Contemporary Mathematics in Context: A Unified Approach

Levels: 9-12

Developers: Arthur Coxford, James Fey, Christian Hirsch, Harold Schoen, Gail Burrill, Eric
Hart, Brian Keller, Ann Watkins, Mary Jo Messenger, and Beth Ritsema

Publisher: Glencoe/McGraw-Hill

Review Materials: Courses 1, 2, and 3 were reviewed in published form. Course 4 is now in
published form, but was not reviewed. The Teacher’s Guide (which includes student facing
pages), Teaching Resources, and Assessment Resources for Courses 1-3 were reviewed. Also
available are: Implementing the Core-Plus Mathematics Curriculum, Scope and Sequence for
Courses 1-4, Calculator Software and Guide, Reference and Practice Books for Courses 1-3, and
Assessment and Maintenance Worksheet Builder CDs.

Format/Description: This is a complete four-year secondary school mathematics curriculum
for all students consisting of a three-year core program plus a flexible fourth-year course
continuing the prepa ration of students for college mathematics. The curriculum is centered
around four strands: algebra/functions, geometry/trigonometry, statistics/probability, and discrete
mathematics. Mathematical modeling is a central focus throughout the curriculum. Each of
Courses 1-3 contains six units which focus on different mathematical topics together with a
seventh Capstone unit. Course 4 consists of ten units that formalize and extend the core program
with a focus on the mathematics for students who plan to continue their study of mathematics in
college. The units integrate mathematical topics from different mathematical disciplines (see
Content Overview below). In addition, the text is written in a way that often connects the
current mathematics and contextual settings with mathematics and contents seen previously in
the curriculum . Units take from four to six weeks to complete. Each unit consists of three to six
lessons. Each lesson is organized around several multi-day investigations. Each lesson consist
of five parts: the Launch- presentation of a problem situation with several general questions for
the entire class to think about; the Exploration- students investigate focused problems in small
collaborative groups; the Checkpoint- groups share and summarize findings with the full class;
and On Your Own- students work individually on problems. The fifth part is the MORE
(Modeling, Organizing, Reflecting, Extending) out-of-class activities intended to be done
individually. In this section in particular, tasks are differentiated in difficulty in order to
accommodate variation in student ability and interest. The Capstone is a thematic two-week
project-oriented unit that explores a context involving reference to many of the mathematical
topics in that year’s curriculum.

The Teacher’s Guide includes instructional notes for each page of student materials and
suggestions for implementation of the materials in the classroom including specifics on the role
of the teacher, unit overviews, learning objectives, suggested timelines, lists of materials needed,
suggested assignments, solutions to student activities and problems. Teacher Resources include
blackline masters and transparencies. Quizzes, end-of-unit assessments, take-home assessments,
projects, and mid- term and final exam tasks are included in the Assessment Resources. For
additional information concerning the format and other valuable information, see: Implementing
the Core-Plus Mathematics Curriculum, and Contemporary Mathematics in Context Sampler.

Pedagogy: The curriculum delivery focus is on student learning. The student materials are
written in a way that promotes active student participation. Students spend most of their time in
class working collaboratively in small groups or individually with the teacher as facilitator or
“coach.” However, as indicated above, some time is devoted to full class discussion with teacher
as director (Launch) or moderator (Share and Summary). Students are consistently asked to
explain their reasoning in each lesson. The core mathematical topics are accessible to all
students. Differences in interest and performance are accommodated by the depth and level of
abstraction to which the topics could be pursued and by the choices of homework tasks and
projects. The booklet Implementing the Core-Plus Mathematics Curriculum contains
descriptions and suggestions with regard to the collaborative learning and classroom discourse.
Included in the Teachers’ Guide is information on and suggestions for the personal Math
Toolkits that students construct as they progress through each course. These toolkits provide
students with a summary of the important mathematical concepts and methods that they have
learned. Each student retains his/her toolkit as a reference in the next course.

Technology: Graphing calculators are an essential component of this curriculum and are used
heavily in most units. In particular, the TI-82, TI-83, and TI-92 series are supported in the
student and teacher materials. Downloadable programs for the TI-82-83-92 are available, as is
the booklet, Calculator Software Guide. Students use the table, graph, programming, list,
statistics, plot, (recursive) sequence, and matrix capabilities of these calculators. CPMP has
developed spreadsheet software for the TI-92. Calculator or computer spreadsheet software is
necessary for Unit 10 of Course 4.

