Download Math Analysis and Trig.

Math Analysis/Trigonometry
Conceptual Category(s)
Domain
Cluster
Alignments:
CCSS: See below
Performance: 1.6, 1.10, 3.3
Knowledge: (MA) 2
MACLE: See below
NETS: 3d; 4c; 6b
DOK: 1,2
• Functions
• Trigonometric Functions (F-TF)
• Extend the domain of trigonometric functions using the unit circle (F-TF.1-3)
Standards
Learning Targets
F-TF.1-3
1. Understand radian measure of an angle as the length of the arc on
the unit circle subtended by the angle
2. Explain how the unit circle in the coordinate plane enables the
extension of trigonometric functions to all real numbers,
interpreted as radian measures of angles traversed
counterclockwise around the unit circle
3. Use special triangles to determine geometrically the values of sine,
! !
!
cosine, and tangent for ! , ! , and ! , and use the unit circle to
express the values of sine, cosine, and tangent for ! − !, ! +
!, and 2! − ! in terms of their values for x, where x is any real
number
Unit A: Trigonometric Functions – Unit Circle Approach: Know
and understand radian measure and how it relates to the unit circle.
Develop the trigonometric functions from the special angles on the unit
circle
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Sketch angles in standard position in degrees and radian
Name coterminal angle degrees and radians
Convert degrees to radians and vice versa
Find the length of a circular arc given the radius and subtended
angle
Find the linear and angular speeds of an object traveling in a
circular motion
CCSS: F-TF.1
MACLE: N/A
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Find coordinates associated with “familiar” angles
Find values of the six trig functions using the unit circle and a
calculator
Find values of the six trig functions for points on terminal rays of an
angle
CCSS: F-TF.1-3
MACLE: N/A
Approved 7-15-13
Revised 2013
1
Math Analysis/Trigonometry
Instructional Strategies
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Lecture enhanced with:
• SMART Notebook
• PowerPoint
• the Internet
Drill and guided practice
Demonstrations: e.g., Twisty-Tie activity to explain the concept of radians
Problem solving such as finding:
• the distance between cities
• linear and angular speeds of rotating objects
Reflective discussion
Class discussion
Computer assisted instruction
Games: e.g., “Unit Circle Jeopardy” to practice finding trignonometric values
Assessments/Evaluations
The students will be assessed on the concepts taught using a variety of modalities:
• Direct teacher observations
• Quizzes: Function Friday Quiz #1
• Homework assignments: See attached pacing guide
• Formal common assessment: Unit A test
Mastery Level: 80%
Instructional Resources/Tools
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Textbook: Prentice Hall, Precalculus Graphing and Data Analysis, Prentice Hall, Inc., Sullivan and Sullivan 2001.
Website(s): Finding trig values practice: http://www.mathmistakes.info/facts/TrigFacts/index.html
Graphing calculator
Approved 7-15-13
Revised 2013
2
Math Analysis/Trigonometry
Conceptual Category(s)
Domain
Cluster
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Functions
Trigonometric Functions (F-TF)
Extend the domain of trigonometric functions using the unit circle (F-TF.4)
Model periodic phenomena with trigonometric functions (F-TF.5)
Alignments:
CCSS: See below
Performance: 1.6, 1.10
Knowledge: (MA) 2,4
MACLE: See below
NETS: 3d; 4c; 6b
DOK: 1,2
Standards
F-TF.4, 5
4. Use the unit circle to explain symmetry (odd and even) and
periodicity of trigonometric functions
5. Choose trigonometric functions to model periodic phenomena
with specified amplitude, frequency, and midline
Learning Targets
Unit B: Graphing the Six Basic Trigonometric Functions: Graph and
interpret graphs of the six basic trigonometric functions. Understand
periodicity, amplitude, and midline
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Graph the six basic trigonometric functions
Determine the domain, range, period, and amplitude for the six basic
trigonometric functions
Graph the six trigonometric functions with transformations
Determine the transformations for any trigonometric function
Write the equation of a sinusoidal function given a graph or specified
information
CCSS: F-TF.4,5
MACLE: N/A
Approved 7-15-13
Revised 2013
3
Math Analysis/Trigonometry
Instructional Strategies
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Lecture enhanced with:
• SMART Notebook
• PowerPoint
• the Internet
Drill and guided practice
Demonstrations: Ann Fibian activity – students will create a sinusoid function using collected data
Problem solving: e.g., Ferris Wheel problem
Reflective discussion
Class discussion
Computer assisted instruction
Station review activity
Assessments/Evaluations
The students will be assessed on the concepts taught using a variety of modalities:
• Direct teacher observations
• Class activities:
• Biorhythms lab: Students will determine their biorhythms for the current month and make predictions
• Stayed – Tuned lab (optional: if time allows)
• Quizzes: Function Friday quizzes #2 and #3
• Homework assignments: see attached pacing guide
• Formal common assessment: Unit B test
Mastery Level: 80%
Instructional Resources/Tools
Textbook: Prentice Hall, Precalculus Graphing and Data Analysis, Prentice Hall, Inc., Sullivan and Sullivan 2001.
