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Introduction and Implementation Strategies for the Interactive Mathematics Program: A Guide for Teacher-Leaders and Administrators


Appendix A:
A Unit-by-Unit Summary of
the IMP® Curriculum

Year 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 Interactive Mathematics Program®, which for many students involves changes in how they learn mathematics and what they think of as mathematics. In this unit, major emphasis is placed on developing the ability to think about and explore mathematical problems.

Some important mathematical ideas and concepts are introduced and worked with in this unit, especially function tables, the use of variables, positive and negative numbers, and some basic geometrical concepts.

Another major theme is the idea of proof. This is not developed as a formal process but rather as part of the larger theme of reasoning and explaining. Students? ability to create and understand proofs will develop over the course of the four-year IMP? curriculum, and their work in this unit is just a beginning.

The Game of Pig

A probability game called Pig forms 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 optimal strategy for playing the game. In order to find a good strategy and prove that it is optimal, students work with the concept of expected value and develop a mathematical analysis of the game based on an area model for probability. They can also use a computer to simulate both Pig and a simpler version of the game so that they can compare strategies and check the theoretical probabilities experimentally.

Probabilistic thinking frequently runs counter to our intuitions. For this reason, the activities in this unit are based in concrete experiences. Students? belief in luck is terribly persistent; it takes a great deal of experience before they become willing to base their predictions on probabilistic notions. The gambler's fallacy?that the next roll of the dice depends on previous rolls?is held with conviction even by well-informed adults. One goal of this unit is for students to recognize this fallacy, both in dice games and in real-life situations. More broadly, they will come to understand theoretical probability, and see how and when it can be used to model and give insight into situations that occur every day.

The Overland Trail

This unit looks at the mid-nineteenth century western migration across the United States in terms of the many linear relationships involved. These relationships grow out of the study of planning what to take on the 2,400-mile trek, estimating the cost of the move, and studying rates of consumption and of travel.

Students construct mathematical models and draw graphs by hand and with a graphing calculator. They interpret graphs in terms of the "stories" the graphs tell, and create graphs from "stories." They write algebraic expressions that represent situations, use manipulatives to represent variables, and solve systems of equations using graphs made by hand and by graphing calculator. In the process of graphing equations, they see the need to solve equations for one variable in terms of another, and learn techniques for doing so. 

The Pit and the Pendulum

This unit opens with an excerpt from The Pit and the Pendulum, by Edgar Allan Poe. In the story, 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. When the pendulum has about 12 swings left, the prisoner creates a plan for escape and executes it. Students are presented with the problem of whether the prisoner would have enough time to escape. To resolve this question, students construct pendulums and conduct experiments to find out what variables determine the length of the period of a pendulum and what the relationship is between the period and these variables.

In the process, students are introduced to the normal distribution and the standard deviation as tools for determining whether a change in one variable really does affect another. They make and refine conjectures, analyze data collected from experiments, and use graphing calculators to learn about quadratic equations and to explore curve fitting. Finally, after deriving a theoretical answer to the problem, students actually build a 30-foot pendulum to test their theory.

Shadows

This unit opens with the question "How can you predict the length of a shadow?" Students experiment with flashlights to isolate the important variables and try to predict the length of the shadow in terms of one of those variables. In order to understand shadows and the data they have found, students learn some geometry.

Students work with a variety of concrete objects to come to an understanding of similar polygons, especially similar triangles. They then return to the problem of the shadow, applying their knowledge of similar triangles and using informal methods for solving proportions to develop a general formula.

In the last part of the unit, students learn about the three primary trigonometric functions?sine, cosine, and tangent?as they are defined for acute angles, and apply these functions to problems of finding heights and distances.

Year 2

Do Bees Build It Best?

In this unit, students work on the following problem: Bees store their honey in honeycombs that consist of cells made from wax. What is the best design for a honeycomb?

To analyze this problem, students begin by learning about area and the Pythagorean theorem. Then, using the Pythagorean theorem and trigonometry, they find a formula for the area of a regular polygon with fixed perimeter. They discover that the greater the number of sides, the larger the area of the polygon.

Students then turn their attention to volume and surface area, focusing on prisms whose bases are regular polygons. They discover that for such prisms, if the honeycomb cells are to fit together, the mathematical "winner" in terms of maximizing volume for a given surface area is a regular hexagonal prism. This is essentially the choice that bees make as well.

Cookies

This unit focuses on graphing systems of linear inequalities and solving systems of linear equations. Although the central problem is in the field of linear programming, the major goals of the unit are for students to learn how to manipulate equations and how to reason using graphs.

Students begin by considering a classic linear programming problem, in which they are asked to maximize the profits of a bakery that makes two kinds of cookies. The constraints are the amounts of ingredients, oven time, and labor time available.

First, students work toward a graphical solution of the problem. They see how the linear function can be maximized or minimized by studying the graph. Since the maximum or minimum point they are looking for is often at the intersection of two lines, they are motivated to investigate a method for solving two equations in two unknowns. They then return to work in groups on the cookie problem. Each group presents both a solution and a proof showing that its solution does maximize profits. Finally, each group invents its own linear programming problem and presents the problem and its solution to the class.

