*Picture this: You're on a planning team, consulting to the city manager. Your task is to come up with a reasonable plan for the use of 550 acres of land recently obtained by the city. The acreage includes a recently closed army base, a 300-acre farm, and abandoned mining land.*

*a maximum of 200 acres from the army base and the mining land will be used for recreation, and**the amount of army land used for recreation plus the amount of farm land used for development will together total 100 acres.*

*Not only are you dealing with opposing factions, but with improvement costs ranging from $50 to $2,000 per acre, depending on which parcel of land is involved and how it will be used. You have to satisfy everyone while minimizing the total cost for improvements. To arrive at a reasonable allocation plan will demand careful analysis and attention to detail.*

*Meadows or Malls*? You're going to be working on it for the next six to eight weeks, learning and using algebra, geometry, and matrix operations.

## About the Program

*Curriculum and Evaluation Standards for School Mathematics*(National Council of Teachers of Mathematics 1989). Since 1989, IMP has been used in classrooms throughout the United States to develop and test the kinds of tasks sought by NCTM, embedding those tasks within a larger vision of a complete mathematics program. It is one of five programs funded by NSF to develop new comprehensive curriculums at the high school level.

## Thinking About Mathematics

## Learning the Basics

*Algebra.*Solving systems of linear equations for unknowns is an important skill in traditional algebra classes. In IMP, this topic is presented both in Meadows or Malls? and in a second-year unit called Cookies, which deals with maximizing profits from a bakery. Students don't just learn one method, but develop their own approaches in groups and then share ideas with one another. In*Meadows or Malls*, they also see how to use matrices and the technology of graphing calculators to solve such systems.*Geometry.*Basic concepts in traditional geometry include similar triangles and the Pythagorean theorem. In Shadows, a first-year IMP unit, students learn about similar triangles, develop proportion equations to solve similar triangles, and apply the concept of similarity to predict the lengths of shadows. They also extend their knowledge to see how similarity is used as the foundation of trigonometry.In*Do Bees Build It Best?*students develop the Pythagorean theorem experimentally, prove it algebraically or geometrically, and apply it to see why the hexagonal prism of the bees' honeycomb design is the most efficient regular prism possible. Several units later, students apply the Pythagorean theorem to develop other principles, such as the distance formula in coordinate geometry.*Trigonometry.*In traditional programs, sine, cosine, and tangent are introduced in the 11th or 12th grade. In IMP, students begin working with these functions in 9th grade (in Shadows), and learn their value and application over the years. Right triangle trigonometry is used in several units in the second and third year.In a fourth-year unit,*High Dive*, students extend trigonometry from right triangles to circular functions, defining the trigonometric functions for angles of more than 90 degrees. The problem-solving context in this unit is a circus act in which a performer jumps off a Ferris wheel into a moving tub of water. Not only are students developing and applying general ideas from trigonometry, but they are also learning principles of physics, developing laws for falling objects, and using vectors to find vertical and horizontal elements of velocity.

## Word Problems with a Difference

*Meadows or Malls*? resembles the word problems of traditional algebra courses. Both take students out of the realm of pure mathematics, requiring them to see the mathematical structure in a real-life situation. Let's ignore the fact that traditional word problems often strike students as contrived and artificial (two trains going in opposite directions from the station, for example). What's more important is the way they are presented.

*Meadows or Malls*? involves solving systems of equations, understanding the geometry of how two-dimensional planes intersect in three-dimensional space, and developing and working with abstract notions such as identity element and inverse.

## What About Drill and Practice?

## What About Results?

After her first semester of a traditional college math course, Theresa, from San Antonio, said, I always ask questions. The others don't. I thought it was because they knew it already, but then after class, they would ask me questions. I realized that they are just scared to ask. They don't know what is going on. Oh yes, I got an A.

Teachers also appreciate the curriculum. Over the years, we have worked with many talented teachers across the country who are now master IMP teacher/trainers in their own communities. Reflecting on her experience, a California teacher (Bussey 1992) wrote: Too many kids are flunking out of Algebra I or getting turned off to mathematics. Algebra I acts as a sieve, keeping only a select few, while filtering out many talented kids. Isn't there more than one way to "learn" mathematics and "do" mathematics?...When I'm in my classroom and witness my students working in groups, debating mathematical principles, and developing their own ideas to solve meaningful problems, that is when I feel most successful. My students have proven to me that they all can learn, that learning can be meaningful and relevant—and fun. What more can one ask of the educational process?