Meaning and Importance

Traditionally, geometry in schools has been taught as the prime example of a formal deductive system. While this view of the content is important, its domination has led to the exclusion of other, less readily formalized topics and applications. Geometry instruction should not be limited to formal deductive proof and simple measurement activities, but should include the study of geometric transformations, analytic geometry, topology, and the connection of geometry with algebra and other areas of mathematics. Posing and solving problems in this more richly defined geometry allows students to use geometric intuition to develop more generic mathematical problem-solving skills.

K-12 Development and Emphases

• visual recognition of shapes by their appearances as a whole (level 0)
• analysis and description of shapes in terms of their properties (level 1)
• higher "theoretical" levels involving informal deduction (level 2)
• formal deduction involving axioms and theorems (level 3), and
• work with abstract geometric systems (level 4).

Although the levels are not completely separate and the transitions are complex, the model is very useful for characterizing levels of children's thinking. One particularly pertinent finding that shows up consistently in the research is that appropriately targeted instruction is critical to children's movement through these levels. Stagnation at early levels is the frequent result of a geometry curriculum that dwells on identification of shapes and their properties.

It is not difficult to conceive of a curriculum that adheres to the van Hiele model. By virtue of living in a three-dimensional world, having dealt with space for five years, children enter school with a remarkable amount of intuitive geometric knowledge. The geometry curriculum should take advantage of this intuition in expanding and formalizing the students' knowledge. In early elementary school, a rich, qualitative, hands-on study of geometric objects helps young children develop spatial sense and a strong intuitive grasp of geometric properties and relationships. Eventually they develop a comfortable vocabulary of appropriate geometric terminology. In the middle school years, students should begin to use their knowledge in a more analytical manner to solve problems, make conjectures, and look for patterns and generalizations. Gradually they develop the ability to make inferences and logical deductions based on geometric relationships. In high school, the study of geometry expands to include coordinate geometry, trigonometry, and both inductive and deductive reasoning.

The study of geometry should make abundant use of experiences that require active student involvement. Constructing models, folding paper cutouts, using mirrors, pattern blocks, geoboards, and tangrams, and creating geometric computer graphics all allow opportunities for students to learn by doing, to reflect upon their actions, and to communicate their observations and conclusions. These activities and others of the same type should be used to achieve the goals in the seven specific areas of study that comprise this standard and which are described below.

In their study of spatial relationships, young students should make regular use of concrete materials in hands-on activities designed to develop their understanding of objects in space. The early focus should be the description of the location and orientation of objects in relation to other objects. Additionally, students can begin an exploration of symmetry, congruence, and similarity. Older students should study the two-dimensional representations of three-dimensional objects by sketching shadows, projections, and perspectives.

In the study of the properties of geometric figures, students deal explicitly with the identification and classification of standard geometric objects by the number of edges and vertices, the number and shapes of the faces, the acuteness of the angles, and so on. Cut-and-paste constructions of paper models, combining shapes to form new shapes and decomposing complex shapes into simpler ones are useful exercises to aid in exploring shapes and their properties. As their studies continue, older students should be able to perform classic constructions with straight edges and compasses as well as with appropriate computer software and to formulate good mathematical definitions for the common shapes, eventually being able to make deductions and solve problems using their properties.

The standard geometric transformations include translation, rotation, reflection, and scaling. They are central to the study of geometry and its applications in that these manipulations of figures offer the most natural approach to understanding congruence, similarity, symmetry, and other geometric relationships. Younger children should have a great deal of experience with flips, slides, and turns of concrete objects, figures made on geoboards, and paper-and-pencil figures. Older students should be able to use more formal terminology and procedures for determining the results of the standard transformations. An added benefit of experience gained with simple and composite transformations is the mathematical connection that older students can make to functions and function composition.

Coordinate geometry provides another strong connection between geometry and algebra. Students can be informally introduced to coordinates as early as kindergarten by locating entries in tables and finding points on maps and grids. In later elementary grades, they can learn to plot figures on a coordinate plane, and still later, study the effects of various transformations on the coordinates of the points of two- and three-dimensional figures. High-school students should be able to represent geometric transformations algebraically and interpret algebraic equations geometrically.

Measurement and geometry are interrelated, and understanding the geometry of measurement is necessary for the understanding of measurement. In elementary school, students should learn the meaning of such geometric measures as length, area, volume and angle measure and should be actively involved in the measurement of those attributes for all kinds of two- and three-dimensional objects, not simply the most standard, uniform ones. Throughout school, they should use measurement activities to reinforce their understanding of geometric properties. Eventually all students should understand such principles as the quadratic change in area and cubic change in volume that occurs with a linear change of scale. Another of the interdependencies between geometry and measurement is seen in high school when students learn to use trigonometry to make indirect measurements.

