(These units comprise IMP Year 2, and are also available for purchase individually.)

Unit 1: 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 a 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.

Unit 2: 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: 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. 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 find a method for solving two equations in two unknowns. Students then complete their work on the cookie problem, presenting a solution and a proof showing that the solution maximizes profits. Finally, groups invent their own linear programming problem, and present the problem and its solution to the class.

Unit 3: 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 project in which they propose a hypothesis about two populations they think differ in some respect. They collect sample data about the two populations and analyze their data using bar graphs, tables, and the χ2 statistic.

Unit 4: 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 focuses 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.

Unit 5: 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.

From Algebra

• Expressing real-world situations in terms of equations and inequalities 
• Understanding and using the distributive property 
• Developing and using several methods for solving systems of linear equations in two variables 
• Defining and recognizing dependent, inconsistent, and independent pairs of linear equations 
• Solving non-routine equations using graphing calculators 
• Writing and graphing linear inequalities in two variables 
• Developing and using principles of linear programming for two variables 
• Creating linear programming problems with two variables 
• Solving quadratic equations by factoring 
• Studying the number of roots of a quadratic equation and relating this number to the graph of the associated quadratic function
• Using the method of completing the square to analyze the graphs of quadratic equations and to solve quadratic equations 
• Understanding and using exponential expressions, including zero, negative, and fractional exponents 
• Developing and using the laws of exponents
• Using scientific notation 
• Using the concept of the order of magnitude in estimation 

From Geometry

• Developing the meaning of area using both standard and nonstandard units 
• Developing and using several methods for finding areas of polygons, including developing formulas for the area of triangles, rectangles, parallelograms, trapezoids, and regular polygons 
• Understanding and finding surface area and volume for three-dimensional solids, including prisms and cylinders 
• Discovering and using the Pythagorean theorem 
• Understanding and explaining a proof of the Pythagorean theorem 
• Finding figures with the maximum area for a given perimeter 
• Understanding the relationship between the areas and volumes of similar figures 
• Using and developing methods for creating tessellations 

From Trigonometry

• Applying right-triangle trigonometry to area and perimeter problems 

From Probability and Statistics

• Drawing inferences from statistical data 
• Designing, conducting, and interpreting statistical experiments 
• Making and testing statistical hypotheses 
• Formulating null hypotheses and understanding their role in statistical reasoning 
• Understanding and using the χ2 statistic 
• Understanding and appreciating that tests of statistical significance do not lead to definitive conclusions 
• Solving problems that involve conditional probability 

From Logic

• Working with indirect proof and proof by contradiction
• Using "if, then" statements