Geometry Strand

The Geometry Strand: Developing Geometric Reasoning in Connected Mathematics

Connected Mathematics develops four mathematical strands: Number and Operation, Geometry and Measurement, Data Analysis and Probability and Algebra. Within the geometry strand are inter-related and complementary ideas: shape and transformations of shape, measurement and visualization. Each geometry unit may focus on just one of these, but in fact these ways of thinking about geometry are not distinct. For example, understanding a transformation requires students to be able to visualize a shape in another orientation, and identify what distances or angle measures are defining parts of the transformation.

The formal study of geometric ideas starts in CMP in grade 6 with Shapes and Designs. This is the first unit in the geometry strand that will develop students' ability to recognize, display, analyze, measure, and reason about the shapes and visual patterns that are important features of our world. The last explicitly geometric unit is Kaleidoscopes, Hubcaps and Mirrors, in which students use the concept of transformation to reason in increasingly formal ways about properties of specific geometric shapes, and conditions for similarity and congruence of shapes.

The first unit to deal explicitly with measurement is Covering and Surrounding, in which area and perimeter are the foci. Measurement (of surface area and volume) continues to be a focus in Filling and Wrapping. Lengths of sides of a triangle, and areas of squares, are important ideas in Looking for Pythagoras; measurement of distances from an axis or from a center of rotation play an crucial role in Kaleidoscopes, Hubcaps and Mirrors. 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. Ruins of Montarek develops students' spatial visualization skills through rich, hands-on problem situations. Students continue to use visualization skills as they develop understanding of similarity and transformations.

Overall goals for CMP Geometry and Measurement Strand

The mathematical learning goals below signify what students should be able to do in Geometry by the end of eighth grade.

Shapes and their Properties

• Generate important examples of angles, lines and 2 and 3 dimensional shapes (6)

• Categorize, define, and relate figures in a variety of representations (6,7)

• Understand principles governing the construction of shapes with reasons why certain shapes serve special purposes (e.g. triangles for trusses) (6)

• Build and visualize three-dimensional figures from various two-dimensional representations and vice versa (6)

• Recognize and use shapes and their properties to make mathematical arguments and to solve problems (6, 7, 8)

• Use the Pythagorean Theorem and properties of special triangles (e.g. isosceles right triangles) to solve problems (8)

• Use a coordinate grid to describe and investigate relationships among shapes (7, 8)

• Recognize and use standard, essential geometric vocabulary. (6, 7, 8)

Transformations - Symmetry, Similarity, and Congruence

• Visually identify line, rotational and translational symmetries and use them to solve problems (6, 8)

• Use scale factor and ratios to create similar figures or determine whether two or more shapes are similar or congruent (7)

• Visualize and predict the effects of similarity and congruence transformations on lengths, angle measures, perimeters, areas, volume and orientation (7, 8)

• Investigate the effects of combining one or more transformations on a shape (8)

• Identify and use congruent triangles and/or quadrilaterals to solve problems about shapes and measurement (6, 8)

• Use properties of similar figures to solve problems about shapes and measurement (7)

• Use a coordinate grid to explore and verify similarity and congruence relationships (7, 8)

Measurement

• Understand what it means to measure an attribute of a figure or a phenomenon (6)

• Estimate and measure angles, line segments, areas and volumes using tools and formulas (6, 7)

• Relate angle measure and side lengths to the shape of a polygon (6)

• Find area and perimeter of rectangles, parallelograms, triangles, circles, and irregular figures (7)

• Find surface area and volume of rectangular solids, cylinders, and prisms as well as cones and spheres (7)

• Relate units within and between the customary and metric systems (6, 7)

• Use ratios and proportions to derive indirect measurements (7)

• Use measurement concepts to solve problems (6, 7, 8)

Geometric Connections

• Use geometric concepts to build understanding of concepts in other areas of mathematics (6, 7, 8)

• Connect geometric concepts to concepts in other areas of mathematics (6, 7, 8)

Geometry Units

There are 7 Geometry Units in CMP. In addition, there are units which develop geometric thinking, though their focus is largely on number, or algebra. For example, Shapes of Algebra is largely algebraic in focus, but gets its impetus from the strong connections between Algebra and Geometry.

• Shapes and Designs (6th grade)

• Covering and Surrounding (6th grade)

• Ruins of Montarek (6th grade)

• Stretching and Shrinking (7th grade)

• Filling and Wrapping (7th grade)

• Looking for Pythagoras (8th grade)

• Kaleidoscopes, Hubcaps and Mirrors (8th grade)

• Shapes of Algebra (8th grade)

Every important idea addressed in the CMP Geometry Strand is placed carefully to make initial development appropriate to student developmental level, and also to connect productively to other units already studied. For example, the idea of "properties of a shape" is introduced in Shapes and Designs, where students informally compare regular and non-regular figures and begin to identify properties, such as equal or parallel sides, of some polygonal shapes. In Stretching and Shrinking they investigate the effect of similarity transformations on properties of shapes. In Covering and Surrounding and Filling and Wrapping they use properties of a rectangle to efficiently calculate surface areas and volumes of various polygons or prisms, The 8th grade unit Looking For Pythagoras is about a particular property of right triangles. And finally, in Hubcaps students learn to recognize symmetry as another property a shape might have, and create formal definitions for symmetries.

It is important to note that many goals are revisited in later units, in the same grade level or later, either within classroom problems or in the Connections problems in the ACE homework assignments. Meanwhile, units that are not geometric in focus are interspersed between these geometric units, and connections and distributed practice of geometric ideas continue to occur. Indeed, geometric ideas are often tools used to further understanding on non-geometric units. For example, "area of a rectangle" is a useful model for understanding multiplication of polynomials (Frogs and Fleas), or probability of the occurrence of two independent events (What Do you Expect).

Goals for each unit are available in the Mathematical Help section.

Relating Geometry in CMP to Geometry in High School

Students who have been successful in CMP Geometry units will enter high school knowing a lot about measurement of attributes of two- and three- dimensional objects. They will have more than knowledge of formulas for areas or volumes of common shapes; they will know what attributes might be measurable, and how to go about finding these measures, even for unfamiliar shapes. They will know that some one dimensional attributes relate to two dimensional attributes of a shape, but that there are limits to how closely one and two dimensional measures are related.

They will also relate measurement to congruence and similarity, knowing how a linear scale factor relates to changes in area or in volume. Students will explore congruence, symmetry, and transformations in greater depth in future mathematics classes.

CMP Students will have reasoned at increasingly formal levels, using both Euclidean and Transformational approaches, about common geometric shapes, particularly about triangles and special quadrilaterals. Higher levels of reasoning can be expected from them as they develop and mature in high school.

While there are topics in High School Geometry that are not addressed in CMP, most topics will appear very familiar to students; the biggest change for most students will be increased level of formality in the reasoning and recording of reasoning.