MATH Connections: A Secondary Core Curriculum
Year 3:
Algebra/number/function: Students will know the concept of function and
its representation by sets
of ordered pairs , graphs, input/output tables, and symbols. They will recognize
the terms dependent
and independent variable and the usual functional notation and language, such as
"y is a function of x."
They will be able to apply the vertical line test to a graph in order to
determine whether or not the graph
represents functions. They will know the terms domain and range and be able to
find both implicit
domains of functions and domains determined by context (e.g the variable length
won't be negative).
They will be able to do this work with polynomials, especially quadratics, and
functions involving
absolute value in real world settings. They will have seen both the general form
(y=ax2+bx+c) and
"vertex form" (y=a(x-h)2+k) of quadratic functions and know the term
parabola as a description of the
shape of the graph of a quadratic function. They will know the formula for the
vertex of a parabola .
They will know the concept of functional composition and the notation for it.
They will be able to
compose several functions and consider the question as to whether or not a
composition exits.
Conversely, they will be able to separate many functions into compositional
components. They will
know the term one-to-one function and be able to apply the horizontal line test
to the graph of a
function in order to determine whether or not it is one-to-one. They will
utilize some compositions to
shift ("transform") the graph of some functions and/or study the properties of
functions. For example, if
f(x)= f(-x) for all x, then the graph is symmetric around the y-axis. Similarly,
if g(x)=x+b, and f is a
polynomial, then students will know that the graph of g·f is a vertical shift of
the graph of f. Students will
know the term inverse function and have strategies to determine the symbolic
representation of the
inverse of some simple functions when the symbolic representation of the initial
function is given. They
will know some properties of inverse functions such as the domain of the inverse
function is the range of
the initial function and the range of the inverse function is the domain of the
initial function. They will also
know properties such as f(f -1(x))= x and f -1(f(x))=x.
They will understand and be able to determine the
graph of an inverse function from the initial function by reflection of the
graph over the line y=x, when the
inverse function exists. Similarly, they will be able to create the table of an
inverse function by
interchanging the values of x, or input, and y, or output, when the inverse
exists. They will know that
some values of a function are more significant than others relative to
describing the behavior of the
function. For instance, they will know that maxima and minima are important
descriptors, as are "zeros"
of a function. They will be able to connect finding the zeros of a function to
solving an equation. They
will have several strategies, including guess and check, for finding zeros of
some functions. Moreover,
they will be able to use the quadratic equation to find the zeros of a quadratic
function. They will
understand the term discriminant and understand the relation of the discriminant
to the number of
(distinct) zeros of a function and roots of the corresponding quadratic
equation. They will also be able
to use factoring to find the roots of an equation in simple cases. Finally, they
will be able to use the
technology of graphing calculators to find approximations of zeros of a
function. They will also realize
the limitations of such technology. For example, to use the trace feature of a
graphing calculator, it is
necessary to have an appropriate viewing window, first. Students will recognize
an exponential function
symbolically as a " function which has a variable as an exponent." They will be
able to recognize and
identify the basic shape of the graph of an exponential growth function and
differentiate it from the graph
of an exponential decay function. They will be able to determine symbolic
representations (in base 2) of
exponential situations with respect to doubling time (in growth situations) and
halving time (in terms of
decay situations). They will understand and have developed the basic laws of
integer exponents
(including zero) and extended the laws to all real exponents. They will
understand the meaning of
rational exponents in terms of roots and powers. They will recognize the terms
base and power as
applied to an exponential expression. They will explore many applications of
exponential models
including half-life situations and compound interest. They will develop an
intuitive sense of continuous
versus discrete situations; where continuous, implies, for example, that a
process goes on " virtually"
without interruption (e.g. bacteria are virtually always growing). They will
understand logarithms with
base b>0 as the inverse of the exponential function y=bx and, hence be able to
determine its graph, etc.
They will see the value of logarithms as functional models in many situations.
They will be able to use
the concept of logarithms to solve exponential equations and analyze data. For
example, they will be
able to apply a linear fit to a log plot (a plot of (x, log y)) of data points
as a method for determining a
functional model which is an exponential function. They will be able to apply a
linear fit to a log-log plot
( a plot of (log x, log y)) of data points in order to determine a functional
model which is a power
function. In the latter situation, they will see that the slope of the " fit"
line is the exponent of x in the
power function model. (So, they will see at least one purpose for an irrational
exponent.) They will
know the number e intuitively as the " limiting" value of the expression
(1+1/n)1/n as n gets large. They
will know the terms and symbols for common log, symbolized as log, and natural
log, symbolized as ln.
They will have developed the basic properties of logarithms and see their
benefit in many situations.
They will be able to determine estimates for some logarithms and use graphing
calculators to find
approximate values of other logarithms. They will be introduced to the intuitive
concept of a series
(Maclaurin expansion) and the limit of a series, and see how polynomials can be
used to approximate
the trigonometric and exponential functions, thus providing a link between these
three families of
functions.