Geometric modeling is a powerful problem-solving skill and should be modeled for and frequently used by students. A simple diagram, such as a pie-shaped graph, a force diagram in physics, or a dot-and-line illustration of a network, can illuminate the essence of a problem and allow geometric intuition to aid in the approach to a solution. Visualization skills and understanding will both improve as students are encouraged to make such models.

The relationship between geometry and deductive reasoning originated with the ancient Greek philosophers, and remains an important part of the study of geometry. A key ingredient of deductive reasoning is being able to recognize which statements have been justified and which have been assumed without proof. This is an ability which all students should develop, in all areas, not just geometry, or even just mathematics! At first, deductive reasoning is informal, with students inferring new properties or relationships from those already established, without detailed explanations at every step. Later, deduction becomes more formal as students learn to state all permissible assumptions at the beginning of a proof and all subsequent statements are systematically justified from what has been assumed or proved before. The idea of deductive proof should not be confused with the specific two-column format of proof found in most geometry textbooks. The object of studying deductive proof is to develop reasoning skills, not to write out arguments in a particular arrangement. Note that proof by mathematical induction is another deductive method which should not be neglected.

IN SUMMARY, students of all ages should recognize and be aware of the presence of geometry in nature, in art, and in woman- and man-made structures. They should believe that geometry and geometric applications are all around them and, through study of those applications, come to better understand and appreciate the role of geometry in life. Carpenters use triangles for structural support, scientists use geometric models of molecules to provide clues to understanding their chemical and physical properties, and merchants use traffic-flow diagrams to plan the placement of their stock and special displays. These and many, many more examples should leave no doubt in students' minds as to the importance of the study of geometry.

Reference: Geddes, Dorothy, & Fortunato, Irene. "Geometry : Research and Classroom Activities" in D. T. Owens (Ed.) Research Ideas for the Classroom: Middle Grades Mathematics. New York: Macmillan, 1993.

This introduction duplicates the section of Chapter 8 that discusses this content standard. Although each content standard is discussed in a separate chapter, it is not the intention that each be treated separately in the classroom. Indeed, as noted in Chapter 1, an effective curriculum is one that successfully integrates these areas to present students with rich and meaningful cross-strand experiences. Many of the activities provided in this chapter are intended to convey this message; you may well be using other activities which would be appropriate for this document. Please submit your suggestions of additional integrative activities for inclusion in subsequent versions of this curriculum framework; address them to Framework, P. O. Box 10867, New Brunswick, NJ 08906.

In their study of spatial relationships, students focus on developing their understanding of objects in space. They discuss and describe the relative positions of objects using phrases like "in front of" and "on top of." They describe and draw three-dimensional objects in different relative locations. They compare and contrast shapes, describing the shapes of the faces and bases of three-dimensional figures. They discuss symmetry and look for examples of symmetry in their environment. They look for shapes that are the same size and shape (congruent) or the same shape but different sizes (similar). They use mirrors to explore symmetry.

In beginning their study of properties of geometric figures, students look for shapes in the environment, make models from sticks and clay or paper and glue, and draw shapes. They sort objects according to shape. They recognize, classify, sort, describe, and compare geometric shapes such as the sphere, cylinder, cone, rectangular solid, cube, square, circle, triangle, rectangle, hexagon, trapezoid, and rhombus. They describe the angle at which two edges meet in different polygons as being smaller than a right angle, a right angle, or larger than a right angle. They discuss points, lines, line segments, intersecting and non-intersecting lines, and midpoints of lines.

Students begin looking at geometric transformations by using concrete materials such as paper dolls to model slides (translations), flips (reflections), and turns (rotations). Students put shapes together to make new shapes and take shapes apart to form "simpler" shapes. Students work on spatial puzzles, often involving pattern blocks or tangrams. They look for plane shapes in complex drawings and explore tilings. They divide figures into equal fractional parts, often by folding along one, two, or three lines.

Coordinate geometry in grades K-2 involves describing the motion of an object. Students make maps of real. imaginary, or storybook journeys. They describe the location of an object on a grid or a point in a plane using numbers or letters. They give instructions to an imaginary "turtle" to crawl around the outline of a figure.

Students in these grades also begin to explore the geometry of measurement. In kindergarten, students discuss and describe quantitative properties of objects using phrases like "bigger" or "longer." They order objects by length or weight. In first and second grade, they quantify properties of objects by counting and measuring. They determine the areas of figures by cutting them out of grid paper and counting the squares. They measure the perimeter of a polygon by adding the lengths of all of the sides.

Students begin to explore geometric modeling by constructing shapes from a variety of materials, including toothpicks and clay, paper and glue, or commercial materials. They use templates to draw designs and record what they have done with pattern blocks and tangrams. They fold, draw, and color shapes. They copy geoboard figures, construct geoboard figures from memory, and construct geoboard figures by following oral or written instructions. They may also use geometric models, such as the number line, for skip counting or repeated addition.

