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 5-6
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.
Informal geometry and spatial visualization are vital aspects of a
mathematics program for grades 5 and 6. Middle school students
experience the fun and challenge of learning geometry through creating
plans, building models, drawing, sorting and classifying objects, and
discovering, visualizing, and representing concepts and geometric
properties. Students develop the understanding needed to function in
a three-dimensional world through explorations and investigations
organized around physical models.
Studying geometry also provides opportunities for divergent
thinking and creative problem solving while developing students'
logical thinking abilities. Geometric concepts and representations
can help students better understand number concepts while being
particularly well-suited for addressing the First Four Standards:
problem solving, reasoning, making connections, and communicating
Students' experiences in learning geometry should help them
perceive geometry as having a dynamically important role in their
environment and not merely as the learning of vocabulary, memorizing
definitions and formulas, and stating properties of shapes. Students,
working in groups or independently, should explore and investigate
problems in two and three dimensions, make and test conjectures,
construct and use models, drawings, and computer technology, develop
spatial sense, use inductive and deductive reasoning, and then
communicate their results with confidence and conviction. They should
be challenged to find alternative approaches and solutions.
In their study of spatial relationships, students in grades
5 and 6 further develop their understanding of projections (e.g., top,
front, and side views), perspectives (e.g., drawings made on isometric
dot paper), and maps. They also consolidate their understanding of
the concepts of symmetry (both line and rotational), congruence, and
Students expand their understanding of properties of geometric
figures by using models to develop the concepts needed to make
abstractions and generalizations. They focus on the properties of
lines and planes as well as on those of plane and solid geometric
figures. Students at this age begin to classify geometric figures
according to common properties and develop informal definitions.
Still using models, drawings, and computer graphics, students
expand their understanding of geometric transformation,
including translations (slides), reflections (flips), rotations
(turns), and dilations (stretchers/shrinkers). At these grade levels,
the connections between transformations and congruence, similarity,
and symmetry are explored. Students also begin to use coordinate
geometry to show how figures change orientation but not shape
under transformations. For these investigations they use all four
quadrants of the coordinate plane (positive and negative numbers).
Students develop greater understanding of the geometry of
measurement as they develop strategies for finding perimeters,
areas (of rectangles and triangles), volumes, surface areas, and angle
measures. The emphasis at this level should be on looking for
different ways to find an answer, not simply on using formulas.
Students use models for many problems, look for patterns in their
answers, and form conjectures about general methods that might be
appropriate for certain types of problems. Students apply what they
are learning about areas to help them develop an understanding of the
Students continue to use geometric modeling to help them
solve a variety of problems. They explore patterns of geometric
change as well as those involving number patterns. They use geometric
representations to assist them in solving problems in discrete
mathematics. They use concrete materials, drawings, and computers to
help them visualize geometric patterns.
Students in these grade levels are beginning to develop more
sophisticated reasoning skills. In studying geometry, they
have many opportunities to make conjectures based on data they have
collected and patterns they have observed. This inductive reasoning
can then be related to what they already know; students should be
encouraged to explain their thinking and justify their responses.
Throughout fifth and sixth grade, students use concrete materials,
drawings, and computer graphics to increase the number of geometric
concepts with which they are familiar and to explore how these
concepts can be used in geometric reasoning. Students' natural
curiosity about the world provides ample opportunities for linking
mathematics with other subjects. The continued experience with two-
and three-dimensional figures provided at these grade levels helps
students build the firm foundation needed for the more formal geometry
of the secondary school.
Standard 7 - Geometry and Spatial Sense - Grades 5-6
Indicators and Activities
The cumulative progress indicators for grade 8 appear below in
boldface type. Each indicator is followed by activities which
illustrate how it can be addressed in the classroom in grades 5 and
Building upon knowledge and skills gained in the preceding grades,
experiences in grades 5-6 will be such that all students:
11. Relate two-dimensional and three-dimensional
geometry using shadows, perspectives, projections, and
- Students use centimeter cubes to construct a building such
as the one pictured below. They then represent their building by
drawing the base and telling how many cubes are stacked in each
portion of building.
- Students put three or four cubes together to make a solid
and draw two different projective views of the solid on triangle dot
paper, such as those shown below.
- Students copy pictures of solids drawn on triangle dot
paper such as the one below, build the solids, and find their
- Students use circles and rectangles to make
3-dimensional models of cylinders, cones, prisms, and other
- Students predict and sketch the shapes of the faces of a
pyramid, or, given a flat design for a box, predict what it will look
like when put together.
