**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. The 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 have just had in *High Dive* with trigonometric functions and polar coordinates, and leads them to see the need for and 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, which students work on in pairs: They write a program to make the cube turn, and they program an animated graphic display of their own design.