Website(s):
• This java applet graphs functions based on unit circle values and the students can explore the effects of “a” values
• http://www.intmath.com/trigonometric-graphs/1-graphs-sine-cosine-amplitude.php
• Graphing calculator
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Approved 7-15-13
Revised 2013
4
Math Analysis/Trigonometry
Conceptual Category(s)
Domain
Cluster
Alignments:
CCSS: See below
Performance: 1.6, 3.5
Knowledge: (MA) 4
MACLE: See below
NETS: 3d; 4c; 6b
DOK: 1-3
• Functions
• Trigonometric Functions (F-TF)
• Prove and apply trigonometric identities. (F-TF.8, 9)
Standards
Learning Targets
F-TF.8, 9
8. Prove the Pythagorean identity !"#! ! + !"# ! ! = 1 and use it
to find !"# ! , !"# ! , !" !"# ! given
!"# ! , !"# ! , !" !"# ! and the quadrant of the angle
9. Prove the addition and subtraction formulas for sine, cosine, and
tangent and use them to solve problems
Unit C: Analytic Trigonometry: Develop and use trigonometric identities
to simplify trigonometric expressions. Verify trigonometric identities
algebraically
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Use trigonometric identities to simplify expressions
Verify/prove trigonometric identities algebraically
Use sum/difference, double angle, and half angle identities to determine
exact values of trig functions
Use trigonometric identities and algebraic techniques to solve
trigonometric equations
CCSS: F-TF.8,9
MACLE: N/A
Instructional Strategies
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Lecture enhanced with:
• SMART Notebook
• PowerPoint
• the Internet
Drill and guided practice
Approved 7-15-13
Revised 2013
5
Math Analysis/Trigonometry
Demonstrations:
• Sum and Difference Identity activity
• “Sinbad and Cosette” story
• Problem solving: e.g., Use identities to find exact values of non-familiar angles
• Reflective discussion
• Class discussion
• Computer assisted instruction
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Assessments/Evaluations
The students will be assessed on the concepts taught using a variety of modalities:
• Direct teacher observations
• Identities Matching activity
• Quizzes: Function Friday quiz #4 and #5
• Homework assignments: See attached pacing guide
• Formal common assessment: Unit C test
Mastery Level: 80%
Instructional Resources/Tools
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Textbook: Prentice Hall Precalculus Graphing and Data Analysis, Prentice Hall, Inc., Sullivan and Sullivan 2001.
