Students can develop strong spatial sense from consistent experiences in classroom activities that use a wide variety of manipulatives and technology. The key components of spatial sense, as identified in the K-12 Overview, are spatial relationships, properties of geometric figures, geometric transformations, coordinate geometry, geometry of measurement, geometric modeling, and reasoning.

Students in grades 7 and 8 learn geometry by: engaging in activities and spatial experiences organized around physical models, modeling, mapping, and measuring; discovering geometric relationships by using mathematical procedures such as drawing, sorting, classifying, transforming, and finding patterns; and solving geometric problems.

Building explicit linkages among mathematical topics is especially important with respect to geometry, since geometric concepts contribute to students' understanding of other topics in mathematics. For example, the number line provides a way of representing whole numbers, fractions, decimals, integers, lengths, and probability. Regions are used in developing understanding of multiplication, fraction concepts, area, and percent. The coordinate plane is used to relate geometry to algebra and functions. Similar triangles are used in connection with ratio and proportion.

Students continue to develop their understanding of spatial relationships by examining projections (viewing objects from different perspectives), shadows, perspectives, and maps. They apply the understanding developed in earlier grades to solve problems involving congruence, similarity, and symmetry.

Students begin to explore the logical interrelationships among previously-discovered properties of geometric figures at these grade levels. They extend their work with two-dimensional figures to include circles as well as special quadrilaterals. They continue to work with various polygons, lines, planes, and three-dimensional figures such as cubes, prisms, cylinders, cones, pyramids, and spheres.

The study of geometric transformations continues as well at these grade levels, becoming more closely linked to the study of algebraic concepts and coordinate geometry in all four quadrants. Students begin to represent transformations and/or their results symbolically. They also continue to analyze the relationships between figures and their transformations, considering congruence, similarity, and symmetry.

The geometry of measurement is extended to circles, cylinders, cones, and spheres in these grades. Students learn about pi and use it in a variety of contexts. They explore different ways to find perimeters, circumferences, areas, volumes, surface areas, and angle measures. They also develop and apply the Pythagorean Theorem. The emphasis is on understanding the processes used and on recording the procedures in a formula; students should not simply be given a formula and be expected to use it.

Students continue to use geometric modeling to represent problem situations in different areas. Drawings of various types are particularly useful to students in understanding the context of problems. Number lines, coordinate planes, regions, and similar triangles help students to visualize numerical situations. Especially important are the patterns produced by change processes, including growth and decay.

Students further develop their reasoning skills by making conjectures as they explore relationships among various shapes and polygons. For example, as students learn about the midpoints of line segments, they can make guesses about the shapes produced by connecting midpoints of consecutive sides of quadrilaterals. By testing their hypotheses with drawings they make (by hand or using a computer), the students come to actually see the possibilities that can exist. The informal arguments that students develop at these grade levels are important precursors to the more formal study of geometry in high school.

The emphasis in grades 7 and 8 should be on investigating and using geometric ideas and relationships, not on memorizing definitions and formulas. A special feature of these grade levels is that students are preparing to take the New Jersey Early Warning Test (EWT). Many of the items in the Measurement and Geometry Cluster of the EWT will ask students to use those geometric ideas and relationships to solve problems - not simply to recall formulas. Indeed, the formulas needed for the problems are given to them on the Reference Sheet that accompanies the test. In their general classroom activity, as well as in preparation for the EWT, students should use a variety of concrete materials to model and analyze situations in two and three dimensions. They should use drawings that they make, either by hand or with the aid of a computer, to further examine geometric situations or to record what they have done. Geometry approached in this way can be fun and challenging for students.

The cumulative progress indicators for grade 8 appear below in boldface type. Each indicator is followed by activities which illustrate how it can be addressed in the classroom in grades 7 and 8.

Building upon knowledge and skills gained in the preceding grades, experiences in grades 7-8 will be such that all students:

11. Relate two-dimensional and three-dimensional geometry using shadows, perspectives, projections, and maps.

• Students build and draw solids made of cubes. They learn to build solids from drawings and to make their own drawings of solids. Among the drawings with which they should be familiar are the two-dimensional flat view from top, front, and side; the three-dimensional corner view; and the map view showing the base of the building with the number of cubes in each stack. For example, they can build the solid below; presented here are a three-dimensional corner view, a flat view, and a map view.

• Students predict what the intersections of a plane with a cylinder, cone, or sphere will be. Then they slice clay models to verify their predictions.

• Students use cubes made of clear plexiglas and partially filled with colored water to investigate cross sections of a plane with a cube. They try to tilt the cube so that the surface of the water forms various shapes, such as a square, a rectangle that is not a square, a trapezoid, a hexagon, and others.

12. Understand and apply the concepts of symmetry, similarity, and congruence.

• Students create three-dimensional symmetric designs using cubes, cylinders, pyramids, cones, and spheres.

• Students build scale models of the classroom, using similarity to help them determine the appropriate measures of the models.

• Students use compasses and straight-edges to construct congruent line segments and angles.

