Download BCSS Mathematics Pacing Guide 2012

BCSS Mathematics Pacing Guide 2012 - 2013
Precalculus
Domain
Interpreting
Functions
Content
Standard # and
Identifier
18c
Content Standard Description
Graph rational functions, identifying
zeros and asymptotes when suitable
factorizations are available, and showing
end behavior.
AHSGE
1st 9 Weeks
ACT/Quality Core Standards
Description
Standards for
Mathematical
Practice
5, 6
Quality Core
Prerequisites
Skill
Textbook Resources
McGraw Hill Precalculus
1.2, 1.3
[F-IF7d]
Limits
4
Interpreting
Functions
16
Interpreting
Functions
17
1|Page
Determine numerically, algebraically,
and graphically the limits of functions at
specific values and at infinity.
For a function that models a relationship
between two quantities, interpret key
features of graphs and tables in terms of
the quantities, and sketch graphs
showing key features given a verbal
description of the relationship. (Key
features include intercepts; intervals
where the function is increasing,
decreasing, positive, or negative;
relative maximums and minimums;
symmetries; end behavior; and
periodicity.)* [F-IF4]
Calculate and interpret the average rate
of change of a function (presented
symbolically or as a table) over a
specified interval. Estimate the rate of
change from a graph.* [F-IF6]
1.3
2, 4, 5, 6, 7, 8
1.4
2, 4, 5
1.4
BCSS Mathematics Pacing Guide 2012 - 2013
Precalculus
Domain
Interpreting
Functions
Content
Standard # and
Identifier
18, 18a
Content Standard Description
Graph functions expressed symbolically,
and show key features of the graph, by
hand in simple cases and using
technology for more complicated cases.*
[F-IF7]
AHSGE
1st 9 Weeks
ACT/Quality Core Standards
Description
Standards for
Mathematical
Practice
5, 6
Quality Core
Prerequisites
Skill
Textbook Resources
McGraw Hill Precalculus
1.2, 1.3, 1.4, 1.5
a. Graph square root, cube root, and
piecewise-defined functions, including
step functions and absolute value
functions. [F-IF7b]
Building
Functions
19
Building
Functions
20
Determine the inverse of a function and a
relation.
Building
Functions
21
Verify by composition that one function
is the inverse of another.
Building
Functions
22
Building
Functions
23
Interpreting
Functions
18c
2|Page
Compose functions.
[F-BF1c]
1, 2, 3, 4, 5, 6,
7, 8
1.6
1.7
2, 4, 5, 7
1.7
Read values of an inverse function from
a graph or a table, given that the function
has an inverse. [F-BF4c]
2, 4, 5, 7
1.7
Produce an invertible function from a
non-invertible function by restricting the
domain. [F-BF4d]
Graph rational functions, identifying
zeros and asymptotes when suitable
factorizations are available, and showing
end behavior. [F-IF7d]
2, 4, 5, 7
1.7
[F-BF4b]
5, 6
2.1, 2.2, 2.4, 2.5 (examples 1-4)
BCSS Mathematics Pacing Guide 2012 - 2013
Precalculus
Domain
Interpreting
Functions
Content
Standard # and
Identifier
18b
Content Standard Description
Graph polynomial functions, identifying
zeros when suitable factorizations are
available, and showing end behavior. [FIF7c]
Building
Functions
25
Compare effects of parameter changes
on graphs of transcendental functions.
Interpreting
Functions
18d
Graph exponential and logarithmic
functions, showing intercepts and end
behavior, and trigonometric functions,
showing period, midline, and amplitude.
[F-IF7e]
Building
Functions
24
Understand the inverse relationship
between exponents and logarithms, and
use this relationship to solve problems
involving logarithms and exponents. [F-
BF5]
3|Page
AHSGE
1st 9 Weeks
ACT/Quality Core Standards
Description
Standards for
Mathematical
Practice
5, 6
Quality Core
Prerequisites
Skill
Textbook Resources
McGraw Hill Precalculus
2.2
3.1 (examples 1-3)
5, 6
3.1 (examples 1-3), 3.2
3.2, 3.3, 3.4
BCSS Mathematics Pacing Guide 2012 - 2013
Precalculus
Domain
Trigonometric
Functions
Content
Standard # and
Identifier
29
Content Standard Description
AHSGE
2nd 9 Weeks
ACT/Quality Core Standards
Description
Standards for
Mathematical
Practice
Quality Core
Prerequisites
Skill
Textbook Resources
McGraw Hill Precalculus
4.3
Use special triangles to determine
geometrically the values of sine, cosine,
and tangent for 3π, 4π, and 6π, and use
the unit circle to express the values of
sine, cosine, and tangent for π – x, π + x,
and 2π – x in terms of their values of x,
where x is any real number.
