MATH Connections: A Secondary Core Curriculum
Review Materials:
Field-test materials for all three grades were reviewed. A
Teacher Resource
Package will be available and was reviewed. The package includes a Teacher
Edition for each grade, which contains student pages, Teacher Commentary,
suggestions about pacing, blackline masters, and solutions to all problems. The
package also contains "Books in Brief," which describe the role present
materials and its relation to past and to future material. Also included in the
package is a set of student assessments. A second set of assessments is also
available. Ancillary materials are under development including booklets with
challenging problems and projects.
Format/Description: This is a complete three-year secondary school
mathematics curriculum
intended for all students. The curriculum is integrated; meaning that material
from several areas of
mathematics are included in any given year. For our purposes, we have centered
our description
around six strands: algebra/number/function, geometry, trigonometry,
probability and statistics,
discrete mathematics and logic and reasoning. Not all strands appear in each
year. The student
material is content-centered. That is, mathematical topics take center stage in
each section usually
followed by concepts applied or connected to various contextual and "real-world"
situations. Each
grade level is divided into two books, a. and b, for ease of using; especially
with block scheduling. The
a. section is meant to precede the b. section. Each section is divided into
chapters and each chapter is
divided into several sub-sections. Each sub-section begins with stated learning
objectives for that subsection
and several student activities within explorations followed by a problem set.
The activities are
coded with icons indicating either a discussion topic, a writing topic, or an
activity that should be done
before proceeding. Some sub-sections contain ideas for longer student projects.
The margins of the
student materials contain Thinking Tips, About Symbols,
and About Words
(notes that detail how
some everyday words have more specific meanings in mathematics). Appendices for
each level detail
technology information helping students learn to use a TI-82 (83) Graphing
Calculator, use a
spreadsheet, program a TI-82 (83), etc.
The Teacher Commentary includes an overview and general comments on the student
materials as well
as suggestions concerning the implementation of the material in the classroom.
It contains comments
and answers to the activities and problems contained in each sub-section.
Blackline Masters are
included, also.
Pedagogy: The curriculum delivery in the classroom
is based on directed student activities where
students "try out" approaches suggested by the materials or apply concepts with
which they are
working. There are follow-up homework problems. The classroom activities may be
directed by the
teacher using the text along with a more conventional teaching approach or
students can read the text
and do activities on their own with the teacher acting as both a guide and
provider of additional material
or explanation. These materials have been implemented with students working
individually as well as
with students working in groups. Sometimes the Teacher Commentary or student
materials suggest
student interaction such as comparing answers to activities or general
discussion. Other times, choices
are left to the teacher.
Technology: Graphing calculators are essential for the delivery of the
curriculum. Such calculators
are used in the majority of the material. The TI-82 (83) series is supported by
the student materials. In
addition, spreadsheet capabilities are used in some units. For example,
spreadsheets are used in the
material on linear programming. (The use of Excel is supported in an appendix to
that material.) In
addition, students will use technology to produce scatterplots, fit regression
lines, graph functions,
observe patterns, and do tedious calculations. Some programming on graphing
calculators is included.
The availability of a dynamic geometry program such as the Geometer's Sketchpad
is suggested for
some of the geometry material.
Assessment: Assessments are provided for each sub-section of each book.
These assessments
include "quizzes" for each chapter and end-of-section assessments.
Content Overview: MATH Connections is an integrated curriculum in the
sense that content
associated with algebra, geometry, and data analysis appears at each level.
(Note that some coordinate
geometry regarding straight lines appears in the first year of the curriculum
and is listed under the
algebra/number/function section for Year 1 below.) In addition, this
review identifies several strands
that occur in the curriculum taken as a whole: algebra/number/function,
geometry, trigonometry,
probability and statistics, discrete mathematics and logic and reasoning. Not
all strands appear in each
year. See below for details concerning the appearance of these strands in the
various years. It may be
noted that Year 1 material is heavily concentrated in algebra, Year 2 material
is heavily concentrated in
geometry, and Year 3 contains considerable material in pre-calculus and discrete
mathematics. While
the above emphases suggest a more traditional division of material, it must be
noted that many of the
concepts within these topics are non-traditional. Moreover, "traditional" topics
associated with these
emphases may appear elsewhere in the curriculum. For example, axiomatic proof
involving geometry is
covered in Year 3 despite the emphasis on geometry in Year 2. As a second
example, while the
concept of function may be more traditionally included in a course on
pre-calculus, the topic of functions
appears throughout the entire MATH Connections curriculum. In the words
of the developers, "a
major goal of the curriculum is the development of higher-order thinking
skills." The
developmental approach most often used is for the students to actively explore a
concept in order to
develop experimental evidence and/or recognize a pattern, be assured the pattern
is accurate, represent
the pattern by a symbolic formula, and then apply the formula in real-world
situations. There is a strong
symbolic component throughout the curriculum. It is the philosophy of the
developers that a great deal
of mathematical power comes from being able to " understand formulas and be able
to transfer use them
to a variety of contexts." Graphical and tabular representations are also
included throughout. Many
times students are asked to think about and discuss or write about what they are
doing and why they
are doing it. The problem set reinforces and extends, to some degree, the
material in the sub-section.
