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Introduction and Implementation Strategies for the Interactive Mathematics Program: A
Guide for Teacher-Leaders and Administrators
Appendix B:
Concepts and Skills for
the IMP® Curriculum
The
Interactive Mathematics Program has developed an integrated four-year high
school mathematics sequence, designed to replace the traditional Algebra I -
Geometry-Algebra II/Trigonometry-Precalculus sequence.
The
following year-by-year 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 year-by-year 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 angle-sum formulas
- Defining and developing criteria
for establishing similarity and congruence
- Using properties of similar
polygons to solve real-world problems
From Trigonometry
- Using
similarity to define right-triangle trigonometric functions
- Applying
right-triangle trigonometry to real-world 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 real-world 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 real-world
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 non-routine 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 three-dimensional 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 χ2 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 point-slope
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 three-dimensional 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
right-triangle trigonometry to real-world situations
- Extending the
right-triangle 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
real-world 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 3-dimensional 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
real-world 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
least-squares approximation and using a calculator's regression capability
to do curve-fitting
- Extending the
number system to include complex numbers to solve certain quadratic
equations
From Trigonometry
- Extending the
right-triangle 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.
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