Website(s): e.g., Trig Identities: http://calculustricks.com/lessons/trig-identities/
Graphing calculator
Approved 7-15-13
Revised 2013
6
Math Analysis/Trigonometry
Conceptual Category(s)
Domain
Cluster
Alignments:
CCSS: See below
Performance: 1.6, 1.10, 3.7
Knowledge: (MA) 4
MACLE: See below
NETS: 3d; 4c; 6b
DOK: 1-3
• Functions
• Trigonometric Functions (F-TF)
• Model periodic phenomena with trigonometric functions. (F-TF.6, 7)
Standards
F-TF.6, 7
6. Understand that restricting a trigonometric function to a domain
on which it is always increasing or always decreasing allows its
inverse to be constructed
7. Use inverse functions to solve trigonometric equations that arise in
modeling context; evaluate the solutions using technology, and
interpret them in terms of the context
Learning Targets
Unit D: Solving Equations and Inequalities: Solve trigonometric
equations and inequalities. Apply the concepts of reference angles,
principal values, and general solutions to solve trigonometric equations
within a modeling context
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Solve trigonometric equations over one period, and solve trigonometric
equations or inequalities for all real solutions
Determine the reference angle for any given angle
Use trigonometric identities and algebraic techniques to solve
trigonometric equations
Find the exact values of the six trigonometric functions given a point on
the terminal side of an angle, then find the angle and its reference angle
Use a calculator to find the approximate value of an inverse
trigonometric function
CCSS: F-TF.7
MACLE: N/A
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Evaluate inverse trig expressions and give principal and/or general exact
values
CCSS: F-TF.6
MACLE: N/A
Approved 7-15-13
Revised 2013
7
Math Analysis/Trigonometry
Instructional Strategies
Lecture enhanced with:
• SMART Notebook
• PowerPoint
• the Internet
• Drill and guided practice
• Demonstration – Graphing:
• sin -1 x
• cos -1 x
• tan -1 x
on a calculator to visualize principal values
• Problem solving:
• Yo-Yo problem
• Plant germination
• Reflective discussion
• Class discussion
• Computer assisted instruction
• Activity:
• Mission Impossible – Students will work collaboratively to solve application problems
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Assessments/Evaluations
The students will be assessed on the concepts taught using a variety of modalities:
• Direct teacher observations
• Quizzes
• Homework assignments: See attached pacing guide
• Formal common assessment: Unit D test
Mastery Level: 80%
Instructional Resources/Tools
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Textbook: Prentice Hall, Precalculus Graphing and Data Analysis, Prentice Hall, Inc., Sullivan and Sullivan 2001.
Website: http://mathworld.wolfram.com/InverseTrigonometricFunctions.html
Graphing calculator
Approved 7-15-13
Revised 2013
8
Math Analysis/Trigonometry
Conceptual Category(s)
Domain
Cluster
Alignments:
CCSS: See below
Performance: 1.6, 1.10, 3.7
Knowledge: (MA) 4
MACLE: See below
NETS: 3d; 4c; 6b
DOK: 1-3
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Geometry
Number and Quantity
Similarity, Right Triangles, and Trigonometry (G-SRT)
Vector and Matrix Quantities (N-VM)
Apply trigonometry to general triangles. (G-SRT.9-11)
Represent and model with vector quantities. (N-VM.1-3)
Standards
G-SRT.9-11
!
9. Derive the formula ! = ! !"#$% ! for the area of a triangle by
drawing an auxiliary line from a vertex perpendicular to the
opposite side
10. Prove the Laws of Sines and Cosines and use them to solve
problems
11. Understand and apply the Law of Sines and the Law of Cosines to
find unknown measurements in right and non-right triangles (e.g.,
surveying problems, resultant forces)
N-VM.1-3
1. Recognize vector quantities as having both magnitude and
direction. Represent vector quantities by directed line segments,
and use appropriate symbols for vectors and their magnitudes
(e.g., v, ! , ! , v)
2. Find the components of a vector by subtracting the coordinates of
an initial point from the coordinates of a terminal point
3. Solve problems involving velocity and other quantities that can be
represented by vectors
Learning Targets
Unit E: Solving Triangles: Solve right and non-right triangles using the
Law of Sines or Cosines. Recognize and use vector quantities to solve
problems
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Solve right triangles using trigonometry as it applies to real world
situations
Use the Law of Sines and Cosines to solve oblique triangles
Apply the Law of Sines and Cosines to real world situations
CCSS: G-SRT.10,11
MACLE: N/A
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Determine the area of triangles
CCSS: G-SRT.9
MACLE: N/A
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Represent and use vectors appropriately
Find vector components given coordinates of initial and terminal points
Approved 7-15-13
Revised 2013
9
Math Analysis/Trigonometry
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Solve real world problems that can be represented by vectors
CCSS: N-VM.1-3
MACLE: N/A
Instructional Strategies
Lecture enhanced with:
• SMART Notebook
• PowerPoint
• the Internet
• Drill and guided practice
Demonstration: Show how two triangles can be formed from 2 sets of congruent sides and a set of non-included congruent angles (S-S-A)
• Problem solving – Find:
• heights of objects directly and indirectly
• resultant vectors in situations
• Reflective discussion
• Class discussion
• Computer assisted instruction
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Assessments/Evaluations
The students will be assessed on the concepts taught using a variety of modalities:
• Direct teacher observations
• Projects with scoring guides:
• Flag Pole lab – using direct and indirect measurement to determine heights of objects
• Trig theme based problems: (optional- if time allows) Students will create three word problems using:
• SOH-CAH-TOA
• Law of Sines
• Law of Cosines
• Quizzes: Appropriate Function Friday (with inverses) quizzes
• Homework assignments: see attached pacing guide
• Formal common assessment: Unit E test
Mastery Level: 80%
Approved 7-15-13
Revised 2013
10
Math Analysis/Trigonometry
Instructional Resources/Tools
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Textbook: Prentice Hall, Precalculus Graphing and Data Analysis, Prentice Hall, Inc., Sullivan and Sullivan 2001.