Geometry provides a rich context in which to begin to develop students' reasoning skills. Students apply thinking skills in geometric tasks from identifying shapes to discovering properties of shapes, creating geometric patterns, and solving geometric puzzles and problems in a variety of ways. They create, describe, and extend geometric patterns. They use attribute blocks to focus on the properties of objects.

Geometry provides a unique opportunity to focus on the process standards, especially the connections standard. Many geometric ideas are used to help students understand concepts in other strands. For example, students often use their understanding of familiar shapes to help build an understanding of fractions. Teachers in grades K-2 need to plan classroom activities that involve several mathematical processes and interrelate geometry with other topics in mathematics. Geometry should not be taught only in isolation; it should be a natural and integrated part of the entire curriculum.

Experiences will be such that all students in grades K-2:

A. explore spatial relationships such as the direction, orientation, and perspectives of objects in space; their relative shapes and sizes, and the relations between objects and their shadows or projections.

• Blindfolded students are given real objects to touch and then must select the object from a collection of visible objects.
• Students imagine being a circle and draw, tell, or write about what they could do and how they might feel.
• Students predict what shape will result when a folded piece of paper is cut and then opened.
• Students compare the sequence of objects seen from different points of view. For example, from the classroom window, the swings are to the left of the monkeybars, but the relationship is reversed if you are standing on the blacktop.
• Students predict and draw what the shadow of an object placed between a light and a screen will look like.

• Students look for examples of congruent figures (same size and shape) in the environment.
• Students explore symmetry by using mirrors with pattern blocks or by folding paper or by making inkblot designs. Students find the lines of symmetry in the letters of the alphabet. They fold paper and cut out symmetric designs. They identify the symmetry in wallpaper or giftwrap designs.
• Students use scale models of cars as an introduction to the concept of similarity (same shape, different size).
• Self-similar shapes are ones which have miniature versions of the whole within themselves. For example, a head of broccoli is self-similar. The head itself looks like a "tree," as does each stalk and each branch from each stalk. These self-similar shapes are fractals. Other examples include cauliflower, Queen Anne's lace, and trees.

• Students identify the "footprint" made by pressing a face of a three-dimensional solid into clay. Later they identify the blocks that might have been used to make a specific "footprint." They also build, draw, or paint a picture of the creature that might have made a footprint.
• Students outline a triangle, a square, and a circle on the floor with string or tape. Then they walk around each figure, chanting a rhyme, such as "Triangle, triangle, triangle, 1, 2,3, I can walk around you as easy as can be," and counting the sides as they walk.
• Some students use Muppet Math to work with Kermit's geometric paintings, while others use Shape Up! to compare everyday objects to geometric shapes.

• Students make shapes with their fingers and arms.
• Students sort a collection of shapes into groups, explaining their reasoning. Some groups to consider include "all right angles" or "four-siders." The teacher encourages the students to invent appropriate group names and to use informal language to describe the properties.
• Students sort pictures cut from magazines according to shape. They then discuss other ways that the pictures could be sorted.
• Students make class books shaped like a triangle, a rectangle, a square, and a circle. They fill each book with pictures of objects that have the shape of the book.
• Students turn a geometric shape into a picture. For example, a triangle might become a tower, a clown face, or the roof of a house.

• Students use tangram pieces to construct triangles, rectangles, and squares.
• Students investigate which pattern block shapes can be formed from the equilateral triangles, recording their results in pictures and on a chart.
• Students work in groups to decide how to divide a rectangular candy bar among four people. The students then compare the various ways that each group solved the problem.

• Students use Unifix cubes or pattern blocks to create colorful designs. They then discuss how many blocks they used (area) and the distance around their design (perimeter). They also discuss why these polygon shapes fit together like a puzzle.
• Students use different shapes to make quilt patterns.
• During free play time, students use pattern blocks to make different space-filling designs. They record any patterns that they especially like, using templates or drawing around the blocks.

• Students look at the world around them for examples of changes in position that do not change size or shape. For example, a child going down a slide illustrates a slide, a merry-go-round illustrates a turn, and a mirror illustrates a flip.
• Students investigate the shapes that they can see when they place a mirror on a square pattern block.

• Students use maps of their community to find various ways to get from school to the park. They use letters and numbers to describe the location of the school and that of the park.
• Students create a map based on The Little Gingerbread Man, showing where each of the people in the story lives.
• Students describe how to get from one point in the school to another and try to follow each others' directions.

• In connection with a unit on dinosaurs in science, students discuss the different ways in which the size of dinosaurs can be described. They decide to measure the size of a dinosaur's footprint in two ways: by using string to measure the distance around it and by using base ten blocks to measure the space inside it.
• Pairs of students investigate the many different designs that they can make using unit squares and 1/2-unit right triangles. They record their results on dot paper.

• Working in committees (e.g., Bureau of Streets, Housing Commission), the class designs and builds a "model city."
• Students take a "geometry walk" through their school, looking for examples of specific shapes and concepts.
• Students create geometric patterns using potato prints.
• Students decorate their classroom for the winter holidays using geometric shapes.