Is There Really a Difference?

In this unit, students collect data and compare different population groups to one another. In particular, they concentrate on this question: If a sample from one population differs in some respect from a sample from a different population, how reliably can we infer that the overall populations differ in that respect?

Students begin by making double bar graphs of some classroom data, and they explore the process of making and testing hypotheses. They realize that there is variation even among different samples from the same population and see the usefulness of the concept of a null hypothesis as they examine this variation. They build on their understanding of standard deviation from the Year 1 unit The Pit and the Pendulum and learn that the chi-square (χ2) statistic can give the probability of seeing differences of a certain size between samples when the populations are really the same.

Students' work in this unit culminates in a two-week project in which they propose a hypothesis about two populations they think really differ in some respect. They collect sample data about the two populations and analyze their data using bar graphs, tables, and the χ2 statistic.

Fireworks

The central problem of this unit involves sending up a rocket to create a fireworks display. The rocket's trajectory is a parabola. This unit has a focus on quadratic expressions, equations, and functions. Students see that they can use algebra to find the vertex of the graph of a quadratic function by writing the quadratic expression in a particular form. This unit deepens students' knowledge of graphing, forging a connection between graphs of functions and solutions of equations.

All About Alice

This unit opens with a model based on Lewis Carroll's Alice's Adventures in Wonderland, in which Alice's height is doubled or halved by eating or drinking certain magical items. 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 work with exponents, the unit covers scientific notation and the manipulation of numbers written in scientific notation.

Year 3

Orchard Hideout

The central problem of this unit concerns two people who have planted an orchard on a circular lot. They want to know how long it will take before the trees grow so large that someone outside the orchard cannot see into the center of the orchard. Answering this question requires students to study circles and coordinate geometry. They develop the formulas for the circumference and the area of a circle, as well as the distance and midpoint formulas, and learn to find the distance from a point to a line. Another theme of the unit is geometric proof.

Throughout this unit, students are applying knowledge they acquired in earlier units about similar triangles, trigonometry, and the Pythagorean theorem.

This unit also includes a sequence of supplemental problems on equations for conic sections. These activities are a natural extension of the unit's focus on coordinate geometry and also form a natural follow-up to the work with parabolas in the Year 2 unit Fireworks.

Meadows or Malls?

The title problem of this unit concerns a decision that must be made about land use. This problem can be expressed using a system of linear inequalities, which lends itself to a solution by means of linear programming, a topic introduced in the Year 2 unit Cookies. Building on their work in that unit, students see that a key step in solving the system of linear inequalities is to find various points of intersection of the graphs of the corresponding equations. This, in turn, leads to the need to solve systems of linear equations. Along the way, students learn about graphing equations in three variables, see that the graph of a linear equation in three variables is a plane, and study the possible intersections of planes in space.

Because graphing calculators allow students to find inverses of square matrices (when the inverses exist), matrices are a good tool for solving systems of linear equations with several variables. So, in addition to strengthening their skills with traditional methods, students learn to express linear systems in terms of matrices and develop the matrix operations required to understand the role of matrices in the solution process.

Small World, Isn't It?

This unit opens with a table of world-population data over the last thousand years; it asks the following question: If population growth continues to follow this pattern, how long will it be until people are squashed up against each other?

In order to attack this problem, students study a variety of situations involving rates of growth. Based on these examples, they develop the concept of slope, and then generalize this to the idea of the derivative, the instantaneous rate of growth. In studying derivatives numerically, they discover that an exponential function has the special property that its derivative is proportional to the value of the function, and see that, intuitively, population growth functions ought to have a similar property. This, together with simplified growth models, suggests that an exponential function is a reasonable choice to use to approximate their population data. They also learn that every exponential function can be expressed in terms of any positive base (except 1) and that scientists use as a standard base the number for which the derivative of the exponential function equals the value of the function. They find this base, e, experimentally. Along the route of their study of exponential functions, they review logarithms, are introduced to the natural log function, and see that logarithms are a useful tool for answering questions raised by exponential functions.

Pennant Fever

One team has a three-game lead over its closest rival for the baseball pennant. Each team has seven games to go in the season (none of which are between these two teams). The central problem of the unit is to find the probability that the team that is leading will win the pennant.

Students use the teams' current records to set up a probability model for the problem. Their analysis of that model requires an understanding of combinatorial coefficients and uses probability tree diagrams. In the course of their analysis, students work through the general topics of permutations and combinations, and develop the binomial theorem and properties of Pascal's triangle. Their general understanding of the binomial distribution is also applied to several decision problems involving statistical reasoning.

High Dive

The central problem of this unit involves a circus act in which a diver will fall from a turning Ferris wheel into a tub of water that is on a moving cart. Students' task is to determine when the diver should be released from the Ferris wheel in order to land in the moving tub of water.

The geometry of the Ferris wheel generates the need to express the diver's position in terms of the angle through which the Ferris wheel has turned. Students are led to extend right-triangle trigonometric functions to the circular functions. They learn about the graphs of the sine and cosine functions and apply them both to geometric situations and to other contexts. In particular, they see how the graph of a sinusoidal function changes as various parameters such as period and amplitude are changed.

