New Jersey Mathematics Curriculum Framework

## STANDARD 7 - GEOMETRY AND SPATIAL SENSE

 All students will develop spatial sense and an ability to use geometric properties and relationships to solve problems in mathematics and in everyday life.

## Standard 7 - Geometry and Spatial Sense - Grades 9-12

### Overview

Students can develop a strong spatial sense from consistent experiences in classroom activities which use a wide variety of manipulatives and technology. The key components of this 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.

Geometry has historically held an important role in high school mathematics, primarily through its focus on deductive reasoning and proof; developing skills in deductive reasoning, learning how to construct proofs, and understanding geometric properties are important outcomes of the high school geometry course. Equally important, however, is the continued development of visualization skills, pictorial representations, and applications of geometric ideas since geometry helps students represent and describe the world in which they live and answer questions about natural, physical, and social phenomena.

Deductive reasoning is highly dependent upon understanding and communication skills. In fact, mathematics can be considered as a language - a language of patterns. This language of mathematics must be meaningful if students are to discuss mathematics, construct arguments, and apply geometry productively. Communication and language play a critical role in helping students to construct links between their informal, intuitive geometric notions and the more abstract language and symbolism of high school geometry.

Geometry describes the real world from several viewpoints. One viewpoint is that of standard Euclidean geometry - a deductive system developed from basic axioms. Other widely used viewpoints are those of coordinate geometry, transformational geometry, and vector geometry. The interplay between geometry and algebra strengthens the students' ability to formulate and analyze problems from situations both within and outside mathematics. Although students will at times work separately in synthetic, coordinate, transformational, and vector geometry, they should also have many opportunities to compare, contrast, and translate among these systems. Further, students should learn that certain types of problems are often solved more easily in one specific system than another specific system.

Visualization and pictorial representation are also important aspects of a high school geometry course. Students should have opportunities to visualize and work with two- and three-dimensional figures in order to develop spatial skills fundamental to everyday life and to many careers. By using physical models and other real-world objects, students can develop a strong base for geometric intuition. They can then draw upon these experiences and intuitions when working with abstract ideas.

The goal of high school geometry includes applying geometric ideas to problems in a variety of areas. Each student must develop the ability to solve problems if he or she is to become a productive citizen. Instruction thus must begin with problem situations - not only exercises to be accomplished independently but also problems to be solved in small groups or by the entire class working cooperatively.

Applications of mathematics have changed dramatically over the last twenty years, primarily due to rapid advances in technology. Geometry has, in fact, become more important to students because of computergraphics. Thus, calculators and computers are appropriate and necessary tools in learning geometry.

Students in high school continue to develop their understanding of spatial relationships. They construct models from two-dimensional representations of objects, they interpret two- and three-dimensional representations of geometric objects, and they construct two-dimensional representations of actual objects.

Students formalize their understanding of properties of geometric figures, using known properties to deduce new relationships. Specific figures which are studied include polygons, circles, prisms, cylinders, pyramids, cones, and spheres. Properties considered should include congruence, similarity, symmetry, measures of angles (especially special relationships such as supplementary and complementary angles), parallelism, and perpendicularity.

In high school, students apply the principles of geometric transformations and coordinate geometry that they learned in the earlier grades, using these to help develop further understanding of geometric concepts and to establish justifications for conclusions inferred about geometric objects and their relationships. They also begin to use vectors to represent geometric situations.

The geometry of measurement is extended in the high school grades to include formalizing procedures for finding perimeters, circumferences, areas, volumes, and surface areas, and solving indirect measurement problems using trigonometric ratios. Students should also use trigonometric functions to model periodic phenomena, establishing an important connection between geometry and algebra.

Students use a variety of geometric representations in geometric modeling at these grade levels, such as graphs of algebraic functions on coordinate grids, networks composed of vertices and edges, vectors, transformations, and right triangles to solve problems involving trigonometry. They also explore and analyze further the patterns produced by geometric change.

Deductive reasoning takes on an increasingly important role in the high school years. Students use inductive reasoning as they look for patterns and make conjectures; they use deductive reasoning to justify their conjectures and present reasonable explanations.