Assessment: The assessment program is explained in the booklet Implementing the Core-Plus
Mathematics Curriculum. The program includes curriculum-embedded informal assessment by
the teacher when students are working collaboratively or individually. There are several points
in each investigation where the teacher can more formally assess students either orally or in
writing. In particular, the Checkpoints, On Your Own, and MORE tasks provide such
opportunities. Student journals and portfolios are also suggested. Supplementary materials
incorporating end-of- lesson quizzes, end-of-unit exams (in-class and take- home), and projects
are contained in the Assessment Resources. There are Forms A and B for all quizzes and tests.
Scoring rubrics and suggestions for grading are also included. Assessments are available on CD
for Courses 1-3 in order to allow teachers to customize assessments for their classes.

Content Overview: The first four NCTM Standards: Mathematics as Problem Solving,
Mathematics as Communication, Mathematics as Reasoning, and Mathematical Connections are
addressed heavily in every unit. Students are consistently asked thought-provoking questions
and required to explain their reasoning. Many of the questions are open-ended. The
mathematical topics are strongly integrated. Connections among the units and the strands
mentioned above are made by repeating contexts, such as bungee jumping or population growth;
revisiting mathematical topics from a different point of view, such as symmetry or matrices; by
habits of mind, such as visual or recursive thinking; and by returning to the consistent themes of
data, multi-representation, shape, and change. The development of mathematical content is done
largely through mathematical modeling of real-world contexts and reflection on underlying
mathematical structures. The contexts have been carefully chosen to appeal to students without
diminishing the importance and usefulness of the mathematics. Students become acquainted
with mathematical definitions and terminology in the course of solving problems.

As stated above, the curriculum is centered around four “strands.” As the developers express it,
algebra and functions “is de signed to develop student ability to recognize, represent, and solve
problems involving relations among quantitative variables.” Geometry and trigonometry is
aimed at developing “visual thinking and students’ ability to construct, reason with, interpret,
and apply mathematical models of patterns in visual and physical contexts.” Statistics and
probability develops “student ability to analyze data intelligently, to recognize and measure,
variation, and to understand the patterns that underlie probability situations.” Discrete
mathematics “develops student ability to model and solve problems involving sequential
change, enumeration, decision making in finite settings, and relationships among a finite number
of elements.” These descriptions are taken from the booklet Implementing the Core-Plus
Mathematics Curriculum. This booklet includes other details concerning these strands. Another
source of information regarding content in the curriculum is the Scope and Sequence. Our
description of what students completing this curriculum will know and be able to do is given in
terms of these strands. Some topics, for example, NOW-NEXT relations, could be listed in more
that one strand; in algebra and functions and also in discrete mathematics, for instance. The
decision has been made to list most topics only once within each course. It should be
emphasized that these strands are interwoven throughout units and throughout the courses.
Moreover, the present description does not necessarily describe topics in the order in which they
are encountered in the curriculum. For the most part, topics which are briefly mentioned but not
emphasized to any degree (e.g. “indegree” of a vertex in a digraph) are omitted. In some places,
we simply state that students are “exposed” to such topics (e.g. the concept of a permutation in
Course 1). In contrast, in what follows, when it is said that a student will be able to do a task
or has a strategy to solve a problem, it is implied that the student will also be able to
explain what they are doing and why they are doing it. One additional facet of the curriculum
appertains to the use of technology; in particular, graphing calculators. In each course, emphasis
is placed on students’ ability to recognize and employ the capabilities of the graphing calculator
in relation to the mathematical topics and contexts presented. The technology is utilized in ways
which facilitate the student’s insight into the mathematical concepts and which provide
additional strategies for solving problems. The use of technology does not replace understanding
with dependence upon senselessly applying a calculator algorithm.

