Math - Level 4: The Course/Image.iso/Sample Study Guide/m30 Study Guide-92

Text File | 1992-12-17 | 32KB | 858 lines

             Program of Studies                         (Question Bank Module-Objective)

Polynomial Functions

1. Students will be expected to demonstrate an                   Rinehart and Winston
understanding that a polynomial function is a                      90-91, 104-107
function of the form
f(x) = anx^n + a(n – 1)x^(n – 1) + a(n – 2)x^(n – 2) +                          Nelson
. . . + a2x^2 + a1x + a0,                                                                  142-144
where a0, a1, a2,... an are real numbers and n                           151-156
are elements of N.
                                                                                                 Addison Wesley
 (mod-11.obj-1 & 2)                                                                   42-48, 63-78
 (mod-12.obj-1)

2.    Students will be expected to demonstrate
an understanding that a polynomial function
can be graphed on a Cartesian plane and that
such graphs will have particular characteristics
depending on the function.

2.1    Students will be expected to sketch the
graphs of integral polynomial functions

(mod-14.obj-1 to 4)

2.1.1    Students will be expected to draw
the graphsof integral polynomial functions
using calculators or computers.

(mod-14.obj-1)

2.1.2    Students will be expected to investigate
the characteristics of the graphs of
polynomial functions of different degrees
and determine the effects of a multiplicity of
zeros on the graphs of polynomial functions.

(mod-14.obj-2 & 3)

2.1.3    Students will be expected to find
approximations for the zeros of integral
polynomial functions using calculators
or computers.

(mod-14.obj-4)

2.1.4    Students will be expected to analyze
points on the graphs of polynomial functions
using calculators or computers.

2.1.5    Students will be expected to solve
problems that can be represented by
polynomial functions.


3.    Students will be expected to demonstrate
an understanding that many polynomial functions
can have the same zeros.

(mod-13.obj-4)

3.1    Students will be expected to derive an
equation of an integral polynomial function given
its zeros.


3.2    Students will be expected to derive the
equation of an integral polynomial function
given its zeros and an ordered pair that satisfies it.

3.2.1    Students will be expected to find the equation
of a polynomial function given its zeros and any other
information that will uniquely define it.


4.    Students will be expected to demonstrate                               Rinehart and Winston
an understanding of the following form of                                        92-95, 98-103
the division algorithm for polynomials:
If any polynomial P(x) is divided by a binomial                                       Nelson
of the form   (x – a) (called D(x)), the result                                         149-153
will be a polynomial quotient Q(x) and a remainder R.
                                                                                                      Addison Wesley
(mod-11.obj-3 & 4)                                                                              49-63

4.1    Students will be expected to divide integral
polynomial functions in one variable by a binomial.

4.2    Students will be expected to write the
division operation on a polynomial function
by a binomial in the form of the Division
Algorithm:  P(x) = D(x)Q(x) + R.


5.    Students will be expected to demonstrate
 an understanding that when a polynomial P(x)
is divided by a binomial of the form (x – a),
the remainder R is equal to P(a) (Remainder Theorem).

5.1    Students will be expected to use the
Remainder Theorem to evaluate polynomial
functions for rational values of the variable.

(mod-12.obj-3)

5.1.1    Students will be expected to prove
the Remainder Theorem.

(mod-12.obj-2)

5.1.2    Students will be expected to use the
Remainder Theorem to prove that if a number
a is a zero of a polynomial function P(x) then
(x – a) will be a factor of P(x) (Factor Theorem).

5.2    Students will be expected to use the
Factor Theorem to factor an integral polynomial
 function completely and to determine all of its
real zeros.

(mod-12.obj-4)
(mod-13.obj-1 to 3)

5.2.1    Students will be expected to use a
technology to factor polynomial functions.

5.2.2    Students will be expected to recognize
that all rational zeros of a polynomial function
will be of the form  p/q where p is a factor of
a0 and q is a factor of an.


Trigonometric and Circular Functions

1.    Students will be expected to demonstrate
an understanding that the radian measure of
an angle is the ratio of the arc it subtends
to the radius of a circle in which it is a
central angle, and that one radian is the
measure of a central angle subtended in a
circle by an arc whose length is equal to the
radius of the circle.

