each shape in general form and then verifying that
the sum of the areas of the shapes on the legs does equal the area
of the shape on the hypotenuse. For example, in the case of the
semicircles, students will make an argument similar to the one
described below.
The sum of the areas of the two smaller semicircles
may be written as
�
�
� ��π���� �� � ��� � ��π���� �� � ���π ������
� ����� ��
�
�
�
�
�
�����
�����
70 – Grades 4–8 teacher tasks
Pythagoras (c.585–c.497 B.C.E.) was a
Greek mathematician best known for the
theorem that bears his name. However,
most historians doubt that Pythagoras was
the first to discover the theorem. In fact,
tablets uncovered reveal that the Babylonians were aware of this right triangle result
for specific triangles at least 1000 years
earlier.
S3MTP • Chapter 3
The area of the largest semicircle is
Then using the fact that AC2 + BC2 = AB2 (Pythagorean Theorem), they conclude that the two area expressions are equal. For other regular shapes, students
proceed in a similar manner.
A more sophisticated proof involves the notion of proportionality. Once students have given proofs for several different specific
types of regular shapes, they may be encouraged to think more
generally.

It is important for prospective teachers to recall that because the
figures are all similar, their linear dimensions are proportional and
their areas vary as the square of that proportion.
Then a more general proof based on the idea of proportionality
would be similar to the following argument.
If we call the figure on AC , “F,” the figure on �� , “G,” and
the figure on AB , “H,” with areas f, g, and h, respectively, then,
using the constants of proportionality between the linear dimensions of figures F and H, and figures G and H, respectively, we
��
have that �� � � �����
and
��
�
�
Then
.
Grades 4–8 teacher tasks – 71
S3MTP • Chapter 3
SECTION 3.3
GRADES 4–8 TEACHER TASK
MEASURES OF CENTRAL TENDENCY AND SPREAD: DESIGNING DATA
1) Design two sets of data to represent grades on
Test 1 of two different classes of a freshman
mathematics course. Each class has 25 students.
The mean of Class A is 75, the mode is 71, and the
median is 74. The mean of Class B is 79, the mode,
69, and the median, 75.
•
For each set of data, calculate the range, variance, and standard deviation.
•
Create a double stem-and-leaf plot and a
double-box plot to display your data for both
classes.
2) Add 5 to each data value in Class A. Then calculate the mean, mode, median, range, variance, and
standard deviation. Note the manner and amount
of change in each measure.
•
Make some general observations about what
you expect to happen when each data value is
increased (or decreased) by a fixed amount.
3) Compare the following method of “curving” to
the results in step 2) above: multiply each score by
100/95. Then calculate the mean, mode, median,
range, variance, and standard deviation. Note the
manner and amount of change in each measure.
•
•
Which measures were more affected by this
method of “curving” than by adding a constant
value of 5? Which were less affected?
When might this method of “curving” be
appropriate? What is the significance of the
fraction 100/95?
4) Decrease each data value in Class B by 10%. Calculate the mean, mode, median, range, variance,
and standard deviation. Note the manner and
amount of change in each measure.
•
Make some general observations about what
you expect to happen when each data value is
decreased (or increased) by a fixed percentage.
72 – Grades 4–8 teacher tasks
STATE BOARD FOR EDUCATOR
CERTIFICATION STANDARD V.
MATHEMATICAL PROCESSES
Grades 4–8 Teacher Skills
The beginning teacher of mathematics is
able to:
5.11s investigate and explore problems that
have multiple solutions.
SBEC STANDARD IV.
PROBABILITY AND STATISTICS
Grades 4–8 Teacher Skills
The beginning teacher of mathematics is
able to:
4.11s develop and justify concepts and
measures of central tendency (e.g., mean,
median, mode) and dispersion (e.g., range,
interquartile range, variance, standard deviation) and use those measures to describe
a set of data;
4.12s calculate and interpret percentiles and
quartiles.
SBEC STANDARD VII.
MATHEMATICAL LEARNING AND
INSTRUCTION
Grades 4–8 Teacher Knowledge
The beginning teacher of mathematics
knows and understands:
7.5k the process by which students construct
mathematical knowledge;
7.8k how individual and group instruction
can promote learning and create a learning
environment that actively engages students
in learning and encourages self-motivation;
7.9k a variety of instructional methods,
tools, and tasks that promote students’ confidence, curiosity, and inventiveness while
using mathematics described in the TEKS.
S3MTP • Chapter 3
Supporting Discussion
The primary focus of this task is developing an understanding of
the statistical concepts of central tendency and spread for a data
set. There is also an emphasis on the visual representation of data.
DISCUSSION TOPIC: WHY AVERAGE GRADES?
In working with prospective teachers,
it would be worthwhile to ask them to
consider why they would average grades for
students in their classes. Generally, teachers
average grades because that is what they are
instructed to do in their teacher training.
Prospective teachers need to reason statistically about the mean and what it represents
in their assessment of what students understand. They are often hesitant to consider
the use of mode or median as an appropriate
measure of student learning.
Discussions similar to these can be very
productive in developing mathematical
concepts.
ADDITIONAL TOPICS FOR DISCUSSION
• Explain how the mean and the median
are related in a normal distribution.
• A normal curve is sometimes used in
discussing a statistics principle known as
the Central Limit Theorem. Consult a
textbook or other resources to investigate
the meaning of this theorem. Explain the
meaning in your own words.
• Consider the effects on a data set of
increasing (or decreasing) each data
point by a fixed amount or percentage.
Compare these effects with the
geometric impact of translations or
size transformations.
Data analysis represents one of the most prominent uses of
mathematics in our everyday lives. Measures of the data are an
important way to think about the “shape” of the data.

Prospective teachers need opportunities to explore basic concepts involving statistical reasoning, with an emphasis on developing conceptual knowledge in addition to computational skill or
knowledge of formulas.
Prospective teachers often struggle at first with the idea of creating their own data sets with given constraints. They will develop
different strategies for constructing these sets and the set-construction step may seem rather time-consuming. It is important
to discuss with them the idea of modeling a data set with another data set that has the same number of data points and the
same sum, a conceptual representation for the mean.
Creating a visual display of the data sets will help prospective
teachers make comparisons between the two sets of data and will
also help them solidify ideas of central tendency as they observe
the “placement” of mean and median in the stem-and-leaf plot
and double-box plot.

Before prospective teachers make the indicated adjustments to
each data set, it is important that they take time to think about
what they expect to happen when these changes are made.
They will notice that when 5 is added to each data point, the
mean, median and mode also each increase by 5, but the range,
variance, and standard deviation remain the same. In general,
when each data point is increased (or decreased) by a fixed
amount, the mean, median, and mode will by affected in the
same manner (increased or decreased). However, the range, variance, and standard deviation will not be affected by the increase
or decrease.
In contrast, when each data point of the set is changed by a constant scale factor, the mean, median, mode, range, and standard
deviation are changed by that scale factor also. In general, when
each data point of a set is increased (or decreased) by a fixed factor, the mean, median, mode, range, and standard deviation will
each be affected in the same manner (increased or decreased by
that factor). In the case of the variance, Var(aX) = a2Var(X).
Grades 4–8 teacher tasks – 73
S3MTP • Chapter 3
SECTION 3.4
GRADES 4–8 TEACHER TASK
THE DISTRIBUTIVE PROPERTY:
PATTERNS IN POWERS
1) Consider the squares of whole numbers for which
the digit in the units place is 5. Complete the table
below.
Number (n)
Square of Number (n)
5
25
15
225
25
35
45
STATE BOARD FOR EDUCATOR
CERTIFICATION MATHEMATICS STANDARD I.
NUMBER CONCEPTS
Grades 4–8 Teacher Skills
The beginning teacher of mathematics is
able to:
1.17s analyze and describe relationships
between number properties, operations,
and algorithms for the four basic operations
involving integers, rational numbers, and
real numbers;
1.21s extend and generalize the operations
on rationals and integers to include exponents, their operations, their properties, and
their applications to the real numbers.
55
65
75
SBEC STANDARD II.
PATTERNS AND ALGEBRA
85
Grades 4–8 Teacher Skills
95
The beginning teacher of mathematics is
able to:
2) Look for a pattern that will help you determine
a shortcut method for finding the square of the
whole numbers for which the units digit is five.
Explain and illustrate your method. Then extend it
to see if your shortcut works for larger numbers.
3) Expand each of the numbers in the first column
using powers of 10. Use the distributive property
of multiplication over addition to square these
numbers. Use the result to justify the method you
developed in step 2).
4) Suppose we wanted to prove a similar rule for base
four numbers. What would the units digit be in the
numbers we want to square? Why? Develop a proof
to validate your conjecture.
Supporting Discussion
This task involves using basic ideas of place value and expanded
notation while connecting to the properties of whole number
operations.
74 – Grades 4–8 teacher tasks
2.1s use inductive reasoning to identify,
extend, and create patterns using concrete
models, figures, numbers, and algebraic
expressions;
2.8s apply all skills specified for teachers in
grades EC–4, using content and contexts
appropriate for grades 4–8.
S3MTP • Chapter 3

ADDITIONAL TOPICS FOR DISCUSSION
• Visual representations (rectangular arrays) of the expressions (a + b)2, (a - b)2,
and (a + b)(a - b).
• Whether division distributes over addition (from the left and from the right).
• Whether the “power” operation is commutative. For example, since 24 = 42, does
this mean that the power and base will
always commute?
• A proof that multiplication distributes
over subtraction. That is, for all whole
numbers a, b, and c, with b ≥ c, show
that a (b - c) = a . b – a . c.
It is important for teachers of middle grades students to be
prepared to point out the series of mathematical steps, supported
by number properties, that develop into the shortcuts that middle
grades students use.
The distributive property is introduced in the middle grades, and
these students need the property and its connection to addition
and multiplication to be more fully explained and developed.
They also need a deeper and more complete understanding of
place value and more experience in mental computation. Middle
grades students are intrigued by shortcuts, but are often unable
to connect the shortcut or algorithm to the mathematics that is
taking place; that is, too frequently they memorize the shortcut
without understanding the underlying concepts.