Course 1:

Algebra and functions : Students will be able to identify key variables in a variety of situations.
Students will be able to identify patterns of change and model them using NOW-NEXT
recurrence relations, tables, graphs, verbal descriptions and symbolic function rules relating the
variables. Both linear and nonlinear relationships are encountered. However, linear and
exponential relationships are featured in Course 1. The linear situations students will be able to
model include many real- world contexts as well as using lines to describe the shape of data on a
scatterplot. In addition to using the linear regression capability of the calculators to fit lines to
linear data patterns, students will be able to write an equation of a line given the line’s slope and
y- intercept or given two points on the line. They will be able to interpret the meaning of the
coefficients of a linear equation in real world terms given the context of the problem. For
instance, they will be able to relate the concept of slope to the geometric inclination of the graph
of a straight line and they will be able to interpret the same coefficient as the rate of change of
one variable with respect to the other in a different context. They will be able to recognize a
linear relationship represented by a table, by a graph, and by symbolic forms, particularly y =
ax + b, and construct one representation from another. They will be able to solve linear
equations in one variable using standard balancing or “undoing” algorithms. They will be able to
use the commutative property of addition and the distributive property of multiplication over
addition to rewrite linear equations in one variable in mathematically equivalent forms. They
will be able to write linear inequalities in one variable from contexts and be able to solve linear
inequalities in one variable by inspection, by using tables, and by using some graphical methods.
They will also be able to determine equivalence of linear equations with two variables using
tables and graphs. Students will be able to recognize exponential growth (or decay) by
examining tables and graphs of situations where something is growing (shrinking) at an ever-increasing
(decreasing) rate. They will understand the equation y=a(b^x), where 1< b and also
where 0<b<1. They will use graphs and tables to solve exponential equations. They will be able
to use exponential properties to model compound growth (e.g. compound interest). They will be
able to model many real- world situations using exponential growth and decay models, and be
able to use technology to fit an exponential curve to data on a scatterplot. They will be able to
compare exponential models for different positive values of a and b and make comparisons
between linear and exponential models.

Geometry and trigonometry: Students will enhance their planar and spacial visualization
skills. They will be able to define, recognize, and classify shapes such as squares, rectangles,
parallelograms, rhombi, trapezoids, kites, regular polygons, prisms, pyramids, (and possibly
tetrahedra, octahedrons, dodecahedrons, and icosahedrons), cones, and cylinders. They will
understand and be able to use mathematical terms such as face, edge, base, solid, shell, skeleton,
and rigid (and possibly oblique and adjacent face) to describe three-dimensional objects. They
will be able to create face-views and isometric and perspective drawings of objects made of
cubes. They will understand how nets may be used to describe three-dimensional objects. They
will be able to recognize reflection symmetry in both two and three dimensions. Students will be
able to describe the symmetry of plane tessellations, polygons, and frieze (strip) patterns using
the language of isometric transformations (translations, rotations, reflections, and glide
reflections). They will be able to use polygons and three-dimensional shapes to model some
real-world situations and reason about some spatial situations. They will enhance their skill of
measuring and describing the size of an object through the use of length, perimeter, diagonal
measure (of a rectangle), area, volume, and surface area (of planar, locally flat three-dimensional
objects, and the cylinder.) They will understand area as the number of square units necessary to
fill a plane region. They will understand volume as the number of cubes needed to fill a three-
dimensional region. They will know and be able to use several standard formulas for area (e.g.
triangle, parallelogram, circle) and volume (e.g. cylinder, prism). They will know (have seen at
least suggestions toward a proof of) and be able to use the Pythagorean Theorem. In Course 1
students develop their reasoning abilities by using inductive reasoning, providing counter
examples, and explaining or justifying their responses throughout all strands of mathematics
(formal proof construction is begun in Course 3).

Statistics and probability: Students will enhance their skills in collecting, organizing,
displaying, and analyzing data. They will understand, be able to find, be able to use, and be able
to compare the three measures of central tendency: the mean, median and mode. They will be
able to display and interpret distributions of data using histograms, plots over time, bar graphs,
standard and back-to-back stem and leaf plots, box plots, frequency tables, and scatterplots.
They will be able to choose appropriate representations of data and compare representations.
They will understand and be able to compute several measures of variation including the mean
absolute deviation, percentiles, and the range and interquartile range. They will be able to use
the five number summary (minimum, lower quartile, median, upper quartile, and maximum) to
describe a distribution. They will be able to recognize reflective symmetry and outliers in a
distribution and be able to explain the significance of each. They will know the effect of a linear
transformation (not given in those terms) of data on the mean, median, and mean absolute
deviation of a data set. They will be able to use technology (involving random number
generators), random digit tables , and experimental methods to simulate “real world” and other
probabilistic situations. For example, they will be able to simulate the distribution and expected
value of a geometric random variable, and, hence establish empirical probabilities. (The term
geometric random variable is not used.) They will understand what a trial of a simulation is.
They will understand what equally- likely probabilities are and establish some simple theoretical
probabilities, such as getting a head when flipping a fair coin. They will informally understand
expected value in many instances. They will be able to estimate expected value by computing
averages in simulations. They will have explored properties of random digits and should
informally be able to explain what “random” means. They will understand the Law of Large
Numbers as meaning, for example, that as the number of trials increases in a binomial random
variable experiment, the actual percentage of successes approaches the theoretically expected
percentage of successes. (The term binomial random variable is not used.) They will be exposed
to the concept of a permutation.

