# Units

#### IMP Year 1

**THE OVERLAND TRAIL** Students look at mid-19th-century Western migration in terms of the many linear relationships involved.

**THE PIT AND THE PENDULUM** Exploring an excerpt from this Edgar Allan Poe classic, students use data from experiments and statistical ideas, such as standard deviation, to develop a formula for the period of a pendulum.

**SHADOWS** Students use principles about similar triangles and basic trigonometry to develop formulas for finding the length of a shadow.

**COOKIES** In their work to maximize profits for a bakery, students deepen their understanding of the relationship between equations and inequalities and their graphs.

**ALL ABOUT ALICE** The unit starts with a model based on Lewis Carroll’s Alice’s Adventures in Wonderland, through which students develop the basic principles for working with exponents.

#### IMP Year 2

**FIREWORKS** The central problem of this unit involves sending up a rocket to create a fireworks display. This unit builds on the algebraic investigations of Year 1, with a special focus on quadratic expressions, equations, and functions.

**GEOMETRY BY DESIGN** provides students with historical knowledge about how people created mathematics, and in particular, geometry. Students use the ancient tools of straightedge and compass to do constructions, and ruler and protractor to make accurate drawings. The classical deductive system consisting of Euclid’s postulates and theorems is introduced to prove theorems about triangles and quadrilaterals.

**THE GAME OF PIG** Students develop a mathematical analysis for a complex game based on an area model for probability.

**DO BEES BUILD IT BEST?** Students study surface area, volume, and trigonometry to answer the question, “What is the best shape for a honeycomb?”

**SMALL WORLD, ISN'T IT?** Beginning with a table of population data, students study situations involving rates of growth, develop the concept of slope, and then generalize this to the idea of the derivative.

#### IMP Year 3

**PENNANT FEVER** Students use combinatorics to develop the binomial distribution and find the probability that the team leading in the pennant race will ultimately win the pennant.

**ORCHARD HIDEOUT** Students study circles and coordinate geometry to determine how long it will take before the trees in a circular orchard grow so large that someone standing at the center of the orchard cannot see out.

**HIGH DIVE** Using trigonometry, polar coordinates, and the physics of falling objects, students model this problem: When should a diver on a Ferris wheel aiming for a moving tub of water be released in order to create a splash instead of a splat?

**THE WORLD OF FUNCTIONS** In this unit, students explore families of functions in terms of various representations—tables, graphs, algebraic representations, and situations they can model; they also explore ways of combining functions using arithmetic operations and composition.

**IS THERE REALLY A DIFFERENCE?** Students build on prior experience with statistical ideas from IMP Years 1 and 2, expanding their understanding of statistical analysis.

#### IMP Year 4

**MEADOWS OR MALLS?** This unit concerns making a decision about land use and builds on skills learned in Cookies about graphing systems of linear inequalities and solving systems of linear equations.

**HOW MUCH? HOW FAST?** This unit adds integrals to the derivative concepts explored in Year 2. Students solve accumulation problems using a version of the Fundamental Theorem of Calculus. They find that the derivative of the function that describes the amount of accumulation up to a particular time is the rate of accumulation, and that the function describing accumulation is an anti-derivative of the function describing the rate of accumulation.

**THE POLLSTER'S DILEMMA** The central problem of this unit concerns an election poll, and students use normal distributions and standard deviations to find confidence intervals and see how concepts such as margin of error are used in polling results.

**AS THE CUBE TURNS** Students study the fundamental geometric transformations—translations, rotations, and reflections—in two and three dimensions, in order to create a display of a cube rotating around an axis in three-dimensional space.

**KNOW HOW** In this unit, students independently research mathematical concepts and skills that they either have not yet learned or may have forgotten. Students reflect on their future needs for independent learning, and consider what it means to really know something.