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ESSENTIAL
QUESTION |
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Begin High Dive Unit (Circular Trigonometry)
How can I use math to model this circus problem?
How can I break it down into manageable
units? What
factors do I need to consider that will affect the
path of the diver? |
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Use similarity to develop sine, cosine functions and their
relationships
Graphing trig functions
Define Inverse trig functions |
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Review basics of circles and angles
Delineate info needed to solve unit problem
Find speed of object moving at constant angular
speed
Find height (h(t)) for specific values of t, given a
circular path
Extend sine function to all angles.
Use the term reference angle.
Derive general formula for h(t) for circular motion.
Use similarity to analyze domain and range of sine
function.
Graph h(t)
Change parameters and analyze affect on h(t)
(amplitude, period, etc.)
Use POW 1 to review recursive formulae and introduce
math induction)
Define inverse trig functions and principal values
Interpret area under graph of speed function as
total distance traveled. |
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| Homework, Classwork, Observation |
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| Continue with High Dive Unit |
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Develop trig identities
Develop polar coordinate system
Physics concepts for falling bodies are developed.
Vectors are used to represent forces
Quadratic formula |
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Introduce periodic nature of sine function.
Derive formula for height of object falling from
rest as a function of time.
Find x coordinate of platform for specific cases.
Define cosine for all angles and relate to x coord.
Introduce polar coordinates.
Change from rectangular coordinate expression to
polar form.
Develop tangent function for all angles.
Derive odd/even and co-function identities.
Be able to explain and use Pythagorean Identities
Use polar coordinates to express rectangular
expressions and vise versa.
Develop quadratic expression to describe position of
falling object wrt time.
Solve quadratic equations by completing the square.
Derive quadratic formula and use to solve quadratic
equations.
Evaluate and interpret significance of 2 solutions,
a negative velocity, etc.
Extend concepts to object with non-zero initial vel.
Analyze components of velocity as vertical and
horizontal vectors.
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Finish High Dive Unit
Begin As the Cube Turns Unit (Programming)
How do I draw an object on my calculator?
How can I animate it?
Do I really need math to do that?? |
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Synthesis of math and physics
Using a technical manual
Use draw function of the calculator
Synthesis of coordinate geometry, matrices,
synthetic geometry principles and trig to create a
program |
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Use skills developed in unit to solve the unit problem.
Use manual to explore drawing
capabilities of calculator.
Write plain language programs
Work with programming loops
Use delay loops in a program
Analyze what a specific program does
Use a loop to create animation of an object
Be able to use transformations to move an object
Use translation vectors to move objects
Synthesize skills to use loops and translation
vectors in a program.
Review algebra of matrices.
Find area of triangle using A = 1/2absinC
Review polar coordinates
Derive coordinate formulas for rotation in a plane.
Derive and prove trig sum and difference formulas.
Compare and contrast sine and cosine functions.
Use formulas to rotate an object. |
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Homework, Classwork, Observation, Presentations, POW's,
Assessments, portfolio
Homework, classwork, observation, presentations,
POW's |
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| Continue with As the Cube Turns |
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Use a matrix to express rotation
Use rotation matrix in a program
Use line of sight to describe projection of a point
onto a plane.
Use similar triangles to find fractional distances
along a line.,
Analyze programs combining rotation and translation
matrices
Review 3D graphing
Extend projection concept to 3D
Use matrices to express reflections
Find intersection of line segment and plane.
Express projections in terms of coordinates. |
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| Homework, classwork, POW's presentations |
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Finish As the Cube Turns
Introduce Know How Unit (Algebra) |
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Synthesize concepts of unit to solve unit problem.
Independent learning/review of concepts they may
have missed or not remember. |
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Express rotation in 3D in terms of coordinates
Express rotation in 3D using matrices.
Synthesize concepts to solve unit problem.
Use factoring to solve equations
Radian measure of angles
Learn and use Law of Sines and Cosines to solve
problem stations in 3D using matrices
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| Homework, classwork, POW's presentations, portfolio, assessments |
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Finish Know How Unit
Introduce The World of Functions Unit (Algebra) |
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Develop concept of families of functions that can be
represented as tables, graphs, algebraic
expressions, and models for real world situations.
Families to be studied include polynomials,
exponential, logarithmic, sine, reciprocal and
rational. |
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Develop key ideas about the ellipse
Develop a proof of the quadratic formula
Finding patterns from examples in a problem
situation
Learn about complex numbers
Formally define function
Represent situation with a function
Sketch graphs of functions based on situations
Create a table
Define the family of linear functions, sinusoidal
functions and exponential growth and decay
functions.
Learn about parametric equations.
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Homework, Classwork, observations, presentations POW's,
assessments, portfolio, individual projects
Homework, etc. |
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| Continue The World of Functions |
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Prove second differences of quadratic equations are constant.
Prove patterns for the tables of various functions
using their algebraic form. (exponential, cubic,
etc.)
Work with functions having vertical asymptotes.
Identify family of rational functions.
Examine end behavior of functions and horizontal
asymptotes.
Be able to locate vertical and horizontal asymptotes
of rational functions.
Adjust parameters to find a specific function from a
family.
Fit an exponential function to two data points.
Fit a linear equation to two set of data points.
Develop a measure of "Quality of fit"
Use least squares approximation
Use regression feature on calculator to find curve
of best fit.
Use absolute value and step functions as models.
Develop arithmetic and algebra of functions,
especially composition of functions.
Use composition of functions to model a situation.
Investigating commutativity of composition of
functions.
Given a function be able to decompose it into two
functions. Recognize that this is a non-unique
operation.
Explore inverse functions and their
graphs.
Investigate the identity of function families. |
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Finish The World of Functions
Begin Pollster's Dilemma Unit (Data)
How confident should a candidate be, based on the
results of a particular poll? |
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See content above
General concepts for sampling
Sampling with replacement
Central Limit Theorem and normal distribution
Mean and standard deviation
Confidence levels and margin of error |
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See skills above
Review use of combinatorial
coefficients for probabilities.
Develop concept of theoretical distribution.
Use probability bar graph to illustrate a
distribution.
Be able to distinguish sampling with replacement and
without replacement.
Use true proportion and sample proportion
Review binomial distribution.
Investigate changing poll size to increase
reliability.
Review normal distribution and standard deviation.
Review probabilities for a given number of standard
deviations about the mean.
Introduce Central Limit Theorem.
Use recursion to examine a complex problem.
Estimate area under a normal curve.
Be able to use Normal Table.
Use linear interpolation to estimate probabilities.
Apply central limit theorem to polling situation. |
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| Continue Pollster's Dilemma |
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Introduce concept of variance.
Review key ideas of binomial distribution
Find formulas for mean and standard deviation of
poll results in terms of poll size and true
proportion.
Apply to binomially distributed situations other
than polling.
Establish confidence intervals.
Examine how variance depends on true proportion.
Review mean, median mode, margin of error.
Distinguish between confidence and probability.
See that variance is max when pop is evenly divided.
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