In developing a shortcut to use with their students, prospective
teachers should work from a strong intuitive base, as shortcuts
can be meaningful if they arise naturally from an understanding of
the mathematical concepts, but detrimental if not.
In working through the task, prospective teachers are usually
quick to determine the shortcut, but have some difficulty deciding what to do to justify their result. Although they can rather
easily write a number in expanded form, they are usually uncomfortable using the distributive property in squaring the number
written in expanded form. They tend to make mistakes in using
the distributive property and want to use the shortcut they have
uncovered to justify their discovery.
Completed tables are used to uncover a shortcut:
Number (n)
Square of Number (n)
5
25
15
225
25
625
35
1225
45
2025
55
3025
65
4225
75
5625
85
7225
95
9025
Grades 4–8 teacher tasks – 75
S3MTP • Chapter 3
Prospective teachers may readily observe that each of the numbers ends in 25 and that those numbers have leading digits of 0,
2, 6, 12, 20, 30, 42, 56, 72, and 90. After a little thought, they
may notice that these are just 0(1), 1(2), 2(3), 3(4), 4(5), 5(6),
6(7), 7(8), 8(9), and 9(10). So, they generate the digits in the
square of the number by taking the digit in the tens place (of
the original number) and multiplying by one greater than that
digit, then “attaching” a 25. For instance, the square of 85 could
be found by multiplying 8(9) = 72 and “attaching” 25, yielding
7225.

Generalizing results is often a pitfall for many students. It is important to have them verbalize what they did, write down what they
are saying, and then try to generalize their findings.
Students may be prompted by the instructor to formalize the
patterns they have observed and develop a series of written steps
such as:
Let a5 represent a number ending in 5. That is, the
leading digits are represented by a. It appears that we
can square a number ending in 5 as follows:
(a5)2 = 100a(a + 1) + 25,
where a(a + 1) represents the leading digits
of the square.
We can justify this as follows: if the leading digits
of a number ending in 5 are represented by a, then
the number a5 can be written as 10a + 5. Squaring
this gives
(10a + 5)2 = 100a2 + 2(10a)(5) + 25
= 100a2 + 100a + 25
= 100a(a + 1) + 25.
The factor of 100 in the first term in this description guarantees
that the first term will have no effect on the last two digits of the
square, so the square will end in 25. Also, the leading digits will
be represented by a(a + 1) since that same factor of 100 has the
effect of shifting the product a(a + 1) exactly two places to the
left, so that those digits terminate right before the 25 begins.
To facilitate an understanding of properties of addition and multiplication, it is often effective to have the students think and
work in other number bases.
76 – Grades 4–8 teacher tasks
IDEAS ABOUT PROOF
Many students get their first ideas about
mathematical proof in a high school geometry course. Unfortunately, students in high
school geometry often think of a proof as
a numbered list of statements and reasons.
Mathematical proof, however, involves a
measure of discovery.
Once students have studied proofs as a series
of simple steps and looked for clues that lead
from one step to the next, they are able to
recognize techniques used and move on to
construct proofs on their own.
S3MTP • Chapter 3

Having the prospective teacher work in a base other than base
ten puts the operations and properties in the forefront.
Working in base four, the prospective teachers must focus on
the operations and properties and how they affect the outcome.
They will start from a very investigative perspective (similar
to the base ten procedure above), but will be less comfortable
actually thinking in base four. The discomfort they experience
is similar to that of their students when confronted with unfamiliar material in base ten. However, as the prospective teachers
proceed with their “squaring process” they will discover that the
shortcut is similar to that for base ten. Justification of this shortcut will once again emphasize the importance of place value.
Grades 4–8 teacher tasks – 77
S3MTP • Chapter 3
SECTION 3.5
GRADES 4–8 TEACHER TASK
GEOMETRY, MEASUREMENT, AND MODELING:
THE PAPER STACKING PROBLEM
Suppose you have a very large sheet of paper. You cut
the paper in half and stack one of the sheets on top
of the other. Then you cut the stack in half again and
place one stack on top of the other. Assume that you
continue this cutting and stacking process.
1) If you were to continue to cut the stack in half
and restack 25 times, how tall would the stack be?
Make a guess before you attempt any calculations.
2) Find an expression for the number of sheets in the
stack after n cuts.
3) How many cuts would be required to make the
stack at least one mile high if the paper is 7/1000
of an inch thick?
x
x
4) Graph y = 2 and solve equations like 2 = 6
graphically.
5) Use this idea of cutting and stacking paper to
x
model the exponential function f(x) = 3 . How
–x
could we model the function h(x) = 2 or
–x
g(x) = 3 using paper cutting, stacking, or both?
6) Use graphing calculators and/or spreadsheets to
x
n
investigate definitions of expressions like a , x and
n! using recursion.
7) Connect what you have learned in this exercise to
the idea of exponential and logarithmic functions.
STATE BOARD FOR EDUCATOR
CERTIFICATION MATHEMATICS STANDARD
III. GEOMETRY AND MEASUREMENT
Grades 4–8 Teacher Skills
The beginning teacher of mathematics is
able to:
3.9s investigate and prove geometric relationships within the axiomatic structure of
Euclidean geometry;
3.10s analyze and solve problems involving
one-, two-, and three-dimensional objects
such as lines, angles, circles, triangles, polygons, cylinders, prisms, and spheres;
3.11s analyze the relationship among threedimensional figures and related two-dimensional representations (e.g., projections,
cross-sections, nets) and use these representations to solve problems.
SBEC MATHEMATICS STANDARD II.
PATTERNS AND ALGEBRA
Grades 4–8 Teacher Skills
The beginning teacher of mathematics is
able to:
2.10s use linear and nonlinear functions and
relations, including polynomial, absolute
value, trigonometric, rational, radical,
exponential, logarithmic, and piecewise
functions, to model problems;
2.11s use a variety of representations and
methods (e.g., numerical methods, tables,
graphs, algebraic techniques) to solve linear
and nonlinear equations, inequalities, and
systems.
Supporting Discussion
An accurate direct measurement is often difficult or impossible
with a given tool; for example, a bathroom scale will not accurately weigh a canary. Similarly, a ruler will not accurately
measure the thickness of a piece of paper. In these cases, indirect
measurements may be necessary. The techniques used in this
problem and the suggested extension require the integration of
many geometrical concepts, including measurement, the use of
formulas, and the additivity of volume.
78 – Grades 4–8 teacher tasks
S3MTP • Chapter 3

EXTENSION IDEA:
The Paper Towel Problem
A roll of paper towels is made up of two coaxial right circular cylinders, one of which is
a hollow core.
• Measure the length and the inner and
outer radii of a roll of paper towels.
• Compute the volume of paper towels in
the full roll.
• Measure the length and width of a single
paper towel and compute its area.
• Set up an equation relating the thickness
of a paper towel to the volume of the paper towels in the roll. (Assume that there
is not airspace between paper towels.)
• Solve the equation for the thickness of
one paper towel.
• The wrapper on the paper towels may
indicate the thickness of a sheet. If so,
determine the percent error in your
calculated thickness.
ASSESSMENT
Beginning in whole class discussions and
moving to students working in cooperative groups will help to facilitate much of
the discussion regarding table building,
measurement, and overall problem-solving
strategies.
The Paper Towel Problem provides not only
a nice extension to this problem, but also a
useful individual assessment item to measure
student learning.
It is important that prospective teachers of the middle grades
have a variety of experiences with measurement so that they can
help their students develop an understanding of direct and indirect measurement, as well as measurement error and precision.
Prospective teachers should guess the height of the paper stack
before actually beginning their calculations. They will most
likely be surprised at the difference between their guess and their
final answer. They should then start with a table-building activity to decide how many sheets they have after each cut and then
develop a strategy to determine the thickness of a single sheet
of paper. As this measurement is problematic, they may resort
to measuring the thickness of a ream of paper to determine the
thickness of a single sheet, thus using indirect measurement as
needed in a real context. Prospective teachers are often amazed
to discover how tall the stack would actually be after only 25
cuts.