[F-TF3]
Trigonometric
Functions
30
Interpreting
Functions
18d
Trigonometric
Functions
26
Trigonometric
Functions
31
Trigonometric
Functions
32
4|Page
Use the unit circle to explain symmetry
(odd and even) and periodicity of
trigonometric functions. [F-TF4]
Graph exponential and logarithmic
functions, showing intercepts and end
behavior, and trigonometric functions,
showing period, midline, and amplitude.
[F-IF7e]
Determine the amplitude, period, phase
shift, domain, and range of trigonometric
functions and their inverses.
Understand that restricting a
trigonometric function to a domain on
which it is always increasing or always
decreasing allows its inverse to be
constructed. [F-TF6]
Use inverse functions to solve
trigonometric equations that arise in
modeling contexts; evaluate the
solutions using technology, and interpret
them in terms of the context. [F-TF7]
4.3
5, 6
4.4, 4.5 (examples 1-4)
4.4, 4.5 (ex. 1-4), 4.6 (ex. 1-5)
4.6 (ex. 1-5)
4.3-4.6
BCSS Mathematics Pacing Guide 2012 - 2013
Precalculus
Domain
Similarity,
Right
Triangles, and
Trigonometry
Content
Standard # and
Identifier
35
Content Standard Description
(+) Derive the formula A = (1/2)ab sin(C)
for the area of a triangle by drawing an
auxiliary line from a vertex perpendicular
to the opposite side. (Apply formulas
previously derived in Geometry.) [G-SRT9]
Prove the Pythagorean identity sin2(θ) +
cos2(θ) = 1, and use it to find sin(θ),
cos(θ), or tan(θ) given sin(θ), cos(θ), or
tan(θ) and the quadrant of the angle. [FTF8]
Prove the addition and subtraction
formulas for sine, cosine, and tangent,
and use them to solve problems. [F-TF9]
AHSGE
2nd 9 Weeks
ACT/Quality Core Standards
Description
Standards for
Mathematical
Practice
Quality Core
Prerequisites
Skill
Textbook Resources
McGraw Hill Precalculus
4.7 example 8
Trigonometric
Functions
33
Trigonometric
Functions
34
Trigonometric
Functions
27
Use the sum, difference, and half-angle
identities to find the exact value of a
trigonometric function.
5.4, 5.5 (example 5)
Vector and
Matrix
Quantities
11
(+) Work with 2 x 2 matrices as
transformations of the plane, and interpret
the absolute value of the determinant in
terms of area. [N-VM12]
6.2 (page 384 #45-48), page 387, page 393 #
37
Reasoning
with
Equations
and
Inequalities
14
(+) Represent a system of linear equations
as a single matrix equation in a vector
variable. [A-REI8]
6.3, Enrichment: ex. 3 & 4
5|Page
3, 7, 8
5.1
5.4
BCSS Mathematics Pacing Guide 2012 - 2013
Precalculus
Domain
Conic
Sections
Content
Standard # and
Identifier
15, 15a
Content Standard Description
AHSGE
3rd 9 Weeks
ACT/Quality Core Standards
Description
Standards for
Mathematical
Practice
Quality Core
Prerequisites
Skill
Textbook Resources
McGraw Hill Precalculus
7.1 (skip ex. 5), 7.2, 7.3
Create graphs of conic sections, including
parabolas, hyperbolas, ellipses, circles,
and degenerate conics from second
degree equations.
Ex. Graph x2 – 6x + y2 –
12y + 41 = 0 or y2 – 4x +
2y + 5 = 0.
a.Formulate equations of conic
sections from their determining
characteristics.