( mod-41.obj-1 to 3)
( mod-42.obj-1)

1.1    Students will be expected to identify
the radian measure of a central angle in a
circle.

(mod-42.obj-2)

1.2    Students will be expected to convert
angle measurements between degree and
radian measure and vice versa.

(mod-42.obj-3)

1.3    Students will be expected to determine
the exact values of the trigonometric ratios
for angles coterminal with  (nπ)/6, (nπ)/4,
(nπ)/3, (nπ)/2 and n is an element of I.

(mod-42.obj-4)


2.    Students will be expected to demonstrate
an understanding that identities are statements
of equality that are true for all values of the
variable and that trigonometric identities are
equations that express relations among
trigonometric functions that are valid for all
values of the variables for which the functions
are defined.

2.1    Students will be expected to use the
following fundamental trigonometric identities

Reciprocal Identities
csc a = 1/sin a
sec a = 1/cos a
cot a = 1/tan a

Quotient Identities
tan a = sin a/cos a
cot a = cos a/sina

Pythagorean Identities
sin^2 a + cos^2 a = 1
tan^2a + 1 = sec^2a
cot^2 a + 1 = csc^2 a

(mod-46.obj-1)

2.1.1    Students will be expected to derive                              Rinehart and Winston
the quotient and Pythagorean identities                                 296-301, 320-324,
using logical processes.                                                          327-329

(mod-46.obj-1)                                                                                Nelson
                                                                                                      191-211, 221-226,
2.1.2    Students will be expected to use                                    230-234
the fundamental trigonometric identities
to simplify, evaluate and prove                                                    Addison Wesley
trigonometric expressions involving identities.                       178-225, 236-240,
                                                                                                  251-255
(mod-46.obj-1)

2.2    Students will be expected to use the
addition and subtraction identities (formulas):

cos (a +/– b) = cos a cos b +/– sin a sin b
sin (a +/– b) = sin a cos b +/– cos a sin b

(mod-47.obj-1 to 2)

3.    Students will be expected to demonstrate
an understanding that trigonometric functions
can be graphed on a Cartesian plane.

3.1    Students will be expected to graph the
following forms of the sine, cosine and
tangent functions:

y = a sin [b(q + c)] + d
y = a cos [b(q + c)] + d
y = tan q

(mod-45.obj-2)

3.1.1    Students will be expected to use
calculators or computers to draw and
analyze the graphs of trigonometric
functions.

(mod-45.obj-1)

3.1.2    Students will be expected to
investigate the effects of the parameters
a, b, c and d on the graphs of trigonometric
functions using calculators or computers.

(mod-45.obj-1)

3.1.3    Students will be expected to state
the domain and range of all the trigonometric
functions.

(mod-45.obj-1)


4.    Students will be expected to demonstrate
an understanding of the methods used to solve
trigonometric equations.

4.1    Students will be expected to solve first
and second degree trigonometric equations
involving multiples of angles on the
domain 0 ≤ q < 2π.

(mod-43.obj-1 to 3)

4.1.1    Students will be expected to use
calculators or computers to solve
trigonometric equations by evaluating
the graphs of trigonometric functions.

(mod-44.obj-1)

4.2    Students will be expected to demonstrate
the relationship between the root of a
trigonometric equation and the graph of
the corresponding function.

Statistics                                                                              Rinehart and Winston
                                                                                                      393-403
1.    Students will be expected to demonstrate
an understanding that a bivariate distribution                               Nelson
involved two variables that may have some                                   449-460
relationship to each other.
                                                                                                     Addison Wesley
1.1    Students will be expected to plot sets                                    477-498
of bivariate data on a scatter plot.

(mod-62.obj-1)

1.2    Students will be expected to plot a
line of best fit on a scatter plot using
the median fit method.

(mod-62.obj-2)

1.3    Students will be expected to develop and
use prediction equations of the line of best
fit to make inferences for populations.

(mod-62.obj-3)
        1.4    Students will be expected to recognize
and describe the apparent correlation between
the variables of a bivariate distribution from
a scatter plot.

(mod-62.obj-4)

1.5    Students will be expected to collect,
organize and analyze sets of bivariate data.

(mod-62.obj-5)

1.5.1    Students will be expected to apply
statistical processes and statistical
reasoning in investigations involving
bivariate data.

2.    Students will be expected to demonstrate
an understanding that data can be distributed
normally, and that a normal distribution has
particular characteristics that can be used
to describe and analyze many situations.

2.1    Students will be expected to find and
interpret the mean and standard deviation
of a set of normally distributed data.

(mod-61.obj-1)

2.1.1    Students will be expected to use
calculators or computers to calculate the
mean and standard deviation of sets of
normally distributed data.

