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.