Ex. Write the equation of
an ellipse with center
(5, -3), a horizontal major
axis of length 10, and a
minor axis of length 4.
Expressing
Geometric
Properties
with
Equations
Expressing
Geometric
Properties
with
Equations
6|Page
36
Derive the equations of a parabola given a
focus and a directrix.
2, 3, 7, 8
7.1 (skip ex. 5)
[G-GPE2]
37
Derive the equations of ellipses and
hyperbolas given the foci, using the fact
that the sum or difference of distances
from the foci is constant.
[G-GPE3]
7.2, 7.3
BCSS Mathematics Pacing Guide 2012 - 2013
Precalculus
Domain
Trigonometric
Functions
Content
Standard # and
Identifier
28, 28a, 28b
Content Standard Description
Utilize parametric equations by graphing
and by converting to rectangular form.
AHSGE
3rd 9 Weeks
ACT/Quality Core Standards
Description
Standards for
Mathematical
Practice
Quality Core
Prerequisites
Skill
Textbook Resources
McGraw Hill Precalculus
7.5
a. Solve application-based problems
involving parametric equations.
b. Solve applied problems that include
sequences with recurrence relations.
Vector and
Matrix
Operations
5
Vector and
Matrix
Operations
6
Vector and
Matrix
Operations
7
7|Page
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|, ||v||, v). [N-VM1]
Find the components of a vector by
subtracting the coordinates of an initial
point from the coordinates of a terminal
point. [N-VM2]
Solve problems involving velocity and
other quantities that can be represented by
vectors. [N-VM3]
8.1, 8.2
8.1, 8.2
8.1, 8.2
BCSS Mathematics Pacing Guide 2012 - 2013
Precalculus
Domain
Vector and
Matrix
Operations
Content
Standard # and
Identifier
8, 8a, 8b, 8c
Content Standard Description
(+) Add and subtract vectors. [N-VM4]
a. (+) Add vectors end-to-end,
component-wise, and by the
parallelogram rule. Understand that the
magnitude of a sum of two vectors is
typically not the sum of the magnitudes.
[N-VM4a]
b. (+) Given two vectors in magnitude
and direction form, determine the
magnitude and direction of their sum. [NVM4b]
c. (+) Understand vector subtraction v - w
as v + (-w), where -w is the additive
inverse of w, with the same magnitude as
w and pointing in the opposite direction.
Represent vector subtraction graphically
by connecting the tips in the appropriate
order, and perform vector subtraction
component-wise. [N-VM4c]
8|Page
AHSGE
3rd 9 Weeks
ACT/Quality Core Standards
Description
Standards for
Mathematical
Practice
Quality Core
Prerequisites
Skill
Textbook Resources
McGraw Hill Precalculus
8.1, 8.2
BCSS Mathematics Pacing Guide 2012 - 2013
Precalculus
Domain
Vector and
Matrix
Operations
Content
Standard # and
Identifier
9, 9a, 9b
Content Standard Description
(+) Multiply a vector by a scalar. [NVM5]
AHSGE
3rd 9 Weeks
ACT/Quality Core Standards
Description
Standards for
Mathematical
Practice
Quality Core
Prerequisites
Skill
Textbook Resources
McGraw Hill Precalculus
8.1
a. (+) Represent scalar multiplication
graphically by scaling vectors and
possibly reversing their direction;
perform scalar multiplication componentwise, e.g., as c(vx, vy) = (cvx, cvy). [NVM5a]
b. (+) Compute the magnitude of a scalar
multiple cv using ||cv|| = |c|v. Compute
the direction of cv knowing that when |c|v
≠ 0, the direction of cv is either along v
(for c > 0) or against v (for c < 0). [NVM5b]
Vector and
Matrix
Operations
10
The Complex
Number
System
1
9|Page
Multiply a vector (regarded as a matrix
with one column) by a matrix of suitable
dimensions to produce another vector.
