Download COURSE OVERVIEW

A One Semester Course on
the Introduction of Mathematical Thought
for Secondary Students
by
Edward Keyes
prepared for the
Senior Seminar in Mathematics
at Seattle Pacific University
Autumn 2002
Table of Contents
I. Course Rationale....................................................................................................3
II. Course Overview...................................................................................................6
III. Course Outline.....................................................................................................7
IV. Unit Rationales
A. Probability...............................................................................................11
1. Unit Outline for Probability........................................................13
2. Lesson Plans for Probability........................................................17
B. Combinatorics..........................................................................................46
C. The Binomial Theorem............................................................................48
D. Series and Limits.....................................................................................50
E. Graph Theory...........................................................................................53
F. Logical Proof ...........................................................................................56
G. Number Theory........................................................................................58
H. General Forms..........................................................................................60
I. Geometric Construction............................................................................62
J. Triangle Oddities......................................................................................63
Appendix: Washington State Essential Academic Learning Requirements (EALR's)
.....................................64
Bibliography...............................................................................................................66
2
Course Rationale
The purpose of this course is to provide advanced math students an alternative to
AP calculus courses in their final year and still challenge them at the college prep level.
Since many students who enroll in AP Calculus do not always take the test, and those
who do, do not always receive college credit for their efforts, and since college professors
express some desire that students take calculus at the college level anyway, this course
will provide some unique opportunities.
Traditional high school math programs usually leave little in the way of options
for students who may not wish to take calculus yet, but wish to continue their
mathematical education. Where to go?
Students should expect, but were not intimidated by the idea of a challenging
course. The variety of topics, many of which will be new to them, should be counter to
the assumption that math classes were boring and predictable. Also attractive is the
exploratory nature of the course where conceptual challenges predominate and busy work
is minimal.
Although all math instructors attempt to highlight mathematical methods of
thought, calculation and formulas often become the driving emphasis to elicit topical
progress. This course focuses on that thought process and no special skills outside of
algebra and a most basic familiarity with geometry are prerequisites. While critical
thinking is applicable as cross-disciplinary skill, this course is unique in that honing and
defining the students' abilities is the primary focus.
This leads directly into the challenging aspects of the course. Writing well will be
expected, not just computing. Problem solving approaches will be left up to the student
in many cases, and some problems may be left unsolved, or proved unsolvable,
something many have not yet encountered in the math classroom. Creativity and finding
more that one correct answer will be encouraged, rather than just re-capping an
instructor's presentation.
While being good preparation for college level mathematics, the course will not
cover any one subject to a depth of understanding expected in any single college course.
This course is intended to give students a mere taste of a wide variety of mathematical
disciplines, so that when they arrive at college and open the catalog, the math courses
listed will have meaning. What the course does offer are the basic concepts and
definitions for each topic to give each student a beginner's vocabulary and some
expectations about what direction each area will take them.
The scheduling of the units takes the Seattle Public School calendar into account
concerning: the length of the units, so as to have the course fit into a single semester; the
placement of the introduction of ideas, so these do not conflict with holidays or in-service
days; the conclusions, generally falling shortly before holidays or vacations to avoid
splitting a topic around an extended school absence.
It is easier to list the concepts not covered in this course that are listed in the
academic standards in math in the EALR's (see Appendix). Not covered are
measurement tools and systems (from 1.2), geometric transformation (the last item in
1.3), and statistics (from 1.4). Estimation will also play a minimal role. Otherwise,
almost every investigation in every unit will require the student to investigate a real-life
situation, search for patterns in an attempt to define and solve the problem. They will
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attack the problem using all the components of mathematical reasoning and need to
understand the concepts and procedures to utilize these tools. A final component of each
section is the clear communication to classmates of understanding, either through written
or oral presentations.
The wide content variety encompasses the spirit of the EALR's as well as the
specificities.
The order of the course is intentionally cyclic, with each unit opening with a
further example from Pascal's Triangle to reinforce observations and conclusions already
reached. Then a new direction for investigation is introduced. With the exception of the
final unit, there is some room for re-ordering. I would recommend keeping units six and
seven in the designated order for simplicity's sake. Although students may have
previously had some experience with geometric proof and the manipulation in number
theory may be executable, the introduction to logical proof will be a boon to their
toolboxes for handling proofs.
Investigatory exercises are included at the beginning of each unit to establish the
students' confidence and knowledge levels, so appropriate scaffolding can be provided.
Each unit provides group learning activities. The emphases on creative and critical
thinking in how to approach the individual problems take precedent over the actual
problem solving. Direct instruction will be used primarily in providing an overview at
the beginning of each unit, and in some independent practice through homework. All
these are a direct influence of Vygotskyan organizational structures wherein the zone of
proximal development is investigated built upon.
Using the higher levels of thinking in Bloom's Taxonomy is the very engine
behind the progress to be made in the subject matter. Definitions and theorems will be
developed by the students themselves. The students will need to use abstractions (e.g.
ideas) in particular and concrete situations, break down these abstractions into their
constituent elements or parts, put together those elements or parts to form a new theory,
and make judgments about the value of those statements or methods for a given purpose.
Providing student's with a foundation upon which to build and which can be
returned to for reference, commonly known as scaffolding, is developed in this course
through the use of Pascal's Triangle in each of the units. Pascal's triangle, known also as
Yang Hui's triangle provides a wonderful tool with which to introduce a rich variety of
mathematical topics. These include: Geometry, Number Theory, Combinatorics, Graph
Theory, and Probability. Introductions to the writing of proofs, the idea of general forms,
and mathematical thought processes contribute to the overall development of the student
in mathematics.
Once students enter any university program, calculus will be required. Then they
will be inundated by a choice of mathematical topics to investigate, some required, some
optional. Knowledge is the power to make informed choices, and, especially at larger
universities, the course descriptions in the catalogs provide little substantive information.
Exposure to a variety of topics can help students make those choices.
The broad variety of topics enables the student to explore more freely many
aspects of the subject less thoroughly covered in standard curricula, and introduce some
important new ones.
Exploration encouragement is key in the development of each topic and in the
connections presented in the course itself.
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I think that this course can be either a dream course, or a nightmare. If the topics
are familiar to the instructor and a bit of refreshing is done prior to each unit beginning,
the content will not be too overwhelming. However, the teacher must be active in the
group organization, and re-organization if necessary, and provide as much out-of-class
independent practice as individuals who require extra reinforcement will need. It has
been said a mathematician is someone who spends 99% of the time stuck. It is too easy
to just hand students the formulae if they are stuck, and this may be the most challenging
part of the course. The struggle should be real, even if the road has already been welltraveled.
The end result, I believe, will be a rewardingly personal relationship between the
students and mathematics.
5
Course Overview
Teaching Area: Mathematics
Grade Level: 12th Grade
Course Goal: To introduce high school seniors to a variety of mathematical disciplines
including: combinatorics; graph theory; number theory; series; proof; and the binomial
theorem. It will also provide an introduction to mathematical thought where, in a
supportive environment, students will be encouraged to furnish carefully reasoned
arguments that would meet the standards of the broader mathematics community. .
Organizational structure: The course is divided into 10 sections of different lengths,
each covering a different mathematical topic. The sections may be taught sequentially as
a one term course, or individual lessons may be extracted for use as enrichment topics in
related courses. An alternative is that the course may be allowed to evolve naturally
based on student investigations, re-ordering the units based on the instructor's discretion.
The exception is the final unit in which the students are asked to apply a cumulative
knowledge of the topics covered in the previous units in the presentation of an original
observation (a generic topic is provided for students not able to devise their own).
Resources used: A comprehensive bibliography is attached. Suggested resources on
individual topics and a specific bibliography accompany each unit plan.
http://www.seattleschools.org/area/acastan/stan/math/math.xml, which is the Seattle
Public School’s web page defining the EALR’s for mathematics (see Appendix A).
A full description by grade level can be obtained in PDF format at
http://www.seattleschools.org/area/acastan/full/math.pdf .
http://www.seattleschools.org/area/main/calendar.dxml, which is the Seattle Public
Schools year calendar for 2002, used as a time template for the course.
6
Course Outline
Date: 9/4-9/17
Unit One: Probability
Generalization: Determining the probability of an event depends
on proper data gathering techniques and careful observation.
Knowing the odds aids in informed decision making.
Concepts: Included are: proper data gathering and observation
techniques; basic computational formulas; experimentation;
decision trees, conditional probability.
Activities: Grade lottery, a game to introduce the concept of
probability and determining fairness, 1 die/two dice to define
sample size, "a trip to the movies," a game to define dependent and
independent variables.
Projects: Creating Pascal's Triangle, students will create a
graphical display to help interpret topics presented throughout the
rest of the semester. Data gathering and interpretation project:
students will select a topic of personal interest (jobs, sports, health)
and gather statistical information designed to help with better
decision making by showing wise choices can improve one's odds.
EALR's: 1.1, 1.4, 1.5, 2.1, 2.2, 2.3, 3.1, 3.2, 3.3, 4.1, 4.2, 4.3, 5.1,
5.3
Date: 9/18-9/26
Unit Two: Combinatorics
Generalization: The application of different counting and grouping
methods can supplement probabilistic techniques.
Concepts: Included are: combinations from the probability unit;
the pigeonhole or Dirichlet's Box Principle, the Fubini Principle.
9/25
Activities: Members of the Student Council/Prom committee, a
2 hr. early dismissal project to synthesize the definitions needed to determine if order
matters in a probability question. Demonstrate how these
combinations in can be determined with Pascal's Triangle.
Projects: Create and analyze pigeonhole sorting system for use in
determining combinations as group work Students will assemble
an oral presentation with visual aids to demonstrate combinatorics
applications in everyday life.
EALR's: 1.1, 1.4, 1.5, 2.1, 2.2, 2.3, 3.1, 3.2, 3.3, 4.1, 4.2, 4.3, 5.1,
5.3
Date: 9/27-10/10
10/2
2hr. early dismissal
10/11
Unit Three: The Binomial Theorem
Generalization: The development and understanding of the
binomial theorem can be traced from Euclid, through Newton, to
the present day and is used in a broad variety of mathematical
disciplines.
Concepts: Newton's formula, algebraic meaning, the random walk.
Activities: The random walk, a classic mathematical problem, will
be demonstrated in the hallway using tiles and tape to show the
7
development day
no school
outcomes based on various directional parameters. Creating a
modified version of Pascal's Triangle using fractional and negative
numbers ala Newton.
Projects: Biography of a 20th century mathematician.
EALR's: 1.1, 1.5, 2.1, 2.2, 2.3, 3.1, 3.2, 3.3, 4.2, 4.3, 5.1
Date: 10/14-10/25
Unit Four: Series
Generalization: Series and limits led to better measuring and
estimating techniques and the development of Calculus. They
have many applications in everyday life.
Concepts: Infinite series and limits. Defining different types of
convergence, geometric series, harmonic series.
Activities: Fibonacci numbers in Pascal's Triangle. Demonstrate
why projections are dangerous using statistics in the news
(i.e. given three year trend in Schwarzenegger's salary history and
projecting, he would make more than the GNP of all the world's
nations in less than a decade). Figure the reasons behind a daily
dosage of a prescription drug.
Projects: Finding series/limits in the natural world using predator/
prey or other ecological balances. Fibonacci number hunt in the
natural world. Golden ratio designs found in nature or in
architecture. Do as group presentations.
EALR's: 1.1, 1.5, 2.1, 2.2, 2.3, 3.1, 3.2, 3.3, 4.2, 5.1, 5.3
Date: 10/28-11/13
10/30
2hr. early dismissal
Unit Five: Graph Theory
Generalization: The multiple ways points on a graph can be
interconnected are used in multiple disciplines and industries when
efficiency is at stake.
Concepts: Included are: applications for garbage men, mail
delivery, airlines. Eulerian trails, Eulerian circuits, trees.
Hamiltonian circuits in Platonic solids (see unit nine).
Activities: The taxicab problem will be presented to review
combinatorics and present definitions using Pascal's triangle. The
Konigsberg bridge problem, another classical problem will be used
to elicit student analysis of how to determine whether the above
circuits or trails are possible.
Projects: Students will develop practical applications for
implementation of graph theory such as garbage truck/postal
carrier routing, airline scheduling, running the errands of a 'soccer
mom.' Drawing Platonic solids using Pascal's triangle and
applying graph theory to a three dimensional object and comparing
it to two dimensional objects.
EALR's: 1.1, 1.2, 1.3, 1.5, 2.1, 2.2, 2.3, 3.1, 3.2, 3.3, 4.1, 4.2, 4.3,
5.1, 5.2, 5.3
11/1
End of Quarter
11/11
Veteran’s Day
no school
8
Date: 11/14-11/26
Unit Six: Logical Proof
Generalization: Recognizing axiomatic statements and the truth or
falsity in combination to determine the logical validity of a given
statement is basic to mathematical proof at any level.
Concepts: Included are: truth tables; tautologies; basic symbols;
sets and Venn diagrams; the idea of axioms.
Activities: Creating truth tables and Venn diagrams to
demonstrate the inclusion or exclusion of statements or elements of
set and analyze observations of various statistical groupings
Projects: Axiomatic high school: students try to devolve their
high school experiences into a set of minimal axioms for success.
They will construct graphical works demonstrating the 'theorems'
necessary to achieve certain goals (lettering, valedictorian,
'coolness,' graduation). Predict a self-chosen quality observable in
Pascal's Triangle and prove or disprove the general case. Both are
group oriented.
EALR's: 1.1,1.3, 1.5, 2.1, 2.2, 2.3, 3.1, 3.2, 3.3, 4.2, 4.3, 5.1, 5.2,
5.3
Date: 12/2-12/20
Unit Seven: Number Theory
Generalization: Proofs, discoveries and generalizations can be
made through manipulation of the integers.