2.2    Students will be expected to apply
the characteristics of a normal distribution.

2.2.1    Students will be expected to solve
problems involving data that are normally
distributed.

(mod-61.obj-2)

2.3    Students will be expected to find and
apply the standard normal curve and the
z-scores of data that are normally distributed.

(mod-61.obj-3 & 4)

2.3.1    Students will be expected to apply
z-scores to solve problems involving
probability distributions.


3.    Students will be expected to demonstrate
an understanding that the results of a survey
can be interpreted with measurable degrees
of confidence.

3.1    Students will be expected to distinguish
between a population and a sample and assess
the strengths, weaknesses and biases of
given samples.

(mod-63.obj-5)

3.2    Students will be expected to collect
and organize the results of yes/no surveys
taken from defined samples.

(mod-63.obj-1)

3.2.1    Students will be expected to design
and administer a simple survey.

3.2.2    Students will be expected to collect
and organize the results of a simple survey.

3.3    Students will be expected to draw box
plots of the results of multiple samples.

(mod-63.obj-2)

3.3.1    Students will be expected to carry
out investigations involving multiple samples
taken from populations with known and
unknown proportions of yes responses.

3.4    Students will be expected to use chars
of 90 per cent box plots to find the
confidence interval within which such
conclusions and inferences are made
based on the results of yes/no surveys.

(mod-63.obj-3)

3.4.1    Students will be expected to use
statistical inferences to solve problems.

3.5    Students will be expected to draw
statistical conclusions, make inferences
to populations and explain the confidence
with which such conclusions and inferences
are made based on the results of yes/no surveys.

(mod-63.obj-4)

3.5.1    Students will be expected to design
and administer a survey to a random sample
of a population, collect and organize the
responses, and analyze the results,
including making inferences to the
population and evaluating the results
for the confidence with which they may
be held.


Quadratic Relations

1.    Students will be expected to demonstrate
an understanding of the physical properties
of the conic sections with respect to the
intersection of a plane and a cone.

( mod-51.obj-1-5)
( mod-52.obj-1-6)

1.1    Students will be expected to describe
the conic section formed by the intersection
of a plane and a cone.

1.1.1    Students will be expected to identify
the point at which each of the conics becomes
degenerate.

2.    Students will be expected to demonstrate
an understanding of the general quadratic
relation
Ax2 + Bxy + Cy2 + Dx + Ey + F = 0 as the
algebraic representation of any conic.

(mod-53.obj-1-5)
(mod-54.obj-1-6)

2.1    Students will be expected to describe
the conics that would be generated by
various combinations of values for the
numerical coefficients.

2.1.1    Students will be expected to
investigate and describe the effects of
the numerical coefficients on the graphs
of quadratic relations, using calculators
or computers.

3.    Students will be expected to demonstrate                            Addison Wesley
an understanding of the effects of the                                       Master Grapher,
 numerical coefficients in the general quadratic                       30 Grapher,
relation Ax2 + Bxy + Cy2 + Dx + Ey + F = 0                                  Computing
where B = 0 on the curves of the resulting                                 Graphing
conics.                                                                                        Experiments 3

(mod-55.obj-1-9)                                                                       IBM Tool Kit

3.1    Students will be expected to analyze                                   Zap-a-Graph
the graphs of ellipses, parabolas, and
hyperbolas, given their equations.

3.1.1    Students will be expected to use
calculators or computers to draw the
graphs of ellipses, parabolas and hyperbolas.

3.1.2    Students will be expected to
recognize which conditions are required
for an ellipse to become a circle.

3.1.3    Students will be expected to
investigate and describe how the length
of the axes affects the orientation, size
and shape of the graph.

    4.    Students will be expected to demonstrate
an understanding that a locus is a system of
points that satisfies a given condition.

(mod-56.obj-1-7)

4.1    Students will be expected to recognize
that each conic can be described as a locus
of points.

4.1.1    Students will be expected to use the
locus definition to verify the equations that
describe the conics.

4.1.2    Students will be expected to solve
problems that involve analyzing and
determining the characteristics of a body
that follows a conical path.

4.1.3    Students will be expected to solve
problems that involve analyzing and
determining the characteristics of a conical
surface.

5.    Students will be expected to demonstrate
an understanding that any conic can be described
as the locus of point, such that, the ratio of
the distance between any point and a fixed
point to the distance between the same point
and a fixed line is a constant.