Work with matrices as transformations of
vectors. [N-VM11]
Represent complex numbers on the
complex plane in rectangular and polar
form (including real and imaginary
numbers), and explain why the
rectangular and polar forms of a given
complex number represent the same
number. [N-CN4]
Page 517
9.5 (ex. 1-6)
BCSS Mathematics Pacing Guide 2012 - 2013
Precalculus
Domain
The Complex
Number
System
Content
Standard # and
Identifier
2
Content Standard Description
AHSGE
3rd 9 Weeks
ACT/Quality Core Standards
Description
Standards for
Mathematical
Practice
Quality Core
Prerequisites
Skill
Textbook Resources
McGraw Hill Precalculus
Pages P6-P7 (examples 1-3), 9.5 (ex. 1-6)
Represent addition, subtraction,
multiplication, and conjugation of
complex numbers geometrically on the
complex plane; use properties of this
representation for computation.
[N-CN5]
The Complex
Number
System
3
Seeing
Structure in
Expressions
12
Ex. (-1 + i)3 = 8 because
(-1 + i) has modulus 2 and
argument 120°.
Calculate the distance between numbers
in the complex plane as the modulus of
the difference, and the midpoint of a
segment as the average of the numbers at
its endpoints.
9.5 (ex. 1-6)
[N-CN6]
Derive the formula for the sum of a finite
geometric series (when the common ratio
is not 1), and use the formula to solve
problems.* [A-SSE4]
3, 4, 7, 8
10.3 (examples 6-8)
Example: Calculate mortgage payments.
Arithmetic
with
Polynomials
and Rational
Expressions
10 | P a g e
13
(+) Know and apply the Binomial
Theorem for the expansion of (x + y)n in
powers of x and y for a positive integer n,
where x and y are any numbers, with
coefficients determined, for example, by
Pascal's Triangle. (The Binomial
Theorem can be proved by mathematical
induction or by a combinatorial
argument.) [A-APR5]
10.5
BCSS Mathematics Pacing Guide 2012 - 2013
Precalculus
Domain
Limits
Content
Standard # and
Identifier
4, 4a
Content Standard Description
AHSGE
4th 9 Weeks
ACT/Quality Core Standards
Description
Standards for
Mathematical
Practice
Quality Core
Prerequisites
Skill
Textbook Resources
McGraw Hill Precalculus
Review 1.3; 12.1, 12.2 (skip 12.2 ex. 7)
Determine numerically, algebraically, and
graphically the limits of functions at
specific values and at infinity.
a. Apply limits in problems involving
convergence and divergence.
Expressing
Geometric
Properties
with
Equations
38
Interpreting
Categorical
and
Quantitative
Data
Interpreting
Categorical
and
Quantitative
Data
39
11 | P a g e
40
Give an informal argument using
Cavalieri’s principle for the formulas for
the volume of a sphere and other solid
figures. [G-GMD2]
Use statistics appropriate to the shape of
the data distribution to compare center
(median, mean) and spread (interquartile
range, standard deviation) of two or more
different data sets. (Focus on increasing
rigor using standard deviation). [S-ID2]
Interpret differences in shape, center, and
spread in the context of the data sets,
accounting for possible effects of extreme
data points (outliers). (Identify unifrom,
skewed, and normal distridutions in a set
of data. Determine the quartiles and
interquartile range for a set of data.) [SID3]
http://www.cut-theknot.org/Curriculum/Calculus/Cavalieri.shtml
http://www.answers.com/topic/method-ofindivisibles
1, 2, 3, 4, 5, 7
11.1, 11.2
1, 2, 3, 4, 5, 7
11.1
BCSS Mathematics Pacing Guide 2012 - 2013
Precalculus
Domain
Interpreting
Categorical
and
Quantitative
Data
Content
Standard # and
Identifier
41
Content Standard Description
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, and tables
to estimate areas under the normal curve.
[S-ID4]
AHSGE
4th 9 Weeks
ACT/Quality Core Standards
Description
Standards for
Mathematical
Practice
1, 2, 3, 4, 5, 7
Quality Core
Prerequisites
Skill
Textbook Resources
McGraw Hill Precalculus
11.2, 11.3
Using
Probability
to Make
Decisions
50
(+) Define a random variable for a
quantity of interest by assigning a
numerical value to each event in a sample
space; graph the corresponding probability
distribution using the same graphical
displays as for data distributions. [S-MD1]
11.2
Using
Probability
to Make
Decisions
51
(+) Calculate the expected value of a
random variable; interpret it as the mean
of the probability distribution. [S-MD2]
11.2
12 | P a g e
BCSS Mathematics Pacing Guide 2012 - 2013
Precalculus
Domain
Using
Probability
to Make
Decisions
Content
Standard # and
Identifier
54, 54a, 54b
Content Standard Description
(+) Weigh the possible outcomes of a
decision by assigning probabilities to
payoff values and finding expected
values. [S-MD5]
a. Find the expected payoff for a game
of chance. [S-MD5a]
Examples: Find the expected winnings
from a state lottery ticket or a game at a
fast-food restaurant.