With respect to spatial relationships, students in these grade levels continue to examine direction, orientation, and perspectives of objects in space. They are aware of relative positions of objects: Which walls are opposite to each other? What is between the ceiling and the floor? Students also expand their understanding of congruence, similarity, and symmetry. Now they can identify congruent shapes, draw and identify a line of symmetry, and describe the symmetry found in nature.

Students are also extending their understanding of properties of geometric figures. Now they are ready to discuss these more carefully and to begin relating different figures to each other more carefully. By experimenting with concrete materials, drawings, and computers, they are able to discover properties of shapes such as that all squares have four equal sides. They use the language of properties to describe shapes and to explain solutions for geometric problems, but they are not yet able to deduce new properties from old ones or consider which properties are necessary and sufficient for defining a shape. They recognize the concepts of point, line, line segment, ray, plane, intersecting lines, radius, diameter, inside, outside, and on a figure. They extend the shapes they can identify to include ellipses, pentagons, and octagons.

Students continue to explore geometric transformations. Using concrete materials, pictures, and computer graphics, they explore the effects of transformations on shapes.

Coordinate geometry continues to be another focus of study in these grades. Students create and interpret maps, using information found in tables and charts. Some grids use only numbers at these grade levels, while others use a combination of letters and numbers.

The geometry of measurement begins to take on more significance in grades 3 and 4, as students focus more on the concepts of perimeter and area. Students learn different ways of finding the perimeter of an object: using string around the edge and then measuring the length of the string, using a measuring tape, measuring the length of each side (if it is straight!) and then adding the measures together, or using a trundle wheel. They also develop strategies for finding the area of a figure.

Students extend their use of geometric modeling in these grades. For example, they may use geometric shapes split into congruent regions to build understanding of fraction concepts. They may draw diagrams consisting of points and lines to show who plays who in a chess tournament. They continue to build three-dimensional models of shapes, to draw two- and three-dimensional shapes with increasing accuracy, and to use computers to help them analyze geometric properties.

Students' use of reasoning continues to provide opportunities to connect geometry to the process standards, to other areas of mathematics and to the real world. Students explain how they have approached a particular problem, share results with each other, and justify their answers.

Students in third and fourth grade are still dealing with geometry in a qualitative way but are beginning to adopt more quantitative points of view toward geometry. They are able to use their natural curiosity about the world to expand their understanding of geometric concepts and spatial sense.

Building upon the K-2 expectations, experiences in grades 3-4 will be such that all students:

A. explore spatial relationships such as the direction, orientation, and perspectives of objects in space; their relative shapes and sizes, and the relations between objects and their shadows or projections.

• Students compare the sizes of the many shapes found in the classroom, such as the heights of students or the areas of students' hands.
• The teacher holds up a shape or describes a shape. Students locate the shape hidden in a box or bag, without looking at the shapes.
• Students explore what happens to a shadow when a square is held at various angles to a beam of light. They continue their investigation with other two- and three-dimensional figures.
• Students measure the length of the shadow of a stick at half-hour intervals.
• Students trace the faces of a solid on a transparency and then challenge each other to identify the solid. They check their guess by bringing the solid to the overhead and placing it on each face in turn.
• Students predict the positions of three students from different points of view (perspective). For example, from the front of the room, they might see Joe on the left, Rhonda in the middle, and Carly on the right. From the back of the room, the positions would be reversed.

• Students make a collection of real natural shapes, including a wide variety of three-dimensional shapes such as fruits and vegetables, shells, flowers, and leaves. They describe the symmetry found in these shapes. They also look for spheres, spirals, helices, three-way junctions, meanders, tubes, and branching patterns (explosions).
• Students find objects that are self-similar, i.e., that contain copies of a basic motif which is repeated endlessly at even smaller sizes. These smaller replicas are the same shape. Fractal structures can be found in cauliflower, broccoli, Queen Anne's lace, and trees.
• Students look for examples of congruent figures (same size and shape) in the environment.
• Students use scale models of airplanes as an introduction to the concept of similarity. They should recognize that figures that have the same shape but are different sizes are similar figures.

• Students look for a "Shape of the Day" throughout the school day, recording the number of times that the shape is seen.
• Students look for lines in the classroom, identifying pairs of lines that are parallel, that intersect, or that are perpendicular.
• Students use Logo to describe the path made by a turtle as it goes around different geometric shapes.

• Students make a chart or bar graph showing how many squares, rectangles, triangles, etc. they find in their classroom.
• Students "walk" a shape and have other students guess the shape.
• Students classify shapes according to whether they contain right angles only, all angles smaller than a right angle, or at least one angle larger than a right angle.
• One person thinks of a shape. The others ask questions about its properties, trying to guess it. For example, "Does it have a right angle?"