• Students work through the Sketching Similarities lesson that is described in the First Four Standards of this Framework. Students use a computer program and various similar figures to discover that corresponding angles have equal measures and corresponding sides have equal ratios.

13. Identify, describe, compare, and classify plane and solid geometric figures.

• Students use toothpicks to construct as many different types of triangles as possible, where each side of the triangles consists of between one and five toothpicks. They record their findings in a table, showing how many triangles are scalene, isosceles, equilateral, right, and obtuse. They also indicate which combinations of sides are impossible.

• Students sort collections of quadrilaterals according to the number of lines of symmetry that each has.

• Students play clue games designed to help them distinguish between necessary and sufficient conditions in describing a shape. For example: If you want to challenge your friend to identify a square by giving a set of clues, which minimum set of clues would you select from the list below? Explain your selection. Is it possible to select a different minimum set of clues? Explain.

  - 4 right angles	- all angles are 90 degrees
  - 4 sides	- all sides of equal length
  - all angles congruent	- opposite angles congruent
  - opposite sides parallel	- simple closed curve

• Openended assessment items like those used on the Early Warning Test can always be used to provoke discussion and classroom activity. One of the sample items in the New Jersey Department of Education's Mathematics Instructional Guide (MG3) shows several figures and asks which of them can be put together to form a square. The developmental and extension activities provided there offer good suggestions for manipulative and transformation tasks.

• Students work through the A Sure Thing !? lesson that is described in the Introduction to this Framework. They investigate the relationship among the measures of the interior angles of a triangle by cutting out arbitrary triangles, tearing them into three pieces so that each corner is intact, and fitting the corners around a single point to make a straight angle.

• Students use diagrams to demonstrate the relationships among properties. For example, they might draw a diagram to show the logical relationship of ideas leading to the angle sum for a quadrilateral, or, as below, to clarify the relationship among different types of quadrilaterals.
  

14. Understand the properties of lines and planes, including parallel and perpendicular lines and planes, and intersecting lines and planes and their angles of incidence.

• Students build a model of a cube, connect a midpoint of an edge with a midpoint of another edge, and then connect two other midpoints of edges to each other. They describe the relationships of the segments they have constructed. They change one of the line segments to another location and repeat the activity.

• Students identify congruent angles on a parallelogram grid, using their results to develop conjectures about alternate interior angles and corresponding angles of parallel lines and about opposite angles of a parallelogram.

• Working in groups, students review geometric vocabulary by sorting words written on index cards into groups and explaining their reasons for setting up the groups in the way that they did.

15. Explore the relationships among geometric transformations (translations, reflections, rotations, and dilations), tessellations (tilings), and congruence and similarity.

• Students use the "nibble" technique to create a shape which will tessellate the plane, that is, copies of this shape will fit together to cover a planar surface like a sheet of white oak-tag. Start with a square, cut off a "nibble" along the top or bottom edge of the square and translate the nibble vertically to the opposite edge of the square; the "nibble" will then be outside the boundary of the original square. Take a "nibble" from the right or left edge of the square and translate it horizontally to the opposite edge of the square. Trace this shape repeatedly onto a sheet of white oak-tag, by interlocking the pattern, and decorate the copies of the shape. Attempt this process several times until a pleasing shape is created.

• Students analyze the patterns found in Arabic designs such as tiled floors and walls in Spain, identifying figures that represent translations, reflections, and rotations. Then they generate their own tile designs using basic geometric shapes. They can create their own tile patterns using Tesselmania! software.

• Students apply transformations to figures drawn on coordinate grids, record the coordinates of the original figure and its image, and look for patterns. They express these patterns verbally and symbolically. For example, flipping a point across the x-axis changes the sign of the y-coordinate so that the point (x,y) moves to (x,-y).

• Students practice doing geometric transformations mentally by using the Geometric Golfer or similar computer software. These programs present a series of puzzles in which there is an object shape and a target shape. The task is to use the fewest transformations possible to change the object shape so that it is congruent to the target shape. In the golf game, the object is a ball and the target is a hole.

16. Develop, understand, and apply a variety of strategies for determining perimeter, area, surface area, angle measure, and volume.

• Students use a paper fastener to connect two models of rays to form angles of different sizes. They estimate the correct position, then measure their guess with a protractor to see how close they were.

• Students are given a parallelogram-shaped piece of oak-tag and asked to cut it apart and arrange the parts so that it is easy to find its area. Their solutions are expressed verbally and symbolically. This same process is repeated for a trapezoid.

• Students bring cans from home, arrange them by estimated volume from smallest to largest, determine the actual volumes by measuring and computing, and compare these results to their estimates.

• Good conceptual assessment items designed to measure students' understanding of area frequently ask the students to find the area remaining in one figure after the area of another figure has been removed. One sample item from the New Jersey Department of Education's Mathematics Instructional Guide, for example, asks students to find the area of a circular path that surrounds a circular flower garden (MG1). Problems like this one are not only good practice for tests like the Early Warning Test but can also be used as informal assessments by teachers who listen carefully to their students' discussions about them.

• Students work through the Rod Dogs lesson that is described in the First Four Standards of this Framework. Students determine the effects of increasing the dimensions of an object on its surface area and volume.