Concepts: Review prime numbers. Euclid's proof of the infinity
of primes; examples of easily understood yet unsolved problems;
experimentation, hypothesis and proof/disproof of observed
patterns.
Activities: Create prime triangle, related in form to Pascal's
triangle, but requiring situational analysis and proof. Correction of
worksheet proofs to demonstrate analytical skills in critical
thinking. Computer simulation exploration of properties of
numbers and synthesize and evaluate conjectures.
Projects: Two proofs, one about Pascal's Triangle, attempted in
groups. The first will incorporate the computer simulations used in
the activities section.
EALR's: 1.1, 1.5, 2.1, 2.2, 2.3, 3.1, 3.2, 3.3, 4.2, 4.3, 5.1, 5.2, 5.3
11/27
2hr early dismissal
12/21-1/5
winter break
no school
Date: 1/6-1/14
Unit Eight: General Forms
Generalization: Many mathematical formulas are very specific and
discovering the more general forms can bring together a variety of
ideas and subjects within mathematics .
Concepts: Using Pascal's Triangle, create and explore different
generalizations of the triangle in two and multiple dimensions.
Activities: Binomial and other radices. Create computer displays
utilizing binary and other base interpretations of Pascal's triangle.
Review axiomatic high school projects and evaluate.
9
Projects: Sierpinski's Sieve construction from binomial Pascal's
Triangle: a result of the computer graphic activity, students will
create fractal designs and present a technology-based
demonstration. Students will also gather and present practical
applications of fractal designs in various industries (entertainment,
ecological and economic, self-chosen).
EALR's: 1.1, 1.3, 1.4, 1.5, 2.1, 2.2, 2.3, 3.1, 3.2, 3.3, 4.1, 4.2, 4.3,
5.1, 5.2, 5.3
Date: 1/15-1/21
1/20
Martin Luther King
Day
no school
Date: 1/22-1/24
1/24
last day of quarter
Unit Nine: Geometric Constructions
Generalization: Algebraic patterns from the binomial theorem and
Pascal's Triangle can be used to define and create Platonic
polygons and solids.
Concepts: Polygon construction using the triangle, figurate
numbers.
Activities: Use Pascal's Triangle to determine geometric forms
within a circle. Relate figurate numbers to the triangle. Review of
graph theory concepts and generalization to applications and
evaluation of the methods to implementation.
Projects: Make Platonic solids (as drawn in Unit 5). Creating
origami figures based on published patterns, then analyze the
forms and create original figures and describe the pattern required
to construct the form.
EALR's: 1.1, 1.2, 1.3, 1.5, 2.1, 2.2, 2.3, 3.1, 3.2, 3.3, 4.1, 4.2, 4.3,
5.1, 5.2, 5.3
Unit Ten: Triangle Oddities
Generalization: Students use the topics of logical proof, number
theory and the techniques used in the previous nine units to justify
compositions and presentations, or by addressing the 'question of
the 11's.'
Concepts: In Pascal's (Yang Hui's) triangle, an interesting
pattern can be discerned concerning the powers of 11. Algebraic
manipulation, the previous nine units in review.
Activities: Algorithm discovery by group to determine powers of
11. Presentation and discussion of any outstanding conjectures yet
unproved in the previous units. Full class exploration and attempts
at resolving and proving or disproving these.
Projects: Relate student discoveries in the triangle.
Writing a conjecture and proof for publication.
EALR's: 1.1, 1.5, 2.1, 2.2, 2.3, 3.1, 3.2, 3.3, 4.2, 4.3, 5.1
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Probability Unit Rationale
Although included in the EALR's, probability needs the most additional practice
time to be allocated of all the essentials. Everyday people must make decisions which
can potentially impact the rest of their lives. Insurance, investments, advertising
campaigns, school selection, health statistics, manufacturing, the natural sciences and
other daily occurrences require us to evaluate information and make a choice. Many
people, however, are fooled into unwise choices by those that would take advantage of
them. The old adage, "if it sounds too good to be true, it probably is," can still be great
advice, but giving students the knowledge to make good choices is a better option.
The history of the mathematical study of probability is usually begun with a study
of Girolamo Cardano's Liber de ludo Aleae, a book on games of chance. A series of
letters between Blaise Pascal and Pierre de Fermat studied the nature of dice gaming.
Jacob Bernoulli is generally considered the founder of probability theory, merging pure
mathematics with the more empirical experimental studies of his time.
To begin the unit, students must acquire a vocabulary to discuss probability. The
initial lessons let students explore through experimentation and synthesize a definition for
sample space, as the set of all possible outcomes of an experiment, referred to as a
probability space in [4] (p171).
This will be followed by another group learning experience to formulate a
definition of permutation, the re-arrangement of the elements of a sample space into a 1to-1 correspondence with itself ([4] p111). This approach is taken for simplicity's sake,
enabling students to deal with the entire set initially, and then develop ways of dealing
with subsets. Introduction of factorial notation (!) will be introduced ([4] p102).
Combinations, number of ways of picking r unordered outcomes from n
possibilities. Also known as the binomial coefficient or choice number and read "n
choose r."
n!
n
  
 nCr
 r  r! (n  r )!
where n! is a factorial, will be the introduced notation (both the parenthetical and the
subscripted versions on their calculators) for combinations and then back tracking,
number of ways of obtaining an ordered subset of r elements from a set of n elements
n!
will be shown to be
 n Pr , ([4], p103,
(n  r )!
At this point, students will be introduced to Pascal's Triangle and provided with
some cardstock to create a version to use for the rest of the term.
Review set notation, and Venn diagrams to demonstrate the definitions where S is
a sample space and A and B are events from S.
A  B is the union of the events and is defined as an event of which the outcome is an event
belonging to either set or both
A  B is the intersecti on of the events and is defined as an event of which the outcome is an event
belonging to both.
Ac is the complement of the outcomes in the event set, i.e. not in the set.
11
If A  B  {} , the empty set, then the sets are mutually exclusive, or disjoint ([4], p9 - 10).
The notation for the probability of an event is P(A).
The addition rule for two events will be examined first through demonstration that
the proposition P(A  B)= P(A)+ P(B) is false because the intersection of the two sets is
counted twice. The correct equation is P( A  B )= P(A)+ P(B)-P( A  B ) as in [4]
(p172).
The addition rule for three events is given as:
P(A  B  C )= P(A)+ P(B)+ P(C)- P(A  B)- P( A  C )- P( B  C )+ P(A  B  C)
and is again best demonstrated through the use of Venn diagrams.
The unit will conclude with examples of conditional probability which is defined
in [4] (p170) for an event A given an event B as: P(A|B)= P( A  B ) / P(B) if P(B)>0.
And finally, a really tough one, independence. As an example; in countries with
high television ownership, life expectancies are longer. Does owning a tv help in lifeexpectancy? Students need to learn that while a relationship may not be independent, this
does not imply causality.
Introduce the definition given in [4] (p175) that two events are independent iff
P(A  B)=P(A)P(B).
Suggested reading for this unit would be to review any basic probability text, and
the topics covered here are also available in most standard discrete and statistics texts.
The text referenced in the unit frequently is an excellent general text, and is listed in the
bibliography as [4]John P. D'Angelo and Douglas West, Mathematical Thinking:
Problem Solving and Proofs, 2nd ed., Prentice Hall, 2000.
12
Probability Unit Plan
Unit Title: Fundamentals of Probability
Organizational Structure: Specific to general
Unit Objective: This unit's purpose is to enable students through the analysis of various
data sources, synthesis of discovered definitions and evaluation of a variety of problem
solving methods, to understand and apply the basic concepts of probability including:
 understanding the concepts of sample space and probability distribution and
construct sample spaces and distributions in simple cases;
 using simple simulations to construct empirical probability distributions;
 computing and interpreting the expected value of random variables;
 understanding the concepts of conditional probability and independent events;
 and understand how to compute the probability of a compound event.
Unit Generalization: Through the use of in-class experimentation, computer simulation,
and internet and interview data gathering the students will define the terms sample space,
permutation, combination, dependent, independent, conditional, and joint as used in the
study of probability. Using these terms the students will examine various situations and
develop generalized methods of problem solving utilizing the most logically appropriate.
The recognition of everyday situations as interpretable using these tools can then be
applied to good decision making.
Washington EALR's:
1.1
Understand and apply concepts and procedures from number sense, including
number and numeration, computation, and estimation.
1.4
Understand and apply concepts and procedures from probability and statistics,
including probability, statistics, and prediction and inference.
1.5
Understand and apply concepts and procedures from algebraic sense, including
relations and representations, and operations.
2.1
Investigate situations by searching for patterns and exploring a variety of
approaches.
2.2
Formulate questions and define the problem.
2.3
Construct solutions by choosing the necessary information and using the
appropriate mathematical tools.
3.1
Analyze information from a variety of sources; use models, known facts, patterns,
and relationships to validate thinking.
3.2
Predict results, make inferences and make conjectures based on analysis of
problem situations.
3.3
Draw conclusions and verify results, support mathematical arguments, justify
results, and check for reasonableness of solutions.
4.1
Gather information, read, listen, and observe to assess and extract mathematical
information.
4.2
Organize and interpret information.
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4.3
5.2
5.3
Represent and share information; share, explain, and defend mathematical ideas
using terms, language, charts, and graphs that can be clearly understood by a
variety of audiences.
Relate mathematical concepts and procedures to other disciplines; identify and
apply mathematical thinking and notation in other subject areas.
Relate mathematical concepts and procedures to real-life situations; understand
the connections between mathematics and problem-solving skills used every day
at work and at home.
Resources Used:
Edward Brooks, The Philosophy of Arithmetic as Developed from the Three Fundamental
Processes of Synthesis, Analysis, and Comparison, 571pp, Normal Publishing Company,
1876
Dewey C. Duncan, Ten Mathematical Refreshments: Mathematics in Secondary School
Classrooms, pp 118-124, Rising/Wiesen, eds., Thomas Y. Crowell Co., 1972
Ernest R. Ranucci, Mathematics on the Ceiling: Mathematics in Secondary School
Classrooms, pp 84-90, Rising/Wiesen, eds., Thomas Y. Crowell Co., 1972
Various Authors, In Developing Mathematical Reasoning in Grades K-12, Lee V. Stiff
and Frances R. Curcio, editors, National Council of Teachers of Mathematics, 1999
Various Authors, In Learning Mathematics for a New Century, Maurice J. Burke and
Frances R. Curcio, editors, National Council of Teachers of Mathematics, 2000
Principles and Standards for School Mathematics. National Council of Teachers of
Mathematics, 2000
http://www.seattleschools.org/area/acastan/stan/math/math.xml, which is the Seattle
Public School’s web page defining the EALR’s for mathematics (see Appendix A).
A full description by grade level can be obtained in PDF format at
http://www.seattleschools.org/area/acastan/full/math.pdf .
http://www.seattleschools.org/area/main/calendar.dxml, which is the Seattle Public
Schools year calendar for 2002, used as a time template for the course.
14
Date
9/4
Key Concept(s)
Define “sample
space,” the total
number of possible
outcomes of an
experiment.
EALR’s 1.4, 2.1, 4.2,
4.3
9/5
Permutation is the rearrangement of the
elements of a sample
space into a 1-to-1
correspondence with
itself.
EALR’s 1.4, 2.1, 2.3,
4.1, 4.3, 5.3
9/6
Combination is the
number of ways of
picking a chosen
number of unordered
outcomes from a
sample space set.
EALR's 1.4, 2.1, 2.3,
4.1, 4.2, 4.3, 5.3
9/9
Review definitions.
EALR's 1.1, 1.4, 2.1,
2.2, 2.3, 4.3, 5.3
9/10
Venn Diagram
review/introduction
Probability is the
total number of
specific outcomes
Teacher and/or Learner Activities
Anticipatory Set: “Is the lottery a fair way to be
graded?” Students compile results of various random
outcomes of a variety of shapes and numbers of dice.
In a guided discovery learning group investigation
session, students share their results and formulate a
definition to provide a detailed definition of “sample
space” as the list of all possible outcomes of an
experiment. Define probability, P.
Review set notation {} in direct instruction and state
the sample space is a set. In a group discovery session,
“elect” four people from class to serve as “pres.,”
“v.p.,” “sec.,” and “treas.” and have students determine
all possible combinations in mock election, or provide
similar groupwork investigations (provided). Introduce
factorial notation (!) to determine number of
permutations.
Homework: three to five examples from daily
observations citing the sample space and why this is a
permutation.
Student input is used to compare homework examples
and include or discount them as permutations.
Following up on the previous lesson, this time we will
elect a committee in which order does not matter and
compare the number of possible committees. Students
will try various sizes of committee.
Given heavy stock paper and markers, students will
create a personal Pascal's Triangle for use throughout
the term. Given the formulas for finding basic
permutations and combinations to include on the cards,
guided discovery to the connection with the triangle
will be done.
Since this is the first full week after summer break, a
recap of the first three days will help reinforce the
previous days' work.
Drawing on previous learning the students will select 4
lunch bags to be mixed up in carpool and determine the
probability that each person got the right lunch,
combining sample space, combination and permutation
definitions.
Review set notation, union, intersection, complement,
with graphical interpretation on the Venn diagram.
Students will complete a Venn work sheet to produce
an analytic interpretation of a given set of parameters.
Given the visual aid of the diagrams, working in
15
divided by the total
number of outcomes
in a sample space.
EALR's 1.1, 1.4, 1.5,
2.1, 3.1, 3.2, 3.3, 4.2,
4.3
9/11
9/12
Venn Diagram cont.