Website(s):
• This website has an interactive demonstration of the law of cosines – http://www.mathwarehouse.com/trigonometry/law-of-cosinesformula-examples.php
• This website uses “vector addition” and “find resultant vector” (magnitude and direction) –
http://www.youtube.com/watch?v=8iMix7klXbs
Graphing calculator
Approved 7-15-13
Revised 2013
11
Math Analysis/Trigonometry
Conceptual Category(s)
Domain
Cluster
Alignments:
CCSS: See below
Performance: 1.6, 3.5
Knowledge: (MA) 4
MACLE: See below
NETS: 3d; 4c; 6b
DOK: 1-3
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Numbers and Quantity
Algebra
Vector and Matrix Quantities (N-VM)
Reasoning with Equations and Inequalities (A-REI)
Perform operations on matrices and use matrices in applications (N-VM.6-10)
Solve systems of equations (A-REI.9)
Standards
Learning Targets
N-VM.6-10
6. Use matrices to represent and manipulate data, e.g., to represent
payoffs or incidence relationships in a network
7. Multiply matrices by scalars to produce new matrices, e.g., as
when all of the payoffs in a game are doubled
8. Add, subtract, and multiply matrices of appropriate dimensions.
9. Understand that, unlike multiplication of numbers, matrix
multiplication for square matrices is not a commutative operation,
but still satisfies the associative and distributive properties
10. Understand that the zero and identity matrices play a role in matrix
addition and multiplication similar to the role of 0 and 1 in the real
numbers. The determinant of a square matrix is nonzero if and
only if the matrix has a multiplicative inverse
Unit F: Matrices: Perform basic operations on matrices, understand which
basic properties are satisfied with matrices, and use matrices to represent
and manipulate data. Find and use an inverse matrix to solve a system of
equations
A-REI.9
9. Find the inverse of a matrix if it exists and use it to solve systems
of linear equations (using technology for matrices of dimension
3x3 or greater)
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Use matrices to represent real world data and problem solve
CCSS: N-VM.6,7
MACLE: N/A
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Add, subtract, and multiply matrices and understand which properties of
equality are satisfied or not satisfied with each operation
CCSS: N-VM.8,9
MACLE: N/A
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Understand and use zero and identity matrices, and determinants
appropriately with technology
CCSS: N-VM.10
MACLE: N/A
Approved 7-15-13
Revised 2013
12
Math Analysis/Trigonometry
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Use matrices to represent a system of linear equations
Use inverse matrices to solve systems of linear equations
CCSS: A-REI.9
MACLE: N/A
Instructional Strategies
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Lecture enhanced with:
• SMART Notebook
• PowerPoint
• the Internet
Drill and guided practice
Reflective discussion
Class discussion
Computer assisted instruction
Assessments/Evaluations
The students will be assessed on the concepts taught using a variety of modalities:
• Direct teacher observations
• Quizzes
• Homework assignments: see attached pacing guide
• Formal common assessment: Unit F test
Mastery Level: 80%
Instructional Resources/Tools
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Textbook: Algebra 2 – Common Core (Pearson)
SMART Board resource – “Matrices and Vectors” downloadable at:
http://exchange.smarttech.com/search.html?q=matrices%20and%20vectors&lang=en
Graphing calculator
Approved 7-15-13
Revised 2013
13
Math Analysis/Trigonometry
Conceptual Category(s)
Domain
Cluster
Alignments:
CCSS: See below
Performance: 1.6, 1.10
Knowledge: (MA) 2,4
MACLE: See below
NETS: 3d; 4c; 6b
DOK: 1-3
• Geometry
• Expressing geometric properties with equations (G-GPE)
• Translate between the geometric description and the equation for a conic section (G-GPE.1-3)
Standards
G-GPE.1-3
1. Derive the equation of a circle of given center and radius using the
Pythagorean Theorem; complete the square to find the center and
radius of a circle given by an equation
2. Derive the equation of a parabola given a focus and directrix
3. Derive the equations of ellipses and hyperbolas given the foci,
using the fact that the sum or difference of distance from the foci
is constant
Learning Targets
