Students can develop strong spatial sense from consistent experiences in classroom activities that use a 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 kindergarten through second grade, the emphasis is on qualitative, not quantitative, properties of geometric objects. Students are at the visualization level of geometric thinking, where they perceive figures as "wholes". They recognize squares and rectangles, but perhaps not that squares are a special case of rectangles. To enrich and develop their geometric thinking, children at these grade levels need to explore geometry using a variety of physical objects, drawings, and computer tools. They work with solids, pattern blocks, templates, geoboards, and computer drawing tools to develop their understanding of geometric concepts and their spatial sense. They construct models and drawings to experiment with orientation, position, and scale, and to develop visualization skills. Students begin to develop a geometric vocabulary. A sample unit on geometry for the second-grade level can be found in Chapter 17 of this Framework.

In their study of spatial relationships, students focus on developing their understanding of objects in space. They discuss and describe the relative positions of objects using phrases like "in front of" and "on top of." They describe and draw three-dimensional objects in different relative locations. They compare and contrast shapes, describing the shapes of the faces and bases of three-dimensional figures. They discuss symmetry and look for examples of symmetry in their environment. They look for shapes that are the same size and shape (congruent) or the same shape but different sizes (similar). They use mirrors to explore symmetry.

In beginning their study of properties of geometric figures, students look for shapes in the environment, make models from sticks and clay or paper and glue, and draw shapes. They sort objects according to shape. They recognize, classify, sort, describe, and compare geometric shapes such as the sphere, cylinder, cone, rectangular solid, cube, square, circle, triangle, rectangle, hexagon, trapezoid, and rhombus. They describe the angle at which two edges meet in different polygons as being smaller than a right angle, a right angle, or larger than a right angle. They discuss points, lines, line segments, intersecting and non-intersecting lines, and midpoints of lines.

Students begin looking at geometric transformations by using concrete materials such as paper dolls to model slides (translations), flips (reflections), and turns (rotations). Students put shapes together to make new shapes and take shapes apart to form simpler shapes. Students work on spatial puzzles, often involving pattern blocks or tangrams. They look for plane shapes in complex drawings and explore tilings. They divide figures into equal fractional parts, for example, by folding along one, two, or three lines.

Coordinate geometry in grades K-2 involves describing the motion of an object. Students make maps of real, imaginary, or storybook journeys. They describe the location of an object on a grid or a point in aplane using numbers or letters. They give instructions to an imaginary "turtle" to crawl around the outline of a figure.

Students in these grades also begin to explore the geometry of measurement. In kindergarten, students discuss and describe quantitative properties of objects using phrases like "bigger" or "longer." They order objects by length or weight. In first and second grade, they quantify properties of objects by counting and measuring. They determine the areas of figures by cutting them out of grid paper and counting the squares. They measure the perimeter of a polygon by adding the lengths of all of the sides.

Students begin to explore geometric modeling by constructing shapes from a variety of materials, including toothpicks and clay, paper and glue, or commercial materials. They use templates to draw designs, and record what they have constructed out of pattern blocks and tangrams. They fold, draw, and color shapes. They copy geoboard figures, and construct them both from memory and by following oral or written instructions. They may also use geometric models, such as the number line, for skip counting or repeated addition.

Geometry provides a rich context in which to begin to develop students' reasoning skills. Students apply thinking skills in geometric tasks from identifying shapes to discovering properties of shapes, creating geometric patterns, and solving geometric puzzles and problems in a variety of ways. They create, describe, and extend geometric patterns. They use attribute blocks to focus on the similarities and differences of objects.

Geometry provides a unique opportunity to focus on the First Four Standards, especially Standard 2 which stresses the importance of making connections to other mathematical topics. For example, students often use their understanding of familiar shapes to help build an understanding of fractions. Teachers in grades K-2 need to plan classroom activities that involve several mathematical processes and relate geometry to other topics in mathematics. Geometry should not be taught in isolation; it should be a natural and integrated part of the entire curriculum.

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 kindergarten and grades 1 and 2.

Experiences will be such that all students in grades K-2:

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.

• Blindfolded students are given real objects to touch and then, with the blindfolds removed, select the objects from a collection of visible objects.

• Students work through the Will a Dinosaurs Fit? lesson that is described in the First Four Standards of this Framework. They discuss the size of the different dinosaurs and arrange them from smallest to largest.

• Students predict what shape will result when a small piece is cut out of a folded piece of paper in different ways (along a diagonal, across a fold or a corner, or in the center) and the paper is then unfolded.

• Students compare the sequence of objects seen from different points of view. For example, from the classroom window, the swings are to the left of the monkeybars, but the relationship is reversed if the objects are viewed from the blacktop facing the classroom.

• Students predict and draw what the shadow of an object placed between a light and a screen will look like.