This type of task provides opportunities for table building, determining a closed form rule, using direct and indirect measurement,
and using a graphing calculator appropriately.
Extensions to other graphing calculator exercises and to inverse
relationships such as exponential and logarithmic functions provide excellent class discussion. Students can start these problems
in class and continue them in cooperative groups.
Prospective teachers may present their findings as follows:
Each cut and stack doubles the height of the stack, so
25
after 25 repetitions, the height of the stack would be 2
times the thickness of the sheet. In fact, after n repetin
tions, there would be 2 sheets. If we assume that the
thickness of a sheet of paper is 7/1000 inch, we would
25
have a stack 2 (7/1000) inches high, or about 234,881
inches. That’s approximately 3.7 miles high!
To get a stack 1 mile high, we would need n steps,
n
where 2 (7/1000) = 63360 inches. (There are 12
inches in a foot and 5280 feet in a mile, so there are
12(5280)=63360 inches in a mile.) Thus, we need
, so n
���� ������� ���
About 23 iterations would make this slightly less than a
mile high. This is reasonable: since 25 iterations made
a stack 3.7 miles high, 24 iterations would yield a stack
1.85 miles high, and 23 iterations would yield a stack
0.925 miles high.
Grades 4–8 teacher tasks – 79
S3MTP • Chapter 3
Working through the Paper Towel extension to the problem
requires developing and solving equations along with calculating
percent error.

The Paper Towel extension provides additional depth to the problem and requires that prospective teachers consider the importance of tracking measurement error.
For instance, meaningful class discussion may center on “tolerable” error in this type of measurement (with sheets of paper or
paper towels) compared to the “tolerable” error in laser surgery
for the eye.
80 – Grades 4–8 teacher tasks
S3MTP • Chapter 3
SECTION 3.6
STATE BOARD FOR EDUCATOR
CERTIFICATION MATHEMATICS STANDARD
IV. PROBABILITY AND STATISTICS
GRADES 4–8 TEACHER TASK
PROBABILITY AND STATISTICS: THE SPICY GUMBALL3
Grades 4–8 Teacher Skills
Sally and her brother are at the grocery store with
their dad. On the way out of the store, Sally sees
a gumball machine and asks her dad for enough
money to buy two gumballs, one for her and one for
her brother. There are only six gumballs left in the
machine—four white and two red. Sally loves the red
ones because they are spicy. She wonders what her
chances are of getting at least one red gumball from
the machine.
The beginning teacher of mathematics is
able to:
4.10s investigate real-world problems by
designing, conducting, analyzing, and interpreting statistical experiments;
4.14s explain and use precise probability
language to make observations and draw
conclusions from single variable data and
to describe the level of confidence in the
conclusion;
4.15s determine probability by constructing
sample spaces to model situations.
1) Conjecture: Make a conjecture regarding the
answer to Sally’s question (the chance of getting at
least one red gumball). Describe methods to model
or simulate this situation. Discuss the underlying
assumptions and/or constraints associated with the
different methods.
SBEC MATHEMATICS STANDARD V.
MATHEMATICAL PROCESSES
2) Sample Space: Consider the different outcomes
possible from Sally’s purchase of two gumballs.
Describe different sample spaces and in each case,
determine the probabilities associated with the
sample points. (Try to include at least one sample
space with equally likely outcomes and at least one
where the outcomes are not equally likely.)
Grades 4–8 Teacher Skills
The beginning teacher of mathematics is
able to:
5.9s use physical and numerical models to
represent a given problem or mathematical
procedure;
5.10s recognize that assumptions are made
when solving problems and identify and
evaluate those assumptions;
5.12s apply content knowledge to develop
a mathematical model of a real-world situation and analyze and evaluate how well the
model represents the situation;
5.13s develop and use simulations as a tool
to model and solve problems.
This task was adapted from the Dana Center’s
TEXTEAMS Institute Statistical reasoning across
the TEKS (2002). TEXTEAMS (Texas Teachers
Empowered for Achievement in Mathematics
and Science, www.utdanacenter.org/texteams)
is a Dana Center–managed statewide teacher
professional development program that offers a
comprehensive system of professional development
for K–12 mathematics and science teachers,
delivered through a statewide network of trainers.
The program’s institutes provide a core set of
professional development materials and skills and
help teachers develop a common understanding of
important mathematics and science content and
the state’s curriculum standards (the Texas Essential
Knowledge and Skills).
3
Supporting Discussion
This task is designed to help prospective middle grades teachers
formalize the idea of probability, beginning from their intuition
about what will happen.

Middle grades teachers should be prepared to give their students
numerous opportunities to engage in probabilistic thinking about
simple situations so that their students can develop ideas about
chance and probability.
1) Conjecture
When your students make a conjecture about this problem,
the conjecture—regardless of the modeling method suggested—should be based on the simulated experiment of two
draws from the gumball machine without replacement of any
gumballs. After the first experiment is complete, the machine is
Grades 4–8 teacher tasks – 81
S3MTP • Chapter 3
then reloaded and the experiment repeated. The important idea
is that a series of repeated, identical experiments must generate
the data.
Possible techniques for the simulation include:
• Physically replicate the experiment with an actual gumball machine.
• Use colored counters or chips to represent gumballs in
the gumball machine.
• Use a six-sided number cube to simulate the first gumball (since there are six options). In this case, discuss
how the choice of the second gumball will be handled.
• Construct a tree diagram.
Possible assumptions and/or constraints that might
surface include:
• Are the items that represent the gumballs the same size?
The same shape?
• What issues surface if they are different sizes or shapes?
• Does the device that represents the machine provide
sufficient shuffling (i.e., randomization) after each trial?
How can this be accomplished?

Prospective teachers need experiences working in groups and
thinking about how to set up and carry out a simulation to accomplish the task at hand.
They may start their investigation by using counters to simulate
the purchase of two gumballs ten different times and observing
the number of these simulated purchases in which the event R
(red) occurred.
i
10
Using ratio form (where i is a natural number less than or
equal to 10), they can express the number of times the event R
occurred compared to the total of ten purchases. As they simulate five additional gumball purchases, they should then record
the cumulative total number of times the event R occurred compared to the total of fifteen trials. Continuing this process, they
may find it helpful to record their results in tabular form:
SBEC MATHEMATICS STANDARD VII.
MATHEMATICAL LEARNING AND
INSTRUCTION
Grades 4–8 Teacher Skills
The beginning teacher of mathematics is
able to:
7.18s use mathematics labs, simulations,
open-ended investigations, research projects, and other activities when appropriate
to guide students’ learning;
7.19s apply appropriate technology to
promote mathematical learning.
POSSIBLE CLASS DISCUSSION ABOUT
APPROPRIATE CALCULATOR USE
• How a calculator or computer random
number generator can be used to produce
a random natural number between 1 and
6.
• How the second draw can be modeled on
the calculator.
• The logic necessary to create a simulation.
• The program display for the color of
the two gumballs. One example of such
output might be RR, RW, WR, or WW.
82 – Grades 4–8 teacher tasks
S3MTP • Chapter 3
POSSIBLE CLASS DISCUSSION ABOUT AN
APPROPRIATE GRAPHICAL REPRESENTATION
• Appropriate scaling is important.
• The horizontal axis represents the
number of purchases and the vertical axis
represents the cumulative frequency of
the occurrence of R.
Total Number
of Purchases
Cumulative Relative
Frequency for the
Occurrence of R
10
• Points in such plots are often connected
to illustrate the overall trend, but is that
an appropriate representation of the data?
15
• If the plot were continued to include
more simulated purchases, the cumulative frequencies would “stabilize” around
a particular value. This limit value is the
theoretical probability of R.
:
• If students plot the data from columns
one and two, the graph will be approximately linear. The slope of this linear
approximation will represent the probability of R.
Cumulative Frequency
for the Occurrence
of R
20
.
In extending the table to keep track of the cumulative ratio of
occurrence of the event R for a total of 500 purchases, prospective teachers may opt to use the random number generator on a
graphing calculator (or computer) to perform this simulation.