b. Evaluate and compare strategies on
the basis of expected values. [S-MD5b]
Example: Compare a high-deductible
versus a low-deductible automobile
insurance policy using various, but
reasonable, chances of having a minor
or a major accident.
13 | P a g e
AHSGE
4th 9 Weeks
ACT/Quality Core Standards
Description
Standards for
Mathematical
Practice
Quality Core
Prerequisites
Skill
Textbook Resources
McGraw Hill Precalculus
11.2
BCSS Mathematics Pacing Guide 2012 - 2013
Precalculus
Domain
Using
Probability
to Make
Decisions
Content
Standard # and
Identifier
52
Content Standard Description
AHSGE
4th 9 Weeks
ACT/Quality Core Standards
Description
Standards for
Mathematical
Practice
Quality Core
Prerequisites
Skill
Textbook Resources
McGraw Hill Precalculus
11.3
(+) Develop a probability distribution for
a random variable defined for a sample
space in which theoretical probabilities
can be calculated; find the expected value.
[S-MD3]
Example: Find the theoretical probability
distribution for the number of correct
answers obtained by guessing on all five
questions of a multiple-choice test where
each question has four choices, and find
the expected grade under various grading
schemes.
Using
Probability
to Make
Decisions
53
11.3
(+) Develop a probability distribution for
a random variable defined for a sample
space in which probabilities are assigned
empirically; find the expected value. [SMD4]
Example: Find a current data distribution
on the number of television sets per
household in the United States, and
calculate the expected number of sets per
household. How many television sets
would you expect to find in 100 randomly
selected households?
Making
Inferences
and
Justifying
Conclusions
14 | P a g e
44
Understand statistics as a process for
making inferences about population
parameters based on a random sample
from that population. [S-IC1]
2, 4, 6
11.4
BCSS Mathematics Pacing Guide 2012 - 2013
Precalculus
Domain
Making
Inferences
and
Justifying
Conclusions
Making
Inferences
and
Justifying
Conclusions
Making
Inferences
and
Justifying
Conclusions
Making
Inferences
and
Justifying
Conclusions
Making
Inferences
and
Justifying
Conclusions
15 | P a g e
Content
Standard # and
Identifier
45
Content Standard Description
Decide if a specified model is consistent
with results from a given data-generating
process, e.g., using simulation. [S-IC2]
AHSGE
4th 9 Weeks
ACT/Quality Core Standards
Description
Standards for
Mathematical
Practice
1, 2, 3, 4, 5, 6,
7, 8
Quality Core
Prerequisites
Skill
Textbook Resources
McGraw Hill Precalculus
11.4, 11.5
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?
46
Recognize the purposes of and differences
among sample surveys, experiments, and
observational studies; explain how
randomization relates to each. [S-IC3]
2, 3, 4, 6
11.4
47
Use data from a sample survey to estimate
a population mean or proportion; develop
a margin of error through the use of
simulation models for random sampling.
[S-IC4]
1, 3, 4, 5, 6
11.4
48
Use data from a randomized experiment to
compare two treatments; use simulations
to decide if differences between
parameters are significant. [S-IC5]
1, 3, 4, 5, 6, 8
11.6
49
Evaluate reports based on data. [S-IC6]
1, 2, 3, 4, 5, 6,
7, 8
11.1-11.7
BCSS Mathematics Pacing Guide 2012 - 2013
Precalculus
Domain
Interpreting
Categorical
and
Quantitative
Data
Interpreting
Categorical
and
Quantitative
Data
16 | P a g e
Content
Standard # and
Identifier
42
43
Content Standard Description
Compute (using technology) and interpret
the correlation coefficient of a linear fit.