It usually does not contain traditional simple drill and practice problems. It
often does contain additional
material and/or cases not covered in the preceding narrative as well as
applications. Our description of
what students completing this curriculum will know and be able to do is given in
terms of the strands
mentioned above. Some topics, such as coordinate geometry, could be listed under
more than one
strand. However, when topics are closely integrated, the decision has been made
to list them only
under one strand in each year. Moreover, the description below does not
necessarily describe the
order in which topics are encountered in the curriculum within each year. For
the most part, topics
which are mentioned or worked with briefly but not emphasized to any degree are
omitted. However, in
some places we say, simply, that students are "exposed" to such topics.
Year 1
Algebra/number/function: Using appropriate variables and constants,
students will be able to write an
equation to represent one quantity in terms of one or more other quantities.
They will be able to
evaluate equations or formulas, including exponential equations, for specified
values. Given a table of
values, they will look for any patterns and, if found, write a formula to
represent a generalization of the
pattern. They will be able to use the correct order of operations to compute an
arithmetic expression or
to simplify an algebraic expression. Students will be able to solve an equation
in one variable, by
applying the appropriate laws of algebra, including the commutative and
associative laws of addition and
multiplication and the distributive law. They will be able to write an
exponential expression to represent
a real-world situation involving exponential growth or decay. Students will be
able to construct a
rectangular coordinate system and plot points that represent ordered pairs of
coordinates. Given the
coordinates of two points, they will be able to compute the slope of a line.
They will recognize that a
vertical line has no slope. They will be able to write a linear equation in the
form y=mx+b when given
two points or the slope and y-intercept. Using a graphing calculator, students
will investigate the effect
of a change in a or b on the graph of a linear equation y=ax+b. They will also
explore the effect of
integral changes in integral exponents on the graph of y=xa, for a
·1. They will be able to graph a linear
equation in the form y=mx+b (with and without the aid of a calculator). They
will compare graphs of
linear equations in terms of their slopes. Given a real-world situation, they
will be able to write a linear
equation of the form y=mx+b, draw its graph, and use both in the analysis of the
problem. Using a tree
diagram to organize possible outcomes, students will develop an algorithm for an
efficient way to play a
number guessing game and apply it in solving real-world problems. They will
graph inequalities in one
variable on a number line. Students will utilize graphs and/or tables of values
to compare two sets of
data. By plotting points, they will graph two linear equations on the same set
of axes, find the point of
intersection, and make comparisons. They will also use the calculator to graph
pairs of equations (some
non-linear) and find their point(s) of intersection. They will solve systems of
two linear equations of the
form y=mx+b algebraically (substitution method). They will graph equations of
the form ax+by=c by
plotting points (including the x- and y-intercepts) and solve systems of these
equations graphically.
Students will be introduced to a linear programming problem and its terminology
(e.g. constraints, region
of feasible solutions). They will graph a system of linear inequalities and
investigate where maximum
profit occurs within the feasible region.
Students will be able to determine whether or not a situation describes a
function. They will be able to
identify the domain and range of many functions, use function notation, and
evaluate a function at
specified values. They will write a formula for a sequence of numbers and use it
to find the nth term in
the sequence. They will use a spreadsheet to generate a sequence. They will
determine whether or not
two functions are equal on the domain of counting numbers. Students will be able
to make a table
and/or graph to represent a function, including step functions. They will write
a function and create a
simple program for the calculator to convert temperature scales (Celsius to
Fahrenheit). They will write
a linear or exponential growth (involving compound interest or population
growth) function for real-world
situations and graph it on the coordinate plane. They will also graph a function
on the calculator
and use the trace feature to evaluate it at specific values. Given two or more
functions, students will be
able to find and evaluate a composite function. They will be able to write a
function as the composite of
two or more simpler functions. They will be able to describe some real-world
processes as composite
functions. They will examine the laws of commutativity and associativity with
respect to the operation of
composition (e.g. f·g·g·f).