Students are also introduced to several additional trigonometric concepts, such as polar coordinates, inverse trigonometric functions, and the Pythagorean identity.

Students then study the physics of falling objects and develop an algebraic expression for the time of the diver's fall in terms of his position. In this unit, students solve a simplified version of the unit problem, in which they do not take into account the diver's initial velocity, which is imparted by the movement of the Ferris wheel itself. The first unit of Year 4 returns to this situation, and students then learn how to analyze the diver's motion in terms of its vertical and horizontal components.

Year 4

The Diver Returns

Year 4 starts with a new unit entitled The Diver Returns. This builds upon the High Dive unit from Year 3, in which a circus diver is dropped from a Ferris wheel into a tub of water that is on a moving cart. Now, the Ferris wheel is in motion!

In Year 3, the geometry of the Ferris wheel generated the need to express the diver's position in terms of the angle through which the Ferris wheel has turned. Students then studied the physics of falling objects and developed an algebraic expression for the time of the diver's fall in terms of his position.

In The Diver Returns, students must learn how to analyze the diver's motion in terms of its vertical and horizontal initial components. This more realistic—and more complex—problem leads to the study of quadratic equations and the need to express a solution in terms of the coefficients. That work culminates in the development of the quadratic formula and the introduction of complex numbers and vector components.

The World of Functions

This unit builds on students' extensive previous work with functions. They explore some basic families of functions (linear, quadratic, general polynomial, exponential, trigonometric, logarithmic, reciprocal, general rational, and power) in terms of various representations: tables, graphs, algebraic representations, and situations that they model.

Students use functions to understand a variety of problem situations. They see that finding an appropriate function to model a situation sometimes involves recognizing a pattern in the data and at other times requires insight into the situation itself. Then students explore ways of combining functions, in various representations, using arithmetic operations and composition. They conclude the unit by returning to the population-growth problem in the Year 3 unit Small World, Isn't It? Then they use their new knowledge to find a function that fits the data better than the simple exponential ones they used in the third year.

The Pollster's Dilemma

The central limit theorem is the cornerstone of this unit on sampling. Through a variety of situations, students look at the process of sampling, with a special focus on how the size of the sample affects the variation in sample results. The opening problem concerns an election poll that shows 53% of the voters favoring a certain candidate. This question is posed: How confident should the candidate be about her lead, based on this poll?

Students conduct sampling experiments and are led to conclude that there is a theoretical probability distribution for the results from a sample of a given size. They review ideas from the Year 3 unit Pennant Fever to see how to find the theoretical probabilities.

By experimentation, they see that the results from a set of polls of a given size are approximately normally distributed. They are given the statement of the central limit theorem, which confirms the experimental observation. Building on work in the Year 1 unit The Pit and the Pendulum, students learn how to use normal distributions and standard deviations to find confidence intervals. They also see how concepts such as margin of error are used in reporting polling results.

In addition to putting the new concepts to work on the unit problem, students work in pairs on a sampling project for a question of their own. They write reports and make presentations showing how they chose their sample size and what their results mean.

How Much? How Fast?

In this unit, students learn about integrals. How Much? How Fast? has two central problems. The first involves determining the volume of a pyramid and introduces the idea of approximation by easy-to-find pieces. The second, more complex problem involves a solar energy collector. In the first phase of the problem, students do some trigonometric analysis to see how the energy is accumulating.

Ultimately, both problems are solved using a version of the Fundamental Theorem of Calculus. By solving a series of problems, students learn that the derivative of the function that describes the amount of accumulation up to a particular time is the rate of accumulation. The function describing accumulation (such as of volume or of energy) is an antiderivative of the function describing the rate of accumulation.

This unit follows up on the previous development of the derivative as a representation of instantaneous rate of change in the Year 3 unit Small World, Isn't It?, by looking at the issue of accumulation of change. That is, if you have a graph or other means of describing the varying rate of change at which something is changing, how do you determine the total change over time?

As the Cube Turns

This unit opens with an overhead display, generated by a program on a graphing calculator. The two-dimensional display depicts the rotation of a cube in three-dimensional space. Students' central task in the unit is to learn how to write such a program.

Though the task is defined in terms of writing a program, the real focus of the unit is the mathematics behind the program. The unit takes students into several areas of mathematics. They study the fundamental geometric transformations—translations, rotations, and reflections—in both two and three dimensions, and express them in terms of coordinates. The analysis of rotations builds on the experience they had in High Dive with trigonometric functions and polar coordinates, and leads them to develop formulas for the sine and cosine of the sum of two angles. Working with these transformations also provides a new setting in which students can work with matrices, which they previously studied in connection with systems of linear equations.

Another complex component of their work is analyzing the way to represent a three-dimensional object on a two-dimensional screen. They have an opportunity to see how projection onto a plane is affected by both the choice of the plane and the choice of a viewpoint or center of projection.

The unit closes with two major projects on which students work in pairs: They write a program to make the cube turn, and they program an animated graphic display of their own design.


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