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UNIT + TOPICS + OBJECTIVES

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 problem-solving abilities through a 'real-world' 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 in-out tables; use variables to express generalizations; find, analyze, and generalize geometric and numerical patterns; work with order-of-operations 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 mid-nineteenth 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 2400-mile 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 real-life 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, in-out 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 thirty-foot 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 exponents--positive, negative, zero, and even fractional--and 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 functions--sine, cosine, and tangent--as 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.
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