Scholastic Area that Course Is Active In: Mathematics
Students will explore rational functions.
Investigate and explain characteristics of rational functions, including domain, range, zeros, points of discontinuity, intervals of increase and decrease, rates of change, local and absolute extrema, symmetry, asymptotes, and end behavior.
Find inverses of rational functions, discussing domain and range, symmetry, and function composition.
Solve rational equations and inequalities analytically, graphically, and by using appropriate technology.
Students will understand and use vectors.
Represent vectors algebraically and geometrically.
Convert between vectors expressed using rectangular coordinates and vectors expressed using magnitude and direction.
Add and subtract vectors and compute scalar multiples of vectors.
Use vectors to solve realistic problems.
Students will use complex numbers in trigonometric form.
Represent complex numbers in trigonometric form.
Find products, quotients, powers, and roots of complex numbers in trigonometric form.
Students will explore parametric representations of plane curves.
Convert between Cartesian and parametric form.
Graph equations in parametric form showing direction and beginning and ending points where appropriate.
Students will explore polar equations.
Express coordinates of points in rectangular and polar form.
Graph and identify characteristics of simple polar equations including lines, circles, cardioids, lima?ons, and roses.
Students will use the circle to define the trigonometric functions.
Define and understand angles measured in degrees and radians, including but not limited to 0°, 30°, 45°, 60°, 90°, their multiples, and equivalences.
Understand and apply the six trigonometric functions as functions of general angles in standard position.
Find values of trigonometric functions using points on the terminal sides of angles in the standard position.
Understand and apply the six trigonometric functions as functions of arc length on the unit circle.
Find values of trigonometric functions using the unit circle.
Students will investigate and use the graphs of the six trigonometric functions.
Understand and apply the six basic trigonometric functions as functions of real numbers.
Determine the characteristics of the graphs of the six basic trigonometric functions.
Graph transformations of trigonometric functions including changing period, amplitude, phase shift, and vertical shift.
Apply graphs of trigonometric functions in realistic contexts involving periodic phenomena.
Students will investigate functions.
Compare and contrast properties of functions within and across the following types: linear, quadratic, polynomial, power, rational, exponential, logarithmic, trigonometric, and piecewise.
Investigate transformations of functions.
Investigate characteristics of functions built through sum, difference, product, quotient, and composition.
Students will establish the identities below and use them to simplify trigonometric expressions and verify equivalence statements. tan θ = sin θ / cos θcot θ = cos θ / sin θsec θ = 1 / cos θ csc θ = 1 / sin θsin² θ + cos² θ = 1cot² θ + 1 = csc² θsin(∝ ± β) = sin ∝ cosβ ± cos ∝ sinβcos(∝ ± β) = cos ∝ cosβ ± sin ∝ sinβ sin(2θ) = 2sin θ cos θcos(2θ) = cos² θ - sin² θ
Students will solve trigonometric equations both graphically and algebraically.
Solve trigonometric equations over a variety of domains, using technology as appropriate.
Use the coordinates of a point on the terminal side of an angle to express x as r cosθ and y as r sinθ.
Apply the law of sines and the law of cosines.
Students will verify and apply ½ab sin C to find the area of a triangle.
Students will investigate and use inverse sine, inverse cosine, and inverse tangent functions.
Find values of the above functions using technology as appropriate.
Determine characteristics of the above functions and their graphs.
Students will use sequences and series.
Use and find recursive and explicit formulae for the terms of sequences.
Recognize and use simple arithmetic and geometric sequences.
Investigate limits of sequences.
Use mathematical induction to find and prove formulae for sums of finite series.
Find and apply the sums of finite and, where appropriate, infinite arithmetic and geometric series.
Use summation notation to explore series.
Determine geometric series and their limits.
Strand: Data Analysis and Probability
Using simulation, students will develop the idea of the central limit theorem.
Using student-generated data from random samples of at least 30 members, students will determine the margin of error and confidence interval for a specified level of confidence.
Students will use confidence intervals and margins of error to make inferences from data about a population. Technology is used to evaluate confidence intervals, but students will be aware of the ideas involved.
Strand: Process Standards
Students will solve problems (using appropriate technology).
Build new mathematical knowledge through problem solving.
Solve problems that arise in mathematics and in other contexts.
Apply and adapt a variety of appropriate strategies to solve problems.
Monitor and reflect on the process of mathematical problem solving.
Students will reason and evaluate mathematical arguments.
Recognize reasoning and proof as fundamental aspects of mathematics.
Make and investigate mathematical conjectures.
Develop and evaluate mathematical arguments and proofs.
Select and use various types of reasoning and methods of proof.
Students will communicate mathematically.
Organize and consolidate their mathematical thinking through communication.
Communicate their mathematical thinking coherently and clearly to peers, teachers, and others.
Analyze and evaluate the mathematical thinking and strategies of others.
Use the language of mathematics to express mathematical ideas precisely.
Students will make connections among mathematical ideas and to other disciplines.
Recognize and use connections among mathematical ideas.
Understand how mathematical ideas interconnect and build on one another to produce a coherent whole.
Recognize and apply mathematics in contexts outside of mathematics.
Students will represent mathematics in multiple ways.
Create and use representations to organize, record, and communicate mathematical ideas.
Select, apply, and translate among mathematical representations to solve problems.
Use representations to model and interpret physical, social, and mathematical phenomena.
The Guiding Sub-questions are related, relevant, and connected to exploring the Essential Question. They are higher level questions and are specific enough to guide the work of the unit. (Subquestions must be entered one at a time and updated . . . they are numbered automatically.)
Begin writing a unit by establishing what you want students to know and be able to do and planning how you will know "what they know". This Assessment Plan is a general plan (specific assessment instruments are in the teaching procedures); this section should both help you to plan and to give teachers an idea of the varied types of assessment that will be used in the unit. Be sure to include informal checks of understanding, student self-assessment, and authentic assessment. Include pre and post assessment.
Preparation for students includes notes on preparing the learner such as possible misconceptions students may have, ideas of pre-exposure for learners, and prerequisite lessons. It includes ideas for accelerated learning.
Unit Resources include general, global resources that might include bookmarks, books, periodicals, media and software. URLs need to be provided for each resource to identify a source from which it can be obtained. Resources might include those purchased as part of an adoption. More specific resources will be referenced within the teaching procedures.