Introduction and Implementation Strategies for the Interactive Mathematics Program: A
Guide for TeacherLeaders and Administrators
Appendix A:
A UnitbyUnit 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 fouryear 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
wellinformed adults. One goal of this unit is for students to recognize this
fallacy, both in dice games and in reallife 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 midnineteenth
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 2,400mile 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 30foot 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 chisquare (χ^{2})
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 twoweek 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 χ^{2}
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 followup 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 worldpopulation 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 threegame 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 righttriangle
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 populationgrowth 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
easytofind 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 twodimensional display depicts the rotation of a cube in threedimensional
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 threedimensional object
on a twodimensional 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.
