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Level 1 | Level 2 | Level 3 | Level 4 | Level 5 | Level 6
Objectives and Content - Level 6
What Is the Shape of Your
World? (polar coordinates)
In this module, students will:
- graph points with polar coordinates
- determine points with cylindrical
coordinates
- determine the image of a point
using a stereo graphic projection
- determine the image of a point
using a cylindrical projection
Naturally Interesting (natural
logarithms)
In this module, students will:
- define e as the limit of
a sequence
- solve problems involving compound
interest
- examine the relationship between
common and natural logarithms
Functioning on a Path
(polynomial and rational functions)
In this module, students will:
- identify characteristics of odd-
and even-degree polynomial functions
- identify the maximum total number
of peaks and valleys in the graph of an nth-degree polynomial
function
- reflect graphs of polynomial functions
in the x- and y-axes
- translate the graphs of polynomial
functions
- stretch the graphs of polynomial
functions
- identify transformations of the
graphs of polynomial functions
- write rational expressions as
sums of polynomial and rational expressions
- identify the relationship between
the end behaviors of the rational function r(x) and the
polynomial function f(x), where
- identify the asymptotic behavior of rational
functions
- examine the graphs of rational
functions of the form
for various values of k and different degrees of p(x)
Changing the Rules Changes
the Game (finite geometry)
In this module, students will:
- 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
To Null or Not to Null (hypothesis
testing)
In this module, students will:
- 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
Ostriches Are Composed (operations
on functions)
In this module, students will:
- represent functions using set
diagrams
- identify domains and ranges of
polynomial, logarithmic, and trigonometric functions, and their inverses
- explore composite functions algebraically
- examine inverse functions graphically
and algebraically
Mathematics in Motion
(parametric equations)
In this module, students will:
- develop parametric equations for
a parabola
- develop parametric equations for
an ellipse
- review and extend parametric equations
for a circle
- use parametric equations to model
motion
- use vectors and trigonometry to
determine parametric relationships
- examine the difference between
speed and angular speed
Here We Go Again! (periodic
functions)
In this module, students will:
- examine the sine and cosine function
- explore the effects of parameters
a, b, h, and k on functions of the form
g(x) = af (b(x-h)) + k
- make predictions based on periodic
models
- examine functions created by adding
periodic functions
- explore the tangent, cotangent,
cosecant, and secant functions as other examples of periodic functions
The Sequence Makes the Difference
(finite differences)
In this module, students will:
- 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
Brilliant Induction (proof
by induction)
In this module, students will:
- write proofs using the principle
of mathematical induction
Cards and Binos and Reels,
Oh My! (probability)
In this module, students will:
- design simulations
- determine conditional probabilities
- perform binomial experiments
- develop the binomial probability
formula
- determine expected values
- represent elements in Pascal's
triangle using combinatorics
An Imaginary Journey Through
a Real World
(complex numbers)
In this module, students will:
- represent complex numbers in multiple
forms
- perform operations on complex
numbers using multiple representations
- determine complex roots of polynomials
- evaluate roots and powers of complex
numbers
What Did You Expect Big
Chi? (chi square test)
In this module, students will:
- 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
Slow Down! You're Deriving
over the Limit (derivatives)
In this module, students will:
- investigate the relationship between
average rate of change and the slope of a line
- investigate the relationship between
instanta neous 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
Total Chaos
(iteration)
In this module, students will:
- investigate iterative processes
- use iteration with functions
- study the following terms
associated with iteration: orbit, fixed point, attractor, repeller,
period, and cycle
- investigate iterative functions
using web plots
- classify orbits as fixed, periodic, or chaotic