Disjoint or mutually
exclusive events are
if outcomes A and B
cannot happen at the
same time.
EALR's 1.1, 1.4, 1.5,
2.3, 3.2, 4.2, 4.3, 5.3
Quiz, The addition
rule must be used to
avoid double
counting.
EALR's 1.1, 1.4, 1.5,
2.3, 3.2, 4.2, 4.3, 5.3
9/13
Conditional
probability is the
likelihood of one
event given another.
EALR's 1.4, 2.1, 2.3,
4.1, 4.2, 4.3, 5.3
9/16
Two events are
independent if
P(A  B)=P(A)P(B)
EALR's 1.1, 1.4, 1.5,
2.1, 3.1, 3.2, 3.3, 4.2,
4.3
9/17
Unit Review
EALR's 1.1, 1.4, 1.5,
2.1, 3.1, 3.2, 3.3, 4.2,
4.3, 5.2, 5.3
groups, students will discover probability is a fraction
of the total possible outcomes.
Examples to be used are all mutually exclusive so
P(A)  P(~A) will equal one, demonstrating the
definition to the left.
Homework will be another Venn diagram sheet with
problems anticipating the lesson for 9/11.
Computing probability 'P' if events are mutually
exclusive will be discovered with student input from
the homework assignment. Direct instruction will
define the complement rule.
Using two of the diagrams from the previous
homework, a guided group discovery will use specific
values to determine probabilities of having a particular
video game.
A short quiz, including definitions, determining sample
spaces, and Venn diagram completion to check concept
understanding.
Using the movie diagrams a guided discovery session
to synthesize an addition rule for P(A  B) to avoid
double counting.
Make sure the special case for disjoint sets is included
in the student conclusions.
Through direct instruction, introduce the notation for
conditional probability.
Using the video game numbers from 9/11, students
compute the conditional probability of an event and in
group discussion propose a formula to determine
conditional probability.
Homework: computational practice on all topics.
Student/Teacher Input on homework problems as
answers are discussed in call and response format.
Statistics from the news and advertising will be
discussed, especially the humorous type in which
clearly independent events are presented as not
(bandwagon ads, like 'drink beer and lots of pretty
women will find you attractive' are good for this.
Students will design a carnival game of at least two
layers of complexity for a charity fund raiser. They
will determine sample space, whether permutation or
combination is used and if the events are independent
or not. Teacher will model an unfair game. Students
will discuss the moral issues of a fair game. What if
the game is known to be unfair, but it is for a good
cause?
16
Probability Lesson Plan One
Grade level: 11-12
Date: 2002-03 school year
Subject: Mathematics/Probability Unit Orientation
Generalization: In order to effectively calculate probabilities, students must have a
vocabulary to describe the parameters of what they are investigating. In this lesson,
students define sample space, which is the set of possible outcomes of an experiment.
Skills/Attitudes: Observation, comparing, analyzing, working cooperatively in groups.
EALR's/ Standards: 1.3-understand and apply concepts and procedures from
probability and statistics; 2.1-investigate situations; 3.1-analyze information; 3.2-predict
results; 4.1-gather information; 4.2-organize and interpret information; 4.3-represent and
share information; and 5.1-relate concepts and procedures within mathematics. Also,
NCTM Standards and Principles for Data Analysis and Probability for Grades 9-12;
understand and apply basic concepts of probability.
Objective: Given a variety of probabilistic instances in the forms of different sided and
different numbers of dice, the students will analyze the data and demonstrate synthesis by
formulating a definition of "sample space" and methods to determine such a set.
Assessment: Informal
Materials: Pencil and paper, about 20 dice of varying types (include 4,6,8,10,12 and 20sided dice), chalk/whiteboard. Shoe boxes or some other type of box to keep the dice
contained are recommended.
Procedures/activities
Anticipatory Set
(2-5 minutes)
Teacher asks if students are
familiar with lotteries and asks the
students to pick and write down six
numbers from 1-54, inclusive. Ask the
students to rise and write 6 numbers at
random on the board. Write 6, 5, 4, 3, 2
or less in columns on the board. All who
have 0, 1, or 2 of the six chose may sit,
record any names of the remaining
students in the appropriate columns, then
assign them letter grades with 6=A to E/F
for 0,1,2. Then ask if the lottery is a fair
game.
Checks for Understanding
Discuss the concept of
fairness and ask if presented with
choices or decisions what types of
information would be needed to make
good decisions.
Communicate Objective
"Today we are going to develop
some terminology in order to be able to
17
discuss probability effectively by
comparing observations of random events."
Directions
(1-3 minutes)
Have everyone to get into groups of
2 or 3, with one person acting as a recorder
for our probability events. Give each
group some dice and have each group roll
20 times and then record the outcomes on
the table on the board.
Guided Discovery
(3-5 minutes)
Ask the students to write up a
statement regarding a description of the
dice they received and compare it with the
results obtained.
See that the groups are
organized, give each group a variety
of types and number of dice and
confirm the task.
Verify groups go to the board
and record their results.
Walk among the groups and
verify participation and use of
reasoning.
Directions
(1 minute)
Assemble the class into 3 groups,
insure that those with only one die are
paired with those with more than one die.
Guided Discovery
(20-25 min.)
Allow students to discuss the
differences in the expected outcomes of the
various groups.
Have students make a visual
representation of the outcomes for 1-4sided, 1-4 and 1-6-sided pair, and 2-six
sided dice.
Direct Instruction/Transition (2 min.)
Collect the dice. Define for
students the sample space as the total of all
possible outcomes of a random event.
Define Probability, P.
Student Input/Invention
(5-10 min.)
Have students share their findings
and give examples of the total number of
outcomes in the sample spaces of the
various combinations of the dice, record
these next to the previous results.
Verify that students are
discussing the differences between,
for instance, 1-12-sided die and 2-6sided dice. If not, ask if anyone
noticed any distinct differences.
Some questions for
discussion: is there a difference
between a 7, a six and a one, a one
and a six, a three and a three? Yes,
for now.
Verify that the students are
coming up with the correct numbers
for the sample space given the dice
provided. Ask for reasoning if the
total is not accurate.
18
Teacher Clarification (1 minute)
As can be seen from our data, the
total number of outcomes rolling 1-4-sided
die is four, of rolling 1-12 sided die is
twelve, but rolling 2-6-sided dice, the total
number of outcomes is 36.
Closure
(1 minute)
For the next class, assign students
the task of coming up with three different
examples of a sample space that
specifically do not use coins, dice, card, or
such gaming devices.
If a student persists in
insisting that order does not matter,
ask the student to define when this
would be the case and when it would
not. This, of course, is coming in
defining combinations v.
permutations.
If clarification is required, ask
the class, but if lack of generalization
is total, suggest some general sets,
like members of congress, winter
sports, days of the week, etc.
19
Probability Lesson Plan Two
Grade Level: 11-12
Date: 2002-03 school year
Subject: Mathematics/Probability
Time: 50 minutes
Generalization: Permutation is the re-arrangement of the elements of a sample space
into a 1-to-1 correspondence with itself.
Skills/Attitudes: Analysis through the organization of parts, synthesis of previously
learned topics, and group learning.
EALR's/Standards: 1.4 Understand and apply concepts and procedures from
probability and statistics, including probability, statistics, and prediction and inference;
2.1 Investigate situations by searching for patterns and exploring a variety of approaches;
2.3 Construct solutions by choosing the necessary information and using the appropriate
mathematical tools; 4.1 Gather information, read, listen, and observe to assess and
extract mathematical information; 4.3 Represent and share information; share, explain,
and defend mathematical ideas using terms, language, charts, and graphs that can be
clearly understood by a variety of audiences; and 5.3 Relate mathematical concepts and
procedures to real-life situations; understand the connections between mathematics and
problem-solving skills used every day at work and at home.
Objective: With connections to previously learned set notation, the synthesized
definition of sample space, and the introduction of factorial notation, the students will
analyze a pattern in permutation experiments and formulate and evaluate a definition as
above.
Assessment: Informal
Materials: Pencil and paper, chalk/whiteboard, copies of letter and video game
permutation handouts (see attached masters).
Procedures/Activities
Anticipatory Set (2-5 minutes)
Review the definition of sample space
from previous lesson and ask for examples
of sample spaces and questions regarding
probabilities that use that sample space.
Checks for Understanding
Check to see that students make the
connection between sample spaces and the
questions.
Sets such as {Sun., Mon., Tue., Wed.,
Thur., Fri., Sat.} accompanied by a
question like, "What day of the week will
someone be born on?" or {specific sports
division} and "Which team will win the
championship?"
Communicate Objective
Inform students that today they will be
exploring data for a way to define another
term in probability.
20
Guided discovery (20-25 minutes)
Students are given one of two handoutseither the mixed envelope or the mixed
video games scenarios- in groups of two or
three. The scenarios ask to determine the
specific permutation of a set being ordered
in a unique way, the set containing either
three or four elements.
Student Input/Invention (7 minutes)
Have students from remaining groups
demonstrate their particular solutions to the
class. Have students hypothesize a general
formula and definition of a permutation.
Have students investigate 3 or 5 letters,
3 or 4 video games and develop a rule.
Teacher Input
(10 minutes)
Introduce students to factorial (!)
notation. Adapt the students definitions
and rules to define permutation (the new
term to be added to their probability
vocabulary) as the re-arrangement of the
elements of a sample space into a 1-to-1
correspondence with itself.
Verify by walking around that students
are sure of the instructions on the handouts.
Check that probability of the letters is
1/24 and the video games is 1/120.
Verify by walking around that students
are discovering the factorial pattern.
Verify student understanding through
call and response, having students compare
their rule to the factorial notation. Add 2 to
the mix for further demonstration, if
necessary.
Direct Instruction/Transition/Closure
(3 minutes)
Assign homework, have students
investigate permutations by coming up
with three examples of permutations in the
world around us, excluding sports team
standings, as these have already been
introduced.
21
Group Investigation One, Probability Lesson Two
While working at a new job, your employer gives you four different
letters to mail and four different envelopes, each with a different client's
address. The phone rings, and while trying to answer the customer's
question, you put the letters in the envelopes and seal them.
Later, the manager asks if you put the correct letters in the correct
envelopes. You didn't realize the letters had been different, and tell her you
are not sure.
What is the probability that you put each of the four letters in the
correct envelopes?
Explain your conjecture in two of three ways: 1) algebraically
(numerically); 2) by written explanation; and/or 3) pictorially using
drawings or diagrams.
22
Group Investigation Two, Probability Lesson Two
A friend comes to visit you and brings her little brother. To keep him
occupied, you let him play your new video game system. He tries four
different games and finds all of them boring. Then on the fifth game, he's
hooked and plays for an hour, giving you and your friend time to visit.
When they leave, your friend tells her little brother to clean up after himself.
He puts one game back in each box, but does not bother to match the titles of
the games and the titles on the boxes.
What is the probability that all five games end up back in the correct
boxes?
Explain your conjecture in two of three ways: 1) algebraically
(numerically); 2) by written explanation; and/or 3) pictorially using
drawings or diagrams.
23
Probability Lesson Plan Three
Grade Level: 11-12
Date: September 6, 2002
Subject: Mathematics/Probability
Time: 50 minutes
Generalization: Combination is the ways of picking a chosen number of unordered
outcomes from a sample space set.
Skills/Attitudes: Evaluate the difference between a permutation and combination
EALR's/Standards: 1.4 Understand and apply concepts and procedures from
probability and statistics, including probability, statistics, and prediction and inference;
2.1 Investigate situations by searching for patterns and exploring a variety of approaches;
2.3 Construct solutions by choosing the necessary information and using the appropriate
mathematical tools; 4. 1 Gather information, read, listen, and observe to assess and
extract mathematical information; 4.2 Organize and interpret information; 4.3 Represent
and share information; share, explain, and defend mathematical ideas using terms,
language, charts, and graphs that can be clearly understood by a variety of audiences; 5.3
Relate mathematical concepts and procedures to real-life situations; understand the
connections between mathematics and problem-solving skills used every day at work and
at home.
Objective: To determine the possible combinations of unordered groups of four from the
class. Construct a "Pascal's Triangle."
Assessment: Informal
Materials: Pencil and paper, chalk/whiteboard, at least on piece of heavy card stock
paper per student, permanent markers/pens.
Procedures/Activities
Anticipatory Set (2-5 minutes)
Collect homework and review definition
of permutations. Have students give
examples of permutations that they found.
Checks for Understanding
Verify that the sets students came up
with are permutations. Address any that
are not.
Communicate Objective
Some students will invariably have
come up with sports team standings as a
permutation, if not give this example.
Ask the question "What if only the top
four teams make the playoffs in a division
of 10 teams?" Today we investigate how
many combinations are possible when
order does not matter.
Some students will insist that order
does matter for things like home-field
advantage, remind them that any team that
makes the playoffs has a chance of
winning, and all we are interested in is the
combination of possible playoff
participants.
Guided discovery/ Cooperative Learning
(20-25 minutes)
Have students divide up into groups of
24
6, if possible, or 5. They are then to elect a
committee to represent their group to the
class. Have the students determine the
possible number of committees that a
group their size can generate. Repeat for a
four and a two member committee.
Student Input/Invention (10 minutes)
Have students summarize their findings
by group. Write the numerical answers on
the board to each of the combination
experiments so that the students may begin
to see a pattern. Inquire if they can come
up with or have come up with a formula
based on the number of elements in the set
and the number of elements chosen.
Teacher Input/Direct Instruction
(7 minutes)
If student have not already found it,
present the formula for a combination,
along with several computational
examples.