It is also informative to have prospective teachers construct
graphs that include all cumulative data, discuss whether or not
the plotted points should be connected, and make predictions
about what would happen if the plots were continued.
As prospective teachers think about their graphs and discuss the
apparent trend, they become better prepared to make a reasonable conjecture about the probability of the event R.
The apparent variability that they notice will be indicated
graphically by a “jagged” pattern or trend when the number
of simulated purchases is small. In all cases, the plots should
“stabilize” and converge to �� . They may therefore conclude the
following:

The theoretical probability of getting at least one red gumball
is �� . The empirical probabilities when the number of simulated
purchases is large should be very close to �� = 0.6.
Discussion should lead to a long-run frequency definition of
probability:
LONG-RUN FREQUENCY DEFINITION OF PROBABILITY
(LAW OF LARGE NUMBERS)
Suppose an experiment consists of n repeated and identical
trials. Then for any event A, the probability of the event A is
given by
�����������������������������������������������������������
�
�
Grades 4–8 teacher tasks – 83
S3MTP • Chapter 3
2) Sample Space

Prospective middle grades teachers need to understand the
development of the long-run definition of probability and to realize
that there are a variety of legitimate ways to describe sample
spaces for an experiment.
Some sample spaces will include sample points that are equally
likely to occur, and some will not. It is important that prospective teachers realize that there are many ways to model realworld situations and to develop appropriate simulations.
Possible suggestions for an appropriate sample space include:
• Since the number of red gumballs possible is zero, one,
or two, one legitimate sample space might be written as
S = {0,1,2}.
• Another way of describing the sample space is S = {WW,
WR, RW, RR}, where WR represents white on the first
draw and red on the second draw.
• Students may think of labeling the balls so that they are
distinguishable, such as W1, W2, W3, W4, R1, R2 and then
listing the 30 different orderings of gumballs that could
be drawn (such as W1W2, W2W1, R1W1, etc.).
Thinking of the outcomes 0, 1, 2 as equally likely, prospective
teachers might erroneously conjecture that the probability of obtaining exactly zero red gumballs from the machine is �� . While
the sample space S������������ accurately describes the outcomes,
assigning equal probabilities to each sample point does not
reflect the fact that there are several ways in which Sally could
get exactly one red gumball. Similarly, thinking of a sample
space such as S = {WW, WR, RW, RR} presents another situation
in which elements do not have equally likely probabilities. So,
for example, the probability of getting exactly one red is not ��
even though two out of the four sample points have exactly one
red. Although more cumbersome to list, the last sample space
described does consist of sample points that are equally likely.
Certainly, there are real-life situations in which the outcomes
are not equally likely, and prospective teachers should have
experiences in considering these. It is important that they are
not led to think that there is only one correct way to describe
a sample space for an experiment, but to introduce them to
experiments which cannot easily be modeled with a coin, die, or
equally likely area-based spinner.
84 – Grades 4–8 teacher tasks
S3MTP • Chapter 3

Prospective middle grades teachers need to understand that
mathematical models are used to approximate real-life situations
for the purpose of sound decision-making and most often are not
an exact representation of the situation. The more we know about
the situation, the better we can model it.
Grades 4–8 teacher tasks – 85
S3MTP • Chapter 3
86 – Grades 4–8 teacher tasks
Chapter 4
GRADES 8–12 TEACHER TASKS —————————
Teachers of grades 8–12 need a thorough understanding of the
content central to early childhood–12 school mathematics,
including exposure to the mathematics expected of students in
grades below eight and in postsecondary institutions. Currently,
most secondary teacher preparation programs require coursework
that is essentially the same as that required for a traditional
mathematics major, offering students few opportunities to focus
specifically on connections between their coursework and the
State Board for Educator Certification standards, and between
their coursework and the secondary curriculum. Without guidance in making such explicit connections, it is difficult for
prospective teachers to see connections between the mathematics in their advanced courses and the mathematics they will be
expected to teach.
The focus of this chapter’s mathematical tasks is creating ways
to enhance college course content to reflect the mathematical
processes and in-depth knowledge necessary for teaching grades
8–12. Thus we highlight the following knowledge and skills for
8–12 teachers.
• Solve real-world problems by recognizing underlying
assumptions, using concepts of calculus, functions,
measurement, and geometry, and a variety of methods,
including technological methods;
• Communicate mathematical ideas using different types
of representation, including verbal, graphical, numerical, physical, and algebraic;
• Make connections among mathematical concepts
and equivalent representations for expressions of
mathematical ideas;
S3MTP • Chapter 4
• Recognize and generalize mathematical patterns;
• Recognize connections between number theory concepts and related operations and algorithms;
• Use properties of sequences and series to solve problems
involving finite and infinite processes;
• Analyze statistical information, use confidence interval
arguments to formulate and test hypotheses, and recognize misleading uses of statistics;
• Use knowledge of the history of mathematics to enrich
understanding of the development and progression of
mathematical ideas; and
• Use both formal and informal reasoning to explore and
justify mathematical concepts.
It is assumed that before being expected to investigate the tasks
described in this chapter, students will have had mathematical
experiences similar to those for prospective teachers of grades
EC–8 (such as those in chapters 1–3).
Chapter 4 includes six tasks:
• Geometry and Measurement: Rain Gauges;
• Number Concepts: Cantor Sets;
• Mathematical Processes: Using Geometric Models to
Predict Convergence;
• Probability and Statistics: Tests of Significance;
• History of Mathematics: The Life and Contributions of
Pierre de Fermat; and
• Geometry and Calculus Concepts: Using the Monte
Carlo Method to Estimate the Area Under a Curve.
This chart (also included in the preface to this book) suggests
some possible courses for which these tasks might be appropriate.
88 – Grades 8–12 teacher tasks
S3MTP • Chapter 4
Task Correlation Guide 4:
Tasks for grades 8–12 teacher certification level
8–12 Certification Level Tasks
Courses where task may be most appropriate
Geometry and Measurement: Rain Gauges
Foundations of Geometry, capstone course for
secondary teachers, College Algebra
Number Concepts: Cantor Sets
Number theory, capstone course for secondary
teachers
Mathematical Processes: Using Geometric Models to Predict
Convergence
Capstone course for secondary teachers, Calculus
II
Probability and Statistics: Tests of Significance
Statistics, capstone course for secondary teachers
History of Mathematics: The Life and Contributions of Pierre
de Fermat
History of Mathematics course, capstone course
for secondary teachers
Geometry and Calculus Concepts: Using the Monte Carlo
Method to Estimate the Area Under a Curve
Capstone course for secondary teachers, Calculus,
Statistics
Grades 8–12 teacher tasks – 89
S3MTP • Chapter 4
SECTION 4.1
GRADES 8–12 TEACHER TASK
GEOMETRY AND MEASUREMENT: RAIN GAUGES
When measuring rainfall, most forecasters use an
“official” rain gauge that was invented over 100 years
ago. The rain gauge uses a funnel 20 centimeters in
diameter that is placed on a cylinder 50 cm high and
20 cm in diameter. The funnel collects water in a tube
that has a cross-sectional area of the cross-sectional
area of the top of the funnel.
20 cm
STATE BOARD FOR EDUCATOR
CERTIFICATION MATHEMATICS STANDARD III.
GEOMETRY AND MEASUREMENT1
Grades 8–12 Teacher Knowledge
The beginning teacher of mathematics
knows and understands:
3.2k the use of mathematical reasoning
to develop, generalize, justify, and prove
geometric relationships;
3.6k how to use measurement to collect
data, to recognize relationships, and to
develop generalizations, including formulas;
3.7k how to locate, develop, and solve
real-world problems using measurement and
geometry concepts.
50 cm
SBEC MATHEMATICS STANDARD II.
PATTERNS AND ALGEBRA
Grades 8–12 Teacher Knowledge
Tube
1) Construct a scale on the tube that will reflect the
amount of rain (in inches) that has fallen.
2) What are the advantages or disadvantages of using
the “official” rain gauge instead of a cylindrical
gauge with an opening 20 cm in diameter?
3) Explain how we would construct another rain
gauge with a funnel of a diameter greater than 20
cm, so that the water level in its tube after a rain
would be the same as the water level in the tube of
the “official” rain gauge.
4) Construct a simplified version of the “official” rain
gauge by taking a conical funnel of radius R and
cone-height h and resting it on a tube so that the
water level in the tube would be identical to the
water level in an “official” rain gauge. A small portion of the bottom of the cone must be removed so
that the rain can collect in the tube. What are the
maximum dimensions (height and radius) of the
removed cone?
The beginning teacher of mathematics
knows and understands:
2.1k how to use algebraic concepts and reasoning to investigate patterns, make generalizations, formulate mathematical models,
make predictions and validate results.
SBEC MATHEMATICS STANDARD V.
MATHEMATICAL PROCESSES
Grades 8–12 Teacher Knowledge
The beginning teacher of mathematics
knows and understands:
5.4k connections among mathematical
concepts, procedures, and equivalent
representations.
The State Board for Educator Certification
mathematics standards for early childhood–4, 4–8,
and 8–12, with associated knowledge and skills
statements, may be referenced on the web at
1
www.sbec.state.tx.us/SBECOnline/standtest/
standards/ec4math.pdf;
www.sbec.state.tx.us/SBECOnline/standtest/
standards/4-8math.pdf; and
www.sbec.state.tx.us/SBECOnline/standtest/
standards/8-12math.pdf.
90 – Grades 8–12 teacher tasks
S3MTP • Chapter 4
Supporting Discussion
SBEC MATHEMATICS STANDARD V.
MATHEMATICAL PROCESSES
Grades 8–12 Teacher Skills
The beginning teacher of mathematics is
able to:
5.10s recognize that assumptions are made
when solving problems and identify and
evaluate those assumptions;
5.12s apply content knowledge to develop
a mathematical model of a real-world situation and analyze and evaluate how well the
model represents the situation;
By encountering modeling problems that involve algebraic and
geometric reasoning, prospective teachers can see the relevance
of high school mathematics in real-world applications. In this
task, they bring together ideas involving proportional reasoning,
algebra, and geometry.
In reading the description of the “official” rain gauge, students
may not understand the hidden assumptions in accepting rainfall readings from this gauge. For this reason, it is important to
have them work in cooperative groups of three to five. They will
need time to discuss this task.

