Objectives and Content Level 4

Objectives and Content
Level 4

SIMMS Covers Module 1: What Shape Is Your World? (geometry of map projections; polar and cylindrical coordinate systems)
In this module, students will:

Plot points in Cartesian, polar, and cylindrical coordinate systems
Compare distortion in maps made using stereographic and cylindrical projections
Determine the image of a point under a stereographic projection
Identify geometric properties that are preserved (or not preserved) under a stereographic projection
Determine the image of a point under a cylindrical projection
Identify geometric properties that are preserved (or not preserved) under a cylindrical projection

Module 2: Naturally Interesting (natural logarithms)
In this module, students will:

Solve problems involving compound interest
Define e as the limit of a sequence
Use e within mathematical models
Define natural logarithms
Use natural logarithms within mathematical models
Use natural logarithms to solve equations

Module 3: Building Confidence (sampling, normal distribution, central limit theorem, confidence intervals)
In this module, students will:

Apply the law of large numbers to estimate population means
Estimate a population mean using sample means
Examine how sample size affects estimates of population parameters
Identify the relationship between the central limit thoerem and a normal distribution
Apply the 68–95–99.7 rule of normal distributions to create confidence intervals
Use sample statistics to create confidence intervals
Use confidence intervals to estimate population means

Module 4: Functioning on a Path (polynomial, rational, and piecewise functions)
In this module, students will:

Identify the degree, leading coefficient, and continuity of polynomial functions
Determine equations for polynomial functions
Identify absolute maxima and minima in polynomial functions
Determine the maximum number of zeros (roots) for a given polynomial function
Write, graph, and determine the continuity of piecewise functions
Write rational functions as sums of polynomial and rational expressions
Identify asymptotes in the graphs of rational functions
Explore the relationship between the end behaviors of the rational function r(x) and the polynomial function f(x) where
Describe the end behavior and behavior near vertical asymptotes of rational functions

Module 5: Changing the Rules Changes the Game (finite geometries and proof)
In this module, students will:

Determine some basic properties of modular arithmetic
Develop a finite coordinatized geometric system
Use direct proof and proof by exhaustion to prove statements in finite systems
Compare definitions and properties of Euclidean geometry to those of a finite geometry
Examine a finite geometry as an axiomatic system
Use indirect proof to prove statements in finite geometric systems
Apply indirect proof to algebraic systems

Module 6: Ostriches Are Composed (operations on functions)
In this module, students will:

Distinguish between relations and functions
Represent functions using set diagrams and mapping diagrams
Identify domains and ranges of polynomial, rational, logarithmic, and trigonometric functions
Perform arithmetic operations on two given functions
Identify domains and ranges of inverse functions and inverse relations
Examine compositions of functions using mapping diagrams
Create compositions of functions algebraically
Determine inverse functions graphically and algebraically

Module 7: It’s Napped Time (conic sections)
In this module, students will:

Identify conic sections using the definitions based on a double-napped cone
Describe the characteristics of degenerate conic sections
Develop the equation for a circle with center at the origin using the distance formula
Develop the equations for conic sections with centers other than the origin
Identify the coordinates of critical points of a conic section given its equation
Determine the equation of a conic section given critical points and/or points on the conic section
Write equations for conic sections in standard form
Relate conic sections and their equations to real-world situations
Define each conic section as a locus of points
Solve equations containing the square root of a variable
Prove that a given locus of points defines a conic section
Determine the orientation of a conic section given its equation
Determine the equations for the asymptotes associated with hyperbolas
Use asymptotes to determine the graphs and equations of hyperbolas
Identify a parabola’s focus, directrix, vertex, and line of symmetry from its equation
Write equations of parabolas in general form and vertex form
Convert equations of parabolas in general form to vertex form by completing the square

Module 8: The Sequence Makes the Difference (finite differences with polynomial sequences)
In this module, students will:

Generate sequences using polynomial functions
Use the finite-difference process to determine the least degree of a polynomial that generates a polynomial sequence
Determine explicit and recursive formulas for sequences
Determine a polynomial function that generates a given sequence

Module 9: An Imaginary Journal Through the Real World (complex numbers, roots of polynomials, the quadratic formula)
In this module, students will:

Solve quadratic equations by completing the square
Develop the quadratic formula
Use the determinant to identify types of roots for quadratic equations
Use in the representation of complex numbers
Solve quadratic equations using the quadratic formula
Represent solutions to quadratic equations using complex numbers
Determine polynomials given their roots
Identify the numbers and kinds of solutions for polynomial functions by examining graphs
Determine complex roots of polynomials
Represent complex numbers in multiple forms
Simplify and evaluate powers of i
Perform operations on complex numbers using multiple representations
Determine the possible numbers and kinds of roots for polynomial functions of a given degree
Using the fundamental theorem of algebra
Plot complex numbers on the complex coordinate plane
Create graphical transformations using multiplication of complex numbers
Examine the relationship between Julia sets and complex numbers
Convert complex numbers into trigonometric form
Evaluate roots and powers of complex numbers

Module 10: Cards and Binos and Reels, Oh My! (binomial probability)
In this module, students will:

Design simulations
Determine conditional probabilities
Perform binomial experiments
Represent elements in Pascal’s triangle using combinations
Develop a formula for the binomial distribution
Determine expected values

Module 11: Brilliant Induction (proof by mathematical induction)
In this module, students will:

Make conjectures based on observed patterns
Demonstrate that a given conjecture is true for a finite number of cases
Use counterexamples to identify conjectures that are false
Write proofs using the principle of mathematical induction

Module 12: Slow Down! You’re Deriving over the Limit (derivatives)
In this module, students will:

Identify the relationship between average rate of change and the slope of a secant line
Investigate the relationship between instantaneous rate of change and the slope of a tangent line
Explore a graphical interpretation of a derivative
Develop a definition for a derivative
Identify points on a graph for which a derivative does not exist
Examine the derivatives of specific functions

Module 13: Mathematics in Motion (parametric equations and conic sections)
In this module, students will:

Develop parametric equations for a parabola
Use parametric equations to model the paths of moving objects
Use vectors and trigonometry to determine parametric relationships
Examine the difference between speed and angular speed
Extend their knowledge of the parametric equations for a circle (3)
Develop parametric equations for an ellipse
Determine the area of an ellipse

Module 14: How Sure Are You? (hypothesis testing)
In this module, students will:

Express null and alternative hypotheses
Use contrapositive logic
Explore the characteristics of normal curves
Apply the 68-95-99.7 rule for normal distributions
Estimate population parameters using samples and confidence intervals
Determine margins of error in parameter estimations
Use the central limit theorem to evaluate sample means
Compare individual observations to the mean in terms of standard deviations
Interpret and compare statistics using z-scores
Test null hypotheses using specific levels of significance

Module 15: Risky Business (probability and expected value)
In this module, students will:

Design and conduct simulations of probability experiments
Apply the law of large numbers
Calculate weighted means
Use exponential equations as models
Examine random variables and their probability distributions
Determine the expected value of random variables

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