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.

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• 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.