5.17s apply mathematical methods to analyze practical situations; and
5.18s use mathematics to model and solve
problems in other disciplines, such as art,
music, science, social science, and business.
Constructing a scale on the tube requires students to consider
assumptions, such as that rain falls uniformly over a given region, etc.

ASSESSMENT
Students can work on this problem in
groups of three to five, but each student
should be required to submit an individual
solution to be evaluated for mathematical
correctness and mathematical communication.
A scoring rubric could rate aspects of content and form.
A content scoring rubric could use the following point system:
(5) problem brilliantly analyzed
(4) problem successfully analyzed
(3) problem not sufficiently analyzed
(2) problem poorly analyzed
(1) minimal work done on problem
A form scoring rubric could use the following point system:
(5) solid logic structure
(4) logic structure, but includes unnecessary
or unclear passages
Mathematical thinking and inquiry that is important for prospective teachers can be elicited by tasks that require taking a step
back to consider hidden assumptions, that are open-ended, or
that require justification.
Initially, students may not know how to approach the problem.
As a prompt, ask them to talk about what they think it means
when “1 inch” of rain has fallen.
Thus, the most difficult part of question 1) is justifying why the
height of the water collected in a cylinder 20 cm in diameter
gives an accurate representation of the number of inches of rain
that falls.

Students should be allowed the opportunity to discover or formulate the appropriate comparison for determining the scale on their
rain gauge.
Since the opening for the proposed cylindrical rain gauge and
the “official” one are the same, they would collect the same
volume. After setting up the comparison between volumes col�
lected, students will see that �� of an inch of rainfall would fill
one inch of the tube of the “official” rain gauge.
The last observation leads into step 2) of the task. That is, the
tube allows precise measurement of very small amounts of rainfall.
(3) logic structure unsatisfactory
(2) poor logic structure
(1) chaotic logic structure
Grades 8–12 teacher tasks – 91
S3MTP • Chapter 4
In step 3), let R represent the new radius. Using reasoning
similar to that in step 1), we compare the volume collected by
a cylindrical gauge of radius R and the volume collected in the
new tube. Students should arrive at the condition that the crosssectional area of the new tube must be ��� of the cross-sectional
area of the opening.
For step 4), we use the following diagram.
R
h
r
a
The maximum dimensions (height and radius) of the removed
portion of the bottom of the cone would correspond to the radius of the tube (in practice, one could possibly use glue to attach
the tube to the funnel). Above, we showed that we must have
��
�
(1) �� � ��
for the height of water in the tube to correspond to the height of
water in the tube of the “official” rain gauge.

Students may have trouble seeing the similar triangles if they
have not drawn a picture of the cone and labeled it properly.
From the diagram, we use similar triangles to get
(2)
�
�
.
=
�
�
Using (1) and (2) we get
�
�����
(3) � �
��
We also see that � � � .
��
92 – Grades 8–12 teacher tasks
EXTENSION IDEA
For the rain gauge constructed in step 4),
determine a scale along the side of the cone
that measures the amount of rain that falls
after the tube fills up.
S3MTP • Chapter 4
SECTION 4.2
GRADES 8–12 TEACHER TASK
NUMBER CONCEPTS: CANTOR SETS
The middle third Cantor set is easily constructed by
performing a sequence of deletion operations on the
unit interval. Begin with the interval [0, 1]. Remove
the middle third of the interval, and you will have the
intervals [0, ] and [ , 1]. Next, remove the middle
third of each of those intervals. Four intervals remain:
[ 0, ], [ , ], [ , ], and [ ,1] . Repeat this process
of removing the middle third indefinitely. The points
that remain comprise the middle third Cantor set.
STATE BOARD FOR EDUCATOR
CERTIFICATION MATHEMATICS STANDARD I.
NUMBER CONCEPTS
Grades 8–12 Teacher Knowledge
The beginning teacher of mathematics
knows and understands:
1.3k the relationship among number
concepts, operations and algorithms, and
the properties of numbers, including ideas of
number theory.
1) What is the length of the removed intervals?
2) Is the middle third Cantor set non-empty? Justify.
3) Show that the middle third Cantor set consists precisely of those numbers in [0,1] whose base three
expansion does not contain the digit 1 (i.e. all
–
–
–
numbers a13 1 + a23 2 + a33 3 + … with ai = 0 or 2
for each i). Reconcile what you show with the fact
that we know that is an element of the middle
third Cantor set.
SBEC MATHEMATICS STANDARD II.
PATTERNS AND ALGEBRA
Grades 8–12 Teacher Knowledge
The beginning teacher of mathematics
knows and understands:
2.5k that patterns are sometimes misleading.
Supporting Discussion
ASSESSMENT
In class:
Show that .3333…=1 in base 4.
Project (groups of 2):
Construct a set that consists precisely of
those numbers whose base five expansion
does not contain the digit 2, by beginning
with an interval, removing intervals, and
repeating the process indefinitely.
This task uses geometric series, set theory, number concepts, and
limits.
Students are often told that 0.99… = 1 or have seen a proof of
it in various formats. Posing a problem with seemingly contradictory results motivates students to extend their reasoning to
include 0.22… = 1 in base three.
In analysis courses, students may encounter the Cantor set and
investigate its denumerability and cardinality, for example.
However, students at various levels can investigate problems
such as this one. Because of the problem’s level of difficulty, it
is important that students have an opportunity to discuss the
problem with each other.

Based upon their answer to question 1), students may decide
that the middle third Cantor set is empty and try to justify this.1
However, the instructor could then propose points (such as 3 )
that they might consider.
Grades 8–12 teacher tasks – 93
S3MTP • Chapter 4
Encourage students to draw various stages of the middle third
Cantor set, make a table, and so on, in order to convince themselves that the middle third Cantor set is (at least) contained in
the proposed set in question 3). After allowing sufficient time
for students to grapple with this, an instructor may prompt students who are having difficulty by asking them to consider the
base three expansions of the elements that remain after the first
iteration and then the second iteration described in the task’s
introduction.
Once the students are convinced that the set proposed in question 3) precisely expresses the elements of the middle third
Cantor set, remind them that they probably used the point �� to
justify that the middle third Cantor set is non-empty (and that
its base three expansion is 0.1).

SBEC MATHEMATICS STANDARD I.
NUMBER CONCEPTS
Grades 8–12 Teacher Skills
The beginning teacher of mathematics is
able to:
1.14s demonstrate a sense of equivalency
among different representations of rational
numbers;
1.15s select appropriate representations
of real numbers (e.g., fractions, decimals,
percents, roots, exponents, scientific notation) for particular situations and justify
that selection;
1.22s apply all skills specified for teachers
in grades EC–8, using content and contexts
appropriate for grades 8–12.
An important aspect in the problem-solving process is revisiting
conclusions and checking for the reasonableness of a solution.
When considering this false counterexample to their findings,
students will have the opportunity to revisit the fact that there is
not a unique way to express rational numbers.
SBEC MATHEMATICS STANDARD II.
PATTERNS AND ALGEBRA
Grades 8–12 Teacher Skills
The beginning teacher of mathematics is
able to:
2.20s analyze the properties of sequences
and series and use them to solve problems
involving finite and infinite processes;
including problems related to simple, compound, and continuous interest rates, as well
as annuities.
94 – Grades 8–12 teacher tasks
S3MTP • Chapter 4
SECTION 4.3
GRADES 8–12 TEACHER TASK
MATHEMATICAL PROCESSES:
USING GEOMETRIC MODELS TO PREDICT
CONVERGENCE
STATE BOARD FOR EDUCATOR
CERTIFICATION MATHEMATICS STANDARD V.
MATHEMATICAL PROCESSES
Grades 8–12 Teacher Knowledge
1) Consider the equilateral triangle given below with
circles inscribed as shown. (Note that there is an infinite
sequence of circles converging into each vertex.)
The beginning teacher of mathematics
knows and understands:
5.4k connections among mathematical
concepts, procedures, and equivalent
representations.
SBEC MATHEMATICS STANDARD V.
MATHEMATICAL PROCESSES
Grades 8–12 Teacher Skills
The beginning teacher of mathematics is
able to:
5.3s use formal and informal reasoning to
explore, investigate, and justify mathematical ideas;
5.7s recognize that a mathematical problem
can be solved in a variety of ways, evaluate
the appropriateness of various strategies, and
select an appropriate strategy for a given
problem;
5.8s evaluate the reasonableness of a solution to a given problem;
5.9s use physical and numerical models to
represent a given problem or mathematical
procedure;
5.15s explore problems using verbal,
graphical, numerical, physical, and algebraic
representations.
Suppose the radius of the largest inscribed circle is 2.
The sum of the areas of the circles forms a series. Also,
the sum of the circumferences forms a series. For each
series:
a) Predict whether the series will converge. Justify in
words.
b)Show whether the series converges.
c) If the series does converge, what is the sum? How can
you check for reasonableness of your answer?
d)Do your answers to a) and b) depend on the value
assigned to the radius of the largest inscribed circle? If
so, in what way?
2) Consider a process similar to the process in part 1),
except that the sets of circles emanating from the first
circle (of radius 2) have successive radii of length 1/k for
k = 1, 2, 3, 4, …. (See figure below.) As before, the sum
of the circumferences of these circles forms a series and
the sum of the areas forms a series. For each series:
a) Predict whether the series will converge. Justify in
words.
SBEC MATHEMATICS STANDARD II.
PATTERNS AND ALGEBRA
b)Show whether the series converges.
Grades 8–12 Teacher Skills
c) Do your results in b) contradict the results you
obtained for the figure above with circles converging
to each vertex? Why or why not?
The beginning teacher of mathematics is
able to:
2.18s apply all skills specified for teachers
in grades EC–8, using content and contexts
appropriate for 8–12;
2.20s analyze the properties of sequences
and series and use them to solve problems
involving finite and infinite processes;
including problems related to simple, compound, and continuous interest rates, as well
as annuities.
Grades 8–12 teacher tasks – 95
S3MTP • Chapter 4
Supporting Discussion

