Download Concepts and Skills for the IMP™ Curriculum

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
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Using variables and algebraic expressions to:
o Represent concrete situations
o Generalize results
o 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
From Geometry
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Understanding the meaning of angles and their measurement
Developing relationships among angles of polygons, including angle-sum formulas
Defining similarity and congruence
Developing criteria for establishing similarity and congruence
Using properties of similar polygons to solve real-world problems
From Trigonometry
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Using similarity to define right-triangle trigonometric functions
Applying right-triangle trigonometry to real-world problems
From Probability and Statistics
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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
Learning about normal distributions and properties of the
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Calculating mean and standard deviation
Using normal distribution, mean, and standard deviation
From Logic
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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
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Developing and using principles for equivalent expressions, including the distributive property
Expressing real-world situations in terms of equations and inequalities
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
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
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
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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
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Applying right triangle trigonometry to area and perimeter problems
From Probability and Statistics
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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 c2 statistic
Understanding and appreciating that tests of statistical significance do not lead to definitive conclusions
Solving problems that involve conditional probability
From Logic
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Working with indirect proof and proof by contradiction
Using "if, then" statements
Content for Year 3
From Algebra
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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
Working with exponential and logarithmic functions:
o Describing their graphs
o Understanding the relationship between logarithms and exponents
o Finding that the derivative of an exponential function is proportional to the value of the function
o Developing general laws of exponents
o Understanding the meaning and significance of e
o 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:
o Developing the operations of matrix addition and multiplication in the context of applied problems
o Understanding the use of matrices in representing systems of linear equations
o Developing the concepts of identity element and inverse in the context of matrices
o Understanding the use of matrices and matrix inverses to solve systems of linear equations
o Relating existence of matrix inverses to uniqueness of solution of corresponding systems of linear
equations
o Using calculators to multiply and invert matrices and to solve systems of linear equations
Extending concepts of linear programming to problems with several variables
From Analytic and Coordinate Geometry
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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:
o The distance formula
o The midpoint formula
o The equation of a circle with arbitrary center and radius
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Finding the distance from a point to a line
Developing and working with equations of planes in three-dimensional coordinate geometry
From Precalculus
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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
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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 in 3-space
From Trigonometry
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Applying right triangle trigonometry to real-world situations
From Probability and Statistics
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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
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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
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Proving and using the quadratic formula
Expressing the physical laws of falling bodies in terms of quadratic functions
From Analytic and Coordinate Geometry
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Defining polar coordinates
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Studying graphs of polar equations
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-space algebraically
From Precalculus
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Studying and using families of functions from several perspectives:
o Through their algebraic representations
o In relationship to their graphs
o As tables of values
o In terms of real-world situations they describe
Studying the effect of changing parameters on functions in a given family
Working with 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 facility to do curve-fitting
From Trigonometry
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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:
o The Pythagorean identity
o Formulas for the sine and cosine of a sum of angles
o The law of sines and the law of cosines
From Probability and Statistics
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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
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Using loops
Writing and interpreting programs
Using graphics facilities on a calculator to create programs involving animation
Performance Skills and Problem Solving
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Working on long-term problems
Drawing on diverse knowledge and methods to solve problems
Applying appropriate technology to problem solving
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Posing questions related to a problem
Generalizing problems
Group Work
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Working cooperatively with others
Sharing ideas
Asking for assistance
Subdividing a task so that group members can work independently on different parts of it
Writing and Communication
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Reading and understanding complex problems
Summarizing the essential ideas of a problem
Describing methods used to approach a problem
Evaluating and improving the quality of written work
Making oral presentations