Geometry: Students will have had a brief historical introduction to
axiomatic geometry. They will
have seen the Common Notions and Postulates as well as proofs of several
Propositions (Theorems)
from the Elements of Euclid. They will have supplied reasons in the steps of
many of Euclid's
Propositions. They will have proved several Propositions themselves. Their
historical tour will have
included a discussion of Postulate 5 (the "Parallel Postulate") as well as some
of Saccheri's and
Riemann's work in non-euclidean geometry. They will have seen how Postulate 5
connects with basic
results concerning parallel lines, parallelograms, the sum of angle measures in
a triangle, and the
existence of a square. They will have had some experience proving statements in
non-euclidean
geometry; especially using Saccheri quadrilaterals. Some of these proofs depend
on diagrams. They
will be able to compare the three geometries: Lobachevskian, Riemannian, and
Euclidean in terms of
axioms about parallel lines and the sums of angle measures in a triangle.
Trigonometry: Students will recognize the graph of a periodic function.
They will be able to
determine the period of a periodic function from its graph and from a table.
They will know that if a is
the period of a function, then f(x+a) = f(x) for all x in the domain of f. They
will see examples of periodic
functions in many contexts. They will work with both degree and radian measure
of angles and be able
to convert from one angle measure to the other. Students will understand the
definition of sine, cosine
and tangent functions as they are related to the unit circle in the coordinate
plane. They will be able to
recognize the graphs of the sine, cosine and tangent functions they will know
the domains (both in terms
of angles and radians) and ranges of these three trigonometric functions, what
the maxima and minima
are with respect to the sine and cosine functions, and where the asymptotes are
with respect to the
tangent function. They will have seen the transformations cf(x); f(cx); f(x)+b;
and f(x+b) and how
combinations of these transformations affect the graphs of each of the three
trigonometric functions
mentioned above. In particular, they will understand how these transformations
affect amplitude and
period of the sine and cosine functions. They will be able to determine the
symbolic representation for a
"sine-type" function given its graph. They will be able to fit a sine function
to a periodic data set. They
will be introduced to the idea that it is possible to find combinations of
trigonometric functions which will
make a reasonable model for any periodic situation. (Students will see that this
means sometimes
modeling a discrete situation by a continuous function.)They will be able to use
the inverse sine, cosine
and tangent functions. They will recognize the graphs of these functions. They
will be able to determine
some values of an inverse trigonometric functions based on the definition of the
parent trigonometric
function and its relation to the unit circle. They will be able to use
technology to determine other values.
Probability and Statistics: Students will enhance their ability to use
counting techniques such as the
Fundamental Counting Principle (sometimes known as the Multiplication Principle )
and the technique of
partitioning a set into disjoint subsets, counting the objects in each subset,
and adding the totals in order
to determine the number of objects in the whole set. They will have seen the
development of basic
counting formulas such as permutation and combination formulas and the general
formula for arranging
N objects in a set which has the property that it can be partitioned into sets
of indistinguishable objects
of several types (like the letters of the word Mississippi). They will be able
to apply these concepts and
techniques both separately and together in counting contexts as well as in the
determination of
theoretical probabilities involving equally likely outcomes. They will know the
concept of conditional
probability both in terms of a restricted sample space in the special case of
equally likely outcomes and
as defined by the formula: P(A|B)=P(A·B)/P(B). They will understand the concept
of independent
events as two events, A and B, which satisfy the formula P(A·B)=P(A)P(B) and be
able to determine
whether or not events are independent in many situations. They will know the
definition of expected
value and be able to compute it in many contexts. They will know the definition
of binomial probabilities
in terms of measuring the probability of success in a sequence of independent
experiments. They will
have seen the development of binomial formulas for the probability of
k successes in N trials. They will know that the expected value in binomial
experiments is equal to the
given probability of success. They will be able to construct bar graphs of
binomial probability
distributions and be able to relate certain probabilities to the areas of the
rectangles in such bar graphs.
They will have seen a formula for the standard deviation in N trails of a
binomial experiment with
probability of success p. They will know that as N increases, the binomial
distribution can be
approximated by a normal curve whose line of symmetry is the expected value of
the binomial situation.
They will be able to utilize the (approximate) area under a normal within one,
two, and three standard
deviations away from the expected value to find probabilities in situations
where outcomes are assumed
to be normally distributed. They also will be able to utilize the area under the
normal curve and its
standard deviation to approximate binomial probabilities. They will have seen an
analytic formula for the
normal curve and they will know that the standard deviation for a normal curve
is a measure of how
"bunched up" the curve is around its expected value. They will know what a
random sample of a
population is and have discussed bias. They will use the normal curve to
evaluate results from a sample
and know what a 95% confidence interval. They will know how the term "margin of
error" is related to
a 95% confidence interval and be able to compute 95% confidence interval is
several situations.
Discrete mathematics: Students will know and be able to use several
strategies to solve a variety of
optimization problems. In particular, they will know, be able to explain, and be
able to use the Greedy
Algorithm and Dynamic Programming to solve (weighted) "block diagram" problems.
