NOTE: This is taken from the Teacher's Edition of Ruins of Montarek ©2002

Ruins of Montarek mathematical and problem solving goals:

Ruins of Montarek was created to help students:

• Read and create two-dimensional representations of three-dimensional cube buildings
• Communicate spatial information
• Observe that the back view of a cube building is the mirror image of the front view and that the left view is the mirror image of the right view
• Understand and recognize line symmetry
• Explain how drawings of the base outline, front view, and right view describe a building
• Construct cube buildings that fit two-dimensional building plans
• Develop a way to describe all buildings that can be made from a set of plans
• Understand that a set of plans can have more than one minimal building but only one maximal building
• Explain how a cube can be represented on isometric dot paper, how the angles on the cube are represented with angles on the dot paper, and how the representations fit what the eye sees when viewing the corner of a cube building
• Make isometric drawings of cube buildings
• Visualize transformations of cube buildings and make isometric drawings of the transformed buildings
• Reason about spatial relationships
• Use models and representations of models to solve problems

The overall goal of Connected Mathematics is to help students develop sound mathematical habits. Through their work in this and other geometry units, students learn important questions to ask themselves about any situation that can be represented and modeled mathematically, such as: How can three-dimensional objects be shown in two-dimensions? How can a better under- standing of space and solid figures be developed? How can imaging skills be developed by study- ing three-dimensional objects—such as buildings made from cubes? What is the value in study- ing isometric drawinga? Symmetric figures? Minimum and maximum figures? How do these ideas help build visualization skills?

The Mathematics in Ruins of Montarek

Spatial visualization is an important aspect of geometry and geometric reasoning, but it is often neglected or ignored completely in the middle-school mathematics curriculum. This unit develops students’ spatial visualization skills through rich, hands-on problem situations. A variety of tools—including cubes, building plans, isometric dot paper, and two-dimensional cube models—are used in the activities. Students work with two- and three-dimensional representations of objects as they learn about the relationships between three-dimensional objects and their two-dimensional representations and about the limitations of representing three-dimensional objects in two-dimensions (for example, the representations don’t always represent unique objects).

The major mathematical goals of this unit are for students to learn to construct, manipulate, and interpret two- and three-dimensional representations of objects and to develop an understanding of the relationships among different representations (such as building plans, isometric dot paper representations, and cube models). In other words, the unit is designed to help students learn to “read” and communicate information about three-dimensional objects from two-dimensional drawings, and to use two-dimensional representations to “write” about three-dimensional objects.

Two major representation schemes are developed in this unit: architectural views (elevations) and isometric dot paper representations. In architectural views, only one face of a building is shown in a drawing; in isometric drawings, three faces are shown. Several steps help to develop students’ understanding of both major forms of representation:

1. Students are challenged to match cube buildings with drawings.
2. Students learn to make drawings that “capture” a building on paper.
3. Students learn to put together a building from a set of plans.
4. Students learn to construct and represent buildings and to evaluate the construction or representation of buildings.

As students work through these steps, they learn about the limitations of two-dimensional representations of three-dimensional objects. The question of uniqueness is raised: Can more than one building be constructed from a single set of plans? Students discover that both building plans and isometric drawings can “hide” cubes, so that it may not be possible to tell what certain parts of a building look like. Because the hidden parts may be completed in more than one way, several buildings may be possible from a set of building plans or an isometric drawing.

Students are also challenged to add constraints to building plans and isometric drawings so that they represent a unique building. They learn that this is possible for a set of building plans (by specifring that the building must be maximal) but impossible for an isometric representation.

Throughout the unit, questions are posed that require spatial reasoning beyond the “reading” involved in moving from cube buildings to drawings and vice versa. For example, in some problems students use clues to reason about how a building might look or to locate a room in a building with a particular view Other problems involve reasoning about the ways incomplete building plans might be completed or about the least and greatest numbers of cubes that can be used to construct a building that fits a specific set of plans. Some students will develop analytic ways of thinking about such problems, such as looking at the building in layers, identi~ring cubes common to all buildings that fit a set of plans, or looking combinatorially at all possibilities. Other students will reason using mental visualization or mental rotation of the building. Through class discussion, students will hear about the spatial reasoning strategies of others and can then experiment with new strategies.

Unlike most other Connected Mathematics units, the assessment pieces for Ruins of Montarek include multiple-choice questions, which are very useful for testing visual discrimination and reasoning. Since students have not encountered multiple-choice questions in other units in the curriculum, an occasional multiple-choice item has been embedded in the ACE questions.