Future mathematics teachers need problem-solving experiences
that integrate ideas, concepts, and knowledge from several areas
of mathematics.
The best use of this task is in a group setting. By design, this task
fosters student collaboration and interaction. The task could be
used in a precalculus or college algebra course, a capstone mathematics course for teachers, or in the study of series in calculus.
However, the task has been used most often in a calculus setting
to encourage students to think conceptually about the convergence or divergence of series.

Tasks that ask for prediction and sense-making are important in
developing mathematical habits of mind.
This task uses geometric series, the harmonic series, geometry,
and elementary trigonometry; further, it extends to ideas that
could lead into a discussion on fractals and fractal dimension.
The questions are structured so that students must first think
about the geometry and then use the algebra to verify their
predictions.

Students need more experiences in comparing analytical
results and geometrical interpretations (and vice versa) as
well as exposure to questioning techniques that promote
good problem-solving skills.
In attempting to solve this problem, calculus-level students
often ignore the geometry of the situation when trying to predict
whether the series converges. They quickly try to set up a series
and determine convergence or divergence without making sense
of the situation. Asking them to predict and then justify this
in words focuses their attention on sense-making in problem
solving. Often, prompting from the instructor is necessary to
pull students away from trying the algebraic approach first. An
instructor may have to direct student groups to base their prediction on nonalgebraic reasoning, or ask the students to support
their prediction using two different approaches.
In attempting to set up the series for part 1), students often
become stumped and puzzled that they are expected to recall
geometric relationships and ideas in order to determine the
terms in the series.

Students are not accustomed to encountering a problem with
series in which the modeling for determining the terms of the
series involves using knowledge from several previous courses.
96 – Grades 8–12 teacher tasks
S3MTP • Chapter 4
The instructor may have to remind students that similarity,
properties of equilateral triangles, etc., may help them determine
the terms in the series.
EXTENSION IDEA
Consider the following process that begins
with a regular hexagon of side length s.
First, remove the middle third of each side
and replace it with two segments, each
of equal length to the segment removed,
directed toward the interior of the hexagon.
Now, remove the middle third of each side
of the new polygon and replace it with
two segments, each of equal length to the
segment removed, directed toward the interior of the polygon. Continue this process
indefinitely.

The instructor should, however, allow time for the students to
discuss and recall the ideas rather than providing a mini-lecture
to review the assumed prior knowledge.
This encourages discussion among the students and gives the
instructor important information about strengths and/or weaknesses in the students’ prior knowledge.
After some time, students may realize that they should use similarity in trying to determine an expression for the radii of the subsequent circles. However, they may attempt something like this:
This produces a fractal called
the Koch snowflake.
a) Find the area of the region bounded by
the Koch snowflake.
b) Find the perimeter of the region (i.e., the
length of the Koch snowflake).
c) Discuss the reasonableness of your findings in parts a) and b). Are your findings
contradictory? Why or why not?
Instead, visualizing the situation in the following way will help
determine an expression for the radii of the subsequent circles.
Using justifications from elementary geometry and trigonometry,
students can derive that the height of the original equilateral triangle is 3r, where r represents the radius of the largest inscribed
circle. Using similarity, one sees that the radii of the inscribed
circles are always 1/3 of the previous radius (in the process of
construction). The resulting geometric series converges for
both series (areas and circumferences). This may be surprising
to some students, who may have predicted that the sum of the
circumferences for part 1) of the task diverges.
In part 2) of the task, students may (incorrectly) predict that,
based on their findings for part 1), the sum of the circumferences
converges.
Grades 8–12 teacher tasks – 97
S3MTP • Chapter 4
However, after classroom discussion they will see that the series
formed by the sum of the circumferences diverges (it is a multiple of the harmonic series) and the series formed by the sum
of the areas converges. In a capstone course, an instructor may
want to probe deeper into the students’ reasoning on whether
their results are reasonable and noncontradictory.
In the extension idea, the hexagon is used to form the Koch
snowflake, to underscore the fact that this curve could be
enclosed by a hexagon. In a capstone course, an instructor may
want to include background material on fractals and/or probe
students’ reasoning and sense-making of an object that has infinite perimeter but finite area.
98 – Grades 8–12 teacher tasks
S3MTP • Chapter 4
SECTION 4.4
GRADES 8–12 TEACHER TASK
PROBABILITY AND STATISTICS:
TESTS OF SIGNIFICANCE
STATE BOARD FOR EDUCATOR
CERTIFICATION MATHEMATICS STANDARD IV.
PROBABILITY AND STATISTICS
The Saxet School District reports that 48.7% of a
sample of 500 tenth-grade students mastered all
objectives on a state assessment.
Grades 8–12 Teacher Knowledge
1) Does this sample result provide evidence that
the proportion of the students from Saxet School
District who mastered all objectives on the state
assessment differs from 50%? Consider an appropriate test and report your conclusion. Discuss any
conditions necessary.
The beginning teacher of mathematics
knows and understands:
4.4k statistical inference and how it is used
in making and evaluating predictions.
2) Determine a sample size for which a sample proportion of pˆ = .487 does differ significantly from .5 at the
α = .05 significance level. Report the details of the
test results for this sample size. Interpret your findings.
SBEC MATHEMATICS STANDARD IV.
PROBABILITY AND STATISTICS
Grades 8–12 Teacher Skills
The beginning teacher of mathematics is
able to:
4.26s analyze and interpret statistical information from the media, such as the results
of polls and surveys, and recognize valid and
misleading uses of statistics;
4.28s use confidence interval arguments to
formulate and test hypotheses.
SBEC MATHEMATICS STANDARD V.
MATHEMATICAL PROCESSES
Grades 8–12 Teacher Skills
The beginning teacher of mathematics is
able to:
5.10s recognize that assumptions are made
when solving problems and identify and
evaluate those assumptions.
Supporting Discussion
Understanding hypothesis testing, confidence intervals, and
p-values empowers students to investigate data and determine
if certain comparisons are statistically significant. Statistical
analysis at this level is found in much of the research literature
in education. In particular, for prospective teachers, this understanding would aid in the interpretation of the applicability, reliability, and relevance of research findings in education,
including data on their own students’ performance and how it
compares to statewide, national, and international data.
In this task, students must decide on an appropriate statistical
test and describe any conditions needed for the test to apply.

Students need experience identifying assumptions they make
when solving problems.
For example, the task does not mention whether the sample is a
random sample. This must be assumed before the hypothesis test
can appropriately be applied.
The p-value for part 1) is very large, so the null hypothesis that
“50% of the students in Saxet School District mastered all objectives on the statewide assessment of academic skills” will not
be rejected. Students should be encouraged to practice stating
statistical assertions. That is, an instructor should hear students
Grades 8–12 teacher tasks – 99
S3MTP • Chapter 4
making correct statements such as “this sample result doesn’t
provide evidence that the proportion of Saxet School District
students who mastered all objectives differs significantly from .5.”
Part 2) of this task does not require difficult computations to
obtain an answer; however, students in a first course in statistics
do not often encounter questions posed in this way.

Providing opportunities for students to investigate the notions of
p-value, sample size, and statistical significance promote development of statistical literacy.
Small-group and full-class discussions about sample size should
follow students’ conclusions that a minimum sample of size 5683
with a mean of .487 would require rejection of the null hypothesis for the given significance level.