Discrete mathematics: Students will be able to create and utilize vertex-edge graphs to model
many real world situations including finding efficient routes (using Euler paths), managing
conflicts (by coloring vertices), and scheduling (using digraphs and PERT charts). Students are
exposed to graphs that contain loops, but most graphs do not contain loops. In this mathematical
context, students will have experienced the three-part mathematic al question: is there an answer
(existence of a solution); if there is an answer, how do you determine an answer (algorithm
formulation); and if there is more than one answer, how do you find the best answer or most
efficient solution method (optimization)? They will have investigated uses and strategies for
coloring the vertices of a graph. They will have investigated criteria for the existence of Euler
circuits and paths and systematic procedures for vertex coloring. They should have discovered
and be able to use Euler’s theorem. They will be able to represent and analyze graphs using
adjacency matrices. They will be able to use at least one strategy (algorithm) for constructing an
Euler path and an Euler circuit and they will be able to Eulerize a connected graph. They will
have at least one strategy for finding critical paths in weighted digraphs. They will be able to use
at least one strategy for scheduling a project (PERT) using the notions of prerequisite tasks, the
EST (earliest start time), EF T (earliest finish time), LST (latest start time), and slack time. They
will have investigated scheduling projects with uncertain task times and been exposed to at least
one algorithm for scheduling such projects. They will understand the degree of a vertex.

Course 2:

Algebra and functions : Students will be able to write systems of linear equations of the form
ax + by = c that model real- life linear relationships between two variables. They will be able to
solve these systems using graphing and linear combinations of equations (which corresponds to
the method of elimination or addition/subtraction method, but is developed by examining the
graph of the sum of non- zero multiples of linear equations). Students will be able to graph
equations of the form ax + by = c by making a table of points and by solving for y (when lines
are not vertical). They will be able to describe patterns of change in power models of direct
variation (in particular, y = ax² and y = ax³) and inverse variation (y = a/x and y = a/x²) by
looking at tables and graphs of those relations and making connections with the symbolic forms.
They will be able to compare these patterns with those in linear and exponential models, as well
as other power models. Students will investigate the properties of quadratic models (y = ax², y =
ax²+c, y = ax² + bx + c, introduced as the sum of power and linear rules). They will be able to
predict the shape of a graph from its symbolic rule. They will be able to write algebraic rules to
model situations involving time/height (e.g. the flight of a springboard diver) and income/profit
in order to answer questions related to the context. Students will compare various quadratic rules
and corresponding tables and graphs. Using a table or graph, they will be able to estimate the
root(s) of a quadratic equation. They will be able to determine the number of roots by looking at
the graph of a quadratic model. They will be able to solve equations of the form
k, ax³ + b = c, and using a table of values, a graph, or by reasoning with the symbolic
form (e.g. ). Students will be able to
use radicals and their properties. For example, they will know that

when a, b > 0.

Similarly, they will know that the expression

is generally false. They will know and be able to use some fractional powers (such as 1/2 and 1/3)
in power models. They will know that, where n is a positive integer. They will know
and be able to use the laws of exponents for integers (and at least the fractional powers they have
studied). They will be able to describe the table and graph patterns of and

Geometry and trigonometry: Using coordinate geometry and programming techniques,
students will investigate some of the mathematics related to geometric shapes which is involved
in computer graphics. They will develop formulas for the distance between two points, midpoint
coordinates, and slope of a line, and then program their calculators to perform these
computations. Using these programs and additional software, they will analyze properties of
geometric shapes to make conjectures and then determine whether they are rectangles, squares,
isosceles triangles, or parallelograms. They will be able to represent transformations
(translations, line reflections, rotations about the origin, glide reflections, and size
transformations centered at the origin) using coordinates, matrices, and geometric descriptions.
They will examine the effects of these transformations and combinations of transformations on
distance, angle measure, parallelism, and area. Students will explore the fundamental theorem of
proportionality (not stated as such) as they investigate relationships between length, perimeter,
surface area, and volume of similar figures. They will recognize that geometric form can be
essential to the functioning of an object (e.g. triangles are rigid, quadrilaterals can pivot at
vertices, circles turn easily at their centers). Using a pantograph, they will be able to create
similar figures with scale factors. They will explore patterns in measures of angles and altitudes
of triangles with a variable-length side.

