STANDARD 7 - GEOMETRY AND SPATIAL SENSE
All students will develop spatial sense and an ability to use
geometric properties and relationships to solve problems in
mathematics and in everyday life.
Standard 7 - Geometry and Spatial Sense - Grades K-2
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
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
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.
Standard 7 - Geometry and Spatial Sense - K-2
Indicators and Activities
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
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,
3. Explore properties of three- and
two-dimensional shapes using concrete objects, drawings, and
- 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
- 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
MuppetTM 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
- 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
- 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
- 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
- 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
- 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
- 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
7. Explore geometric transformations such as
rotations (turns), reflections (flips), and translations
- 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'
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
- Students decorate their classroom for the winter
holidays using geometric shapes.
- Students examine and discuss geometric patterns
found in works of art.
Burns, Marilyn. The Greedy Triangle. New York:
Scholastic, Inc., 1994.
Hutchins, Pat. Changes, Changes. New York: MacMillan,
Rogers, Paul. The Shapes Game. New York: Henry Holt and
MuppetTM Math. Jim Henson Productions.
Shape Up! Sunburst Communications.
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.
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