## Standard 7 - Geometry and Spatial Sense - Grades 9-12

### Indicators and Activities

The cumulative progress indicators for grade 12 appear below in boldface type. Each indicator is followed by activities which illustrate how it can be addressed in the classroom in grades 9, 10, 11 and 12.

Building upon knowledge and skills gained in the preceding grades, experiences in grades 9-12 will be such that all students:

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

• Students find volumes of objects formed by combining geometric figures and develop formulas describing what they have done. For example, they might generate a formula for finding the volume of a silo composed of a cylinder of specified radius and height topped by a hemisphere of the same radius.

• Students construct models to show how the volume of a pyramid with a square base and height equal to a side of the base is related to the volume of a cube with the same base.

• Students develop and use a spreadsheet to determine what the dimensions should be for a cylinder with a fixed volume, in order to minimize the surface area. Similarly, they investigate what should be the dimensions for a rectangle having a fixed perimeter in order to maximize the enclosed area. They discuss how the symmetry of these figures relates to the solutions.

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

• Students use a computer-aided design (CAD) program to investigate rotations of objects in three dimensions.

• Students use The Geometric SuperSupposer to measure components of shapes and make observations. For example, they might construct parallelograms and measure sides, angles, and diagonals, observing that opposite sides are congruent, as are opposite angles, and that diagonals bisect each other.

• Students use The Geometer's Sketchpad to investigate the effects of rotating a triangle about a fixed point.

• Students use commercial materials such as GeoShapes or Polydrons to construct three-dimensional geometric figures. They make tables concerning the number of vertices, edges, and faces in each solid. They measure the figures to determine their surface areas and volumes. They lay the patterns out flat to examine the nets of each solid. [A net is aflat shape which when folded along indicated lines will produce a three-dimensional object; for example, six identical squares joined in the shape of a cross can be folded to form a cube. Tabs added to the net facilitate attaching appropriate edges so that the shape remains three-dimensional.]

• Students work through the Ice Cones lesson that is described in the First Four Standards of this Framework. Students create a variety of paper cones out of circles with radius 10 inches which are cut along a radius. They use graphing calculators to find the maximum volume of such cones.

• Students copy geometric designs using compass and straightedge, and generate their own designs.

• Students investigate wallpaper patterns, classifying them according to the transformations used. They study the structure of crystals from a geometric perspective.

20. Understand and apply properties involving angles, parallel lines, and perpendicular lines.

• Students make tessellations with an assortment of different triangles, noting the variety of geometric patterns that are formed, including parallel lines, congruent angles, congruent triangles, similar triangles, parallelograms, and trapezoids.

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

• Working together, students review geometric vocabulary by sorting words written on index cards into groups and explaining their reasons for creating the groups they did. For example, they might place "parallelogram," "rhombus," "square," and "rectangle" in one group (since they are all parallelograms) and place "kite" and "trapezoid" in another group (since they are not parallelograms).

• Students find a variety of strategies for demonstrating that the sum of the measures of the angles of a triangle is 180 degrees. Some use protractors and measure a pencil-and-paper figure, others create a triangle with Geometer's Sketchpad software and post the measures of the angles before dragging it from a vertex to notice that the sum always remains the same, and still others use a method that requires tearing each of the corners from an oaktag triangle and then fitting them together to make a line.

21. Analyze properties of three-dimensional shapes by constructing models and by drawing and interpreting two-dimensional representations of them.

• Pairs of students work together to describe and draw geometric figures. One student is given a picture involving one or more geometric figures and must describe the drawing to the other student without using the names of the figures. The second student, without seeing the figure, must visualize and represent the picture.

• Students create wind-up posterboard models of rotational three-dimensional solids. They cut out a plane figure such as a circle or a rectangle from posterboard, punch two holes in it near its edges, thread a cut rubberband through the holes, and attach the ends of the rubberband to the ends of a coathanger from which the horizontal wire has been removed. They then twist the rubber band to wind up the figure and release to "show" the solid.

• Students use isometric dot paper to sketch figures made up of cubes. They also sketch top, front, and side views (projections) of the figure.

• One long-term project that some high school teachers use for assessment is the following: Using a variety of means and materials, students begin by constructing models of the Platonic solids and other three dimensional geometric figures. They are then challenged to work in teams to find a relationship among the number of faces, vertices, and edges that holds for all of the solids (Euler's Formula: F + V - E = 2).

