
Introduction and Implementation Strategies for the Interactive Mathematics Program: A
Guide for TeacherLeaders and Administrators
Appendix B:
Concepts and Skills for the IMP® Curriculum
The
Interactive Mathematics Program has developed an integrated fouryear high
school mathematics sequence, designed to replace the traditional Algebra I 
GeometryAlgebra II/TrigonometryPrecalculus sequence.
The
following yearbyyear lists describe the major topics covered in the IMP
curriculum. The lists are formulated in terms of traditional mathematics topic
organization, although the topics listed are covered in an integrated fashion,
in the context of meaningful larger mathematical problems. Generally, topics
taught in a given year are reviewed and extended through the curriculum of
subsequent years. The yearbyyear content descriptions are followed by a list
of performance skills that are an integral part of this curriculum in all four
years.
Content for Year 1
From Algebra
 Using
variables and algebraic expressions to represent concrete situations,
generalize results, and describe functions
 Using
different representations of functions—symbolic, graphical,
situational, and numerical—and understanding the connections between
these representations
 Understanding
and using function notation
 Understanding,
modeling, and computing with signed numbers
 Solving
equations using trial and error
 Interpreting
graphs and using graphs to represent situations
 Relating
graphs to their equations, with emphasis on linear relationships
 Solving pairs
of linear equations by graphing
 Fitting
equations to data, both with and without graphing calculators
 Developing and
using principles for equivalent expressions, including the distributive
property
 Understanding
and using the distributive property
 Developing
principles for equivalent equations and applying these principles to solve
equations
 Solving linear
equations in one variable
 Discovering
and understanding relationships between the algebraic expression defining
a linear function and the graph of that function
From Geometry
 Understanding the meaning of
angles and their measurement
 Developing relationships among
angles of polygons, including anglesum formulas
 Defining and developing criteria
for establishing similarity and congruence
 Using properties of similar
polygons to solve realworld problems
From Trigonometry
 Using
similarity to define righttriangle trigonometric functions
 Applying
righttriangle trigonometry to realworld problems
From Probability and
Statistics
 Developing
basic methods for calculating probabilities
 Constructing
area models and tree diagrams
 Distinguishing
between theoretical and experimental probabilities
 Planning and
carrying out simulations
 Collecting and
analyzing data
 Constructing
frequency bar graphs
 Understanding,
calculating, and interpreting expected value
 Applying the
concept of expected value to realworld situations
 Learning about
normal distributions and their properties
 Calculating
mean and standard deviation
 Using normal
distribution, mean, and standard deviation
From Logic
 Making and testing conjectures
 Formulating counterexamples
 Constructing sound logical
arguments
 Understanding the idea of proof
 Writing proofs
 Developing and describing
algorithms and strategies
Content for Year 2
From Algebra
 Expressing realworld
situations in terms of equations and inequalities
 Understanding and using the
distributive property
 Developing and using several
methods for solving systems of linear equations in two variables
 Defining and recognizing
dependent, inconsistent, and independent pairs of linear equations
 Solving nonroutine equations
using graphing calculators
 Writing and graphing linear
inequalities in two variables
 Developing and using principles
of linear programming for two variables
 Creating linear programming
problems with two variables
 Solving quadratic equations by
factoring
 Studying the number of roots of
a quadratic equation and relating this number to the graph of the
associated quadratic function
 Using the method of completing
the square to analyze the graphs of quadratic equations and to solve
quadratic equations
 Understanding and using
exponential expressions, including zero, negative, and fractional
exponents
 Developing and using laws of
exponents
 Using scientific notation
 Using the concept of order of
magnitude in estimation
From Geometry
 Developing the meaning of area
using both standard and nonstandard units
 Developing and using several
methods for finding areas of polygons, including development of formulas
for area of triangles, rectangles, parallelograms, trapezoids, and regular
polygons
 Understanding and finding
surface area and volume for threedimensional solids, including prisms and
cylinders
 Discovering and using the
Pythagorean theorem
 Understanding and explaining a
proof of the Pythagorean theorem
 Finding figures of maximum area
for a given perimeter
 Understanding the relationship
between the areas and volumes of similar figures
 Using and developing methods
for creating tessellations
From Trigonometry
 Applying right
triangle trigonometry to area and perimeter problems
From Probability and
Statistics
 Drawing inferences from
statistical data
 Designing, conducting, and
interpreting statistical experiments
 Making and testing statistical
hypotheses
 Formulating null hypotheses and
understanding their role in statistical reasoning
 Understanding and using the χ^{2} statistic
 Understanding and appreciating
that tests of statistical significance do not lead to definitive
conclusions
 Solving problems that involve
conditional probability
From Logic
 Working with
indirect proof and proof by contradiction
 Using
"if, then" statements
Content for Year 3
From Algebra
 Working with
exponential and logarithmic functions and describing their graphs
 Understanding
the relationship between logarithms and exponents
 Finding that
the derivative of an exponential function is proportional to the value of
the function
 Developing
general laws of exponents
 Understanding
the meaning and significance of e