• Students investigate the shapes found in their lunches and then discuss how the shapes change as they nibble away. For example, can you change a four-sided sandwich into a triangle?
• Students investigate how to use four triangles from the pattern blocks to make a large triangle, a four-sided figure, and a six-sided figure.

• Students use grids of squares, triangles, and hexagons to create colorful designs. They discuss why these polygon shapes fit together like a puzzle.
• Students use Unifix cubes or pattern blocks to create designs. They then discuss how many blocks they used (area) and the distance around their design (perimeter).

• Students create borders from a single motif, using slides, flips, and turns to repeat the motif.
• Students use stuffed animals or two-sided paperdolls to show movements in the plane: slides, flips, and turns. They discuss how all slides (or flips or turns) are alike.
• Students discuss transformations found in nature, such as the symmetry in the wings of a butterfly (a flip), the way a honeycomb is formed (slides of hexagons), or the petals of a flower (turns).
• Students create quilt designs by using geometric transformations to repeat a basic pattern.

• Students create Logo procedures for drawing rectangles or other geometric figures.
• Students draw maps for stories they have read, using coordinates to identify the locations of critical events or objects.
• Students find the lengths of paths on a grid, such as the distance from Susan's house to school.

• Students in a class discuss how to describe the size of a truck. Some suggestions include the length of the truck, its height (very important if there is an underpass), its cargo capacity (volume), or its weight (important for assessing taxes).
• Each pair of students is given a pattern to cut out of oaktag and fold up into a three-dimensional shape. They are asked to measure the shape in as many ways as they can. They report their findings to the class.

• Students investigate the natural shapes which are produced by the processes of growth and physical change. They identify some of the simple basic shapes that occur over and over again in more complex structures.
• Students make a bulletin board display of "Shapes in the World Around Us."

Furthermore, studying geometry provides opportunities for divergent thinking and creative problem solving as well as for developing students' logical thinking abilities. Geometric concepts and representations also help students to better understand number concepts. Topics in geometry are also particularly well-suited for use when addressing the four process standards.

Students' experiences in learning geometry should help them to perceive geometry as having a dynamically important role in their environment and not merely as learning vocabulary, memorizing definitions and formulas, and stating properties of shapes. Students, working in groups or independently, should explore and investigate problems in two and three dimensions, make and test conjectures, construct and use models, drawings, and computer technology, develop spatial sense, use inductive and deductive reasoning, and then communicate their results with confidence and conviction. They should be challenged to find alternative approaches and solutions.

In their study of spatial relationships, students in grades 5 and 6 further develop their understanding of shadows, projections (e.g., top, front, and side views), perspectives (e.g., drawings made on isometric dot paper), and maps. They also consolidate their understanding of the concepts of symmetry (both line and rotational), congruence, and similarity.

Students expand their understanding of properties of geometric figures by using models to develop the concepts needed to make abstractions and generalizations. They focus on the properties of lines and planes as well as on those of plane and solid geometric figures. Students at this age begin to classify geometric figures according to common properties and develop informal definitions.

Still focussing on models, drawings, and computer graphics, students expand their understanding of geometric transformation, including translations (slides), reflections (flips), rotations (turns), and dilations (stretchers/shrinkers). At these grade levels, the connections between the transformations and congruence, similarity, and symmetry are explored. Students also begin to use coordinate geometry to show how figures change under transformations, using all four quadrants of the coordinate plane (positive and negative numbers).

Students develop greater understanding of the geometry of measurement in these grade levels, as they develop strategies for finding perimeters, areas (of rectangles and triangles), volumes, surface areas, and angle measures. The emphasis at this level is on looking for different ways to find an answer, not on using formulas. Students use actual materials for many problems, look for patterns in their answers, and form conjectures about general methods that might be appropriate for certain types of problems. Students use what they are learning about areas to help them develop understanding of the Pythagorean Theorem.

Students continue to use geometric modeling to help them solve a variety of problems. They explore patterns of geometric change as well as those involving number patterns. They use geometric representations to assist them in solving problems in discrete mathematics. They use the tools of concrete materials, drawings, and computers to help them visualize geometric patterns.

Students in these grade levels are beginning to develop more sophisticated reasoning skills. In studying geometry, they have many opportunities to make conjectures based on data they have collected and patterns they have observed. This inductive reasoning can then be related to what they already know; students should be encouraged to explain their thinking and justify their responses.

Throughout fifth and sixth grade, students use concrete materials, drawings, and computer graphics to increase the number of geometric concepts with which they are familiar and to explore how these concepts can be used in geometric reasoning. Students' natural curiosity about the world provides ample opportunities for linking mathematics with other subjects. The continued experience with two- and three-dimensional figures provided at these grade levels helps students to build the firm foundation needed for the more formal geometry of the secondary school.