They will have
strategies to solve two-variable linear programming problems (including mixture
and transportation
problems, etc.). These strategies include graphical methods and the use of
algebra involving
"dictionaries" (which is essentially the simplex method involving slack, basic,
and non-basic variables).
They will understand the value of technology in solving linear programming
problems with many
variables and will utilize technology (spreadsheets) to solve some linear
programming problems. They
will have some experience solving linear programming problems involving more
than two variables by
hand. They will understand the concept of a graph with vertices and edges. They
will know the
meaning of connected graphs, weighted graphs, trees, spanning trees and cycles.
They will understand
and be able to use Kruskal's Algorithm for finding minimal spanning trees in a
connected weighted
graph. They will have seen two derivations of the basic arithmetic-geometric
mean inequality for two
positive numbers (i.e. the arithmetic mean is less than or equal to the
geometric mean). They will use
this inequality in Geometric Programming to solve several optimization problems,
many of which are
classic calculus problems. Students will be introduced to concepts of infinity.
They will explore some
paradoxes involving the infinite. They will be introduced to several infinite
processes which tend toward
a "limit." Such processes include: unending decimal expansions, general
(convergent) geometric series,
Archimedes •" Method of Exhaustion," and Riemann sums for area under a
curve (integration). They will
intuitively understand limit in a "Bolzano-Weierstrass" sense. (This term is not
used.) That is, students
will " close down" on the limit as a number between bounds which get closer and
closer together. (The
concepts of Maclurin expansions and the limit of (1+1/n)1/n as n gets
large mentioned above could fit
into this context.) Students will be able to work with formulas and algorithms
for finding the limit of a
convergent geometric series and the fractional representation of a repeating
decimal. They will also be
able to use this infinite " closing down" process to position (infinite)
decimals on the number line. They
will know that irrational numbers do not have repeating decimal expansions. Yet,
students will have a
" finite" strategy for positioning the square root of a positive integer on the
number line. They will also
encounter situations with infinity where there is no limit, such as in the case
of the (unending) natural
numbers. Students will see the categorization of numbers in terms of natural
numbers (positive integers),
integers, rational numbers and irrational numbers and investigate how these
categories relate to each
other. They will recognize the term dense set and know, for example, that the
set of rational numbers is
dense, but the set of integers is not. They will know that both the set of
rational numbers and the set of
irrational numbers have " gaps" in them but, taken together to form the real
numbers, there are not gaps.
They will be introduces to the cardinal concept (term not used) which states
that two sets are equivalent
if there is a one-to-one correspondence between them. They will see that the
rational numbers and the
positive integers are equivalent, but that the real numbers and the positive
integers are not. Thus, they
will be introduced to different " sizes" of infinity.
Logic and reasoning: Students will understand what an axiomatic system
is. They will know the role
of undefined terms and the assumed truth values of the axioms. They will know
the difference between
circular and non-circular definitions and have experience determining "
characteristic properties" which
define an object. They will have experienced the danger of hidden assumptions in
situations. They will
understand what a consistent axiomatic system is. They will know the method of
determining
consistency by finding an "instance " (realization) of the system. They will
understand that one can prove
inconsistency by providing a logical contradiction within the system. They will
determine whether or not
several axiomatic systems are consistent. They will know that an axiomatic
system may have many
" instances." They will know what it means for a set of axioms to be independent
or dependent. They
will have strategies for determining dependence or independence of axioms. They
will know the
difference between sentences and " (logical) statements", which can have truth
values attached, and they
will know what theorems are. They will understand the role the Law of
Contradiction and the Law
of the Excluded Middle play in (the usual) logic. They will have considerable
experience rephrasing
complex statements in terms of the connectives and and or, conditionals,
biconditionals and
negations. They will have experience in determining the truth value of a complex
statement from the
truth value of its component parts. They will understand the difference between
universal statements and
existential statements. They will be able to form the negations of quantified
statements and apply
DeMorgan's Laws to quantified statements involving the connectives and and or.
They will be able to
formulate the converse and contrapositive of conditional statements. They will
know what " vacuously
true " means. They will recognize the terms hypothesis and conclusion. They will
understand the
difference between inductive and deductive reasoning. They will have some
experience proving simple
theorems using direct proof methods and using indirect proof methods (i.e. proof
by contradiction).
They will be introduced to the term abstract and the value of an abstract model
as well as understand
the role of simplifying assumptions in mathematical models . They will work with
the axioms for a group
and see many realizations of this axiom system including the integers under
addition, the positive rational
numbers under usual multiplication and modular arithmetic. They will know what a
(binary) operation is.
They may use technology (spreadsheets) to create " multiplication tables" for
finite groups. They will
have experience of formulating simple conjectures and proving them or and other
results in group
theory. Students will apply the method of proof by mathematical induction to
obtain several formulas
concerning the infinite set of positive integers.
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