It is important here to ask students probing questions about why
they think that the sample would have to be so much larger.
Students could also be asked to change � from .487 to .49
and/or to .48. This would allow them to investigate how these
changes affect the sample size necessary to reject or retain the
null hypothesis. Modifying the task or extending it in this way
helps students develop strategies for investigating features or
properties of statistical relationships. It also addresses the importance of statistical literacy for future mathematics teachers.
100 – Grades 8–12 teacher tasks
TECHNOLOGY NOTE
Students can incorporate graphing calculator technology or dynamic statistics software
packages such as Fathom Dynamic Statistics
and use statistical data sets that are easily
obtained from the Internet. They can then
use the technology to investigate a full
range of statistical ideas.
S3MTP • Chapter 4
SECTION 4.5
GRADES 8–12 TEACHER TASK
HISTORY OF MATHEMATICS:
THE LIFE AND CONTRIBUTIONS OF
PIERRE DE FERMAT
STATE BOARD FOR EDUCATOR
CERTIFICATION MATHEMATICS STANDARD VI.
MATHEMATICAL PERSPECTIVES
Write an essay describing the life of Pierre de Fermat
and his contributions to the field of mathematics. Discuss modern applications of his work to cryptography.
Grades 8–12 Teacher Skills
The beginning teacher of mathematics is
able to:
6.1s use key events and knowledge of
specific individuals throughout the history
of mathematics to illustrate age-appropriate
mathematical concepts;
6.3s use the historical developments of
mathematical ideas to illustrate how mathematics progresses from concrete applications to abstract generalizations.
Supporting Discussion

Few students study the history of mathematics in an organized
course, either because one is not offered or because the credit
structure for their program does not encourage it. It is thus the
responsibility of mathematics faculty to provide opportunities
within required courses for these students to learn the historical
significance of the topics studied. Students can investigate the
central mathematical concepts for a given course by researching
the historical foundations that support the concepts. Questions
such as the one above help highlight important accomplishments and diverse areas where mathematics is applied.

ASSESSMENT
It is helpful to provide a rubric to students
so that they know what is expected in their
written essays.
The following is an abbreviated scheme:
(4 points) Well organized, excellent logical
structure, and insightful.
(3) Well organized, good logical structure,
and somewhat insightful.
(2) Somewhat organized, some logical
structure, no insight provided.
(1) Chaotic logical structure and organization, no insight provided.
Understanding the rich and culturally diverse history of mathematics prepares prospective teachers to convey to their students
the contributions by various individuals and cultures to the field of
mathematics.
Historical perspectives may focus on the contributions of individuals such as Fermat or the contributions of several mathematicians
in understanding ideas or concepts such as negative numbers,
functions, symbolic notation, limits, etc.
Often, students study Fermat’s Last Theorem and Fermat’s Little
Theorem in a number theory course. Students may already be
aware that Andrew Wiles proved Fermat’s Last Theorem in
the 1990s. Although understanding Wiles’s proof is beyond the
scope of the undergraduate curriculum, Fermat’s Little Theorem
can be approached at the undergraduate level. Fermat’s Little
Theorem states that if p is a prime and a is a positive integer not
divisible by p, then
. In a basic number theory
class, students may be required to prove this theorem. They
may also be asked to use this theorem to determine the value of
something similar to
. Students can then extend
this theorem to prove another: if p is a prime and a is a positive
Grades 8–12 teacher tasks – 101
S3MTP • Chapter 4
integer not divisible by p, then
. The instructor may also have students investigate cases of odd composite
numbers that satisfy Fermat’s Little Theorem—the Carmichael
numbers.
As students study Fermat’s accomplishments in mathematics,
they also may come to understand that branches of mathematics such as commutative ring theory were discovered through
unsuccessful attempts to prove Fermat’s Last Theorem.

Historical investigations highlight connections between
theoretical mathematics and applied mathematics.
In addition, asking students to investigate connections between
Fermat’s theorems about modular arithmetic and cryptography
underscores how the theoretical mathematics preceded this application by 350 years.

Students often see mathematics as a collection of technical skills
and are not accustomed to investigating the historical struggles
that have led to prevailing understanding and applications.
Faculty may incorporate connections to historical foundations
by assigning brief historical essays either to the whole class or
to individuals who are then required to report their findings.
Faculty may also make assignments such as creating a timetable
or evolution table of mathematical ideas that conveys how the
ideas developed over time and how they are related to contemporary mathematics. It is essential to consider the interplay
between the mathematical understanding and the historical understanding that is gained from these assignments. Assignments
that are too broad in nature will yield poor results. It is important to develop a primary focus for the assignment and then two
to three subquestions that will direct the bulk of the research.

Exposure to the historical development of mathematics helps
students move toward developing broad mathematical literacy.
This includes knowledge that controversial mathematical
ideas still occur in modern times. For example, Cantor’s proof
that there are different sizes of infinity shook the mathematical community in the late nineteenth century. Also, historical
perspectives that focus on understanding the conceptual development of mathematical ideas and their significance to the
mathematical community provide powerful tools for helping
shape students’ views of mathematics. Students get a glimpse of
mathematics as a creative endeavor and as a dynamic field that
is still developing.
102 – Grades 8–12 teacher tasks
OTHER POSSIBLE ESSAY TOPICS
• Archimedes
• Euclid
• The Parallel Postulates
• Pythagoras
• π
• Mathematicians from historically underrepresented groups
• Infinity
• Squaring the circle
• e
• Golden ratio
S3MTP • Chapter 4
SECTION 4.6
GRADES 8–12 TEACHER TASK
GEOMETRY AND CALCULUS CONCEPTS:
USING THE MONTE CARLO METHOD TO ESTIMATE
THE AREA UNDER A CURVE
STATE BOARD FOR EDUCATOR
CERTIFICATION MATHEMATICS STANDARD II.
PATTERNS AND ALGEBRA
1) Write a program that uses the Monte Carlo
Method to estimate an integral of a positive function. Your program should accept input of a function, an interval, and the number of points the user
wishes to use in the estimation. It should also plot
the function and the points in such a way that the
method is illustrated.
Grades 8–12 Teacher Skills
The beginning teacher of mathematics is
able to:
2.25s describe exponential, logarithmic,
and logistic functions algebraically and
graphically, analyze their algebraic and
graphical properties, and use these to model
and solve problems using a variety of methods, including technology;
2) Use your program to estimate
using 1000 points.
3) Accurately approximate
to two decimal places. Make a conjecture and support it with an explanation.
2.28s investigate and solve problems using
techniques of differential and integral calculus along with a variety of other methods,
including technology.
4) Use documents you find on the Internet or elsewhere to write a short paragraph on the history
of the Monte Carlo Method. In addition, discuss
which fields other than mathematics use this
method. Include an example of how it is used in at
least one of those fields.
SBEC MATHEMATICS STANDARD VI.
MATHEMATICAL PERSPECTIVES
Grades 8–12 Teacher Knowledge
The beginning teacher of mathematics
knows and understands:
6.5k how mathematics is used in a variety
of careers and professions.
Supporting Discussion

ASSESSMENT
Why do we need to use the Monte Carlo
Method to approximate
? Could we
evaluate this integral using the methods
taught in calculus?
Applicable mathematics-specific
technology
• Mathematica
• MATLAB
• Maple
• Fathom
Using the Monte Carlo Method to approximate solutions to a variety of mathematical problems provides students with a nontrivial
opportunity to incorporate technology in problem solving.
This task was designed for students who have completed an
introduction to integration. The Monte Carlo Method is often
described using a dartboard analogy. Divide a rectangular-shaped
dartboard into a red part, representing the area beneath the
curve, and a blue part, representing all other space on the board.
Then n darts are thrown at the dartboard. Let m denote the
number of darts that land in the red part. Assuming that darts
have an equal chance of landing anywhere on the board, the following proportion (approximate) will hold:
Area under curve ~ number of darts landing in red part
~
Area of dartboard
total number of darts thrown
Grades 8–12 teacher tasks – 103
S3MTP • Chapter 4
Note that the area under the curve is then the fraction m/n times
the area of the rectangle. Most students follow this analogy, and
thus the exercise could be modified for a student of any level by
replacing the integral with area under a curve.

The prevalence of computers and mathematics-specific technologies makes it critical that prospective teachers gain familiarity
with various uses of technology and programming.
Many mathematics students are required to complete a course
that focuses on using the computer to do mathematics. These
courses often require that the student learn programming skills.
This exercise is an example of what a student might be asked to
do in such a course.
Programming the Monte Carlo Method requires only the knowledge of a random number generator and a for or a while loop.
There is also the issue of determining the bounds for the height
of the rectangle. Because the exercise specifically requires the
function to be positive on the interval, it may be assumed that
the base of the rectangle is on the x-axis. Thus it is only necessary to determine the maximum value of the function on the
given interval. Students who have completed calculus should
have no problem with this. If this task is modified for students
without a calculus background, the instructor might consider
providing the student with instructions on how to overestimate
the maximum value on the interval. Since finding the maximum
value of the function is not the objective of the task and time is
always an issue, the instructor may wish to simply provide the
bounds for the rectangle.

