IMP I students will study 6 units throughout the year.
Each unit is centered around a problem that allows students to work on
and develop various mathematical topics, ideas, foundations, and skills while improving their reading, writing, critical thinking, and problemsolving abilities through a 'realworld' application.UNIT 1: PATTERNS
 "The
primary purpose of this unit is to introduce students to ways of
working on and thinking about mathematics that may be new to them. In a
sense, the unit is an overall introduction to the IMP curriculum, which
involves changes for many students in how they learn mathematics and
what they think of as mathematics. The main mathematical ideas of this
unit include function tables, the use of variables, positive and
negative numbers, and some basic geometrical concepts."
 Students
will analyze and create inout tables; use variables to express
generalizations; find, analyze, and generalize geometric and numerical
patterns; work with orderofoperations rules for arithmetic, use a
concrete model to understand and do arithmetic with positive and
negative integers.
UNIT 2: THE GAME OF PIG
 "A
dice game called Pig forms from the core of this unit. Playing and
analyzing Pig involves students in a wide variety of mathematical
activities. The basic problem for students is to find an optimum
strategy for playing the game. In order to find a good strategy and
prove that it is optimum, students work with the concept of expected
value and develop a mathematical analysis for the game based on an area
model for probability."
 Students
will learn, develop, and analyze strategies; express probability as a
number between zero and one; calculate probabilities based on equally
likely events and area models; decide whether events are independent;
develop concept of expected value; calculate and interpret expected
value; solve problems involving conditional probability; create
situations that fit a given probabilistic model; make and interpret
frequency bar graphs; use simulations to estimate probabilities; compare
the theoretical analysis of a situation with experimental results; and
work with simulations.
UNIT 3: THE OVERLAND TRAIL
 "This
unit looks at the midnineteenth century western migration across what
is now the United States in terms of the many mathematical relationships
involved. These relationships involve planning what to take on the
2400mile trek, estimating the cost of the move, studying rates of
consumption and of travel, and estimating the time to reach the final
goal. A major mathematical focus of the unit is the use of equations,
tables, and graphs to describe reallife situations."
 Students
will compile and organize data; create examples that fit a set of
constraints; interpret ambiguous problems; make estimates and plans for
various situations; use tables of information and lines of best fit to
make predictions and estimates; interpret graphs intuitively; make
graphs from tabular information; quantify graphs with appropriate
scales; use graphs to represent equations and writing equations that
represent graphs; making graphs on graphing calculator; use zoom and
trace facilities to get information from graphing calculator; find lines
of best fit intuitively; use the point of intersection of graphs; work
with rate problems; use graphs, inout tables, algebraic relationships
to describe situations; develop numerical algorithms for problem
situations; express algorithms in words and symbols; develop meaningful
algebraic expressions; using subscript notation; solve equations for one
variable in terms of another; express linear approximations to data
algebraically; and solve problems involving two linear conditions.
UNIT 4: THE PIT & THE PENDULUM / FUNCTIONALITY
 "In Edgar Allan Poe's story, The Pit and the Pendulum,
a prisoner is tied down while a pendulum with a sharp blade slowly
descends. If the prisoner does not act, he will be killed by the
pendulum. Students read an excerpt from the story, and are presented
with the problem of whether the prisoner would have enough time to
escape. To resolve this question, they construct pendulums and conduct
experiments. In the process, they are introduced to the concepts of
normal distribution and standard deviation as tools for determining
whether a change in one variable really does affect another. They use
graphing calculators to learn about quadratic equations and to explore
curve fitting. Finally, after deriving a theoretical answer to the
pendulum problem, students actually build a thirtyfoot pendulum to test
their theory.
 Students
will plan and perform controlled scientific experiments; work with the
concept of period; recognize the phenomenon of measurement variation;
collect and analyze data; express experimental results and other data
using frequency bar graphs; learn about the normal distribution; make
area estimates to understand the normal distribution in problem
contexts; develop concepts of data spread, especially standard
deviation; work with symmetry and concavity in connection with the
normal distribution and standard deviation; distinguish between standard
deviation and sample standard deviation; calculate the mean and
standard deviation of data, both by hand and with calculators; use
standard deviation to decide whether a variation in experiment results
is significant; use function notation; use graphing calculators to
explore graphs; fit a function to data using a graphing calculator; and
make predictions based on curve fitting.
UNIT 5: ALL ABOUT ALICE
 "This unit starts with a model based on Lewis Carroll's Alice in Wonderland,
a story in which Alice's height is doubled or halved by eating or
drinking certain foods she finds. Out of the discussion of this
situation come the basic principles for working with
exponentspositive, negative, zero, and even fractionaland an
introduction to logarithms. Building on the work with exponents, the
unit discusses scientific notation and the manipulation of numbers
written in scientific notation."
 Students
will define the operation for an exponent of zero; define the operation
for negative integer exponents; define operation for fractional
exponents; develop the additive law of exponents; develop the law of
repeated exponentiation; describe the law of repeated exponentiation;
describe the graphs of exponential functions for different bases;
describe the graphs of logarithmic functions; compare graphs of
logarithmic functions for different bases; understand the meaning of
logarithms; make connections between exponential and logarithmic
equations; convert numbers from ordinary notation to scientific
notation, and vise versa; develop principles for computation using
scientific notation; and use the concept of order of magnitude in
estimation.
UNIT 6: SHADOWS
 "The
central question of this unit is, "How can you predict the length of a
shadow?" The unit moves quickly from this concrete problem to the
geometric concept of similarity. Students work with a variety of
approaches to come to an understanding of similar polygons, especially
similar triangles. Then they return to the problem of a shadow,
applying their knowledge of similar triangles and using informal methods
for solving proportions, to develop a general formula. In the last
part of this unit, students learn about the three primary trigonometric
functionssine, cosine, and tangentas they apply to acute angles, and
they apply to these functions to problems of finding heights and
distances."
 Students
will develop intuitive ideas about the meaning of "same shape" and
learning the formal definitions of similarity and congruence; discover
criteria for polygons to be similar and, in particular, for triangles to
be similar; work with the concept of corresponding parts of similar
figures; apply properties of similar triangles to physical situations;
use scale drawings to solve problems; discover the triangle inequality;
investigating the extension of the triangle inequality to polygons;
learn terminology applied to right triangles including hypotenuse, leg,
opposite, and adjacent; learn the right triangle definitions of sine,
cosine, and tangent; use sine, cosine, and tangent to solve problems;
develop formulas relating to sine, cosine, and tangent; develop
equations of proportionality from situation involving similar figures;
develop informal procedures for solving proportions; work with the
concept of the counterexample in understanding the criteria for
similarity; formulate and refine conjectures; prove that vertical angles
are equal; prove the angle sum property for triangles using the
properties of parallel lines; rediscover the angle sum properties of
polygons; discover that vertical angles are equal; discover the
properties of angles formed by a transversal across parallel lines; and
collect and analyze data.