[S-ID8]
Distinguish between correlation and
causation. [S-ID9]
AHSGE
4th 9 Weeks
ACT/Quality Core Standards
Description
Standards for
Mathematical
Practice
4, 5, 6
3, 4, 6
Quality Core
Prerequisites
Skill
Textbook Resources
McGraw Hill Precalculus
11.7
11.7
BCSS Mathematics Pacing Guide 2012 - 2013
Precalculus
Essential Vocabulary
1st 9 Weeks
Chapter 1: zeros, roots, line symmetry, point
symmetry, even function, odd function, continuous
function, limit, discontinuous function, infinite
discontinuity, jump discontinuity, removable
discontinuity, nonremovable discontinuity, end
behavior, increasing, decreasing, constant, critical
point, extrema, maximum, minimum, point of
inflection, average rate of change, secant line, parent
function, constant function, zero function, identity
function, quadratic function, cubic function, square
root function, reciprocal function, absolute value
function, step function, greatest integer function,
transformation, translation, reflection, dilation,
composition, inverse relation, inverse function, oneto-one
Chapter 2: power function, monomial function, radical
function, extraneous solution, polynomial function,
leading coefficient, quartic function, turning point,
repeated zero, multiplicity, Rational Zero Theorem,
lower bound, upper bound, Descartes’ Rule of Signs,
Fundamental Theorem of Algebra, Linear Factorization
Theorem, Conjugate Root Theorem, complex
conjugates, rational function, asymptote, vertical
asymptote, horizontal asymptote, oblique asymptote
Chapter 3: algebraic function, transcendental
function, exponential function, natural base,
continuous function, interest, logarithmic function,
logarithm, common logarithm, natural logarithm
17 | P a g e
2nd 9 Weeks
Chapter 4: quadrantal angle, reference angle, unit
circle, circular function, periodic function, period,
sinusoid, amplitude, frequency, phase shift, vertical
shift, midline, arcsine function, arccosine function,
arctangent function
Chapter 5: identity, trigonometric identity, reduction
identity
Chapter 6: identity matrix, inverse matrix, inverse,
invertible, singular matrix, determinant, square
system, Cramer’s Rule
3rd 9 Weeks
Chapter 7: conic section, degenerate conic, locus,
parabola, focus, directrix, axis of symmetry, vertex,
latus rectum, ellipse, foci, major axis, center, minor
axis, vertices, co-vertices, eccentricity, hyperbola,
transverse axis, conjugate axis, parametric equation,
parameter, orientation, parametric curve
Chapter 8: vector, initial point, terminal point,
standard position, direction, magnitude, quadrant
bearing, true bearing, parallel vectors, equivalent
vectors, opposite vectors, resultant, triangle method,
parallelogram method, zero vector, components,
rectangular components, component form, unit
vector, linear combination
Chapter 9: complex plane, real axis, imaginary axis,
Argand plane, absolute value of a complex number,
polar form, trigonometric form, modulus, argument
Chapter 10: geometric sequence, common ratio,
geometric series, binomial coefficients, Pascal’s
Triangle, Binomial Theorem
4th 9 Weeks
Chapter 11: univariate, negatively skewed
distribution, symmetrical distribution, positively
skewed distribution, resistant statistic, cluster,
bimodal distribution, percentiles, percentile graph,
random variable, discrete random variable,
continuous random variable, probability distribution,
expected value, binomial experiment, binomial
distribution, binomial probability distribution function,
normal distribution, empirical rule, z-value, standard
normal distribution, sampling distribution, standard
error of mean, sampling error, continuity correction
factor, inferential statistics, parameter, point
estimate, interval estimate, confidence level,
maximum error of estimate, critical value, confidence
interval, t-distribution, degrees of freedom,
hypothesis test, null hypothesis, alternative
hypothesis, level of significance, left-tailed test, twotailed test, right-tailed test, p-value, correlation,
bivariate, explanatory variable, response variable,
correlation coefficient, regression line, line of best fit,
residual, least-squares regression line, residual plot,
influential, interpolation, extrapolation
Chapter 12: one-sided limit, two-sided limit, direct
substitution, indeterminate form