Unit G: Conics: Identify and graph conic sections
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Use the distance formula, midpoint formula and complete the square
Write the equation of the perpendicular bisector
Identify, graph, and write the equation of circles, ellipses, hyperbolas,
and parabolas
CCSS: G-GPE.1-3
MACLE: N/A
Approved 7-15-13
Revised 2013
14
Math Analysis/Trigonometry
Instructional Strategies
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Lecture enhanced with:
• SMART Notebook
• PowerPoint
• the Internet
Drill and guided practice
Demonstrations – Websites demonstrating the constructions of:
• conic sections
• wax paper folding
Reflective discussion
Class discussion
Computer assisted instruction
“Sorting Conics” activity: Students will take equations in general form and determine which conic it is
Assessments/Evaluations
The students will be assessed on the concepts taught using a variety of modalities:
• Direct teacher observations
• Quizzes
• Homework assignments: see attached pacing guide
• Formal common assessment: Unit G test
Mastery Level: 80%
Instructional Resources/Tools
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Textbook: Prentice Hall, Precalculus Graphing and Data Analysis, Prentice Hall, Inc., Sullivan and Sullivan 2001.
Website: Conics: http://mathworld.wolfram.com/ConicSection.html
Graphing calculator
Approved 7-15-13
Revised 2013
15
Math Analysis/Trigonometry
Conceptual Category(s)
Domain
Cluster
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Alignments:
CCSS: See below
Performance: 1.2, 3.1, 3.5
Knowledge: (MA) 3
MACLE: See below
NETS: 1d; 3d; 4c
DOK: 1-3
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Statistics and Data Analysis
Statistics and Probability (8.SP)
Conditional Probability and the Rules of Probability (S-CP)
Making Inferences and Justifying Conclusions (S-IC)
Interpreting Categorical and Quantitative Data (S-ID)
Investigate patterns of association in bivariate data (8.SP.4)
Understand independence and conditional probability and use them to interpret data (S-CP.1-5)
Use the rules of probability to compute probabilities of compound events in a uniform probability model
(S-CP.6-9)
Understand and evaluate the random processes underlying statistical experiments (S-IC.1-2)
Make inferences and justify conclusions from sample surveys, experiments, and observational studies
(S-IC.3)
Summarize, represent, and interpret data on a single count or measurement variable (S-ID.4)
Summarize, represent, and interpret data on two categorical and quantitative variables (S-ID.5)
Standards
8.SP.4
4. Understand that patterns of association can also be seen in
bivariate categorical data by displaying frequencies and relative
frequencies in a two-way table. Construct and interpret a two-way
table summarizing data on two categorical variables collected
from the same subjects. Use relative frequencies calculated for
rows or columns to describe possible association between the two
variables
S-CP.1-5
1. Describe events as subsets of a sample space (the set of outcomes)
using characteristics (or categories) of the outcomes, or as unions,
intersections, or complements of other events (“or”, “and”, “not”)
Learning Targets
Unit H: Statistics and Data Analysis: Know and understand both the
experimental and theoretical probability of an event. Determine whether
events are dependent or independent. Recognize normally distributed data
and work with sample surveys and biases
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Determine the experimental probability of an event
Use experimental probability to make predictions
Determine the theoretical probability of an event
Convert between probabilities and odds
CCSS: S-CP.1
MACLE: D.3.A
Approved 7-15-13
Revised 2013
16
Math Analysis/Trigonometry
2. Understand that two events A and B are independent if the
probability of A and B occurring together is the product of their
probabilities, and use this characterization to determine if they are
independent
3. Understand the conditional probability of A given B as P(A and
B)|P(B), and interpret independence of A and B as saying that the
conditional probability of A given B is the same as the probability
of A
4. Construct and interpret two-way frequency tables of data when
two categories are associated with each object being classified.