17. Understand and apply the Pythagorean Theorem.

• Students draw right triangles on graph paper with legs of specified lengths and measure the lengths of their hypotenuses. They record their results in a chart and look for patterns.

• Students create a small right triangle in the middle of a ten-by-ten geoboard or on dot paper and then build squares on each side of the triangle. They record the areas of the squares and look for a relationship involving these areas.

• Students use tangram pieces to build squares on each side of the middle-sized triangular tangram piece. They then describe the relationship among the areas of the three squares.

18. Explore patterns produced by processes of geometric change, relating iteration, approximation, and fractals.

• Students use the reducing and/or enlarging feature on a copier to explore repeated reductions/enlargements by the same factor (iteration).

• Students learn about the natural shapes that are produced by growth. They investigate how nature produces complex structures in which basic shapes occur over and over. For example, spider webs, honeycombs, and snowflakes grow larger in a systematic way (iteration). Students measure the age of a tree by looking at its rings; this illustrates approximation. Students produce geometric designs that illustrate these principles, as well as fractals, where miniature versions of the entire design are evident within a smallportion of the design.

• Students view the slides which accompany the activity book, Fractals for the Classroom, Vol. 1, and determine why each picture might have been included in a book about fractals.

• Students make a table showing the perimeter of a Koch snowflake (a type of fractal) and its area at each stage. They discuss the patterns in the table. This is an example where the perimeter increases without bound but the area approaches a limit.

  Stage Perimeter Area 0 3 1 1 4 4/3 2 16/3 40/27 3 4

19. Investigate, explore, and describe geometry in nature and real-world applications, using models, manipulatives, and appropriate technology.

• Some students read and prepare a report and presentation to the class on String, Straightedge, and Shadow: The Story of Geometry by Julia Diggins. Starting with a chapter about the presence of geometry in nature, this story traces the history of geometric discoveries from the invention of early measuring instruments.

• Students model decay in a bacterial culture by cutting a sheet of grid paper in half repeatedly and recording the area of each rectangle in a table. They then graph the number of cuts versus the area to see an example of exponential decay.

• Students investigate the golden ratio ((1 + sqrt(5))/2) and its application to architecture (such as the Parthenon), designs of everyday objects such as index cards and picture frames, and its occurrence in pinecones, pineapples, and sunflower seeds.

• Students write about why manufacturers make specially designed containers for packaging their products, indicating how the idea of tessellations might be important in the designs.

• Students use a computer program such as The Geometry PreSupposer to investigate the relationship between the lengths of the sides and the measures of the angles in isosceles, scalene, and equilateral triangles.

• Groups of students prepare slide shows using slides from Geometry in Our World to illustrate the connections between geometry, science, and art.

• Pairs of students build kites of different shapes, explaining to the class why they chose a particular shape. Each student predicts which kite will fly highest, writing the prediction in his/her journal. The class flies all of the kites, recording the heights of each by using a clinometer and similar triangles.

• Students watch the video Donald in Mathmagic Land. Although getting a bit dated, thisvideo still thrills most viewers as Donald Duck encounters many animated applications of geometry. Students then form teams which focus on an aspect of the video and do further research on that application.

• Students read and choose projects to make from the book Origami, Japanese Paper Folding by Florence Sakade or some other origami instruction books. The detailed instructions usually given in such books are rich in mathematical language and discussions among the students should provide a setting for the use of much geometric terminology.

References

Diggins, Julia. String, Straightedge, and Shadow: The Story of Geometry. New York: Viking Press, 1965.

Engelhardt, J., Ed. Geometry in Our World. Reston, VA: National Council of Teachers of Mathematics, 1992.

New Jersey Department of Education. Mathematics Instructional Guide. D. Varygiannes, Coord. Trenton: 1996.

Peitgen, Heinz-Otto, et al. Fractals for the Classroom: Strategic Activities, Vol. 1. Reston, VA: National Council of Teachers of Mathematics, and New York: Springer-Verlag, 1991.

Sakade, Florence. Origami, Japanese Paper Folding. Rutland, VT: Charles E. Tuttle Co., 1957.

Software

Geometric Golfer. Minnesota Educational Computing Consortium (MECC).

The Geometry PreSupposer. Sunburst Communications.

Tesselmania! Minnesota Educational Computing Consortium (MECC).

Video

Donald in Mathmagic Land. Walt Disney Studios. Los Angeles: The Walt Disney Company, 1959.

General references

Geddes, D. Curriculum and Evaluation Standards for School Mathematics: Addenda Series: Geometry in the Middle Grades. Reston, VA: National Council of Teachers of Mathematics, 1992.

Owens, D. T., Ed. Research Ideas for the Classroom: Middle Grades Mathematics. New York: MacMillan, 1993.

On-Line Resources

http://dimacs.rutgers.edu/archive/nj_math_coalition/framework.html/

The Framework will be available at this site during Spring 1997. In time, we hope to post additional resources relating to this standard, such as grade-specific activities submitted by New Jersey teachers, and to provide a forum to discuss the Mathematics Standards.