Students will be able to find the sine, cosine, and tangent of an angle in a right triangle. They
will be able to articulate why these ratios are independent of the lengths of the sides of a right
triangle with a given acute angle. Using trigonometric ratios, they will be able to calculate the
length of a side and the measure of an angle (using the inverse trigonometric function keys on
the calculator) of a right triangle, and solve indirect measurement problems. Beginning with an
analysis of situations involving pulleys and sprockets (rotating circular objects) to determine
linear and angular velocity, students move into an investigation of the graphs of the
trigonometric functions and models of periodic motion. They will be able to connect the
trigonometric ratios of sine and cosine with circular motion as they build a model of a Ferris
wheel and explore the periodic nature of its movement. They will be able to sketch graphs of the
sine and cosine functions that model periodic phenomena. They will be able to graph y=AsinBx
and y=AcosBx using the graphics calculator, and examine the effects of the parameters on the
amplitude and period of a given function. They will be able to compare and contrast the graphs
of the sine and cosine functions. Students will use both radian and degree measure with
trigonometric functions.

Statistics and probability: Students will further explore the association between pairs of
variables through interpretation of scatterplots, scatterplot matrices, and correlation matrices.
They will investigate numerical measures to describe the strength of linear association. In
particular, they will be able to compute and interpret the correlation coefficient for ranked and
unranked data. They will recognize the importance of combining graphical and numerical
analysis, since outliers (influential points) can affect results. They will examine the possible
cause-and-effect relationship between two variables, and be able to give possible reasons for
correlation. They will be able to use a regression line for a set of data to make predictions and
compute the error in prediction. They will become familiar with the method of least squares,
used by the calculator to determine the regression line. Students will explore waiting-time
distributions (also known as geometric distributions) through simulation and experimentation, as
well as theoretically. They will be able to construct frequency distributions and histograms for
waiting-time distributions, and will discover they have the same basic shape. They will be able
to estimate average waiting time using histograms and frequency tables. They will be able to
distinguish between independent trials and those that are not independent. They will be able to
find the probability of two independent events both occurring, using an area model and the
Multiplication Rule (p(AandB)=p(A)· p(B)). The y will be able to determine whether or not two
events are independent, using p(A)=p(A|B). They will be able to explain how the word “expect”
is used in terms of probability. Students will be able to construct and analyze probability
distribution tables and graphs. They will be able to use probability notation. They will be able
to determine a rare event (i.e. 0<p(event)•0.05). They will be able to compute the expected
value for games of chance and insurance (called fair price) and for probability distributions.
They will be able to compute the expected value for a waiting-time distribution (E.V.=1/p).

Discrete mathematics: Students will be able to create a matrix model for organizing and
displaying data in many real world situations (business, archeology, sociology, sports,
ecosystems). They will understand the advantages and disadvantages of displaying data in
matrices. They will be able to analyze matrices to draw conclusions about data. They will be
able to perform matrix operations using addition and subtraction of matrices, row sums, and
scalar multiplication, and interpret results. They will be able to use matrix multiplication,
including positive powers of matrices, to help make decisions in a variety of contexts. They will
study algebraic properties of matrix operations. They will be able to represent systems of linear
equations with matrices. They will be able to write a matrix equation of the form AX=C and
solve by multiplying by the inverse matrix A^-1 (using the x^-1 key on the calculator). They will be
able to use this method to solve systems of linear equations (when the inverse of the coefficient
matrix exists). They will be able to use matrix representations for line reflections, translations,
rotations and "size" transformations (dilations) to write calculator programs that simulate
computer animations. Students will extend their knowledge of vertex-edge graphs to represent
and analyze a variety of real- world situations. Given a network of vertices and weighted edges,
they will be able to apply and analyze algorithms that find a minimal spanning tree. They will be
able to find a shortest path from one vertex to another, or a shortest Hamiltonian circuit
(Traveling Salesperson Problem). They will understand why even with a computer a brute-force
method is limited. Given a table or matrix, they will be able to create a weighted graph and given
a weighted graph they will be able to create a distance matrix.

