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 3-4
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
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
Standard 7 - Geometry and Spatial Sense - Grades 3-4
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 grades 3 and
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
- At half-hour intervals, students measure the
length of the shadow of a stick stuck vertically into the
- 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
- 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
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
- 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
- Students use the computer language Logo to
describe the path made by a turtle as it goes around different
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
- 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
- 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
7. Explore geometric transformations such as
rotations (turns), reflections (flips), and translations
- 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
- 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
- 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.
Brown, Marcia. Listen to a Shape. New York: Franklin
Charosh, Mannis. Ellipse. New York: Thomas Y. Crowell,
Yenawine, Phillip. Shapes. New York: Delacourte Press,
Logo. Many versions of Logo are commercially
Burton, G. et al. Curriculum and Evaluation Standards for
School Mathematics: Addenda Series:
ThirdGrade Book. Reston, VA: National Council of
Teachers of Mathematics, 1992.
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