Transition
(2 minutes)
Distribute cardstock for the construction
by the students of Pascal's triangle.
Explain that they will be using this
throughout the term and need to retain it.
Assign completion of the first 20 rows of
the triangle and have students write up a
hypothesis of the connection between the
work they did today and the triangle.
Closure
(3 minutes)
Demonstrate construction of the
triangle and how to obtain each additional
entry through simple addition, have
students continue working on this until the
end of class.
Verify by walking around that students
are taking into account that order does not
matter.
Check for correct numerical answers in
summarizations.
By call and response, verify
understanding of the process of choosing
an unordered subset.
n
n!
  
= nCr and is read n choose r
 r  r!(n  r )!
for picking r unordered outcomes from n
possibilities. These are the three common
representations. The first is for notational
purposes, the second, a computational
algorithm, the third is how it commonly
appears on calculators.
1
1
1
1
1
2
3
1 4 6
are the first few rows.
1
3
1
4
1
Verify by walking around that
students are interpreting the construction
correctly.
25
Lesson Plan Four
Grade Level: 11-12
Date: September 9, 2002
Subject: Mathematics/Probability
Time: 50 minutes
Generalization: Sample space, permutation, and combination are all terms having to do
with investigations in probability.
Skills/Attitudes: Students will evaluate problems from probability and apply methods
learned in three previous lessons to demonstrate understanding of the concepts.
EALR's/Standards: 1.1 Understand and apply concepts and procedures from number
sense, including number and numeration, computation, and estimation; 1.4 Understand
and apply concepts and procedures from probability and statistics, including probability,
statistics, and prediction and inference; 2.1 Investigate situations by searching for
patterns and exploring a variety of approaches; 2.2 Formulate questions and define the
problem; 2.3 Construct solutions by choosing the necessary information and using the
appropriate mathematical tools; 4.3 Represent and share information; share, explain, and
defend mathematical ideas using terms, language, charts, and graphs that can be clearly
understood by a variety of audiences; 5.3 Relate mathematical concepts and procedures
to real-life situations; understand the connections between mathematics and problemsolving skills used every day at work and at home.
Objective: Review terms learned and confirm student comprehension.
Assessment: Formal -- in-class worksheet.
Materials: Pencil and paper, chalk/whiteboard, worksheet.
Procedures/Activities
Anticipatory Set (2-5 minutes)
Collect hypothesis assigned in previous
lesson and discuss. Request students use
formal vocabulary of terms previously
given.
Communicate Objective
Inform students that today will be a
review of the previous work to demonstrate
understanding.
Guided discovery/Cooperative Learning
(10 minutes)
Assign the following problems:
Four lunch bags are mixed up in a carpool,
determine the probability that each student
got the correct lunch.
In a race, there are 12 competitors, the top
4 finishers get free t-shirts, determine how
many combinations there are of t-shirt
winners.
Checks for Understanding
Verify through call and response that
students understand hypothesis and
vocabulary terms. Review exact
definitions if necessary.
Verify that students are labeling terms,
and using the correct algorithms by
walking around.
26
Student Input/Invention (7 minutes)
Have students go to the board to present
answers.
Teacher Input/Transition
(2 minutes)
Pass out worksheet, inform students that
this quiz is not graded, but a check-plus,
check, check-minus valued exercise,
similar to homework.
Have students add to the descriptions of
how answers were derived until all
demonstrate understanding. If necessary,
demonstrate at the board with more and
simpler examples.
Verify by walking around that students
are performing correct algorithms,
understand expectation. If any questions
are pervasive, address these to the entire
class.
Independent Practice
(20 minutes)
See attached worksheet.
Transition/Closure
(1 minute)
Collect worksheets. Wish students a
good day.
27
NAME________________________
Probability Lesson Four Worksheet
For problems 1 through 4, describe the following in your own words and give one
examples of each: (you may use the back of the paper, if necessary)
1) Probability
2) Sample space
3) Permutation
4) Combination
Solve the following:
6) How many permutations are there of a set with
a) 4 elements
b) 5 elements
c) 6 elements
d) 9 elements
7) How many combinations are possible in the following
a) picking 4 elements from a set of 11
b) picking 3 elements from a set of 8
c) picking 19 elements from a set of 20
Solve the following:
8)
5!
9)
3!2!
6!
10)
3!3!
6!
BONUS: List as much of the notation we have covered in this unit as you can and
describe what it represents. You may use the back of the paper.
28
Probability Lesson Plan Five
Grade Level: 11-12
Date: September 10, 2002
Subject: Mathematics/Probability
Time: 50 minutes
Generalization: Probability is the total number of specific outcomes divided by the total
number of outcomes in a sample space.
Skills/Attitudes: Interpretation of sample space using Venn Diagrams, analysis through
the organization of information, and evaluation of predictions drawn from this
representation.
EALR's/Standards: 1.1 Understand and apply concepts and procedures from number
sense, including number and numeration, computation, and estimation; 1.4 Understand
and apply concepts and procedures from probability and statistics, including probability,
statistics, and prediction and inference; 1.5 Understand and apply concepts and
procedures from algebraic sense, including relations and representations, and operations;
2.1 Investigate situations by searching for patterns and exploring a variety of approaches;
3.1 Analyze information from a variety of sources; use models, known facts, patterns,
and relationships to validate thinking; 3.2 Predict results, make inferences and make
conjectures based on analysis of problem situations; 3.3 Draw conclusions and verify
results, support mathematical arguments, justify results, and check for reasonableness of
solutions; 4.2 Organize and interpret information; 4.3 Represent and share information;
share, explain, and defend mathematical ideas using terms, language, charts, and graphs
that can be clearly understood by a variety of audiences.
Objective: Review and create Venn Diagrams and relate these to probability.
Assessment: Informal
Materials: Pencil and paper, chalk/whiteboard
Procedures/Activities
Checks for Understanding
Anticipatory Set (2-5 minutes)
Draw a simple Venn Diagram on the
Verify through call and response
board and elicit from students the meanings understanding of the diagram as a
of the representation.
representation of set notation.
Communicate Objective
Inform student that today will be given
over to the construction of Venn Diagrams
and the role that they play in helping us to
visualize probability.
Guided discovery/Cooperative Learning
(7 minutes)
Have students create several Venn
Diagrams using simple sets. Use U to
represent the universal set or sample space.
Examples:
Refer to example on board to have
students begin their constructions.
29
U={the integers from one to ten, inclusive}
A={odd numbers}
B={even numbers)
U={the integers from one to ten, inclusive}
A={prime numbers}
B={composite numbers}
(remember that 1 will be in neither A nor
B)
Verify by walking around that students
are entering the elements of the sets in the
correct locations on the diagrams.
Add more examples as needed to
confirm mastery.
Teacher Input
(10 minutes)
Define at the board that the probability of
an experiment is the total number of
specific outcomes divided by the outcomes
Verify understanding by call and
in a sample space. Include several
response, addressing any
examples of each using sets that differ from misunderstandings through further
the above examples.
example.
Student Input/Invention (7 minutes)
Have students compute the probabilities
that an outcome in the above examples will
be in A, B, or neither. Ask for any
conclusions based on these numbers.
Students should conclude that all the
probabilities of outcomes are fractions, and
that all the fractions add up to 1.
Teacher Input/Direct Instruction
(10 minutes)
Review the examples from the "Guided
Discovery" section and compute the
probabilities of the specific outcomes.
Include outcomes of individual events as
well, such as the outcome being 2, or 0.
Transition/Closure
(remainder)
Distribute worksheet/homework.
Verify through call and response that
students understand instructions on
homework and allow them to begin
working on it
30
NAME______________________
Probability Lesson Five Worksheet
Enter the elements of the sets into the proper place on the Venn diagrams.
Compute the probabilities given below each diagram.
Let U={dog, cat , goldfish, parrot,
iguana, tarantula, hamster, snake, }
Let A={pets with legs}
Let B={pets without legs}
Compute the probability that a student's
pet has no legs.
Compute the probability that a student's
pet has legs.
Use U as above.
Let A= {mammals}
Let B={pets with legs}
Compute the probability that the pet is
either a member of A or of B, in other
words, is either a mammal or has legs.
Compute the probability that the pet has
no legs.
Let U={the integers from 1 to 20,
inclusive}
Let A={prime numbers}
Let B={odd numbers}
What is the probability that a number is
both odd and prime?
What is the probability that a number is
even and prime?
31
Probability Lesson Plan Six
Grade Level: 11-12
Date: September 11, 2002
Subject: Mathematics/Probability
Time: 50 minutes
Generalization: Disjoint or mutually exclusive events are if outcomes A and B cannot
happen at the same time.
Skills/Attitudes: Students will relate knowledge from previous work to predict the
outcomes of probability events given certain parameters.
EALR's/Standards: 1.1 Understand and apply concepts and procedures from number
sense, including number and numeration, computation, and estimation; 1.4 Understand
and apply concepts and procedures from probability and statistics, including probability,
statistics, and prediction and inference; 1.5 Understand and apply concepts and
procedures from algebraic sense, including relations and representations, and operations;
2.3 Construct solutions by choosing the necessary information and using the appropriate
mathematical tools; 3.2 Predict results, make inferences and make conjectures based on
analysis of problem situations; 4.2 Organize and interpret information; 4.3 Represent and
share information; share, explain, and defend mathematical ideas using terms, language,
charts, and graphs that can be clearly understood by a variety of audiences; 5.3 Relate
mathematical concepts and procedures to real-life situations; understand the connections
between mathematics and problem-solving skills used every day at work and at home.
Objective: Students will recognize mutually exclusive events and compute a
probability.
Assessment: Informal
Materials: Pencil and paper, chalk/whiteboard
Procedures/Activities
Anticipatory Set (2-5 minutes)
Drawing on the previous nights
homework, examine the probabilities that
students computed.
Communicate Objective
Today will cover how to compute
probabilities of mutually exclusive events.
Mutually exclusive events are such that
results A and B cannot happen at the same
time.
Checks for Understanding
Verify that students have come to the
correct conclusions and address through
student/teacher input any questions
regarding interpretation.
Ask students which events were
mutually exclusive in the homework and
which were not. For example, a pet either
had legs or it did not, this is mutually
exclusive, but a number could be both odd
and prime, odd not prime, or even and
prime, not mutually exclusive.
Teacher Input/Direct Instruction
(10 minutes)
32
At the board introduce student to the
notation of the complement of a set. Define
the total probability as P(A)  P( ~A) and
state that we will now us P( ) to note a
particular probability. Review the
worksheet examples using this notation.
Transition
Distribute video game list, wherein
three students have different combinations
of video games.
Cooperative Learning (20-25 minutes)
Using the list, have students compute a
variety of probabilities.
Some examples:
What is the probability, P, that any own a
particular game?
P that any own a sports game?
P that any own a game based on a movie?
Have students create 3 mutually
exclusive examples of probabilities that can
be determined from the information
provided.
Student Input/Invention (7 minutes)
Have students share invented
probabilities by writing at the board.
Closure
(3 minutes)
No homework tonight, but inform
students there will be a short quiz to begin
class tomorrow.
Verify by call and response, writing
various combinations of symbolic notation
on the board and having students verbally
state the meaning.
Notify students that this handout will be
used for other lessons than today's and to
put it in their notebooks when finished.
Verify that students recognize that some
entries are duplicated. See how students
organize the information by walking
around.
Verify that the probabilities are out of
25 possible outcomes.
Check through call and response that
students are correct in their probabilities
and amend as necessary.
Give some opportunity to students to
clarify any definitions or expectations of
what will be on the quiz.
33
Probability Lesson Six Handout
Some students were forming a video game club, the first thing that they did was to make
up a list of the video games that each of them owned.
Ramon
Bond 007: NightFire
Harry Potter and Chamber of Secrets
Madden NFL 2003
Metroid Prime
SEGA Sports NBA 2K3
SEGA Sports NFL 2K3
SOCOM: U.S. Navy SEALs
Tony Hawk ProSkater 4
WWE SmackDown! Shut Your Mouth
Backyard Basketball
Monopoly Tycoon
Grand Theft Auto 3
Sandra
The Sims Hot Date Expansion Pack
Zoo Tycoon
Harry Potter and the Sorcerer's Stone
Harry Potter and Chamber of Secrets
Backyard Basketball
Monopoly Tycoon
SimCity 3000 Unlimited
RollerCoaster Tycoon 2
Ski Resort Tycoon 2
Tuan
Spider-Man: The Movie
DragonBall Z: Goku-GBA
Grand Theft Auto 3
Medal of Honor
Gran Turismo 3
Resident Evil
Halo
Super Mario Advance 2
SEGA Sports NBA 2K3
SEGA Sports NFL 2K3
34
Probability Lesson Plan Seven
Grade Level: 11-12
Date: September 12, 2002
Subject: Mathematics/Probability
Time: 50 minutes
Generalization: The addition rule in probability is used to avoid double counting
elements that are members of more than one set.
Skills/Attitudes: Students will evaluate methods of finding probabilities, synthesizing
previously learned material and formulate an algorithm for events not mutually exclusive.
EALR's/Standards: 1.1 Understand and apply concepts and procedures from number
sense, including number and numeration, computation, and estimation; 1.4 Understand
and apply concepts and procedures from probability and statistics, including probability,
statistics, and prediction and inference; 1.5 Understand and apply concepts and
procedures from algebraic sense, including relations and representations, and operations;
2.3 Construct solutions by choosing the necessary information and using the appropriate
mathematical tools; 3.2 Predict results, make inferences and make conjectures based on
analysis of problem situations; 4.2 Organize and interpret information; 4.3 Represent and
share information; share, explain, and defend mathematical ideas using terms, language,
charts, and graphs that can be clearly understood by a variety of audiences; 5.3 Relate
mathematical concepts and procedures to real-life situations; understand the connections
between mathematics and problem-solving skills used every day at work and at home.
