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."