Use the two-way table as a sample space to decide if events are
independent and to approximate conditional probabilities
5. Recognize and explain the concepts of conditional probability and
independence in everyday language and everyday situations
S-CP.6-9
6. Find the conditional probability of A given B as the fraction of B’s
outcomes that also belong to A, and interpret the answer in terms
of the model
7. Apply the Addition Rule, P(AorB)= P(A) + P(B)- P(AandB), and
interpret the answer in terms of the model
8. Apply the general Multiplication Rule in a uniform probability
model, P(AandB)= P(A)|P(B|A)=P(B)P(A|B), and interpret the
answer in terms of the model
9. Use permutations and combinations to compute probabilities of
compound events and solve problems
S-IC.1-3
1. Understand statistics as a process for making inferences about
population parameters based on a random sample from that
population
2. Decide if a specified model is consistent with results from a given
data-generating process, e.g., using simulation. For example, a
model says a spinning coin falls heads up with probability 0.5.
would a result of 5 tails in a row cause you to question the model?
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Solve problems involving the Fundamental Counting Principle
Solve problems involving permutations and combinations
CCSS: S-CP.9
MACLE: D.4.B
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Find probabilities of overlapping events (Addition Rule)
Find the probability of (disjoint) mutually exclusive events (Addition
Rule)
CCSS: S-CP.7,8
MACLE: D.4.B
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Construct and interpret two-way frequency tables of data when two
categories are associated with each object being classified
Determine whether events are independent or dependent
Find the probability of independent events (Multiplication Rule)
Find the probability of dependent events (Multiplication Rule)
Find and understand the conditional probability of A given B
CCSS: S-CP.2-6
MACLE: D.4.B
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Recognize normally distributed data
Use the characteristics of the normal distribution to solve problems
Use tables to estimate areas under normal curves
Recognize data sets that are not normal
Construct and interpret two-way frequency tables of data when two
categories are associated with each object being classified
CCSS: S-ID.4,5
MACLE: D.1.A
Approved 7-15-13
Revised 2013
17
Math Analysis/Trigonometry
3. Recognize the purposes and differences among sample surveys,
experiments, and observational studies; explain how
randomization relates to each
S-ID.4-5
4. Use the mean and standard deviation of a data set to fit it to a
normal distribution and to estimate population percentages.
Recognize that there are data sets for which such a procedure is
not appropriate. Use calculators, spreadsheets
5. Summarize categorical data from two categories in two-way
frequency tables. Interpret relative frequencies in the context of
the data (including joint, marginal, and a conditional relative
frequency). Recognize possible associations and trends in the data
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Understand and interpret simulations
Describe sample surveys and their biases
Describe experiments and their biases
Describe observational studies and their biases
Explain how randomization relates to each
CCSS: S-IC.1-3
MACLE: N/A
Instructional Strategies
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Lecture enhanced with:
• SMART Notebook
• PowerPoint
• the Internet
Drill and guided practice
Demonstrations – The teacher will use a:
• spinner
• die
• deck of cards
• computer
to model simulations of probability
Problem solving – Students will determine:
• types of sampling given real-life problem situations
• if any biases exist while gathering data
Reflective discussion
Class discussion
Computer assisted instruction
Simulations
Technology enhanced: TI-graphing calculator
Game: Plinko – www.math.psu.edu/dlittle/java/probability/plinko/index.html
Approved 7-15-13
Revised 2013
18
Math Analysis/Trigonometry
Assessments/Evaluations
The students will be assessed on the concepts taught using a variety of modalities:
• Direct teacher observations
• Quizzes
• Homework assignments: see attached pacing guide
• Formal common assessment: Unit H test
Mastery Level: 80%
Instructional Resources/Tools
Textbook(s):
• Geometry and Algebra 2 Common Core (primary source)
• On-Core Mathematics:
• Geometry
• Algebra 2
• Website: www.wolframalpha.com
• Graphing calculator
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Approved 7-15-13
Revised 2013
19