Objective: Students will use the video game list from lesson 6 to discover the addition
rule in probability.
Assessment: Formal, Quiz attached.
Materials: Pencil and paper, chalk/whiteboard, quiz (attached)
Procedures/Activities
Anticipatory Set (0 minutes)
Checks for Understanding
Communicate Objective
Today is a quiz day. The questions on
the quiz are familiar and this should not be
too stressful as they are taken directly from
previous worksheets. After the quiz, the
class will develop a rule to avoid a
common mistake when determining a
probability.
Direct Instruction
Quiz
(25 minutes)
Verify student understand the
instructions and that there are two pages.
Remind them to write their names on the
top of the paper. If any typos are present or
students need general clarification on a task
35
or question, address these with both verbal
and written (on the board)
reminders/answers.
Transition
(2 minutes)
Collect quizzes, have students take out
the video game data sheets.
Guided discovery/Cooperative Learning
(20-25 minutes)
Have students create a diagram using
Ramon's and Sandra's data for sets A and
B. Have them compute P(A) and P(B).
Now have them compute P(A  B).
Have students show how they got their
results at the board and as a class evaluate
the answers.
Teacher Input/Direct Instruction
(10 minutes)
Formally state the addition rule at the
board.
Most students will jump to the P(A  B)
being P(A) + P(B), but this double counts
the elements that are shared by A and B.
If this was not taken into account, have
students reformulate their algorithm. Hint:
have them calculate the compliment of the
union and recall that the probability cannot
exceed 1.
P(A  B)= P(A) + P(B) - P(A  B)
and verify student comprehension through
call an response, use examples from quiz, if
needed.
Closure
(3 minutes)
For homework, have students derive an
addition rule for three events using the
Draw a Venn Diagram with three sets, if
video game data. Have them construct a
this has not already come up in class as an
Venn Diagram demonstrating their findings example.
to be handed in.
36
NAME________________________
Probability Lesson Seven Quiz
For problems 1 through 4, describe the following in your own words and give one
examples of each: (you may use the back of the paper, if necessary)
You may use calculators and your "Pascal's Triangle."
1) Probability
2) Sample space
3) Permutation
4) Combination
Solve the following:
6) How many permutations are there of a set with
a) 3 elements
b) 7 elements
c) 8 elements
d) 10 elements
7) How many combinations are possible in the following
a) picking 5 elements from a set of 11
b) picking 5 elements from a set of 8
c) picking 11 elements from a set of 13
Solve the following:
8)
6!
9)
4!2!
6!
10)
2!4!
8!
37
Probability Lesson Seven Quiz, continued
Enter the elements of the sets into the proper place on the Venn diagrams.
Compute the probabilities given below each diagram.
Let U={bicycle, tricycle, unicycle,
motorcycle, car, skateboard, surfboard,
hang-glider }
Let A={vehicles with only two wheels}
Let B={vehicles without wheels}
Compute the probability that a vehicle has
no wheels.
Compute the probability that a vehicle has
two and only two wheels.
Use U as above.
Let A= {vehicle with wheels}
Let B={vehicles without motors}
Compute the probability that the vehicle is
either a member of A or of B, in other
words, either has no motor or has wheels.
Compute the probability that a vehicle has
no motor.
Let U={the integers from 1 to 20,
inclusive}
Let A={prime numbers}
Let B={even numbers}
What is the probability that a number is
both even and prime?
What is the probability that a number is
even and not prime?
38
Probability Lesson Plan Eight
Grade Level: 11-12
Date: September 13, 2002
Subject: Mathematics/Probability
Time: 50 minutes
Generalization: Conditional probability is the likelihood of one event given another.
Skills/Attitudes: Students will use previously learned concepts to synthesize a method
for determining the algorithm for conditional probability.
EALR's/Standards: 1.4 Understand and apply concepts and procedures from probability
and statistics, including probability, statistics, and prediction and inference; 1.5
Understand and apply concepts and procedures from algebraic sense, including relations
and representations, and operations; 2.1 Investigate situations by searching for patterns
and exploring a variety of approaches; 2.3 Construct solutions by choosing the necessary
information and using the appropriate mathematical tools; 4. 1 Gather information, read,
listen, and observe to assess and extract mathematical information; 4.2 Organize and
interpret information; 4.3 Represent and share information; share, explain, and defend
mathematical ideas using terms, language, charts, and graphs that can be clearly
understood by a variety of audiences; 5.3 Relate mathematical concepts and procedures
to real-life situations; understand the connections between mathematics and problemsolving skills used every day at work and at home.
Objective: Through discussion, determine an algorithm for conditional probability,
introduce notation.
Assessment: Informal
Materials: Pencil and paper, chalk/whiteboard, computational practice
worksheet/homework.
Procedures/Activities
Checks for Understanding
Anticipatory Set (5-7 minutes)
Return and review quizzes from previous
Address and review any major concerns
day.
or general comprehension
misunderstandings by example from the
quiz at the board.
(5-7 minutes)
Collect homework, using student input, P(A  B  C )= P(A)+ P(B)+ P(C) write the addition rule for three events on
P(A  B ) - P( A  C ) - P( B  C )+
the board and review.
P(A  B  C)
Communicate Objective
Using the information on the video club
handout, we will determine an algorithm
for conditional probability.
Guided discovery/Cooperative Learning
(20-25 minutes)
39
Have students try to formulate a
conditional probability. Example are:
The probability choosing one of Ramon's
games, Tuan owns that game.
The probability that all three players own a
game.
The probability that given one of Sandra's
games, neither Tuan nor Ramon own the
games.
Student Input/Invention (7 minutes)
Have students present at the board the
findings regarding conditional probability.
Through class discussion refine these.
Teacher Input/Direct Instruction
(10 minutes)
Introduce formally the notation for the
algorithm to determine a conditional
probability.
Transition/Closure
(3 minutes)
Distribute homework sheet for
collection tomorrow.
Verify by walking around that students
are formulating good questions.
Check that students are using
probability notation to codify their
hypotheses.
Check through call and response that all
students understand the concepts of
conditional probability.
The probability of an event A given an
event B is
P(A|B)= P( A  B ) / P(B) if P(B)>0
Use the probability questions from the
Guided Discovery section to determine and
check student understanding.
Check through call and response that
instructions are clear.
40
NAME__________________________
Probability Lesson Eight Homework Worksheet
Using the video game club list, compute the following:
1) If both Sandra and Ramon own a game, what is the probability that Tuan also owns
the game?
2) What is the probability that any game selected is owned by more than one club
member?
3) What is the probability that if Ramon owns a game, no one else owns that same
game?
4) What is the probability that if Sandra owns a game, that Tuan also owns a copy of that
same game?
5) What is the probability that if a chosen game is a sports game, that Ramon owns it?
that Sandra owns it? that either Sandra or Tuan own it?
41
Probability Lesson Plan Nine
Grade Level: 11-12
Date: September 16, 2002
Subject: Mathematics/Probability
Time: 50 minutes
Generalization: Two events are independent if the probability of the intersection of two
sets is equal to the probability of one multiplied by the other.
Skills/Attitudes: Students will compare and discriminate between conditional and
independent events through analysis and observation of examples integrating previous
knowledge and notation.
EALR's/Standards: 1.1 Understand and apply concepts and procedures from number
sense, including number and numeration, computation, and estimation; 1.4 Understand
and apply concepts and procedures from probability and statistics, including probability,
statistics, and prediction and inference; 1.5 Understand and apply concepts and
procedures from algebraic sense, including relations and representations, and operations;
2.1 Investigate situations by searching for patterns and exploring a variety of approaches;
3.1 Analyze information from a variety of sources; use models, known facts, patterns,
and relationships to validate thinking; 3.2 Predict results, make inferences and make
conjectures based on analysis of problem situations; 3.3 Draw conclusions and verify
results, support mathematical arguments, justify results, and check for reasonableness of
solutions; 4.2 Organize and interpret information; 4.3 Represent and share information;
share, explain, and defend mathematical ideas using terms, language, charts, and graphs
that can be clearly understood by a variety of audiences.
Objective: Use probabilistic notation to determine independence of events.
Assessment: Informal
Materials: Pencil and paper, chalk/whiteboard, enough decks of card to provide one for
each group of four students, large sheets of poster paper for cooperative learning project.
Procedures/Activities
Anticipatory Set (2-5 minutes)
Collect homework begin with answering
any questions that may have come up.
Checks for Understanding
Specifically ask a few students for the
results of their computations and compare
these to the result on the collected sheets
by leafing through them. Look for any
gross difficulties, address these by doing
the problem(s) on the board.
Communicate Objective
Determining whether events are
independent based upon their probabilities
is the topic, and we will develop a
notational algorithm to help us determine
event independence.
42
Teacher Input/Direct Instruction
(10 minutes)
Present some scenarios and ask students
to determine whether the events are
independent. False analogies in advertising
are prime examples.
Formally present the algorithm for
determining event independence.
This is often one of the most confusing
elements in probability. The so-called
gambler's fallacy is an excellent example.
Often gamblers will bet on a number that
has not come up recently, judging that it is
"due." Sports fans also judge that players
who have underperformed recently can also
be "due."
Ask student to determine the odds of a
given outcome and whether this depends
on a previous result.
P(A  B)=P(A)P(B) is the formal
algorithm.
Cooperative Learning (20-25 minutes)
Pass out decks of cards. Ask if the odds
of drawing a red card are dependent on the
last card drawn: if the original draw is
returned to the deck; and if it is not
returned.
Have students determine the odds of
various hands:
2 of a kind (only)
3 of a kind (only)
straight
Student Input/Invention (7 minutes)
Have students present solutions to the
above.
Closure
Collect worksheets.
Have students work the problems on
large sheets, being sure they all sign their
names, to be handed in after presentation.
Verify students use the algorithm to
determine independence algebraically.
Verify students reasoning by walking
around and call and response in
determining the odd of the various hands.
Verify students use correct notation to
make their reasoning clear.
(3 minutes)
43
Probability Lesson Plan Ten
Grade Level: 11-12
Date: September 17, 2002
Subject: Mathematics/Probability
Time: 50 minutes
Generalization: Fairness is not arbitrary, it can be determined by using probability.
Skills/Attitudes: Students will evaluate situations and judge them for fairness,
discussing the moral components and relating this to choices made in life.
EALR's/Standards: 1.1 Understand and apply concepts and procedures from number
sense, including number and numeration, computation, and estimation; 1.4 Understand
and apply concepts and procedures from probability and statistics, including probability,
statistics, and prediction and inference; 1.5 Understand and apply concepts and
procedures from algebraic sense, including relations and representations, and operations;
2.1 Investigate situations by searching for patterns and exploring a variety of approaches;
3.1 Analyze information from a variety of sources; use models, known facts, patterns,
and relationships to validate thinking; 3.2 Predict results, make inferences and make
conjectures based on analysis of problem situations; 3.3 Draw conclusions and verify
results, support mathematical arguments, justify results, and check for reasonableness of
solutions; 4.2 Organize and interpret information; 4.3 Represent and share information;
share, explain, and defend mathematical ideas using terms, language, charts, and graphs
that can be clearly understood by a variety of audiences; 5.2 Relate mathematical
concepts and procedures to other disciplines; identify and apply mathematical thinking
and notation in other subject areas; 5.3 Relate mathematical concepts and procedures to
real-life situations; understand the connections between mathematics and problemsolving skills used every day at work and at home.
Objective: Design a fair carnival game. Discuss the idea of fairness.
Assessment: Informal
Materials: Pencil and paper, chalk/whiteboard, poster paper for project presentation.
Procedures/Activities
Anticipatory Set (2-5 minutes)
As students enter, play carnival barker.
Try to get them to play a game. Use one
that is especially unfair, but make the
reward great. After a couple of student
defeats, begin
Checks for Understanding
Even if students cannot determine why a
game is unfair, it is important that they
perceive the game as unfair. This will help
in the coming discussion.
Communicate Objective
Today is carnival day. Each group will
create a game for a community fund-raiser.
Open a discussion of fairness.
Student Input
(5 minutes)
In the discussion of fairness, let students
44
consider the moral issues of fairness.
When is it necessary? It is always
necessary? Has anyone ever cheated in
order to let someone else win? Is that OK?
As students discuss fairness, ask what
tools they use to judge whether something
is fair or not. Do they use mathematical
tools? Will they now? How? Which
ones?
Transition
(2 minutes)
Pass out poster paper, organize students
into groups.
Guided discovery/Cooperative Learning
(35 minutes)
Have students design a carnival game
which will make money, and yet the
participants have a reasonable chance of
winning, that uses the principles learned in
previous lessons.
The game must appear to be something
that players will be encouraged by looking
at. Salesmanship and general attractiveness
help to sell a game.
Student Input/Invention (15 minutes)
Have students present their carnival
games, demonstrating the probabilities of
winning for the participants and the
expected profit for the charity.
Have students vote on the games they
perceive as fair, the games they would be
attracted to play.
Allow student a lot of leeway in the
construction of their games.
Recommend/require the game have two
or more events to determine the outcome.
Remind students that they must
demonstrate mathematically just how the
game works.
At this point, students should not reveal
the odd in the game, just explain the rules
and encourage play.
Have other student keep notes on the
games, as we will vote on which we would
like to play.
Closure
(3 minutes)
Summarize the actual odds of the
games as figured out by the students
compared with how many students said
they would play the game, i.e., which
would be the most successful charity
games.
45
Combinatorics
Permutation and combination already having been introduced under the
probability umbrella, we will now study in a more general sense the arrangement of
elements from a set.