• Students read Flatland: A Romance of Many Dimensions by Edwin Abbott, a fascinating and imaginative story about life in a two-dimensional world.

• Students use a computer-aided design (CAD) program to investigate rotations of objects in three dimensions.

22. Use transformations, coordinates, and vectors to solve problems in Euclidean geometry.

• Students construct a polygon that outlines the top view of their school. They are asked to imagine that they are architects who need to send this outline by computer to a builder who has no graphics imaging capabilities. They develop strategies for sending this information to the builder. One group locates one corner of the building at the origin and determines the coordinates for the other vertices. Another group uses vectors to tell the builder what direction to proceed from the initial corner located at the origin.

• Students work on the question of where a power transformer should be located on a line so that the length of the cable needed to run to two points not on that line is minimized. They find that if the two points are on the same side of the line, then by using reflections they can construct a straight line that crosses the given line at the desired location.

• Students first determine the coordinates for the vertices of a parallelogram, a rhombus, a rectangle, an isosceles trapezoid, and a square with one vertex at the origin and a side along the x-axis. They then work in groups to determine where the coordinate system should be placed to simplify the coordinate selection for a kite, a rhombus, and a square.

• Students draw two congruent triangles anywhere in the plane and determine the minimum number of reflections needed to map one onto the other.

• Students draw a triangle on graph paper and then find the image of the triangle when the coordinates of each vertex are multiplied by various constants. They draw each resulting triangle and determine its area. They make a table of their results and look for relationships between the constants used for dilation and the ratios of the areas.

• Students use a Mira (Reflecta) to find the center of a circle, to draw the perpendicular bisectors of a line segment, or to draw the medians of a triangle.

• 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).

• Given the equation of a line, students plot the line on a coordinate grid and then shift the line according to a given translation. They then determine the equation of the resulting line. After doing several such problems, students identify patterns that they have found and write conjectures.

• Students work through the Building Parabolas lesson that is described in the First Four Standards of this Framework. They investigate the effects of various coefficients on the general shape of a parabola and connect these to geometric transformations.

23. Use basic trigonometric ratios to solve problems involving indirect measurement.

• Students use trigonometric ratios to determine distances which cannot be measured directly, such as the distance between two points on opposite sides of a chasm.

• Students investigate how the paths of tunnels are determined so that people digging from each end wind up in the same place.

• Students use trigonometry to determine the cloud ceiling at night by directing a light (kept in a narrow beam by a parabolic reflector) toward the clouds. An observer at a specified distance measures the angle of elevation to the point at which the light is reflected from the cloud.

• Students plot the average high temperature for each month over the course of five years to see an example of a periodic function. They discuss what types of functions might be appropriate to represent this relationship.

24. Solve real-world and mathematical problems using geometric models.

• Students visit a construction site where the "framing" step of a building process is taking place. They note where congruence occurs (such as in the beams of the roof, where angles must be congruent). They write about why congruence is essential to buildings and other structures.

• Students use paper fasteners and tagboard strips with a hole punched near each end to investigate the rigidity of various polygon shapes. For shapes that are not rigid, they determine how to make the shape more rigid.

• Students draw a geometric representation and develop a formula to solve the problem of how many handshakes will take place if there are n people and each person shakes hands with each other person exactly once.

• Students work through the On the Boardwalk lesson that is described in the Introduction to this Framework. They determined the probability of winning a prize when tossing a coin onto a grid by having the coin avoid all of the grid lines.

• Students use graph models to represent a situation in which a large company wishes to install a pneumatic tube system that would enable small items to be sent between any of ten locales, possibly by relay. Given the cost associated with possible tubes (edges), the students work in groups to determine optimal pneumatic tube systems for the company. They report their results in letters written individually to the company president.

• Students work through the Making Rectangles lesson that is described in the First Four Standards of this Framework. They use combinations of algebra tiles which they try to arrange into rectangle shapes to help them develop procedures for multiplying binomials and factoring polynomials.

25. Use inductive and deductive reasoning to solve problems and to present reasonable explanations of and justifications for the solutions.

