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, often 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 a plane 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 done with pattern blocks and tangrams. They fold, draw, and color shapes. They copy geoboard figures, construct geoboard figures from memory, and construct geoboard figures 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 properties of objects.

Geometry provides a unique opportunity to focus on the process standards, especially the connections standard. Many geometric ideas are used to help students understand concepts in other strands. 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 interrelate geometry with other topics in mathematics. Geometry should not be taught only in isolation; it should be a natural and integrated part of the entire curriculum.

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

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.

• Blindfolded students are given real objects to touch and then must select the object from a collection of visible objects.
• Students imagine being a circle and draw, tell, or write about what they could do and how they might feel.
• Students predict what shape will result when a folded piece of paper is cut and then opened.
• 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 you are standing on the blacktop.
• Students predict and draw what the shadow of an object placed between a light and a screen will look like.

• 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. They fold paper and cut out symmetric designs. They identify the symmetry in wallpaper or giftwrap designs.
• Students use scale models of cars as an introduction to the concept of similarity (same shape, different size).
• Self-similar shapes are 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.

• Students identify the "footprint" made by pressing a face of a three-dimensional solid into clay. Later they identify the blocks that might have been used to make a specific "footprint." They also build, draw, or paint a picture of the creature that might have made a footprint.
• 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.
• Some students use Muppet Math to work with Kermit's geometric paintings, while others use Shape Up! to compare everyday objects to geometric shapes.

• Students make shapes with their fingers and arms.
• Students sort a collection of shapes into groups, explaining their reasoning. Some groups to consider include "all right angles" or "four-siders." The teacher encourages the students to invent appropriate group names and to use informal language to describe the properties.
• Students sort pictures cut from magazines according to shape. They then discuss other ways that the pictures could be sorted.
• 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.

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

• Students use Unifix cubes or pattern blocks to create colorful designs. They then discuss how 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.

• 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 illustrates a turn, and a mirror illustrates a flip.
• Students investigate the shapes that they can see when they place a mirror on a square pattern block.

• 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 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.

• 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.

• Working in committees (e.g., Bureau of Streets, Housing Commission), the class designs and builds a "model city."
• Students take a "geometry walk" through their school, 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.