Building upon the K-4 expectations, experiences in grades 5-6 will be such that all students:

K. relate two-dimensional and three-dimensional geometry using shadows, perspectives, projections, and maps.

• Students use centimeter cubes to construct a building such as the one pictured below. They then represent their building by drawing the base and telling how many cubes are in each "tower."
  

  2	2	1
  1	1

• Students put three or four cubes together to make a solid and draw two different projective views of the solid on triangle dot paper, such as those shown below.

• Students copy pictures of solids drawn on triangle dot paper such as the one below, build the solids, and find their volumes.
  
• Students predict and sketch the shapes of the faces of a pyramid, or, given a flat design for a box, predict what it will look like when put together.

• Students use square pattern blocks to make different sizes of squares. They record their results and look for patterns. They make conjectures about the relationship between the length of the sides and the number of squares needed to make the square. They repeat the activity using triangles and rhombuses.
• Students compare different Logo procedures for drawing similar rectangles.
• Students look for examples of congruent figures (same size and shape) in the environment.
• Students explore symmetry by looking at the designs formed by placing a mirror on a pattern block design somewhere other than the line of symmetry, by folding paper more than one time. They identify the symmetry in wallpaper or giftwrap designs. They also identify the rotational symmetry found in a pinwheel (e.g., 90). (The figure matches itself by turning rather than by flipping or folding.)
• Students build scale models to investigate similarity. They should recognize that figures that have the same shape but are different sizes are similar figures.
• Students look for examples and create self-similar shapes, ones which have miniature versions of the whole within themselves. For example, a head of broccoli is self-similar. The head itself looks like a "tree," as does each stalk and each branch from each stalk. These self-similar shapes are fractals. Other examples include cauliflower, Queen Anne's lace, and trees. Below is an example of a Fractal shape that a sixth-grader might create.
  

• Students are given a sheet of 3 x 3 dot paper grids. They find and draw as many noncongruent quadrilaterals as they can, using a different set of nine dots for each figure.
  

  They then sort the sixteen quadrilaterals thus formed in different ways, including concave vs. convex.

• Students first trace squares, rectangles, parallelograms, rhombi, trapezoids, kites, and arrowheads onto transparencies. They then draw the lines of symmetry for a given figure. They rotate the transparencies and compare them to the original figure to investigate such properties as: the number of congruent sides in the figure, the number of parallel sides in the figure, whether the diagonals are congruent, whether the diagonals bisect each other, whether the diagonals are perpendicular, and whether the figure has half-turn symmetry. They write about their findings and explain their reasoning.
• Students use Logo to investigate the sum of the measures of the exterior angles of any polygon (360) and the angle measure of each exterior angle of a regular polygon.
• Students are given examples and nonexamples of kites. They compare and contrast the examples and give other examples and nonexamples of kites. Then they write an informal definition of kite.
• Students use straws cut to five different lengths (1 in - 5 in) to form as many triangles as they can, recording the results. They sort the triangles into groups with 0, 2, or 3 equal sides and labeled the groups as scalene, isosceles, and equilateral triangles.

• Students use index cards with slits cut in them to build models of two planes that are parallel or that intersect (in a line)..
• Students use toothpicks to explore how two lines might be related to each other (parallel, intersecting, perpendicular, the same line).
• Students find examples of parallel lines and planes, perpendicular lines and planes, and intersecting lines and planes with different angles in their environment.

• Students build a design on a geoboard, sketch their design, move the pattern to a new spot by using a specified transformation, and sketch the result.
• Students first investigate wallpaper, fabric, and gift wrap designs. They then create a template for a unit figure which they will use to create individual border designs for their classroom. Each student presents her/his design to the class, describing the transformations used to create the design.
• Working in small groups, students tape two pattern blocks together to make a new shape. They then make oaktag copies of their new shape, using these to tile a portion of their desktop. Each group presents their results. The teacher then asks the students to compare the results of the different groups and identify examples of the different transformations.
• Students investigate how transformations affect the size, shape, and orientation of geometric figures. A reflection or flip is a mirror image. A translation or slide moves a figure a specified distance and direction along a straight line. A rotation or turn is a turning motion of a specified amount and direction about a fixed point, the center. These transformations do not change the size and shape of the original figure. However, a dilation enlarges (stretcher) or reduces (shrinker) a figure, producing a figure the same shape but a different size.
• Students look at the world around them for examples of transformations. For example, a child going down a slide illustrates a translation, a merry-go-round illustrates rotation, and a mirror illustrates reflection. Gift wrap and wallpaper designs can also illustrate transformations.