Course 3:

Algebra and functions : Students will be able to work with relations (both equalities and
inequalities) involving several linked variables; sometimes together with additional parameters.
Many relations involve three variables, but several involve more than three. In building or
analyzing models, they will describe linkage s using symbols, tables, and/or graphs. They will be
able to identify how changes in one variable affect the value of another variable. They will be
able to construct linear relationships such as z = ax + by from many contexts. They will be able
to solve equations using the “properties of equality”( a+b = c implies a = b-c and ab=c implies
a=b/c, provided c≠0). They will have at least one strategy for solving quadratic inequalities such
as 2x+3>x² (e.g. by graphing y=2x+3 and y=x² on the graphing calculator and determining
when the first function is higher). They will be able to employ a strategy for solving linear
inequalities in one variable using the “properties of inequality” (a+b<c implies that a<c-b and
ab<c implies a<c/b if b>0.) They will understand what a system of equations is and at least one
method to solve a two-dimensional system. In particular, they will understand a standard linear
programming model with an objective function, constraints, and feasible points. They will know
that the set of feasible points for a linear programming problem in two variables is a plane
region. In particular, they will know that the set of points that satisfies a linear inequality in two
variables is a half-plane, which can be found by graphing the corresponding linear equality and
experimentally determining the half-plane. In addition, students will know that if the objective
function can be optimized, an optimum occurs at a vertex of the feasible region.

Students will use the field properties for real numbers to justify the equivalence of algebraic
equations. They will use algebraic reasoning to derive the quadratic formula. Students will also
develop greater facility with algebraic operations with polynomials, including adding,
subtracting, multiplying, factoring, and solving equations.

Students will be able to identify functions as variations of basic families of functions. Students
will understand a function as a relation where there is exactly one output value (y-value) for each
given input value (x- value). They will understand the domain and range (output values) of a
function. They will be able to determine both practical (contextual) and theoretical (implicit)
domains and ranges of many functions. They will be able to use function notation, such as f(x) =
an expression, and determine when a graph or a table expressing a relation between two
variables represents a function. They will be able to recognize and use functional relationships
in many contexts. They will be able to categorize many functions. Students will understand that
the graph of the function y=- f(x) is a reflection across the x-axis of the graph of f(x), that the
graph of y=f(x)+c is a vertical translation of the graph of y=f(x), that the graph of y=cf(x) is a
vertical stretch of the graph of y=f(x), when c>0, and that the graph of y = f(x - c) is a horizontal
translation of the graph of f(x). They understand the absolute value function and that the sine
and cosine functions are useful in describing periodic motion. They will be able to describe the
effect of the parameter c in y=sin(cx) and y=cos(cx). They will know what polynomials are and
what the degree of a polynomial is. Students will understand the basic operations of adding,
subtracting, multiplying and dividing functions. They will be able to write equivalent symbolic
expressions for functions and understand the value of “unsimplified” expressions as descriptive
of a particular context. They will deepen their skill with the basic properties of addition
(subtraction), and multiplication (division) in terms of associativity, commutativity, and the
distribution of multiplication over addition in the context of operations with functions. They will
be able to factor simple quadratic function expressions without using the quadratic formula.
Students should be able to prove and use the quadratic formula to find the zeros of a quadratic,
understand the relationship between the zeros of a quadratic and its factors, and understand that
the expression

is the distance from the zeros of a quadratic to the axis of symmetry, provided a>0 and

is real. They will have at least two strategies for finding the zeros of a quadratic function using
the graphing calculator. Students should realize the importance of x- intercepts, lines of
symmetry, and maximum or minimum values in describing or determining the graph of a
quadratic function. They should be able to determine the equation of the line of symmetry of a
quadratic function.

Geometry and trigonometry: Students will enhance their skills using both deductive and
inductive reasoning. They will understand the difference between inductive and deductive
reasoning as well as the roles of conjecture, assumptions, mathematical proof, conclusions, and
counterexamples. They will be able to make conjectures and apply simple deductive reasoning
to conditional statements concerning angles formed by intersecting lines and angles formed by
transversals of parallel lines. They will know the terms conditional and implication and how to
use reasoning chains involving modus ponens. They will know, be able to reason with, and be
able to use theorems concerning parallel lines and angle sums of polygons. They will know, be
able to reason with, and be able to use the concepts of vertical and supplementary angles.
Students will know and be able to use the four standard theorems to prove triangles congruent
(SSS, SAS, ASA, and AAS). They will know and be able to use the three standard similarity
theorems to prove triangles are similar (SSS, SAS, and AA). These theorems are derived as
consequences of the Law of Cosine and Law of Sines which are also established. They will
know and be able to use necessary and sufficient conditions to show a quadrilateral is a
parallelogram. They will know and be able to use properties of special quadrilaterals such as
rhombi and kites. They will know and be able to use the midpoint connector theorem. They will
be able to provide a valid deductive argument using previously proved theorems for a given
conjecture.