(mod-57.obj-1-5)
(mod-58.obj-1-7)


Exponential and Logarithmic Functions

    1.    Students will be expected to demonstrate                            Rinehart and Winston
an understanding that an exponential function                                      122-128
is one in which the variable appears in the
 exponent.                                                                                                Nelson
                                                                                                                89-94
(mod-21.obj-1)
                                                                                                             Addison Wesley
1.1    Students will be expected to sketch the                                  262-265, 274-278
graph of exponential functions of the form
y = a^x, a > 0

(mod-21.obj-2)

1.2    Students will be expected to use the
graphs of exponential functions to estimate
the values of roots and powers.

(mod-21.obj-2)

1.2.1    Students will be expected to draw
and analyze the graphs of exponential
functions using calculators or computers.

1.2.2    Students will be expected to determine
the domain and range of the exponential
functions.

1.3    Students will be expected to solve and
verify exponential equations.

(mod-21.obj-3)

2.    Students will be expected to demonstrate                           Rinehart and Winston
an understanding that many real-world                                              125-141
phenomena exhibit exponential properties.
                                                                                                           Nelson
(mod-21.obj-4)                                                                          96-103, 110-116,
                                                                                                   119-128
2.1    Students will be expected to recognize
exponential functions describing situations                                 Addison Wesley
involving exponential growth and decay.                                    262-265, 279-319

2.1.1    Students will be expected to solve
problems involving exponential growth and
decay.

3.    Students will be expected to demonstrate
an understanding of the characteristics and
applications of logarithmic functions.

(mod-22.obj-1)

3.1    Students will be expected to draw the
graphs of logarithmic functions as the
inverses of exponential functions.

(mod-22.obj-2)

3.2    Students will be expected to use the
graphs of logarithmic functions to find
the values of one of the variables, given
the other variable.

(mod-22.obj-2 & 4)

3.2.1    Students will be expected to draw
and analyze the graphs of logarithmic
functions using calculators or computers.

3.2.2    Students will be expected to determine
the domain and range of the logarithmic
functions.

3.3    Students will be expected to convert
functions from exponential form to
logarithmic form and vice versa.

(mod-22.obj-3)

4.    Students will be expected to demonstrate
an understanding that operations with
logarithms are subject to basic properties
and laws.

(mod-23.obj-1)

4.1    Students will be expected to apply
the following laws and properties of
logarithms:

loga mn = loga m + loga n
loga m/n = loga m – loga n
loga m^n = nloga m

(mod-23.obj-2)

4.1.1    Students will be expected to
evaluate logarithmic expressions
using calculators and computers.

4.2    Students will be expected to solve
and verify logarithmic equations.

(mod-24.obj-1 & 2)

4.2.1    Students will be expected to solve
and verify logarithmic equations using
calculators or computers.

5.    Students will be expected to demonstrate                       Rinehart and Winston
an understanding that a logarithm with a                              137-139, 142-144
base of 10 is a common logarithm.
                                                                                                        Nelson
(mod-24.obj-2)                                                                       130-131, 133-137

5.1    Students will be expected to solve                                     Addison Wesley
logarithmic equations and evaluate                                         279-281, 309-319
logarithmic expressions using common
logarithms.

6.    Students will be expected to
demonstrate an understanding that
many phenomena exhibit characteristics
that can be described using logarithmic
functions.

(mod-24.obj-3)

6.1    Students will be expected to recognize
logarithmic functions that describe situations
that have logarithmic characteristics.

6.1.1    Students will be expected to solve
problems that exhibit logarithmic properties
by developing and solving logarithmic equations.

Permutations and Combinations

    1.    Students will be expected to demonstrate                      Rinehart and Winston
an understanding of the Fundamental                                      349-357, 363-371
Counting Principle.
                                                                                                         Nelson
(mod-71.obj-1)                                                                        402-405, 407-410,
                                                                                                    418-432
1.1    Students will be expected to calculate
 the total number of ways that a multiple                                  Addison Wesley
of tasks can be conducted if each task                                             500-512
can be performed in a multiple of ways.

1.1.1    Students will be expected to solve
problems that involve the use of the
fundamental counting principle.

2.    Students will be expected to
demonstrate an understanding that a
permutation is an arrangement in
which the order is important.

(mod-71.obj-2)

2.1    Students will be expected to
calculate the number of permutations
there are of n things taken r at a time
by applying the following formula:
nPr  = n!/(n – r)!