Students will be able to compute the absolute value of a number. They will be
able to calculate a
number written in exponential form. They will recognize the power of exponents
by exploring the classic
doubling problem (1 penny, 2 pennies, 4 pennies, 8 pennies, etc.). They will be
able to convert
numbers expressed in scientific notation to standard notation and vice versa.
Using the calculator, they
will solve problems involving operations with scientific notation. By
investigating patterns in multiplying
integers, they will draw conclusions concerning the multiplication of negative
integers.
Probability and statistics: Students will recognize the need for a
systematic approach to counting and
they will investigate and develop a variety of strategies to solve counting
problems in several contexts.
They will be introduced to set notation and terminology (e.g. intersection,
union, disjoint, empty set).
They will be able to list the elements of a set, count the number of elements,
and identify subsets. They
will realize that ••• and ••• are associative operations. Students will use Venn
diagrams (2 and 3
circles, with a known quantity for the intersection) to partition sets into
disjoint subsets and determine
the number of elements in these subsets. They will use tree diagrams to find all
possible outcomes for an
event. In cases that involve two possible outcomes at each level, they will
become aware of the
symmetry of the branches and the need to create only a partial tree to ascertain
the total number of
outcomes. They will be able to apply the Fundamental Principle of Counting.
Students will utilize these
counting strategies in assigning probabilities to events on the basis of
intuition, past experience,
experimentation, or theory. They will recognize that a probability must be a
number between 0 and 1,
inclusive. They will be able to identify certain and impossible events,
assigning an appropriate
probability to each. Using the definition of probability for events involving
equally likely outcomes (i.e.
P(E)=# of acceptable or favorable ways/total # or possible ways), students will
be able to find the
probability of an event. They will become familiar with expectation as they
compute the expected
frequency of an outcome for a given number of repetitions of an experiment. They
will be introduced to
conditional probability (not stated as such) as they find the probability of an
event given a known
previous occurrence or condition. They will be able to find the probability of
the complement of an
event as well as the intersection or union of two events. They will examine the
generalization
P(A·B)=P(A)+P(B)- P(A·B). They will apply their knowledge of probability in
finding medical indices
that could serve as predictors of survival in trauma cases. Students will
explore simulation as a means of
modeling real-world situations (e.g. waiting times at the post office). They
will become familiar with
random number tables and use the calculator to random generator feature to
simulate the flipping of a
coin N times.
Students will investigate sets of data that have been gathered by them or
another source in terms of
measures of central tendency and spread of data. They will be able to compute
the mean, median, and
mode (using a calculator with statistical capabilities, if they wish) and
determine which is the most
appropriate measure of the center. They will analyze the effect of a change in
one or more data values
(including an extreme value) on the mean, median, and mode. Given a set of data,
they will be able to
construct a bar graph, a dot plot (line plot), a stem-and-leaf plot, and a
histogram. Students will be able
to find the range and mean absolute deviation for a set of data. They will be
able to determine the five
number summary for a set of data, construct a boxplot from these figures, and
analyze the spread of the
data. They will be able to compare and contrast sets of data represented by two
or more boxplots,
histograms, or back-to-back stem-and-leaf plots. They will use a spreadsheet to
calculate the mean,
mean absolute deviation, and the five number summary for a set of data. They
will explore and calculate
the variance and standard deviation for a set of data. They will compare and
contrast two sets of data,
draw conclusions based on an analysis of measures of central tendency and spread
of data, and justify
their conclusions. Students will be able to construct a scatterplot from a table
of values and use it in the
analysis of the data to observe any overall trends and estimate data. They will
determine when a table or
a scatterplot might be the better representation for a set of data. They will
also use linear interpolation
and the least-squares (regression) line to estimate data. In addition, they will
use linear models for
extrapolation (prediction). Using a calculator, they will be able to find the
least-squares line, use it to
make predictions, and determine whether or not their estimates are reasonable.