Students will be able to find the period and amplitude of sine and cosine functions given a graph
or an equation, and move between equations and graphs of sine and cosine functions. They will
also be able to write equations of trigonometric functions that model problem situations. Students
will enhance their understanding of the Pythagorean theorem. For example, they will be able to
prove cases of the Pythagorean identity sin²(x)+cos²(x)=1. As mentioned, students will know
and be able to use the Law of Sine s and the Law of Cosines. Students will be able to use these
laws to help find the area of polygonal regions.

Statistics and probability: Students will enhance their ability to collect, organize, display, and
analyze data. Students will understand the statistical concepts of population and sample.
Further, they will know and be able to use the concepts of a random sample. They will
understand the difference between a sample survey and a census. They will know what bias is
and they will know several ways that bias may occur. They will be able to draw conclusions
about populations from samples. They will be able to create some types of sampling
distributions. They will understand the influence sample size has on statistical conclusions.
They will understand, be able to use, and be able to construct confidence interval for proportions
using 90% box plots. They will understand the term margin of error as it relates to confidence
intervals and sample size.

Students will deepen their understanding of statistical variation. They will know, be able to
compute, be able to estimate from a histogram, and be able to use the concept of standard
deviation for a data set. (They will have used a computational formula for standard deviation
and they will be able to compute the standard deviation using the calculator.) They will know
and be able to use the effect a linear transformation of the data has on the standard deviation.
They will know that the standard deviation is sensitive to extreme values. They will know the
shape of the normal distribution and will be able to use the facts that approximately 68% of the
data lies within one standard deviation of the mean, approximately 95% of the data lies within
two standard deviations of the mean, and approximately 99.7% of the data lies within three
standard deviations of the mean, if the distribution is normal. They will know the relationship
between the median and the mean of a normal distribution. Students will compute z-scores and
use a table of values to determine the proportion of a distribution over specific intervals.
Students will be able to investigate data over time (measurements of the same phenomenon at
different points of time). They will use control charts to determine when processes are out-of control.
Students will analyze tests to measure when a process is out-of control. They will
understand and be able to use the addition rule for the probability of mutually exclusive events.
They will understand factorial notation.

Discrete mathematics: Students will be able to compare and contrast methods of determining
public opinion by voting, surveys, and censuses. With voting, they will know and be able to use
the concepts of majority, plurality, points-for-preference (Borda), pairwise comparison
(Condorcet), sequential-elimination, and approval voting methods. They will understand
insincere (strategic) voting. They will know Arrow’s Theorem which states that there is no
voting method that will always yield a “fair” decision when there are more than two candidates
(choices). (The term “fair” is defined.) Students will be able to model situations which involve
change over time using sequences. They will know what a discrete dynamical system is. They
will understand and be able to use standard sub script notation for sequences. They will
understand and be able to work with sequences that are defined recursively and sequences given
in closed- forms. They will be able to recognize arithmetic and geometric sequences and series in
terms of recursive relations, closed forms, and graphical representations. They will understand
the connection between arithmetic sequences and linear models as well as geometric sequences
and exponential models. Also, they will know and be able to work with sequences of the form
They will know and be able to use finite differences to find closed form formulas
for some sequences and recurrence relations whose closed forms are polynomial expressions.
They will be able to describe the long-term behavior of several sequences. Students will be able
to iterate functions both numerically using a graphing calculator and graphically (using the line
y=x). They will understand what a fixed point is. They will have several strategies for finding
fixed points. They will know what attracting fixed points, repelling fixed points, and a cycle are
and be able to determine these behaviors experimentally, and for iterated linear functions, using
analytic methods.

Course 4:

The Core-Plus Mathematics Project has completed development and field testing of a flexible
fourth-year course continuing the preparation of students for college mathematics. Course 4
consists of four core units for all college-bound students plus sequences of specialized units
based on intended undergraduate major.