Objectives and Content Level 3

Objectives and Content
Level 3

SIMMS Covers Module 1: Colorful Scheduling (coloring theory, topology, and graph theory)
In this module, students will:

Determine the chromatic number of maps and graphs
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
Investigate the four-color theorem for maps drawn on flat surfaces and spheres

Module 2: What’s Your Bearing? (trigonometric ratios)
In this module, students will:

Draw maps to scale
Consider sources of error in measurement
Use right-triangle trigonometry and the Pythagorean theorem to determine unknown lengths and angle measures
Convert among angle measures, bearings, and azimuths
Investigate the relationship between the sine and cosine ratios of central angles
Determine sine and cosine ratios for angles with measures between 90 and 180 degrees
Develop the trigonometric identities and
Derive and use the law of sines
Derive and use the law of cosines

Module 3: Can It! (trigonometric functions and the unit circle)
In this module, students will:

Identify the relationships among angle measure in radians, arc length, and the coordinates of points on a unit circle
Identify circular functions by the shapes of their graphs
Identify the amplitude and period of circular functions
Examine some trigonometric identities
Determine the equations for sine or cosine curves based on their graphs
Determine transformations of the graphs of circular functions
Use sine or cosine functions to model real-world data
Identify the inverse functions for sine, cosine, and tangent
Determine appropriate restrictions on the domain and range of an inverse trigonometric function
Use inverse trigonometric functions to solve trigonometric equations

Module 4: Log Jam (exponential and logarithmic functions)
In this module, students will:

Investigate the characteristics of the pH scale
Compare decimal and rational exponents
Represent expressions with rational exponents as radicals
Rewrite expressions containing negative exponents using positive exponents
Use logarithmic functions to model data
Graph data on logarithmic scales
Convert between exponential and logarithmic equations
Simplify common logarithms
Perform logarithmic arithmetic
Compare properties of logarithms to properties of exponents
Use logarithms to solve exponential equations

Module 5: Motion Pixel Productions (transformational geometry and polar coordinate systems)
In this module, students will:

Identify transformations given the preimage and image
Create rotations on a polar coordinate system
Create rotations on a rectangular coordinate system
Convert coordinates of a point between polar and rectangular systems
Use inverse trigonometric functions to determine angles of rotation
Represent reflections, rotations, translations, and dilations of objects in the plane using 3 x 3 matrices and matrix multiplication
Find the image of a figure under a composition of transformations
Determine single 3 x 3 matrices to represent compositions of transformations
Express a glide reflection as a composition of a reflection and a translation
Define the relationship between a preimage and any congruent image with different orientation as a glide reflection
Determine the reflection line and translation vector for any glide reflection
Express any transformation with congruent preimage and image as a composition of at most three reflections

Module 6: Prove It (logical connectives, truth tables, Venn diagrams,
and proof)
In this module, students will:

Identify the truth value of a conditional
Use Venn diagrams to represent conditionals
Explore the relationship between the connective and and set intersection and the relationship Between the connective or and set union
Use Venn diagrams and truth tables to illustrate compound statements
Use and and or to form compound statements
Write negations of conditional statements
Investigate the negations of compound statements through truth tables, Venn diagrams, and De Morgan’s laws
Create truth tables illustrating conditional, negated, and compound statements
Identify and use logically equivalent forms to rewrite conditional and compound statements
Find the converse, inverse, and contrapositive of a conditional
Write proofs using a chain of if-then statements
Explore proof by exhaustion
Find counterexamples
Use deductive reasoning
Write direct proofs
Develop indirect proofs

Module 7: More or Less (inequalities and limits)
In this module, students will:

Write, interpret, and solve linear inequalities
Use mapping diagrams to represent mathematical relationships
Investigate a graphical representation of a limit
Write, interpret, and solve inequalities containing absolute values
Write, interpret, and solve nonlinear inequalities

Module 8: Big Business (rational functions)
In this module, students will:

Graph and analyze rational functions, including domains, discontinuities, and asymptotes
Evaluate functions around discontinuities
Write rational functions as sums of polynomial and rational expressions
Determine equations of asymptotes
Determine restrictions on domains of rational functions
Graph nonlinear inequalities and systems of relations

Module 9: Strive for Quality (sampling and binomial probability)
In this module, students will:

Distinguish between statistics and parameters
Select simple random samples
Model sampling with binomial experiments
Use tree diagrams to determine conditional probabilities
Identify mutually exclusive and independent events
Develop the binomial probability formula
Determine theoretical binomial probabilities
Determine the expected value of a binomial experiment

Module 10: Fly the Big Sky with Vectors (vectors, law of cosines, and
law of sines)
In this module, students will:

Define and describe vectors
Multiply vectors by scalars
Identify equivalent and opposite vectors
Add vectors using the tip-to-tail method
Use vectors as mathematical models
Apply the law of cosines and law of sines
Examine some trigonometric identities
Identify the ambiguous case for the law of sines
Describe vectors using components
Represent vectors on a Cartesian coordinate system
Add vectors using components

Module 11: It’s All in the Family (transformations of functions)
In this module, students will:

Recognize parent functions for selected exponential, logarithmic, rational, and periodic functions
categorize functions into families
Identify transformations to a parent function
Transform functions graphically
Transform functions algebraically
Use transformed functions as mathematical models

Module 12: Nearly Normal (normal distribution and probability)
In this module, students will:

Organize data using frequency and relative frequency tables, histograms, and polygons
Describe data sets using mean and standard deviation
Use simulations to generate data
Identify binomial experiments
Calculate the mean and standard deviation of binomial distributions
Examine uniform probability distributions
Distinguish between discrete and continuous probability distributions
Calculate probabilities for specific intervals within probability distributions
Examine normal distributions and the 68–95–99.7 rule
Determine probabilities using the properties of normal distributions

Module 13: Controlling the Sky with Parametrics (parametric equations)
In this module, students will:

Identify the domains and ranges of parametric graphs
Describe differences between parametric and nonparametric equations for the same linear graph
Model linear paths with parametric equations
Use component vectors to write parametric equations
Convert between parametric and nonparametric equations for a given linear graph
Use parametric equations and their graphs to model circular paths

Module 14: Having a Ball (non-Euclidean geometry)
In this module, students will:

Identify properties of lines (great circles) on a sphere
Determine how to measure angles on a sphere
Compare the relationships among points and lines in a plane and among points and lines on a sphere
Compare properties of Euclidean geometry with properties of spherical geometry
Determine the sum of the measures of the interior angles of a triangle on a sphere
Describe properties of quadrilaterals on a sphere
Compare similarity in a plane and similarity on a sphere

Module 15: Classical Crystals (polyhedra, Platonic solids, and Archimedean solids)
In this module, students will:

Identify regular polyhedra
Draw nets for and build models of regular polyhedra
Calculate the surface areas of regular polyhedra
Determine the conditions necessary to form the vertex of a regular polyhedron
Determine the measure of dihedral angles for regular polyhedra
Describe relationships among the sides, vertices, and lines of symmetry for regular polygons
Determine planes of symmetry for regular polyhedra
Draw nets for and build models of an Archimedean solid
Examine Euler’s formula for the relationship among the numbers of edges, faces, and vertices of a polyhedron
Determine planar maps and graphs of a regular polyhedron
Identify the dual of a polyhedron

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