Four major topics will be explored in this unit: Dirichlet's Box Principle
(commonly known as the pigeonhole principle); the Fubini Principle; proof by induction;
and the introduction of summation notation.
Dirichlet's Box Principle seems intuitive. Given a number of boxes, a, and a
number of objects, b, such that b>a, then distributing the objects one at a time into the
boxes in order, at least one box must contain more than one object. See [4] (p189).
Suppose we have six distinct objects to distribute into five distinct boxes. Each
box will get one object until five of the original six are used, the remaining one must then
go into a box already occupied by an object.
In general, the number of ways of distributing b distinct objects into a distinct
boxes is ab, but remember that some boxes may now contain no objects.
In a similar vein, Fubini's principle states that if the average number of envelopes
per box is b, then one box will have at least b objects. Similarly, there must be a box with
at most b objects.
It is important to have the students note that in the above examples we are
creating pairwise disjoint subsets, i.e. partitioning the set of objects into a subsets.
We should also include the number of ways to distribute identical objects into distinct
(a  b  1)!  a  b  1
 , where a is the number of boxes and b the
boxes, which is
or 
b!(a  1)!
 b 
number of identical objects.
The Principle of Mathematic Induction states that a statement, Sn, is true for all
positive integers n if S1 is true and if Sk is true, then Sk+1 is true. A good demonstration
proof of this is 2n>n is true for all n ℕ, where ℕ is the set of natural numbers ([4], p51).
Introduction of the Greek letter sigma, used to indicate a sum, will be useful at
this point. For the summation:
8
 2i  x  3
i 0
The sigma is used to indicate the sum, i is called the index, the 0 represents the
initial value of the index and the 8 represents the final value. It is possible that a
summation may have more than one variable, as the x represents in the above, but only
the index is incremented. This is covered in [4], p53.
Have students practice using this notation to represent several sums, such as:
50
the sum of the first fifty positive integers,
i;
i 1
9
the sum of the cubes of the numbers greater than 2, but less than 10,
i
3
;
i 3
n  29
and the sum of the elements of a set A={a:a=2n+2, for 5<n<30},
 2n  2 .
n 6
46
Suggested reading for this unit would be to review any basic probability text, and
the topics covered here are also available in most standard discrete and statistics texts.
The text referenced in the unit frequently is an excellent general text, and is listed in the
bibliography as [4]John P. D'Angelo and Douglas West, Mathematical Thinking:
Problem Solving and Proofs, 2nd ed., Prentice Hall, 2000.
47
The Binomial Theorem
Students are probably more familiar with the binomial theorem than they suspect,
so, begin with what they know. Algebraically multiplying out the terms (a+b)2, and
(a+b)3 to derive the coefficients manually is a comfortable place to start, and then we
begin to put some pieces together.
Allow students to explore the first several powers and through the use of guided
discovery have them attempt to derive the generalized formula for the binomial theorem,
[4], p104. Include both combination and summation notation to arrive at
n
n
( x  y ) n     x n r y r
r 0  r 
Have students gain experience with using the theorem to expand a term such as
(x+y)6 and to find the coefficient of a term such as x11y6 in the expansion of (x+y)17.
The answers to the above should be obtained as follows:
 6
 6
 6
( x  y ) 6    x 6    x 5 y  ...    y 6  x 6  6 x 5 y  ...  y 6 ;
 0
1
 6
and for the second,
17  17! 17  16  15  14  13  12
  

 17  4  14  13  12,376 .
6  5  4  3  2 1
 6  11!6!
Now it is time to return to the outcomes of a binomial experiment, defined as a
compound experiment of n repeated trials in which each trial has one of two possible
outcomes, or, arbitrarily, success or failure. This is also called a Bernoulli Trial, and is
listed in [4] on p175. It shows that the probability of exactly k successes in such an
experiment of n number of trials is:
 n  k nk
  p q
k 
in which the probability of a single success is p and the probability of a failure is 1-p=q.
We can also generalize the binomial theorem using Newton's formula for negative

 n
integers: ( x  y) n    x k a nk . Students can then construct a modified version
k 0  k 
([16], p71) of Pascal's triangle as given here:
n=-4
n=-3
n=-2
n=-1
n=0
n=1
n=2
n=3
n=4
1
1
1
1
1
1
1
1
1
-4
-3
-2
-1
0
1
2
3
4
10
6
3
1
0
0
1
3
6
-20
-10
-4
-1
0
0
0
1
4
35
15
5
1
0
0
0
0
1
-56
-21
-6
-1
0
0
0
0
0
84
28
7
1
0
0
0
0
0
...
...
...
...
...
...
...
...
...
which leads directly in the topic of the next unit.
48
Suggested reading for this unit would be to review any basic probability text, and
the topics covered here are also available in most standard discrete and statistics texts.
The text referenced in the unit frequently is an excellent general text, and is listed in the
bibliography as [4]John P. D'Angelo and Douglas West, Mathematical Thinking:
Problem Solving and Proofs, 2nd ed., Prentice Hall, 2000.
49
Series and Limits
With the introduction of Newton's formula for negative integers in his modified
version of the binomial theorem, we can allow the students to explore what happens
when variables are given values and attempt to determine the numerical value of a given
expression (note: if this unit is taught out of order, sigma summation notation should be
reviewed, otherwise it has been covered in unit three).
Sequences are sets of numbers in a specific order of arrangement and formed
according to a specific rule ([21], p727). Each element of the set is called a term, and
may be finite or infinite. An example of a finite sequence is {2,7,12,17,22,...,47}, an
example of an infinite sequence is {1/2, 1/3, 1/4, ...}. A general term for a sequence, also
called the nth term, is the rule by which any term in the sequence may be determined.
So, if the nth term of xn, a sequence, is given by n2+1, then the first, second and tenth
terms would be 2, 5, and 101, respectively.
A series is a sum of the terms in the sequence ([21], p738-9). Notationally, for
5

1 1 1 1 1
1
1 1 1
whereas  x n     ...
example:  x n for x n  2 is    
1 4 9 16 25
1 4 9
n
n 1
n 1
are examples of finite and infinite series.
An intuitive idea of a limit is a number to which the terms of a sequence get closer
and closer, and can often provide for an initial guess at this value. For instance, the terms
{1, 3/2, 5/3, 7/4, 9/5, 11/6...} give the sequence the appearance of approaching 2. In
order to confirm this, however, we must establish the more rigorous definition ([21],
p729).
L is said to be the limit of an infinite sequence, if, for every  ℕ (i.e., for every
epsilon, no matter how small, an element of the natural numbers) we can find a number,
N such that | xn  L |  for all integers n>N.
It is helpful at this point to show the limit is, indeed, two for the previous
example, and introduce the notation that lim x n  L .
n
If these limits exist, students may begin to manipulate them with the following
theorems on limits ([21], p730):
1) lim ( x n  y n )  lim x n  lim y n
n 
n 
n 
2) lim ( x n  y n )  lim x n  lim y n
n 
n 
n 
3) lim ( xn )  (lim xn ) , where a is any real number, and
a
n
x
4) lim  n
n  y
 n
a
n
 n 
 
provided lim y n  0
n 
yn
 lim
n 
lim x n
x 
note: If lim y n  0 and lim xn  0, then lim  n  does not exist.
n
n
n y
 n
x 
If lim y n  0 and lim xn  0, then lim  n  may or may not exist.
n
n
n y
 n
50
Two relevant investigations of series in relation to the units covered thus far are
the Bernoulli numbers (see [21], p369), which are defined by an initial value of B0=1
 n  1
 n  1
 n  1
 n  1
 B0  
 B1  
 B2  ...  
 Bn  0 and can be expressed in terms of
as 
 0 
 1 
 2 
 n 
Pascal's Triangle thusly
B0
=
1
B0
+
2B1
=
0
B0
+
3B1
+
3B2
=
0
B0
+
4B1
+
6B2
+
4B3
=
0
...
and in the Fibonacci (see [4], p238-241) sequence where the initial values are F0=0, and
F1=1 and the value of any term in the sequence is given by Fn=Fn-1+ Fn-2. This pattern of
numbers can, of course be discovered in the diagonals of Pascal's Triangle. While the
Bernoulli numbers have important applications in analysis and number theory, it may be
of greater value to have the students explore the Fibonacci sequence and report back on
their findings since there are pages and pages of occurrences of the sequence in nature,
architecture, etc.
The next part of the unit deals with infinite series, and whether or not those series
are convergent. Some of the best examples of applications of series and limits that I have
come across are medication levels (which can be important to many students), economic
factors, and ecological studies.
It can be easily shown with an overhead calculator display that it takes
approximately two weeks for a specified dosage of medication to reach the prescribed
level in the human system. Supposing the body loses ~25% of a medication level
through fluid loss in a day, and a full dose is added again the next, demonstrating this
addition and loss on the overhead shows the level of medication in the body approaches a
limit, the desired level. Missing just one day, it will take an additional two weeks to
obtain that level again, and lest one think that taking an extra dose the following day will
work, plugging in this value will show that one is above the recommended dosage and it
will still take an additional two weeks to level off.
As to economics, one great example is the pricing of items to sell the most.
Disregarding profit, what if the local ice cream shop sold cones for one cent? Would you
buy one? Two? Eventually, even the most rabid ice cream fan will stop.
Predator/prey models based on limited available resources also make for great
examples of limits, especially in our more ecologically-minded environment.
If we let Sn be the sum of the first n terms of a series, then lim S n denotes a
n 
convergent series if equal to S, a finite number. A series which is not convergent is
called divergent ([21], p739). Useful in determining the convergence of a series in which
it is difficult to determine if the nth term does not have a limit zero, is the comparison test
for convergence. This states simply, that if from some term on in the series, each term is
then less that or equal to the terms in a known convergent series, the series converges.
Most commonly used in these cases are:
1) The Geometric Series, a + ar + ar2 +...+ arn-1 +... where a and r are constants
is convergent if |r|<1 and divergent otherwise ([21], p740).
51
1
1
1
1
 p  p  ...  p  ... , where p is a constant is convergent
p
1
2
3
n
if p>1 and divergent otherwise (if p=1, this is the divergent harmonic series
1+1/2+1/3...). See [21], p750.
x
The ratio test for convergence states that if we determine lim n 1  R , then if
n  x
n
R<1, then the series converges, if R>1, the series diverges, and if R=1 the test is
indeterminate, and this means we must use a different method to determine convergence
([21]. p767).
Determining whether an alternating series is convergent can be determined if the
nth term has a limit zero, or if after so many terms, the absolute value of each term is less
than that of the preceding term. The series may only be conditionally convergent.
Determining whether the series is convergent by making a series in which all the signs
are positive determines if the series is absolutely convergent (note: the terms of an
absolutely convergent series may be rearranged at will and not affect convergence, but
the terms of a conditionally convergent series may not, as then convergence or
divergence to any desired sum may be derived. See[21], p765-6).
Finally, defining the power series as of the form c0 + c1x + c2x2 +...+ cnxn +...,
where the coefficients are constants is a power series in x. c0 + c1(x-a) + c2(x-a)2 +...+
cn(x-a)n +... is referred to as a power series in (x-a). The set of values for which x is
convergent is defined as the interval of convergence ([21], p776).
2) The p-series,
Suggested reading for this unit would be to review any basic Calculus text. The
text referenced in the unit frequently is an excellent general text, and is listed in the
bibliography as [21]James Stewart, Calculus, 4th ed., Brooks/Cole Publishing Co., 1999.
Also referenced in this section was [4]John P. D'Angelo and Douglas West,
Mathematical Thinking: Problem Solving and Proofs, 2nd ed., Prentice Hall, 2000.
52
Graph Theory
Utilizing Pascal's Triangle, the first step on the journey into graph theory begins
with the simple question of "How many ways are there to get from here to there?" Given
a simple graph of dots (nodes or vertices) and lines (edges) students will seek to
determine an answer under specific parameters.
Tracing paths from A to B, with the only allowable moves being up or right, how
7
many different ways are there to traverse the graph? The answer is   , or 35, and a
 4
graphical demonstration can be done by superimposing the graph on the triangle itself.
The above is an example of a simple graph, on which any edge can connect at
most two vertices. See [4], p202-204 for definitions of the terms included here. The
traditional history of graph theory begins with the Königsberg Bridge problem. This
problem involves a multigraph, a non-simple graph, in which any number of edges may
connect any number of vertices. The third type is a multigraph in which edges are
allowed to connect vertices with themselves, or loop.
The Königsberg Bridge Problem involves the above multigraph, the original
question being whether a route could be taken and each bridge crossed only one time.
53
The question was answered in the negative by Euler and so began the mathematical
excursion into graph theory.
The above problem is one of an Eulerian trail, path, or walk. This is a trail which
traverses a connected graph tracing each edge only once ([4], p205-6). Euler showed that
the graph may have at most two vertices from which an odd number of edges emanate
(called odd degree).
An Eulerian circuit begins and ends at the same node and Euler showed that this
is possible with a graph containing no vertices of odd degree. Note that [4] refers to this
as a closed trail.
Similarly, a Hamiltonian circuit that visits each vertex only once (Hamiltonian
circuits will be explored more fully in unit nine on Platonic solids).
The degree of a vertex is helpful in determining may of a graphs other
characteristics. The degree is simply the number of edges incident with the vertex, and
the degree sequence is a list of the degrees in descending order. For example, the degree
sequence in the Königsberg Bridge Problem is 5,3,3,3.
Directed graphs or digraphs are commonly used in urban and interstate traffic
problems, vertices being connected by a single directed edge for one way streets and
connected by two oppositely directed edges for a two way street. Distances can be
measured and optimized for transportation industry applications by adding weights to
individual edges.
