Students can develop strong spatial sense from consistent experiences in classroom activities that use a wide variety of manipulatives and technology. The key components of spatial sense, as identified in the K-12 Overview, are spatial relationships, properties of geometric figures, geometric transformations, coordinate geometry, geometry of measurement, geometric modeling, and reasoning.

In third and fourth grade, students are beginning to move beyond recognizing whole shapes to analyzing the relevant properties of a shape. They continue to use their own observations about shapes and the relations among these shapes in the physical world to build understanding of geometric concepts. Thus, using manipulative materials to develop geometric concepts and spatial sense remains important at these grade levels. Exploring concepts in a number of different contexts helps students to generalize. Students are extending their understanding of cause and effect and their ability to make conjectures. They are particularly interested in Why? Questions such as Why are most rooms shaped like rectangles? offer interesting points of departure for studying geometric concepts. Connections among geometry, spatial sense, other areas of mathematics, and other subject areas provide many opportunities for students to see how mathematics fits into their lives.

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 the relative positions of objects; you might ask Which walls are opposite each other? What is between the ceiling and the floor? Students also expand their understanding of congruence, similarity, and symmetry. They can identify congruent shapes, draw and identify a line of symmetry, and describe the symmetries found in nature. They search for examples in nature where each part of an item looks like a miniature version of the whole (self-similar).

Students are 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. By experimenting with concrete materials, drawings, and computers, they are able to discover properties of shapes and to make generalizations like 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, octagons, cubes, cylinders, cones, prisms, pyramids, and spheres.

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

Using coordinate geometry students create and interpret maps, sometimes making use of 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 focusmore 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 and then adding the measures together, or using a trundle wheel. They also develop non-formula-based strategies for finding the area of a figure.

Geometric modeling allows students to approach topics visually. For example, geometric shapes allow students to build an understanding of fraction concepts as they cut the shapes into congruent pieces. They can use the problem solving skill of drawing geometric diagrams, such as a polygon with its diagonals, to find out how many matches are played in a round robin 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 Standards 1 - 4, to other areas of mathematics, to other disciplines, 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. They are able to use their natural curiosity about the world to expand their understanding of geometric concepts and spatial sense.

The cumulative progress indicators for grade 4 appear below in boldface type. Each indicator is followed by activities which illustrate how it can be addressed in the classroom in grades 3 and 4.

Building upon knowledge and skills gained in the preceding grades, experiences in grades 3-4 will be such that all students:

1. 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 their hands.

• The teacher holds up a shape or describes a shape. Students locate this shape hidden in a box or bag containing a number of shapes, without looking at the shapes.

• Students explore what happens to the shadow of a square when it is held at various angles to a beam of light. They continue their investigation with other two- and three-dimensional figures.

• At half-hour intervals, students measure the length of the shadow of a stick stuck vertically into the ground.

• 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 projector and placing it on each face in turn.

• Students read Ellipse by Mannis Charosh. This one-concept book illustrates ellipses in all of their possible orientations and describes a variety of experiments that the students can perform to better understand the role of perspective in geometry.

• 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 find a perspective from which Rhonda is on the left, Carly is in the middle and Joe is on the right.

2. Explore relationships among shapes, such as congruence, symmetry, similarity, and self-similarity.

• Students make a collection of 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.

• Students find objects that exhibit self-similarity, i.e., that contain copies of a basic motif which is repeated at smaller sizes of the same shape. Examples of such objects arefeathers, the shape of a coastline, chambered nautilus shells, and plants which branch out such as cauliflower, broccoli, Queen Anne's lace, and ferns.

• Students look for examples of congruent figures (same size and shape) in the environment.

• Students use scale models of cars and airplanes to study similarity. They recognize that figures that have the same shape but different sizes are similar.

3. Explore properties of three- and two-dimensional shapes using concrete objects, drawings, and computer graphics.

• 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 the computer language Logo to describe the path made by a turtle as it goes around different geometric shapes.

4. Use properties of three- and two-dimensional shapes to identify, classify, and describe 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 student thinks of a shape. The others ask questions about its properties, trying to guess it. For example, Does it have a right angle?

5. Investigate and predict the results of combining, subdividing, and changing shapes.

• 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 combine tangram pieces to create a variety of shapes.

6. Use tessellations to explore properties of geometric shapes and their relationships to the concepts of area and perimeter.

• Students use square, triangular, and hexagonal grid paper 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 work through the Tiling a Floor lesson that is described in the First FourStandards of this Framework. They discover that the shapes which can be used for tiling must be able to fit around a point without leaving spaces and without overlapping.

7. Explore geometric transformations such as rotations (turns), reflections (flips), and translations (slides).

• 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 create borders from a single simple design element which is repeated using slides, flips, and turns.

• Students study and describe the use of transformations in Pennsylvania Dutch hex signs, and then they design their own.

• 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.

8. Develop the concepts of coordinates and paths, using maps, tables, and grids.

• Students create Logo procedures for drawing rectangles or other geometric figures.

• A good interdisciplinary assessment in both reading and mathematics is to have students draw maps for stories they have read, using coordinates to identify the locations of critical events or objects in the story.

• Students find the lengths of paths on a grid, such as the distance from Susan's house to school.

9. Understand the variety of ways in which geometric shapes and objects can be measured.

• Students discuss how to describe the size of a truck. Some suggestions include the length of the truck, its height (very important to know when it passes under another road), 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.

10. Investigate the occurrence of geometry in nature, art, and other areas.

• Students investigate the natural shapes that 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 bring examples to class and describe the process in writing. Some interesting examples are honeycombs, pinecones, and seashells.

• Students make a bulletin board display of "Shapes in the World Around Us."

• Students read the beautifully illustrated book Listen to a Shape by Marcia Brown. The color photographs in the book move from the occurrence in nature of simple shapes to more complex ones. Children can be asked to describe and draw their favorite shapes innature as a follow-up.

• Students read Shapes by Phillip Yenawine. This carefully selected collection of works from the Museum of Modern Art is analyzed to show how shapes contribute to the images on the canvas. An interesting open-ended assessment activity would be to ask the students to create their own works of art, combining the geometric shapes they know to make similar striking images.

References

Brown, Marcia. Listen to a Shape. New York: Franklin Watts, 1979.

Charosh, Mannis. Ellipse. New York: Thomas Y. Crowell, 1971.

Yenawine, Phillip. Shapes. New York: Delacourte Press, 1991.

Software

Logo. Many versions of Logo are commercially available.

General reference

Burton, G. et al. Curriculum and Evaluation Standards for School Mathematics: Addenda Series: ThirdGrade Book. Reston, VA: National Council of Teachers of Mathematics, 1992.

On-Line Resources

http://dimacs.rutgers.edu/archive/nj_math_coalition/framework.html/

The Framework will be available at this site during Spring 1997. In time, we hope to post additional resources relating to this standard, such as grade-specific activities submitted by New Jersey teachers, and to provide a forum to discuss the Mathematics Standards.