• Students are given a transparent grid and a worksheet with triangles drawn on it. They first find the base, height, and area of each triangle, using the grid and recording their findings in a table. They then discuss patterns that they see, developing their own formula.
• Students might find perimeter by cutting a string long enough to fit around a figure exactly and then measuring the string. For something large, like the classroom, they might construct and use a trundle wheel. They might also surrounding an object with square inch tiles, or measure each side and then add.
• Students first estimate the perimeter (or area, volume, or surface area) of classroom objects, then sketch the object, measure it, determine the perimeter, and compare their answers to their estimates. Objects which might be used include books, desks, a closet, a door, or windows.
• Students are given a sheet with rectilinear figures (only right angles) on it and a centimeter grid made from a piece of transparency film that they can place over each figure. By counting the squares,, they can find areas. By counting the number of units around a figure, they can determine its perimeter.
• Students use centimeter cubes to build a structure such as the one shown below and then count the cubes to find the volume of the structure.
  

• Students construct squares on each side of a right triangle on a geoboard and find the area of each square. They repeat this process using several different triangles, recording their results in a table. Then they look for patterns in the table.
• Students measure the distance diagonally from first to third base on a baseball field and compare it to the distance run by a player who goes from first to second to third. They note that it is a shorter distance diagonally across the field than it is along the sides. They repeat this type of measuring activity for pictures of squares, noting their results in a table and discussing any patterns they see. For example, students may notice that the diagonal is not quite one and a half times the length of the side of the square.

• 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 whole are evident within it.

• Students design a three-dimensional geometric sculpture.
• Groups of students working together design a doghouse to be built from a 2m x 2m sheet of plywood. They construct a model of their design from oaktag.

• Students use computer programs like The Geometric Pre-Supposer to explore the relationships of sides of polygons or properties of quadrilaterals.
• Students select a country or culture, research the use of specific geometric patterns in that culture, and make a report to the class.
• Specific manipulatives that may be helpful for geometry include pattern blocks, color tiles, centimeter cubes, tangrams, geoboards, links, and templates. Computer programs such as Logo, Shape Up!, Elastic Lines, Building Perspective, or The Factory may also be helpful.

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

Students continue to use geometric modeling to represent problem situations in a variety of different areas. Drawings of various types are particularly useful to students in understanding the context of a 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 by 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. 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 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 to students.

Building upon the K-6 expectations, experiences in grades 7-8 will be such that all students:

K. 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 (two-dimensional flat view from top, front, and side; three-dimensional corner view; and map view showing the base of the building with the number of cubes in each stack) and to make their own drawings of solids.
• 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 circles and rectangles to make three-dimensional models of cylinders.
• Students use clear liter boxes made of 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 water forms a square, a rectangle that is not a square, a trapezoid, a hexagon, and more.

• 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 objects in the room.
• Students use compasses and straightedges to construct congruent line segments and angles.

• Students use toothpicks (1-5 on a side) to construct as many different types of triangles as possible. They record their findings in a table, showing how many triangles are scalene, isosceles, equilateral, and right. 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
  			--  4 sides				--  all sides of equal length
  			--  all angles congruent		--  opposite angles congruent
  			--  opposite sides parallel		--  simple closed curve
  

• Students 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 develop "family trees" to demonstrate the relationships among properties. For example, they might build a family tree to show a logical relationship of ideas leading to the angle sum for a quadrilateral. They might also build a family tree that shows the relationship among different types of quadrilaterals.

• Students build a model of a cube, connect a midpoint of an edge with another midpoint of an 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.

• Students analyze the designs found in tile floors in Spain, identifying figures that represent translations, reflections, and rotations. They then generate their own tile designs using basic geometric shapes.
• Students place transparent grids over line drawings and copy the drawings only grids of different sizes and shapes. They discuss what kinds of grids result in similar figures, what kinds of grids give congruent figures, and what kinds of grids distort the figures.
• 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 (x,y)--> (x, -y).

• Students use a paper fastener to connect two models of rays to form angles of different sizes.
• Students bring in cereal boxes from home and cut them apart to more easily determine the surface area.
• Students are given a cut-out parallelogram 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 process is repeated for a trapezoid.
• Students find the volumes of different backpacks by using familiar solids to approximate their shape. They then compare their results and write about which backpack they think would be "best" and why.

• Students draw right triangles on graph paper with legs of specified lengths and measure the length of the hypotenuse. They record their results in a chart and look for patterns.
• Students create a small isosceles right triangle in the middle of a geoboard and then build squares on each side of the triangle. They record their results on dot paper and look for a relationship among the areas of the three squares.
• Students use tangram pieces to build squares on each side of the middle-sized right triangle. They then describe the relationship among the areas of the three squares.

• Students use the reducing and/or enlarging feature on a copier to explore repeated reductions/enlargements by the same factor (iteration).
• Students investigate the natural shapes that are produced by growth. They look at how nature produces complex structures in which basic shapes occur over and over. For example, spider webs, honeycombs, pineapples, pinecones, nautilus shells, and snowflakes grow larger in a systematic way (iteration).
• Students approximate the age of a tree by looking at its rings.
• 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.