 Approximating
data by an exponential function
 Developing and
using the elimination method for solving systems of linear equations in up
to four variables
 Extending the
concepts of dependent, inconsistent, and independent systems of linear
equations to more than two variables
 Working with
matrices
 Developing the
operations of matrix addition and multiplication in the context of applied
problems
 Understanding
the use of matrices in representing systems of linear equations
 Developing the
concepts of identity element and inverse in the context of matrices
 Understanding
the use of matrices and matrix inverses to solve systems of linear
equations
 Relating the
existence of matrix inverses to the uniqueness of the solution of
corresponding systems of linear equations
 Using
calculators to multiply and invert matrices and to solve systems of linear
equations
 Extending
concepts of linear programming to problems with several variables
 Expressing the
physical laws of falling bodies in terms of quadratic functions
From Analytic and
Coordinate Geometry
 Defining slope
and understanding its relationship to rate of change and to equations for
straight lines
 Developing
equations for straight lines from two points and from pointslope
information
 Developing and
applying various formulas from coordinate geometry, including:
 Distance
formula
 Midpoint
formula
 Equation of
a circle with arbitrary center and radius
 Finding the
distance from a point to a line
 Developing and
working with equations of planes in threedimensional coordinate geometry
 Defining polar
coordinates
 Studying
graphs of polar equations
From Precalculus
 Understanding
and using inverse functions
 Understanding
the meaning of the derivative of a function at a point and its
relationship to instantaneous rate of change
 Approximating
the value of a derivative at a given point
From Geometry
 Developing the
relationship of the area and circumference of a circle to its radius
 Understanding
the definition and significance of using regular polygons to approximate
the area and circumference of a circle
 Discovering
and justifying locus descriptions of various geometric entities, such as
perpendicular bisectors and angle bisectors
 Developing
properties of parallel lines
 Studying the
possible intersections of lines and planes
From Trigonometry
 Applying
righttriangle trigonometry to realworld situations
 Extending the
righttriangle trigonometric functions to circular functions
 Using
trigonometric functions to work with polar coordinates
 Defining
radian measure
 Graphing the
sine and cosine functions and variations of these functions
 Working with
inverse trigonometric functions
 Developing and
using various trigonometric formulas, including:
 The
Pythagorean identity
 Formulas for
the sine and cosine of a sum of angles
 The law of
sines and the law of cosines
From Probability and
Statistics
 Developing and
applying principles for finding the probability for a sequence of events
 Developing
methods for the systematic listing of possibilities for complex problems
 Developing the
meaning of combinatorial and permutation coefficients in the context of
realworld situations, and understanding the distinction between
combinations and permutations
 Developing
principles for computing combinatorial and permutation coefficients
 Understanding
and using Pascal's triangle
 Developing and
applying the binomial distribution
From Logic
 Using "if
and only if" in describing sets of points fitting given criteria
 Defining and
using the concept of the converse of a statement
Content for Year 4
From Algebra
 Proving and
using the quadratic formula
 Expressing the
physical laws of falling bodies in terms of quadratic functions
From Analytic and
Coordinate Geometry
 Expressing
geometric transformations—translations, rotations, and
reflections—in analytic terms
 Using matrices
to represent geometric transformations
 Developing an
analytic expression for projection onto a plane from a point perspective
 Representing a
line in 3dimensional space algebraically
From Precalculus
 Studying and
using families of functions from several perspectives:
 Through
their algebraic representations
 In
relationship to their graphs
 As tables of
values
 In terms of
realworld situations that they describe
 Studying the
effect of changing parameters on functions in a given family
 Working with
end behavior and asymptotes of rational functions
 Working with
the algebra of functions, including composition and inverse functions
 Defining the
leastsquares approximation and using a calculator's regression capability
to do curvefitting
 Extending the
number system to include complex numbers to solve certain quadratic
equations
From Trigonometry
 Extending the
righttriangle trigonometric functions to circular functions
 Defining
radian measure
From Calculus
 Estimating
derivatives from graphs, and developing formulas for derivatives of some
basic functions
 Understanding
accumulation as the antiderivative of a corresponding rate function
From Probability and
Statistics
 Using the
binomial distribution to model a polling situation
 Distinguishing
between sampling with replacement and sampling without replacement
 Understanding
the central limit theorem as a statement about approximating a binomial
distribution by a normal distribution
 Using area
estimates to understand and use a normal distribution table
 Extending the
concepts of mean and standard deviation from sets of data to probability
distributions
 Developing
formulas for mean and standard deviation for binomial sampling situations
 Using the
normal approximation for binomial sampling to assess the significance of
poll results
 Working with
the concepts of confidence interval, confidence level, and margin of error
 Understanding
the relationship between poll size and margin of error
From Programming
 Using loops
 Writing and
interpreting programs
 Using a
graphing calculator to create programs involving animation
Click here to see a grid that shows how the Interactive Mathematics Program
curriculum fits the NCTM standards.