• In a computer-based, open-ended, assessment, groups of students use computer software to draw parallelograms, make measurements, and list as many properties of parallelograms and their diagonals as they can.

• Students prove deductively that a parallelogram is divided into congruent triangle by a diagonal. They also prove that any angle inscribed in a semi-circle is a right angle. (An angle ABC is inscribed in a semi-circle if AC is a diameter and B is any other point on the circle.)

• Students explain in writing to a friend what the formula is for the measure of each interior angle in a regular polygon with n sides and how it is derived.

• Students build staircases from cubes, recording the number of steps and the total number of cubes used for each construction. They look for patterns, expressing them in words and symbolically in equations. They then try to justify their results using deductive reasoning.

• Students use Cabri software to investigate what happens when consecutive midpoints of a quadrilateral are connected in order. They state a conjecture based on their investigation and explain why they think it is true.

• Students investigate the relationship between the number of diagonals that can be drawn from one vertex of a polygon and the number of sides of that polygon. They write about their findings in their journals.

• Students work through the A Sure Thing!? lesson in the Introduction to this Framework. They investigate the number of non-overlapping regions that can be created if they draw all the chords joining n points on the circumference of a circle.

26. Analyze patterns produced by processes of geometric change and express them in terms of iteration, approximation, limits, self-similarity, and fractals.

• Students duplicate the beginning stages of a fractal construction in the plane and analyze the sequences of their perimeters and their areas.

• Students use the reduction and enlargement capabilities of a copy machine to investigate the effects on area. They make a table showing the linear rate of reduction/enlargement and the resulting area for each successive reduction/enlargement. Then they graph the results - an exponential function showing either decay or growth.

• Students use the slides and appropriate activities from Fractals for the Classroom, Vol. 1 to analyze patterns produced by changes in geometric shapes.

• 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 plot the relationship between body height and arm length for people from one year of age through adulthood on coordinate grid paper and on log-log paper. They see that the graph is not a straight line on the coordinate grid paper; it is actually a logarithmic function. They find that the function appears as a straight line on log-log paper.

27. Explore applications of other geometries in real-world contexts.

• Students represent lines using string and pins on styrofoam balls (spheres). They analyze the properties of lines (e.g., all lines intersect) and triangles (e.g., it is possible to have a triangle with three angles of 90 degrees). They apply their results to finding the shortest route between two points on the earth.

• Students investigate the angel and devil drawings of M. C. Escher as examples of geometries in which there may be many "lines" through a given point that do not intersect a given "line." In this case, a "line" is an arc of a circle that is perpendicular to the outside circle of the drawing.

• Students explore another geometry using Non-Euclidean Adventures on the Lénárt Sphere.

• Students determine how many people are needed on a committee if there are to be four subcommittees, with each person on two subcommittees and each pair of subcommittees having one person in common. Most groups use letters to represent the individuals, and represent the four subcommittees by collections of letters, as in the following proposed solution {ABC, ADE, BDF, CEF}. The teacher asks the students to make a diagram of their solution, using "points" for people and "lines" for subcommittees, so that each subcommittee is a line whose points are its members. The rules for subcommittees become axioms about these points and lines; for example, "each person is on two subcommittees" becomes "each point is on two lines." The resulting geometry is an example of a finite geometry.

• ### References

Abbott, Edwin. Flatland: A Romance of Many Dimensions. New York: Viking, 1987.

Lénárt, István. Non-Euclidean Adventures on the Lénárt Sphere. Berkeley, CA: Key Curriculum Press, 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.

### Software

Cabri. Texas Instruments.

Geometric Golfer. Minnesota Educational Computing Consortium (MECC).

The Geometry PreSupposer. Sunburst Communications.

Tesselmania! Minnesota Educational Computing Consortium (MECC).

### General reference

Coxford, A.F. Curriculum and Evaluation Standards for School Mathematics: Addenda Series: Geometry from Multiple Perspectives. Reston, VA: National Council of Teachers of Mathematics, 1991.

### On-Line Resources

http://dimacs.rutgers.edu/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.

* Activities are included here for Indicators 16 and 19, which are also listed for grade 8, since the Standards specify that students demonstrate continued progress in these indicators.