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

  		[Graphic Not Available]

• 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)/2) and its application to architecture (such as the Parthenon), design 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.
• As an interdisciplinary project, students plan a geometry safari to any country in the world where the U.S. dollar is not the major currency. Their budget of $10,000 must cover a three-week trip for four people. They map out a daily itinerary, showing distance in standard and metric units. They create a model of some structure or event for which the country is famous. They make a model of the country's flag and give examples of arts, crafts, and architecture in the country. They discuss transformations found in the folk tunes and music of the country. They relate the topography of the country to fractals. They write brief biographies of famous mathematicians, scientists, writers, artists, and explorers from the country.
• Students take a geometry walk around the school, looking for examples of geometric shapes and concepts. They record their results by sketching the object they find, giving its location, and describing its use. They follow up their walk by discussing why different objects are shaped as they are. For example, why are bike wheels shaped like circles instead of squares? Why are manhole covers circular? Why are traffic signs of different shapes? Finally, they write a note to a friend in their journals about their geometry walk.

• Students use the computer program The Factory to investigate transformations.
• Students use a computer program such as The Geometric 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 then flies all of the kites, recording the heights of each by using a clinometer and similar triangles.

Deductive reasoning is highly dependent upon 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 viewpoints, widely used internationally, are those of coordinate geometry, transformational geometry, and vector geometry. The interplay between geometry and algebra strengthens 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 specific problems are often solved more easily in one or another of these systems.

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. Work with abstract ideas can then draw upon these experiences and intuitions.

The goal of high school geometry includes applying geometric ideas to real 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 relatively simple 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 pertinent to students because of computer graphics. 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 may 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 drawn 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.

Building upon the K-8 expectations, experiences in grades 9-12 will be such that all students:

U. understand and apply properties involving angles, parallel lines, and perpendicular lines.

• Students make tessellations with triangles, observing parallel lines, congruent angles, congruent triangles, similar triangles, etc.
• 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. 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 use the congruence of alternate interior angles and vertical angles to identify all of the congruent angles in a drawing composed of parallel lines and transversals.

• 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 three-dimensional solids. They cut out a plane figure from posterboard, punch two holes in it, 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.
• Students use isometric dot paper to sketch figures made up of cubes. They also sketch top, front, and side views (projections) of the figure.

  		[Graphic Not Available]

• 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 an initial corner (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 by using reflections they can construct a straight line between the two points that crosses the given line at the desired location.
• Students use cutouts of different polygons (different types of triangles, assorted quadrilaterals, and regular polygons) to investigate which ones can be used to cover a plane surface with no overlaps and no gaps (i.e., tessellations). They record their results in a table (including a column to record the measures of the interior angles) and look for patterns.
• 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 write matrix representations for various polygons in the plane. Then they make conjectures concerning how they can tell whether two points in the figure are on the same horizontal or vertical line.
• 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 (x,y)--> (x, -y).

• Students fold paper squares along a diagonal and shade an area less than or equal to one-half of the area of the square. They discuss whether there is a difference in the areas they chose to shade. Then they find the equation of the line along which they folded and describe the shaded area as an inequality.
• 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 investigate and solve linear programming problems, such as the following.
• A lumber company can convert logs into either lumber or plywood. In a given week, the mill can turn out 400 units of production. Each week, 100 units of lumber and 150 units of plywood are required by regular customers. The profit on a unit of lumber is $20. The profit on a unit of plywood is $30. How many units of each type should the mill produce in order to maximize profit?

• Students bring in boxes and cans from home, order them by estimated volume and surface area, determine their actual surface areas and volumes by measuring and computing, and compare their results to their estimates.
• 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 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.

• 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.

• Students visit a construction site where the "framing" step of the 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 in each 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 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 solutions for the company. They report their results in letters written individually to the company president.

• Having developed the idea that a parallelogram is a quadrilateral with two pairs of parallel sides, 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 explain 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 to investigate what happens when consecutive midpoints of a quadrilateral are connected in order. They state one 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 duplicate the beginning stages of a Fractal construction in the plane and analyze the sequence of perimeters and that of 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. They then graph the results, an exponential function showing either decay or growth.
• Students find examples of natural shapes that are produced by growth. They discuss 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). Measuring the age of a tree by looking at its rings illustrates approximation. Clouds, trees, and Queen Anne's lace are examples of fractals, where miniature versions of the whole are evident within it.
• 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 old 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.

• 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 90 angles). 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 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 committees having one person in common. Most groups use letters to represent each individual and made a table showing the members of each committee:
  

  			ABC		ADE		BDF		CEF

  The teacher asks the students whether the rules for making up committees remind them of anything in geometry; several students suggest that they sound like the axioms for a geometric system, with each person representing a point and each subcommittee representing a line. A person is on a committee if the corresponding point is on a line. They then proceed to draw a diagram (model) for their solutions, using these ideas. The resulting geometry is an example of a finite geometry.

• 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.
• Students copy geometric designs using compass and straightedge and generate their own designs.
• Students investigate wallpaper patterns, classifying them according to the transformations used.