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."
  

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• 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.