12. Understand and apply the concepts of symmetry,
similarity, and congruence.
- Students compare different Logo procedures for drawing
- Students look for examples of congruent figures (same size
and shape) in the environment.
- Students explore symmetry by looking at the designs formed
by placing a mirror on a pattern block design somewhere other than the
line of symmetry, or by folding paper more than one time. They
identify the symmetry in wallpaper or giftwrap designs. They also
identify the rotational symmetry found in a pinwheel (e.g., 90 degrees).
(The figure matches itself by turning rather than by flipping or
- Students build scale models to investigate similarity.
They recognize that figures which have the same shape but different
sizes are similar.
13. Identify, describe, compare, and classify plane
and solid geometric figures.
- Students are given a sheet of 3 x 3 dot paper grids. They
find and draw as many noncongruent quadrilaterals as they can, using a
different set of nine dots for each figure; altogether sixteen
different quadrilaterals (pictured below) can be formed.
A nice open-ended approach to assessment of their understanding and
comfort with properties of geometric figures is to ask them to sort
these quadrilaterals in different ways, including concave vs. convex,
by angle sizes, by area, by symmetry, and so on. See how many ways
they can devise.
- Students trace a figure onto several transparencies;
figures such as squares, rectangles, parallelograms, rhombuses,
trapezoids, kites, and arrowheads can be used. Then they draw the
lines of symmetry for the figure. They rotate, translate, and flip
the transparencies and compare them to an original transparency to
investigate such properties as: the number of congruent sides in the
figure, the number of parallel sides in the figure, whether the
diagonals are congruent, whether the diagonals bisect each other,
whether the diagonals are perpendicular, and whether the figure has
half-turn symmetry (180 degrees). They write about their findings and
explain their reasoning.
- Students use Logo to investigate the sum of the measures of
the exterior angles of any polygon (360 degrees) and the angle measure
of each exterior angle of a regular polygon.
- Students select straws cut to five different lengths (for
example, from one inch to five inches) and form as many different
triangles as they can, recording the results. They sort the triangles
into groups with 0, 2, or 3 equal sides and label the groups as
scalene, isosceles, and equilateral triangles.
14. Understand the properties of lines and planes,
including parallel and perpendicular lines and planes, and
intersecting lines and planes and their angles of incidence.
- Students use index cards with slits cut in them to build
models of two planes that are parallel or two planes that intersect
(in a line).
- Students use toothpicks to explore how two lines might be
related to each other (parallel, intersecting, perpendicular, the same
- Students find examples of parallel lines and planes,
perpendicular lines and planes, and intersecting lines and planes with
different angles in their environment.
15. Explore the relationships among geometric
transformations (translations, reflections, rotations, and
dilations), tessellations (tilings), and congruence and
- Students read and examine The World of
M.C. Escher or any other collection of M.C. Escher's work to
find and describe the tessellations in them. Transformations of
tessellating polygons are then performed by the students to make their
- Students create a design on a geoboard, sketch their
design, move the pattern to a new spot by using a specified
transformation, and sketch the result.
- Students investigate wallpaper, fabric, and gift wrap
designs. They create a template for a unit figure which they will use
to create individual border designs for their classroom. Each student
presents her/his design to the class, describing the transformations
used to create the design.
- Working in small groups, students tile a portion of their
desktop using oak-tag copies they have cut of a shape they have
created by taping together two pattern blocks. Each group presents
its results. The teacher then asks the students to compare the
results of the different groups and identify examples of the different
- Students investigate how transformations affect the size,
shape, and orientation of geometric figures. A reflection or flip is
a mirror image. A translation or slide moves a figure a specified
distance and direction along a straight line. A rotation or turn is a
turning motion of a specified amount and direction about a fixed
point, the center. These transformations do not change the size and
shape of the original figure. However, a dilation enlarges
(stretches) or reduces (shrinks) a figure, producing a new figure with
the same shape but a different size.
- Students use Tesselmania! software to
manipulate and transform colorful shapes on the computer screen and
create complex tessellations.
- Students continue to look for and report on
transformations they find in the world around them.
16. Develop, understand, and apply a variety of
strategies for determining perimeter, area, surface area, angle
measure, and volume.
- Students are given a transparent square grid to place over
a worksheet with triangles drawn on it. Using the grid to measure,
they find the base, height, and area of each triangle, recording their
findings in a table. They discuss patterns that they see, developing
their own formula to find areas of triangles.