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

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

• Students explore symmetry by using mirrors with pattern blocks or by folding paper or by making inkblot designs. Students find the lines of symmetry in the letters of the alphabet and in numerals. They fold paper and cut out symmetric designs. They identify the symmetry in wallpaper or giftwrap designs.

• Students use different size dolls and action figures as an introduction to the concept of similarity (same shape, different size).

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

• Students predict what shape they will see when they make various impressions of 3-dimensional objects in sand. For example, the top of a cylinder forms a circle, its side formsa rectangle.

• Students outline a triangle, a square, and a circle on the floor with string or tape. Then they walk around each figure, chanting a rhyme, such as "Triangle, triangle, triangle, 1, 2,3, I can walk around you as easy as can be," and counting the sides as they walk.

• Students work through the Shapetown lesson that is described in the First Four Standards of this Framework. They explore properties of two-dimensional shapes by applying the fundamental concepts of Venn diagrams.

• Some students use Muppet^TM Math to work with Kermit's geometric paintings, while others use Shape Up! to compare everyday objects to geometric shapes.

4. Use properties of three- and two-dimensional shapes to identify, classify, and describe shapes.

• Students make shapes with their fingers and arms.

• Students listen to and look at the book The Shapes Game by Paul Rogers. Each page shows a different shape and many of the things in the world that have that shape. As each page is read, the children find other objects in the room that have the same shape.

• Students listen to and draw illustrations for the story The Greedy Triangle by Marilyn Burns.

• A good open-ended assessment for this critical indicator is to ask students to sort a collection of shapes into groups, explaining their reasoning. Some groups they might consider include "all right angles" or "four-siders." The teacher should encourage the students to invent appropriate group names and to use informal language to describe the properties, and should record the students' responses to look for progress over time.

• A more traditional, but still useful, assessment strategy is to ask students to sort pictures cut from magazines according to shape. This more focused task will generate information about the students' ability to recognize and differentiate among shapes.

• Students make class books shaped like a triangle, a rectangle, a square, and a circle. They fill each book with pictures of objects that have the shape of the book.

• Students turn a geometric shape into a picture. For example, a triangle might become a tower, a clown face, or the roof of a house.

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

• Students use tangram pieces to construct triangles, rectangles, squares, and other shapes.

• Students investigate which pattern block shapes can be formed from the equilateral triangles, recording their results in pictures and on a chart.

• Students work in groups to decide how to divide a rectangular candy bar among four people. The students then compare the various ways that each group solved the problem.

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

• Students use Unifix cubes or pattern blocks to create colorful designs. They then discusshow many blocks they used (area) and the distance around their design (perimeter). They also discuss why these polygon shapes fit together like a puzzle.

• Students use different shapes to make quilt patterns.

• During free play time, students use pattern blocks to make different space-filling designs. They record any patterns that they especially like, using templates or drawing around the blocks.

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

• Students look at the world around them for examples of changes in position that do not change size or shape. For example, a child going down a slide illustrates a slide, a merry-go-round or hands on a clock illustrate a turn, and a mirror illustrates a flip.

• Students look through and discuss the no-text book Changes, Changes by Pat Hutchins. In it, a man and a woman use the same set of building blocks to transform a house into a fire engine, then a boat, a truck, and back to a house. The students tell the story and then draw pictures to show how the blocks changed from one object to another.

• Students investigate the shapes that they can see when they place a mirror on a square pattern block.

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

• Students use maps of their community to find various ways to get from school to the park. They use letters and numbers to describe the location of the school and that of the park.

• Students create a map based on the familiar story of The Little Gingerbread Man, showing where each of the people in the story lives.

• Students describe how to get from one point in the school to another and try to follow each others' directions.

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

• In connection with a unit on dinosaurs in science, students discuss the different ways in which the size of dinosaurs can be described. They decide to measure the size of a dinosaur's footprint in two ways: by using string to measure the distance around it and by using base ten blocks to measure the space inside it.

• Pairs of students investigate the many different designs that they can make using unit squares and 1/2-unit right triangles. They record their results on dot paper.

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

• Students take a "geometry walk" through their school or their neighborhood, looking for examples of specific shapes and concepts.

• Students create geometric patterns using potato prints.

• Students decorate their classroom for the winter holidays using geometric shapes.

• Students examine and discuss geometric patterns found in works of art.

References

Burns, Marilyn. The Greedy Triangle. New York: Scholastic, Inc., 1994.

Hutchins, Pat. Changes, Changes. New York: MacMillan, 1987.

Rogers, Paul. The Shapes Game. New York: Henry Holt and Company, 1989.

Software

Muppet^TM Math. Jim Henson Productions. Sunburst Communications.

Shape Up! Sunburst Communications.

General Reference

Burton, G. and T. Coburn. Curriculum and Evaluation Standards for School Mathematics: Addenda Series: Kindergarten Book. Reston, VA: National Council of Teachers of Mathematics, 1991.

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.