(mod-71.obj-3)

2.1.1    Students will be expected to
calculate the nPr using calculators
and computers.

(mod-71.obj-3)

2.1.2    Students will be expected to
solve problems involving linear
permutations, permutations with
repetitions, circular and ring
permutations.

(mod-71.obj-4)
            2.1.3    Students will be expected to
solve probability questions that involve
the use of permutations.

(mod-71.obj-5)

3.    Students will be expected to demonstrate                     Rinehart Winston
an understanding that a combination is an                          358-364, 368-372,
arrangement in which the order is not
important.                                                                                   Nelson
                                                                                               411-415, 417-425
3.1    Students will be expected to calculate
the number of combinations there are of                                Addison Wesley
n things taken r at a time by applying                                          513-519
the following formula:    nCr = n!/r!(n – r)!

(mod-72.obj-1)

3.1.1    Students will be expected to
calculate nCr using a calculator or
computer.

(mod-72.obj-1)

3.1.2    Students will be expected to solve
problems including probability problems
that involve the use of combinations.

(mod-72.obj-2)

4.    Students will be expected to demonstrate
an understanding that the numerical coefficients
of the terms in a binomial expansion can be
determined using the Binomial Theorem.

(mod-72.obj-3)

4.1    Students will be expected to expand
binomials of the form (x + a)n, n ΠW using
the Binomial Theorem.

4.2    Students will be expected to relate
the numerical coefficients in a binomial
expansion to the terms of Pascal's
Triangle and vice versa.

Sequences and Series

1.    Students will be expected to demonstrate                        Rinehart and Winston
an understanding that a sequence is a set of                          418-420, 432-433
quantities determined by a rule (function)
whose domain is the natural numbers and                                      Nelson
whose range is the terms of the sequence.                             349-353, 364-366

(mod-31.obj-1 & 2)                                                                   Addison Wesley
(mod-32.obj-1 & 3)                                                                         321-328

1.1    Students will be expected to recognize
finite and infinite sequences.

(mod-37.obj-1)

1.2    Students will be expected to write the
terms of a sequence given the function that
defines it.

(mod-31.obj-3)
(mod-32.obj-3 & 4)

1.3    Students will be expected to write the
terms of a sequence given its recursive
definition.

(mod-31.obj-4 & 5)

1.4    Students will be expected to determine
the functions that describes simple sequences.

(mod-32.obj-5)

2.    Students will be expected to demonstrate
an understanding that a series is the sum
of the terms of a sequence.

(mod-34.obj-1 to 3)

2.1    Students will be expected to expand
a series that is given in sigma notation.

3.    Students will be expected to demonstrate
an understanding that a series is the sum of
the terms of a sequence.

(mod-35.obj-2)

3.1    Students will be expected to apply the
general term formula of arithmetic sequences,
tn = a + (n – 1)d

3.1.1    Students will be expected to solve
problems involving the use and application
of the general term formula for arithmetic
sequences.

(mod-34.obj-4)
(mod-35.obj-1)
(mod-37.obj-2 & 3)

3.2    Students will be expected to apply the
sum formula of arithmetic series,
Sn = (n/2)(a + tn); Sn = (n/2)[2a + (n – 1)d].

(mod-35.obj-3)

3.2.1    Students will be expected to solve
problems involving the use and application
of the sum formula for arithmetic series.

(mod-35.obj-4)

3.2.2    Students will be expected to use
technology where applicable.

4.    Students will be expected to demonstrate
an understanding that geometric sequences
are such that each term is equal to the
product of the preceding term and a constant
and that a geometric series is the indicated
sum of the terms of a geometric sequence.

(mod-33.obj-1 to 3)

4.1    Students will be expected to apply the
general term formula of geometric sequences,
tn = ar^(n–1)

(mod-33.obj-4)

4.1.1    Students will be expected to solve
problems involving the use and application
of the general term formula for geometric
sequences.

(mod-33.obj-5 & 6)

4.2    Students will be expected to apply the
sum formula of geometric series,
Sn = (a(r^n – 1))/(r – 1), r ≠ 1'
Sn = (rtn – a)/(r – 1), r≠ 1.

(mod-36.obj-1 to 3)
(mod-37.obj-4 & 5)

4.2.1    Students will be expected to solve
problems involving the use and application
of the sum formula for geometric series.

(mod-36.obj-4 & 5)
(mod-37.obj-6)

4.2.2    Students will be expected to use
technology where applicable.