Programming provides another setting in which students must
revisit mathematical concepts. Success in programming often
indicates understanding of the underlying concepts.
Frequently, the programming process reveals any misconceptions
or weaknesses the student may have in a specific area of mathematics. Thus it is helpful to provide prompts that might further
clarify the student’s understanding. The student is encouraged
to examine the method visually in this task by being asked to
create a plot of the function and the points that were generated.
The instructor might suggest that the points that hit below the
curve be colored red and the points that hit above the curve,
blue. The student should then see that the area below the curve
becomes more filled in as the number of points generated is increased, an illustration of the law of large numbers that underlies
the Monte Carlo Method. The graphs below are examples of
what an instructor should expect. They were created using a
104 – Grades 8–12 teacher tasks
S3MTP • Chapter 4
For more information on Mathematica,
see the Wolfram Research, Inc. website, at
www.wolfram.com.
2
Mathematica2 program that is included at the end of
this task.
Number of points=1000; Approximate Area =0.858
Number of points=5,000; Approximate Area=0.8632
Step 3) of the task is designed to test the student’s understanding of the limiting concept that is illustrated by the graphs. The
student should be aware of the possibility of some fluctuation
because of the nature of the experiment and the random number
generator. For example, rerunning the program multiple times
but using the same number of points results in varying answers.
The list below illustrates this.
1
2
3
4
5
6
0.8808
0.8976
0.8716
0.8596
0.8924
0.8736
Grades 8–12 teacher tasks – 105
S3MTP • Chapter 4
7
8
9
10
11
12
13
14
15
0.8744
0.8752
0.8816
0.8904
0.8596
0.8624
0.8636
0.8872
0.8956
After some experimentation, students can determine that as
they sample more points, their estimates seem to hover around
the value 0.88.
Current trends in early childhood–12 education place increased
emphasis upon introducing mathematical topics via real-world
applications. Requiring students to spend some time researching the Monte Carlo Method provides a historical connection
as well as a connection to its use in contemporary applications.
By inputting “Monte Carlo Method” into any search engine on
the Web, students will see that this method is connected to the
approximation of pi, the thermalization of a molecular dynamics
trajectory, quantum statistics, using radiation therapy to eradicate tumors, molecular excitation, and so on. It is used not only
by mathematicians but also by statisticians, financial analysts,
chemists, astronomers, astrophysicists, and many other professionals in mathematics-based fields.
106 – Grades 8–12 teacher tasks
S3MTP • Chapter 4
Below is the Mathematica program to use the Monte Carlo
Method to estimate the area under a curve:
f[x_]:=Exp[-x^2];
numpts=5000;
xlistb={};
ylistb={};
xlista={};
ylista={};
For [i=1,i<numpts+1,i++,
{xcoord=Random[Real, {0,2}];
ycoord=Random[Real, {0,1}];
If[f[xcoord]>ycoord,
xlistb=Append[xlistb,xcoord],xlista=Append
[xlista,xcoord]];
If[f[xcoord]>ycoord,
ylistb=Append[ylistb,ycoord],ylista=Append
[ylista,ycoord]];
}];
pointsbelow={};
pointsabove={};
For[i=1,i<Length[xlistb]+1,i++,
pointsbelow=Append[pointsbelow,{xlistb[[i]],yl
istb[[i]]}]];
For[i=1,i<Length[xlista]+1,i++,
pointsabove=Append[pointsabove,{xlista[[i]],yl
ista[[i]]}]];
p1=Plot[f[x],{x,0,2},DisplayFunction\
[Rule]Identity];
p2=ListPlot[pointsbelow,PlotStyle\[Rule]{PointSize
[.02],RGBColor[1,0,0]},
DisplayFunction\[Rule]Identity];
p3=ListPlot[pointsabove,PlotStyle\[Rule]{PointSize
[.02],RGBColor[0,0,1]},
DisplayFunction\[Rule]Identity];
Show[{p1,p2,p3},DisplayFunction\
[Rule]$DisplayFunction];
Area=(Length[xlistb]/numpts)*2
N[Area]
Grades 8–12 teacher tasks – 107
S3MTP • References
Supporting and Strengthening Standards-Based Mathematics Teacher Preparation:
Guidelines for Mathematics and Mathematics Education Faculty
References
This references list includes both materials cited in the Guidelines and suggestions for further reading.
Ball, D. L. (1990). Prospective elementary and secondary teachers’ understanding of division. Journal
for Research in Mathematics Education, 21, 132–144.
Ball, D. L. (1991). Research on teaching mathematics: Making subject matter knowledge part of the
equation. In J. Brophy (Ed.), Advances in research on teaching (Vol. 2, pp. 1–48). Greenwich,
CT: JAI Press.
Begle, E. (1979). Critical variables in mathematics education: Findings from a survey of empirical literature.
Washington, DC: Mathematical Association of America.
Charles A. Dana Center. (2003). Advanced Mathematics Educational Support: Support,
recommendations, and resources for facilitating collaboration between higher education mathematics
faculty and Texas public high schools, by Ray Cannon, Richard Parr, and Ann Webb. Austin,
TX: Author.
Charles A. Dana Center and Texas Education Agency. (2001). Rethinking secondary mathematics:
In-depth secondary mathematics. [TEXTEAMS teacher professional development institute].
Austin, TX: Author.
Charles A. Dana Center and Texas Education Agency. (2002). Statistical reasoning across the TEKS.
[TEXTEAMS teacher professional development institute]. Austin, TX: Author.
Charles A. Dana Center, Texas Statewide Systemic Initiative. (1996). Guidelines for the mathematical
preparation of prospective elementary teachers. Austin, TX: Author. Retrieved October 2003
from www.utdanacenter.org/ssi/docs/GuideMath97.pdf.
College Board and Charles A. Dana Center. (1998). Advanced Placement Program mathematics vertical
teams toolkit, by James Epperson, Deborah Holtzman, Susan May, Dara Sandow, and Dick
Stanley. Austin, TX: Author.
Darling-Hammond, L., Wise, A., & Klein, S. (1995). A license to teach: Building a profession for the
21st century schools. Boulder, CO: Westview Press.
Driscoll, M. (1999). Fostering algebraic thinking. Portsmouth, NH: Heinemann.
Ewing, J. (ed.) (1999). Towards excellence: Leading a mathematics department in the 21st century.
Providence, RI: American Mathematical Society Task Force on Excellence. Retrieved
October 2003 from www.ams.org/towardsexcellence/.
109
S3MTP • References
Graeber, A. A., Tirosh, D., & Glover, R. (1989). Preservice teachers’ misconceptions in solving
verbal problems in multiplication and division. Journal for Research in Mathematics Education,
20, 95–102.
Heid, M. K. (1997). The technological revolution and the reform of school mathematics. American
Journal of Education, 106, 5–61.
Kilpatrick, J., Swafford, J., & Findell, B. (Eds.). (2001). Adding it up: Helping children learn
mathematics. Washington, DC: National Academy Press.
Ma, L. (1999). Knowing and teaching elementary mathematics: Teachers’ understanding of fundamental
mathematics in China and the United States. Mahwah, NJ: Lawrence Erlbaum Associates.
Monk, D. A. (1994). Subject area preparation of secondary mathematics and science teachers and
student achievement. Economics of Education Review, 13(2), 125–145.
National Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics.
Reston, VA: Author.
Schifter, D. (1999). Reasoning about operations: Early algebraic thinking in grades K–6. In L. V.
Stiff (Ed.), Developing mathematical reasoning in grades K–12. 1999 Yearbook of the National
Council of Teachers of Mathematics (pp. 62–81). Reston, Va.: National Council of Teachers
of Mathematics.
Schifter, D., Bastable, V., & Russell, S. J. (with Cohen, S., Lester, J. B., & Yafee, L.) (1999). Building
a system of tens: Casebook. Parsippany, NJ: Dale Seymour Publications.
Sowder, J. T., Philipp, R. A., Armstrong, B. E., & Schappelle, B. (1998). Middle grades teachers’
mathematical knowledge and its relationship to instruction. Albany, NY: SUNY Press.
Stigler, J., & Hiebert, J. (1999). The teaching gap. New York: Free Press.
Texas State Board for Educator Certification. (2001). EC–4 Mathematics Educator Standards.
Austin, TX: Author. Retrieved October 2003 from www.sbec.state.tx.us/SBECOnline/
standtest/standards/ec4math.pdf.
Texas State Board for Educator Certification. (2001). 4–8 Mathematics Educator Standards. Austin,
TX: Author. Retrieved October 2003 from www.sbec.state.tx.us/SBECOnline/standtest/
standards/4-8math.pdf
Texas State Board for Educator Certification. (2001). 8–12 Mathematics Educator Standards. Austin,
TX: Author. Retrieved October 2003 from www.sbec.state.tx.us/SBECOnline/standtest/
standards/8-12math.pdf
110
S3MTP • References
Tucker, A., Fey, J., Schifter, D., & Sowder, J. (2001). The mathematical education of teachers.
CBMS Issues in Mathematics Education (11). Providence, R.I.: The American Mathematical
Society, Mathematical Association of America.
U.S. Department of Education, National Center for Education Statistics. (1996). Pursuing Excellence:
A study of U.S. eighth-grade mathematics and science teaching, learning, curriculum, and
achievement in international context, NCES 97–198. Washington, DC: U.S. Government
Printing Office.
U.S. Department of Education, National Center for Education Statistics. (2000). Pursuing excellence:
Comparisons of international eighth-grade mathematics and science achievement from a U.S.
perspective, 1995 and 1999, NCES 2001–028. By Patrick Gonzales, Christopher Calsyn,
Leslie Jocelyn, Kitty Mak, David Kastberg, Sousan Arafeh, Trevor Williams, and Winnie
Tsen. Washington, DC: U.S. Government Printing Office.
111
S3MTP • References
112