Students should examine construction of the graphs, connected and disconnected,
below are the connected graphs for two through five vertices.
Be sure to emphasize isomorphic graphs, defining isomorphism as when two
vertices adjacent in one graph iff the corresponding adjacencies exist in the second graph
as well ([4], p208).
In order to check if graphs are isomorphic, suggest checking: the number of
vertices and edges; the degree sequence; the shortest path between pairs of vertices of a
given degree; and the longest path in the graph.
54
Suggested reading for this unit would be to review any basic graph theory text,
and the topics covered here are also available in most standard discrete texts. The text
referenced in the unit frequently is an excellent general text, and is listed in the
bibliography as [4]John P. D'Angelo and Douglas West, Mathematical Thinking:
Problem Solving and Proofs, 2nd ed., Prentice Hall, 2000.
55
Logical Proof
The purpose of this unit is to serve as an introduction to the propositional
calculus. The fundamental consideration in the study of logic is the statement, a
declarative sentence that is either true or false, but not both. The variables p,q,r, and s are
the most commonly used to represent statements. A compound statement is formed by
joining statements with connectives.
Labeling our statements p and q, we define the three types of connectives: the
conjunction (and) means p and q are both true; the disjunction (or) means p, q, or
possibly both are true; and the negation (not true) mean that p is not true (or, less clearly,
that it is false that p is true). These connectives are denoted as follows:
'and' is represented by '  ' as in p  q , and is read "p and q;"
'or' is represented by '  ' as in p  q , and is read "p or q;"
'not' is represented by '~' as in p ~ q, and is read "not p,"
and the students should use these representations to construct some basic truth tables.
See [4], p32-3.
Conjunction
Disjunction
Negation
pq
pq
p q
p
q
p
~p
T T
T
T
T
T
T
F
T F
F
T
F
T
F
T
F T
F
F
T
T
F F
F
F
F
F
If there are n primary (elementary) statements in a compound statement the truth
table will have 2n rows, so p and q and r and s would need 16 rows in the complete table.
Through the construction of these truth tables have students look for any unique
results. We are interested in two specific results. A compound statement that is always
true no matter what values are assigned to the elementary statements is called a tautology.
One that is always false is called a contradiction. Two important tautologies are named
DeMorgan's Laws:
~ ( p  q) ~ p  ~ q
~ ( p  q) ~ p ~ q
An excellent graphical demonstration can be done by representing DeMorgan's Laws in
terms of set notation:
( A  B )c = Ac  Bc
( A  B )c = Ac  Bc
and having the students represent the relationship in Venn Diagrams. [4] demonstrates
the translation into a logical equivalence about set membership but cautions "Although
relationships between sets correspond to logical statements about membership, the two
expressions tell the same store in different languages. One must not mix them. For
example, A  B is a set, not a statement; it has no truth value (p35)."
Note: The Principle of Mathematical Induction was covered in the second unit, on
combinatorics. If this unit is presented out of order, this should be covered at this point.
Notationally, the introduction of the quantifiers "for every" (denoted  ) and
"there exists" (denoted  ) should be done at this time, if it has not already. These are the
universal quantifier and the existential quantifier, respectively ([4], p28-9).
56
To review the elementary proof techniques, see [4], pages 35-40. The basic
methods are listed here:
Direct proof is exactly what it sounds like; taking a hypothesis and, with any tools
that we have at our disposal, proceed to confirming the statement. Good examples for
presentation are: sums of even numbers are even numbers; sums of an even number of
odd numbers is an even number; or, the sum of an odd and an even number is an odd
number.
Proof by contradiction (which [4] lists in italics rather that boldface) show that if
we assume our hypothesis to be false, at some point in the proof we will reach
contradiction with a known or proven hypothesis, making true equal false, and therefore
our original hypothesis must be true. Traditionally, showing that the square root of two is
irrational is a good example of this method.
Proof by contraposition can be difficult for some, because we are basically
showing that a hypothesis true for a different but equivalent statement. Using the
examples from the paragraph above on direct proof and showing the answers are
equivalent to those in the direct method are good demonstrations.
Counterexamples as a method of disproving a hypothesis in one stroke should be
introduced, as well. This is a concept teenagers should have no problem understanding,
since it is already a major weapon in their arsenals (think: "so-and-so did it, and didn't get
in trouble...").
Harkening back to Euclid's axioms from geometry is necessary to give students
the foundations on which all mathematical proof is built, that there exist certain
undefined true terms and postulates on which theorems are based. A brief history of nonEuclidean geometry's development may be undertaken at this time.
Suggested reading for this unit would be to review any basic introduction to proof
text, and the topics covered here are also available in many logic course texts. The text
referenced in the unit frequently is an excellent general text, and is listed in the
bibliography as [4]John P. D'Angelo and Douglas West, Mathematical Thinking:
Problem Solving and Proofs, 2nd ed., Prentice Hall, 2000. Also recommended is
[8]Stephen Galovich, Doing Mathematics: An Introduction to Proofs and Problem
Solving, Saunders College Publishing, 1993.
57
Number Theory
Of all the mathematical disciplines, number theory has the honor of having the
most amateur practitioners. Many professional mathematicians began their journeys into
the field through this very door. Why? Although the answers to many questions in
mathematics can be mind-bogglingly complex, and require a highly specialized
vocabulary and unique sets of skills, the questions in number theory are often able to be
stated in a way that the person-on-the-street can actually understand.
It has not been that many years (1994) since Andrew Wiles tied the strings
between an elliptic curve which is not modular and it's relation to the then
Shimura/Tanayama conjecture, i.e. proving all semistable elliptic curves with rational
coefficients are modular, and proved Fermat's Last Theorem (aha!). While there are
fewer than a handful of folks on the planet that understand all of Wiles' proof, almost
everyone can understand the question. Interest in questions is all that is required to begin
this particular journey.
Note: mathematical induction was introduced in the combinatorics unit, the
binomial theorem in it's unit and mathematical proof in the previous unit. Some
important notation (combination, summation) was covered at that time. If this unit is
being taught out of sequence, it will be necessary to introduce these at this point.
Obviously, a couple of weeks is not nearly enough time to get a good grasp on the
fundamentals of number theory, but the hope is that an interest will be generated and
some key terms and theorems may be given at least a handshake-type familiarity.
Reintroduce the Fundamental Theorem of Arithmetic (FTA). Students will have
worked with the ideas of greatest common divisor(gcd) and the least common
multiple(lcm) when working with fractions with unlike denominators, but a formal
definition should not be over looked. See [4], pages 124-6, although the lcm is reserved
as an exercise (p.
The FTA states that every positive integer greater that one can be expressed as a
product of prime and that, aside from ordering, this representation is unique.
This is an excellent opportunity to present Euclid's amazingly elegant proof of the
infinitude of primes.
Presentation and proof of the division algorithm ([4], p132) should be included.
The division algorithm is stated: Given integers a and b, such that b>0, then there exist
unique integers q and r which satisfy
a = bq + r
where 0  r < b. The notation is indicative of the fact that q and r are commonly referred
to as the quotient and the remainder, respectively (many proofs are available in basic
number theory texts or online, and I suggest using the one most intuitive to the
instructor).
Students should be asked to participate in proving enough of the basic theorems
about divisibility and also given the definition of the such that they are at least able to
understand the reasoning behind the following two theorems:
1) given integers a and b, not both of which are zero, there exist integers x and y
such that gcd( a, b ) = ax + by;
2)and, let a and b be integers, both of which are not zero, then a and b are
relatively prime iff there exist integers x and y such that gcd( a, b ) = ax + by = 1.
58
Introduction to, and practice exercise in the Euclidean Algorithm for determining
the gcd of a pair of integers ([4], 126-9).
Pascal's triangle contains patterns for the polygonal numbers, and exercises using
these can be given at this time. For example, proving the sum of consecutive triangular
numbers is a perfect square.
Here is a graphical representation of the polygonal numbers:
So, for example, the first three triangular numbers are 1, 3, 6, and 10. Summing
consecutive pairs would give 4, 9, and 16, the first three square numbers. Is this always
true? Finding the numbers in Pascal's triangle does require some summing, except for the
triangular numbers, which can be found easily. To find the squares, one must do the
summing listed above. For the pentagonal numbers, more consecutive numbers must be
added each time, i.e., 1, 2+3, 3+4+5, 4+5+6+7, etc., to get 1, 5, 12, 22.
Suggested reading for this unit would be to review any basic introduction to
number theory text, and the topics covered here are also available in many discrete course
texts. The text referenced in the unit frequently is an excellent general text, and is listed
in the bibliography as [4]John P. D'Angelo and Douglas West, Mathematical Thinking:
Problem Solving and Proofs, 2nd ed., Prentice Hall, 2000. Also recommended is
[3]David M. Burton, Elementary Number Theory, 2nd ed., Wm. C. Brown, Publishers,
1989.
59
General Forms
The version of Pascal's Triangle we began with is a very specific form. In the unit
on the binomial theorem, we expanded the triangle using negative exponents. Is this the
only type of generalization we can make?
This unit begins with a review/introduction of converting numbers of one radix to
another. See [4], p76-8. Numbers in base 10 should be converted into a variety of
different bases. Include sexagesimal for historical purposes (and some Babylonian
notation might be fun), and give our future computer workers a thorough understanding
of binary and hexadecimal.
Base
Numeral
Base ten
100
Base two
110010
Base three
1201
Base sixteen
61
Base sixty
1,40
After assuring understanding on the basic level, have students construct Pascalstyle triangles in other bases. Explore Sierpinski's Sieve construction either manually or
on the internet(recommended). Enter the italicized phrase above into a search engine and
a number of generating programs can be found. Look for ones where the students can set
the parameters such as modulus, color, and number of rows. Ambitious students may
wish to construct their own computer generated graphical display.
A basic introduction to modular math may be introduced at this time. Yet another
way of counting, in terms of remainders, congruence classes are helpful in mathematical
proofs involving divisibility. For more on congruence and modular math, see [4], p1427. The definition of congruence classes modulo n is, if n is a positive integer, and a and b
are integers, then a is congruent to b modulo n if n divides ( b - a) and is written
a  bmod n if n | (b  a) . For help in coming up with some possible proofs in the
triangle, the following theorems should also be given:
1) if a is congruent to b modulo n, then a and b have the same remainder when
divided by n;
2) and, if a is congruent to b modulo n and if c is congruent to d modulo n, then:
a) a  c  b  d (mod n) ;
b) a  c  b  d (mod n) ;
c) and, ac  bd (mod n) .
Ask if students' believe we have done all the generalizing possible. Have students
construct a triangle to determine the coefficients of the terms of the expansion of
( 1 + x + x2 )n and ( 1 + x + x2 + x3 )n
and see if they can synthesize a formula for the general case
( 1 + x + x2 + ... + xk )n.
Important note: Do not expect students to achieve every goal. While we should expect
student to strive to accomplish a task set forth, some may be beyond their knowledge
base or capabilities. This does not mean they should not try. For a comprehensive
exploration of the above topic, consult the article listed in the bibliography by Yuling and
Fengpo [24].
60
Other generalizations can include the idea of extra dimensions. Construct a
regular pyramidal version of the triangle for an exploration in three dimension. What
about four?
Suggested reading for this unit would be to review any basic introduction to
number theory text, and the topics covered here are also available in many discrete course
texts. The text referenced in the unit frequently is an excellent general text, and is listed
in the bibliography as [4]John P. D'Angelo and Douglas West, Mathematical Thinking:
Problem Solving and Proofs, 2nd ed., Prentice Hall, 2000. Also recommended is
[3]David M. Burton, Elementary Number Theory, 2nd ed., Wm. C. Brown, Publishers,
1989.
The listing for the article in the lesson is [24]Gao Yuling and Sun Fengpo,
Formulae for the General Terms of the Generalized Yang Hui's Triangle, International
Journal of Mathematical Education in Science and Technology, Jul./Aug. 1998, Vol. 29,
Issue 4, p587, pp7.
61
Geometric constructions
Students will study the construction of various geometric forms using the
information available in Pascal's Triangle. Explorations of figurate numbers were
presented in the unit on number theory, and some basic shapes will have been explored in
the unit on graph theory. While it is not imperative that these units be taught in order, the
circuits and trails explored in graph theory will be applied to the three dimensional
objects created in this unit and if not taught in order, should be introduced here.
Through an introduction of compass and straightedge constructions, if students
are unfamiliar with them, basic regular polygons will be examined. The constructability
of regular polygons will be examined, for instance why a regular heptagon is not.
A tie to Pascal's Triangle is cited in Conway and Guy [3.5] where Watkins
noticed that the number of sides for constructible polygons with an odd number of sides
are given by the first 32 rows of Pascal's triangle (mod 2) interpreted as binary numbers
(i.e., Seirpinski's Triangle), giving 1, 3, 5, 15, 17, 51, 85, 255.
Base:
two
ten
1
1
1 1
3
1 0 1
5
1 1 1 1
15
1 0 0 0 1
17
1 1 0 0 1 1
51
1 0 1 0 1 0 1
85
1 1 1 1 1 1 1 1 255
1 0 0 0 0 0 0 0 1 257
Students will also construct "Golden Rectangles" using the Golden Ratio as in
[17] and join three golden rectangles perpendicularly to form the two Platonic solids, an
icosahedron and a dodecahedron.
In order to have physical examples of the solids, an introduction to origami will
add some entertainment as well as some math. Students will then look for patterns in the
platonic solids relating to topics already covered. They will trace Eulerian and
Hamiltonian circuits along the edges.