- Students find the perimeter of a figure by taping a string
around it and then untaping and measuring the string. For something
large, like the classroom, they might construct and use a trundle
- Students first estimate the perimeter (or area, volume, or
surface area) of a classroom object, then measure it, determine its
perimeter, and compare their answers to their estimates. Objects
which might be used include books, desks, closets, doors, or
- Students use various same-shape pattern blocks
and arrange as many as are needed around a point to complete a circle.
They discover the size of each angle since there are 360 degrees in one
circle. For example, if it takes six (green) triangles, then each
angle must be 60 degrees (360 degrees ÷ 6).
Students are given a sheet with rectilinear figures (only right
angles) on it, such as the letter "E" at the right, and a
transparent square centimeter grid that they can place over each of
the figures. By counting the squares, they can find the area of each
figure; by counting the number of units around it, they can determine
- Students use centimeter cubes to build a structure such as
the one shown below and then count the cubes to find the volume of the
- Students bring cereal boxes from home, cut them apart, and
determine their surface areas.
- Students find the volumes of different backpacks by using
familiar solids to approximate their shape. They compare their
results and write about which backpack they think would be
"best" and why.
17. Understand and apply the Pythagorean
- Students construct squares on each side of a right triangle
on a geoboard and find the area of each square. They repeat this
process using several different triangles, recording their results in
a table. Then they look for patterns in the table.
- Students measure the distance diagonally from first to
third base on a baseball field and compare it to the distance run by a
player who goes from first to second to third. Theynote that it is a
shorter distance diagonally across the field than it is along the two
sides. They repeat this type of measuring activity for other squares
and rectangles, noting their results in a table and discussing any
patterns they see. They calculate the square of each of the three
sides of each triangle, record their results in a table, and look for
18. Explore patterns produced by processes of
geometric change, relating iteration, approximation, and
- Students use the reducing and/or enlarging feature on a
copier to explore repeated reductions/enlargements by the same factor
- Students investigate the natural shapes that are produced
by growth. They look at how nature produces complex structures in
which basic shapes occur over and over. For example, spider webs,
honeycombs, pineapples, pinecones, nautilus shells, and snowflakes
grow larger in a systematic way (iteration).
19. Investigate, explore, and describe the geometry in
nature and real-world applications, using models,
manipulatives, and appropriate technology.
- Students design a three-dimensional
geometric sculpture. Some may want to find plans for making a
geodesic dome and construct it out of gumdrops and toothpicks.
- Students work through the Two-Toned
Towers lesson that is described in the First Four Standards of
this Framework. They use models to determine how many
different towers can be built using four blocks of two different
- Groups of students working together design a doghouse to
be built from a 4' x 8' sheet of plywood. They construct a scale
model of their design from oaktag.
- Students use computer programs like The Geometry
PreSupposer to explore the relationships of sides of polygons or
properties of quadrilaterals.
- Assessments that make use of manipulatives and
computer software allow students to demonstrate their knowledge and
understanding of geometry. The results of performance tasks such as
the following would be appropriate for a portfolio: Make as many
different sized squares as you can on a
fivebyfive geoboard. Create a tessellation pattern with
pattern blocks or Tessellmania! software that uses slides,
flips, and turns.
- Students select a country or culture, research the use of
specific geometric patterns in that culture, and make a report to the
- Specific manipulatives that may be helpful for geometry
include pattern blocks, color tiles, linking cubes, centimeter cubes,
tangrams, geoboards, links, and templates. Computer programs such as
Logo, Shape Up!, Elastic Lines, Building Perspective, or The
Factory may also be helpful.
Looher, J. L, Ed. The World of M. C. Escher. New York:
Abradale Press, Harry N. Abrams, Inc., 1971, 1988.
Building Perspective. Sunburst Communications.
Elastic Lines. Sunburst Communications.
Logo. Many versions of Logo are commercially
Shape Up! Sunburst Communications.
Tesselmania! Minnesota Educational Computing Consortium
The Factory. Sunburst Communications.
The Geometry PreSupposer. Sunburst Communications.
Diggins, Julia. String, Straightedge, and Shadow: The Story
of Geometry. New York: Viking Press, 1965.
Geddes, D. Curriculum and Evaluation Standards for School
Mathematics: Addenda Series: Geometry
in the Middle Grades. 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