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Math Theory Modules
1st Semester:
Logic:
a.
Level 3: Let the Games Begin
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b.
Level 4: Believe It or Not
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c.
use the logical connectives and and or to form compound statements
use Venn diagrams to illustrate the logical connectives and and or
explore the relationship between and and intersection and the relationship between or and union
develop the truth tables for "p and q" and "p or q"
use the logical operator not to negate statements
develop truth tables using "not" with statements and compound statements, such as "~p and q" and "~p or q"
use the logical conditional p –> q
use tables, the process of elimination, and contrapositive reasoning as problem-solving techniques
identify the hypothesis and conclusion of a conditional statement
identify the truth value of a conditional
use Venn diagrams to represent conditionals
negate conditionals
find the converse, inverse, and contrapositive of a conditional
identify logically equivalent conditionals
write proofs using a chain of if-then statements
explore proof by exhaustion
find counterexamples
use deductive reasoning
write direct proofs
develop indirect proofs
Level 5: Is It Really True?
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define the negation of statements using the quantifiers some, all, and none
recognize and write logically equivalent statements
write negation statements
determine the difference between the inclusive or and the exclusive or through truth tables, Venn diagrams,
and English usage
investigate the negations of compound state ments through truth tables, Venn diagrams, and De Morgan's
laws
determine the truth values of a conditional statement using truth tables
investigate the converse, the inverse, and the contrapositive of the conditional
Theory:
a.
Level 3: Fair is Fair (Fair Division)
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b.
study the properties of fair division
investigate algorithms that result in fair divisions
make fair divisions by dividing an item considered continuous between two or more people
make fair divisions by dividing a set of items considered discrete among two or more people
Level 1: Going in Circuits (Graph Theory)
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use tree diagrams to organize information and solve problems
use the fundamental counting principle
use factorial notation
solve problems involving Hamiltonian circuits
develop algorithms for solving problems
c.
Level 3: Our Town (Graph Theory)
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d.
Level 5: How long Is This Going to Take (Scheduling Theory)
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e.
use geometric models for determining probability
use expected value to determine fair games
explore probabilities of multistage experiments
examine the difference between independent and dependent events
explore conditional probability
Level 5: The Game of Life (Game Theory)
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determine the chromatic number of a map
investigate the four-color theorem for maps drawn on flat surfaces and spheres
create graphs of maps
solve scheduling problems using graphs and coloring theory
identify topologically equivalent graphs
identify planar graphs
determine the relationship between chromatic number and the number of vertices of a complete planar graph
Level 2: Hurry Hurry Step Right Up (Fair Game)
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apply the nearest neighbor and cheapest link algorithms
use network diagrams to analyze scheduling problems
revise schedules given specific constraints
use bin packing to analyze problems
Level 4: Colorful Scheduling
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f.
organize information using graphs
identify different graphs and their modeling uses
use graphs to model real-world situations
identify and create Eulerian circuits
discover two traversability theorem
use digraphs to model real-world situations
use matrices to analyze graphs
investigate the properties of strictly determined games
construct and interpret payoff matrices
find and interpret the saddle points of payoff matrices
find the expected values of games
identify when pure strategies or mixed strategies should be used
find an optimal strategy for each player in a game
Voting Theory packets.
2nd Semester:
Geometry:
a. Level 4: Having a Ball (Non-Euclidian Geometry)
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b.
describe the different types of intrinsic curvature as they relate to the angle-sums of triangles
explain why the Euclidean property of similarity does not hold on spherical surfaces
compare and contrast the Euclidean properties with spherical properties
compare and contrast spherical geometry with hyperbolic geometries from the historical context of Saccheri
quadrilaterals
Level 6: Changing the Rules Changes the Game
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study modular arithmetic systems
coordinatize a finite geometry using modulo 3
reconceptualize many terms of Euclidean geometry in a finite geometry
explore geometries both analytically and synthetically
construct proofs directly, indirectly, and by exhaustion
c.
Level 5: Reinvent the Wheel (Curves of Constant width)
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explore curves of constant width
construct and use lines of support for curves
develop a formula for the perimeter of curves of constant width
determine the area of curves of constant width
explore applications of curves of constant width in real-world situations
Statistics:
a.
Level 4: Confidence Builders
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b.
write null and alternative hypotheses
create confidence intervals using given data and data the students have generated
use confidence intervals to make decisions about null and alternative hypotheses
design a simple experiment that uses simple statistics to investigate a question of interest
Level 2: And the Survey Says
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use a variety of sampling techniques
predict the characteristics of a population based on samples
explore the role that biases play in sampling
use histograms to estimate probabilities and make predictions
investigate how sample size affects a survey's reliability
explore confidence statements and margins of error
c. Level 5: Making Cents of Your Income
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d.
Level 6: To Null or Not.
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e.
estimate a population mean using sample means
estimate the standard deviation of a population using the standard deviation of a sample
use the standard deviation of a sample to construct confidence intervals
use the 68%-95%-99.7% rule to determine the probability that the population mean lies within certain
confidence intervals
formulate null and alternative hypotheses
use confidence intervals to estimate a population mean
use confidence intervals to test a hypothesis
review the differences between statistics and parameters
express null and alternative hypotheses
use contrapositive logic
explore characteristics of a normal curve
examine the 68%-95%-99.7% rule
compare individual observations to the mean in terms of standard deviations
use the central limit theorem to evaluate sample means
interpret and compare statistics using z-scores
test null hypotheses using various levels of significance
Level 6: What Did You Expect, Big Chi?
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calculate chi-square values
use chi-square values to test observed frequencies versus expected frequencies
use the chi-square distribution to determine probabilities
determine and use degrees of freedom when conducting tests on hypotheses
use chi-square tests to determine whether two variables are independent or dependent
Calculus:
a. Level 1: From Rock Bands to Recursion
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b.
Level 2: Take It to the Limit
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c.
analyze number patterns
develop arithmetic and geometric sequences
compare linear equations and explicit formulas for arithmetic sequences
compare the graphs of linear equations and arithmetic sequences
compare exponential equations and explicit formulas for geometric sequences
compare the graphs of exponential equations and geometric sequences
compare the graphs of arithmetic and geometric sequences
evaluate series
identify sequences that are arithmetic, geometric, or neither
develop formulas for finite arithmetic and geometric series
develop a formula for certain infinite geometric series
explore limits graphically and geometrically
Level 6: The Sequence Makes the Difference
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identify and generate polynomial, exponential, and power sequences
use the finite difference process to determine the degree of a polynomial that generates a polynomial
sequence
determine the regression equation that generates a given sequence
determine the explicit and recursive formulas that generate a given sequence
d. Level 6: Brilliant Induction
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e.
Level 4: More or Less
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write proofs using the principle of mathematical induction
interpret and solve linear, absolute value, and polynomial inequalities
create a graphical representation leading to the concept of limit
determine a set of images given a set of pre- images and vice versa
Level 6: Slow Down You’re Deriving over the Limit
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investigate the relationship between average rate of change and the slope of a line
investigate the relationship between instantaneous rate of change and the slope of a tangent line
explore graphical interpretations of derivatives
develop a definition for derivative
examine the derivatives of specific functions