They will label vertices, edges and faces and be introduced to the Euler's Formula,
V - E + F = 2. Using this they can prove that there are only the five platonic solids. See
[4], pages 203 and 226.
Exploration of Euler's Formula and a search for proof should lead students to
discover that it is true only if the solid is the topographical equivalent of a sphere, i.e., has
no holes ([9.5], p448).
Suggested reading for this unit includes the cited sections in [3.4], [4], [9.5], and
[17] or any thorough introduction to geometry should cover most of necessary
background. David Mitchell's Mathematical Origami: Geometric Shapes by Paper
Folding, Tarquin Publications, 1997 is a good intro to for young adults.
62
Triangle Oddities
Students will study various pattern observable in the triangle and attempt to prove
or disprove conjectures that they themselves make.
Presentations of other patterns observable in the triangle presented will be the
appearance of Tchebychev Polynomials, Bessel Polynomials, and Stirling Numbers in
forms similar to Pascal's Triangle.
To encourage students to in their searches, present the pattern of the powers of 11
as included below. Beginning at the rightmost side of a row, transcribe each digit into a
new triangle, if a number has more than one digit, add this to the next number to the left
before transcribing. For example if the numbers in the triangle in row 5, write 1, then 5,
then 0 adding the one to the next number to the left which becomes 11, write 1, repeat the
procedure getting another 11, write one, repeat getting 6, write six and finally write one.
Pascal's Triangle before manipulation
After
Powers of 11
1
1
110
1 1
11
111
1 2 1
121
112
1 3 3 1
1331
113
1 4 6 4 1
14641
114
1 5 10 10 5 1
161051
115
1 6 15 20 15 6 1
1771561
116
1 7 21 35 35 21 7 1
19487171
117
The above can be shown numerically by using 10 and 1 for a and b in a binomial
expansion (thanks Dr. Johnson).
Suggested reading for this unit are [3], [4], [8], [17], and [18], but not as texts for
presentation. Skimming for ideas to supplement those students propose (or, don't
propose) is the goal. Suggestions from one's own "interesting problem file" are welcome.
The idea is for the students to make presentations that are mathematically sound and
understandable by the broader mathematical community.
63
Appendix
Washington State Essential Academic Learning Requirements
Academic Standards
1. The student understands and applies the concepts and procedures of
mathematics. To meet this standard, the student will:
1.1 Understand and apply concepts and procedures from number sense, including
number and numeration, computation, and estimation
1.2 Understand and apply concepts and procedures from measurement, including
attributes and dimensions, approximation and precision, and systems and tools
1.3 Understand and apply concepts and procedures from geometric sense, including
shape and dimension, and relationships and transformations
1.4 Understand and apply concepts and procedures from probability and statistics,
including probability, statistics, and prediction and inference
1.5 Understand and apply concepts and procedures from algebraic sense, including
relations and representations, and operations
2. The student uses mathematics to define and solve problems. To meet this
standard, the student will
2.1 Investigate situations by searching for patterns and exploring a variety of approaches
2.2 Formulate questions and define the problem
2.3 Construct solutions by choosing the necessary information and using the appropriate
mathematical tools
3. The student uses mathematical reasoning. To meet this standard, the student
will
3.1 Analyze information from a variety of sources; use models, known facts, patterns,
and relationships to validate thinking
3.2 Predict results, make inferences and make conjectures based on analysis of problem
situations
3.3 Draw conclusions and verify results, support mathematical arguments, justify
results, and check for reasonableness of solutions
4. The student communicates knowledge and understanding in both everyday and
mathematical language. To meet this standard, the student will
64
4. 1 Gather information, read, listen, and observe to assess and extract mathematical
information
4.2 Organize and interpret information
4.3 Represent and share information; share, explain, and defend mathematical ideas
using terms, language, charts, and graphs that can be clearly understood by a variety of
audiences
5. The student understands how mathematical ideas connect within mathematics,
to other subject areas, and to real-life situations. To meet this standard, the student
will
5.1 Relate concepts and procedures within mathematics; recognize relationships among
mathematical ideas and topics
5.2 Relate mathematical concepts and procedures to other disciplines; identify and apply
mathematical thinking and notation in other subject areas
5.3 Relate mathematical concepts and procedures to real-life situations; understand the
connections between mathematics and problem-solving skills used every day at work
and at home
The above information was obtained at
http://www.seattleschools.org/area/acastan/stan/math/math.xml, which is the Seattle
Public School’s web page defining the EALR’s for mathematics
A full description by grade level can be obtained in PDF format at
http://www.seattleschools.org/area/acastan/full/math.pdf .
65
Bibliography
[1]Thomas F. Banchoff, Dimension(?) On the Shoulders of Giants: New Approaches to
Numeracy, pp25-26, 50-59, Lynn Arthur Steen, ed. National Academy Press, 1990
[2]Edward Brooks, The Philosophy of Arithmetic as Developed from the Three
Fundamental Processes of Synthesis, Analysis, and Comparison, 571pp, Normal
Publishing Company, 1876
A pedagogical work concerned mainly with the teaching and learning of mathematics
emphasizing the achievement of a personal relationship with mathematics through
practical applications that were the norm at the time of publications.
[3]David M. Burton, Elementary Number Theory, 2nd ed., Wm. C. Brown, Publishers,
1989
This is a book on classical number theory intended for a one semester course at the
undergraduate level and designed to be used as supplementary reading in mathematics
survey courses, as well. It provides some historical background and can be a useful tool
to secondary teachers seeking more familiarity with the topic.
[3.5]John H. Conway and Richard K. Guy, The Book of Numbers, Copernicus, 1996
This book lets readers of all levels of mathematical sophistication (or lack thereof)
understand the origins, patterns, and interrelationships of different numbers. Whether it is
a visualization of the Catalan numbers or an explanation of how the Fibonacci numbers
occur in nature, there is something in here to delight everyone. The diagrams and
pictures, many of which are in color, make this book particularly appealing and fun.
[4]John P. D'Angelo and Douglas West, Mathematical Thinking: Problem Solving and
Proofs, 2nd ed., Prentice Hall, 2000
Designed to be a text for both a one semester course in discrete mathematics and a one
semester course in analysis, this book was of high value in describing many of the topics
covered in the course. It includes ample definitions and exercises.
[5]Philip J. Davis and Reuben Hersh, The Mathematical Experience, Houghton Mifflin
Company, 1981
The book tackles problems of mathematical experience which are tough because they fall
into the realm of philosophy: the meaning of proof; the goal of abstraction and
generalization; the existence of mathematical objects and structures; and the necessary
interplay between natural and formal language. What is amazing is that Davis and Hersh
make these ideas not only accessible to an intelligent layman, but also interesting and
vital, without losing the interest of real mathematicians.
[6]Dewey C. Duncan, Ten Mathematical Refreshments: Mathematics in Secondary
School Classrooms, pp 118-124, Rising/Wiesen, eds., Thomas Y. Crowell Co., 1972
[7]William Dunham, The Mathematical Universe, John Wiley & Sons, NY, 1994
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Start with a for Arithmetic and wend your way to z, with a different topic covered for
each, and you will have completed Dunham's wonderfully informative tour of
mathematics.
[8]Stephen Galovich, Doing Mathematics: An Introduction to Proofs and Problem
Solving, Saunders College Publishing, 1993
Galovich's title says it all. Topics covered include propositional calculus, methods of
proof, sets and set operations and strategies for attacking problems. It is clearly written
and gives clear language interpretations to many basic ideas we wish we could express as
succinctly.
[9]Leonard Gillman, Writing Mathematics Well: A Manual for Authors, 49pp, The
Mathematical Association of America, 1987
Gillman gives us a comprehensive guide to writing for publications, or mathematical
manual of style as it were.
[9.5]Jan Gullberg, Mathematics- From the Birth of Numbers, W. W. Norton & Co. 1997
A solid bit of everything. A good reference for a introduction to most classical topics and
history. Brief, but makes up in volume what it lacks in depth.
[10]Reuben Hersh, What is Mathematics, Really?, Oxford University Press, 1997
Hersh takes on the philosophy of math from its Platonic roots to the neo-Platonic and
neo-Fregian present. He supports, quite well, his view of mathematics as a human
cultural and social construct and uses this philosophy to analyze age-old questions of
proof, certainty, and invention versus discovery.
[11]Jaakko Hintikka, The Principles of Mathematics Revisited, Cambridge University
Press, 1996
This book, written by one of philosophy's preeminent logicians, argues that many of the
basic assumptions common to logic, philosophy of mathematics and metaphysics are in
need of change. Jaakko Hintikka proposes a new basic first-order logic and uses it to
explore the foundations of mathematics.
[12]Jeff Holt and John Jones, Discovering Number Theory, W. H. Freeman & Co., 2001
An active exploration text with accompanying cd-rom wherein basic number theory
theorems are discovered through number manipulation and calculation made easier by the
inclusion of software applets. This contains many good theorem development exercises.
[13]Thomas W. Hungerford, Abstract Algebra: An Introduction, second edition, 588pp,
Saunders College Publishing, 1997
If you've already taken some undergrad courses in number theory, discrete mathematics,
or linear algebra, then you'd be more than enough prepared to go through this book on
your own. It's highly readable, and the problems aren't that hard to solve. He also breaks
up the problems in 3 sets with the last being the hardest. Reading through the appendix is
enough to prepare anyone that has taken calculus for the material in the book.
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[14]Philip Kitcher, The Nature of Mathematical Knowledge, Oxford University Press,
1983
This book argues against the view that mathematical knowledge is a priori, contending
that mathematics is an empirical science and develops historically, just as natural
sciences do. Kitcher presents a complete, systematic, and richly detailed account of the
nature of mathematical knowledge and its historical development, focusing on such
neglected issues as how and why mathematical language changes, why certain questions
assume overriding importance, and how standards of proof are modified.
[15]Etta Kralovec and John Buell, The End of Homework, Beacon Press, 2000
This is an opinion piece dedicated to the problem, as seen by parents, that students in
today's classrooms are not getting enough done in the schools, and that teachers fail to
co-ordinate assignments.
[16]Eli Maor, e: The Story of a Number, Princeton University Press, 1994
A wonderful historical narrative account of the history of e. An enjoyable read for those
with no more than a high school calculus background.
[17]T. Pappas, The Joy of Mathematics, World Wide Publishing/Tetra, 1986
Pappas loves to collect problems and share them. There are a wide variety of traditional
lay-person puzzles covering a broad spectrum of mathematical disciplines. The problems
promote critical thinking skills. Some interesting historical side notes are sprinkled
throughout. Each subject is covered in 1-4 pages making the book very easy to read, and
useful to teachers as a resource for enrichment. This book may be too simple or "shallow"
for some of the more serious mathematicians, but the fun feeling of the book makes up
for it.
[18]T. Pappas, More Joy of Mathematics, World Wide Publishing/Tetra, 1991
See above.
[19]Ernest R. Ranucci, Mathematics on the Ceiling: Mathematics in Secondary School
Classrooms, pp 84-90, Rising/Wiesen, eds., Thomas Y. Crowell Co., 1972
[20]Louis E. Raths, et al, Teaching for Thinking, 2nd ed., Teacher College Press,
Columbia University, 1986
[21]James Stewart, Calculus, 4th ed., Brooks/Cole Publishing Co., 1999
An all-around excellent calculus text.
[22]Harrow W. Van Brummelen, The Basis of Mathematics, in Shaping the School
Curriculum: A Biblical View, Geraldine Steensma, editor, Signal Press, 1997
[23]D. Wells, You are a Mathematician, John Wiley & Sons, 1995
This book is a collection of many mathematical discoveries that have occurred down
through the centuries. Some have significant applications, but most would be excellent
fits within the definition of recreational mathematics. Topics such as patterns in numbers,
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mathematical games and mathematics for enjoyment are covered. Problems for
examination and clarification are interspersed throughout the chapters with solutions at
the end of the chapter. The level of difficulty is such that a solid background in algebra is
the only requirement for understanding what is being described.
[24]Gao Yuling and Sun Fengpo, Formulae for the General Terms of the Generalized
Yang Hui's Triangle, International Journal of Mathematical Education in Science and
Technology, Jul./Aug. 1998, Vol. 29, Issue 4, p587, pp7
This is a proof for determining the above and includes a chart for k=3. See the unit on
generalized forms in this paper.
[25]The Oxford English Dictionary, second edition, J.A.H. Murray, et al, original editors,
Oxford University Press, 1989
[26]Various Authors, In Improving Teaching and Learning in Science and Mathematics,
David F. Treagust, Reinders Duit and Barry J Fraser, editors, Teacher College Press,
Columbia University, 1996
[27]Various Authors, In Developing Mathematical Reasoning in Grades K-12, Lee V.
Stiff and Frances R. Curcio, editors, National Council of Teachers of Mathematics, 1999
An NCTM Yearbook, this selection of articles presents multiple views of mathematical
reasoning and its development at all grade levels. It reveals the various perspectives
about the nature of reasoning, and addresses the many issues and concerns involving
mathematical reasoning: how learners reason in mathematics; how communication
promotes reasoning; how teachers gather evidence of student reasoning; which curricular
approaches can be profitably explored; what can be done to ensure success in developing
reasoning; and more. An excellent professional development resource.
[29]Principles and Standards for School Mathematics. National Council of Teachers of
Mathematics, 2000
http://www.seattleschools.org/area/acastan/stan/math/math.xml, which is the Seattle
Public School’s web page defining the EALR’s for mathematics (see Appendix A).
A full description by grade level can be obtained in PDF format at
http://www.seattleschools.org/area/acastan/full/math.pdf .
http://www.seattleschools.org/area/main/calendar.dxml, which is the Seattle Public
Schools year calendar for 2002, used as a time template for the course.
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