Math 580: Geometry for Middle Grade Teachers
Humboldt State University
Spring Semester 2001
(Special Thanks to Dr. Laura Moss who team-taught the first version of this course with me during the Fall
of 1999 and who authored the first version of these notes)
Course Information:
This course is designed to enable teachers and/or prospective teachers of middle school
grades to develop an advanced viewpoint of geometric concepts found in the middle
grades curriculum. This course should enhance a teachers’ understanding of the nature of
geometry as well as laying a foundation for the development of mathematical reasoning
through the study of geometry.
The content of the course includes selected topics in Euclidean geometry, nonEuclidean geometry, and related areas of measurement. Emphasis will be placed on the
concepts of similarity and symmetry within two- and three- dimensional contexts. The
topics studied are connected to the content standards for Geometry and Measurement for
the middle grades, as found in the California Academic Content Standards in
Mathematics.
The nature of the course is one of exploration, analysis, and reasoning. We will be
using the software package, The Geometer’s Sketchpad, published by Key Curriculum
Press, to aid us in our exploration. Learning how to use this tool does take some time,
but I don’t want the focus of the course to be the tool In fact I chose it because doing so
will strengthen your conceptual understanding of the interplay between fundamental
mathematical objects – points, lines, and circles (objects that connect to the straightedge
and compass of Greek Mathematicians) as well as help you visualize mathematical
relationships between visual aspects (perpendicularity, parallelism, similarity, symmetry)
and measurable aspects (length, area, degree) of figures in the Euclidean Plane.
I do expect that all of you – at one time in your life – either took a course in high school
geometry OR that you have learned the foundations of geometry through teaching. This
course is not a course on how to teach geometry. It is a course on improving your
reasoning abilities in geometric contexts.
Course Grade:
Your course grade will be determined by weekly assignments. These assignments will
consist of textbook type questions, responses to readings, activity questions, and
computer lab inquiries.
Course outline:
The goal of the course is to help you build an advanced viewpoint of the ideas that are
taught in middle grades geometry. Not all activities will be applicable to working with
middle grade students. In this 10-mailings course we will be working with the following
topical index:
1
Definitions, classifications of triangles
2
Constructions, classifications of quadrilaterals
3
Properties and relationships of circles
4
Proving Properties Geometrically
5
Similarity
6
Congruence
7
Length, Area, and Volume Relationships
8
Transformations
9
Symmetry
10
Patterns in the Plane
For your reference, I’ve included the following Geometry standards so that you have
them in one place:
California Academic Content Standards in Mathematics
California Geometry Standards: Grade 5 through Geometry
These standards may be obtained under the mathematics framework from the web site:
http://www.cde.ca.gov/cilbranch/eltdiv/cdsmc.htm
Grade Five
1.0 Students understand and compute the volumes and areas of simple objects:
1.1 Derive and use the formula for the area of a triangle and of a parallelogram by
comparing it with the formula for the area of a rectangle. (i.e., two of the same
triangles make a parallelogram with twice the area; a parallelogram is
compared with a rectangle of the same area by pasting and cutting a right
triangle on the parallelogram.)
1.2 Construct a cube and rectangular box from two-dimensional patterns and use
these patterns to compute the surface area for these objects.
1.3 Understand the concept of volume and use the appropriate units in common
measuring systems (i.e., cubic centimeter [cm3], cubic meter [m3], cubic inch
[in.3], cubic yard [yd.3]) to compute the volume of rectangular solids.
1.4 Differentiate between, and use appropriate units of measures for, two- and
three-dimensional objects (i.e., find the perimeter, area, volume).
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Math 580
Geometry for Middle Grade Teachers
2.0 Students identify, describe, and classify the properties of, and the relationships
between, plane and solid geometric figures:
2.1 Measure, identify, and draw angles, perpendicular and parallel lines,
rectangles, and triangles by using appropriate tools (e.g., straightedge, ruler,
compass, protractor, drawing software).
2.2 Know that the sum of the angles of any triangle is 180° and the sum of the
angles of any quadrilateral is 360° and use this information to solve problems.
2.3 Visualize and draw two-dimensional views of three-dimensional objects made
from rectangular solids.
Grade Six
1.0 Students deepen their understanding of the measurement of plane and solid
shapes and use this understanding to solve problems:
1.1 Understand the concept of a constant such as ; know the formulas for the
circumference and area of a circle.
1.2 Know common estimates of  (3.14; 22 ⁄7) and use these values to estimate
and calculate the circumference and the area of circles; compare with actual
measurements.
1.3 Know and use the formulas for the volume of triangular prisms and cylinders
(area of base  height); compare these formulas and explain the similarity
between them and the formula for the volume of a rectangular solid.
2.0 Students identify and describe the properties of two-dimensional figures:
2.1 Identify angles as vertical, adjacent, complementary, or supplementary and
provide descriptions of these terms.
2.2 Use the properties of complementary and supplementary angles and the sum
of the angles of a triangle to solve problems involving an unknown angle.
2.3 Draw quadrilaterals and triangles from given information about them (e.g., a
quadrilateral having equal sides but no right angles, a right isosceles triangle).
Grade Seven
1.0 Students choose appropriate units of measure and use ratios to convert within
and between measurement systems to solve problems:
1.1 Compare weights, capacities, geometric measures, times, and temperatures
within and between measurement systems (e.g., miles per hour and feet per
second, cubic inches to cubic centimeters).
1.2 Construct and read drawings and models made to scale.
1.3 Use measures expressed as rates (e.g., speed, density) and measures expressed
as products (e.g., person-days) to solve problems; check the units of the
solutions; and use dimensional analysis to check the reasonableness of the
answer.
2.0 Students compute the perimeter, area, and volume of common geometric
objects and use the results to find measures of less common objects. They know
how perimeter, area, and volume are affected by changes of scale:
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2.1
Use formulas routinely for finding the perimeter and area of basic twodimensional figures and the surface area and volume of basic threedimensional figures, including rectangles, parallelograms, trapezoids, squares,
triangles, circles, prisms, and cylinders.
2.2 Estimate and compute the area of more complex or irregular two- and threedimensional figures by breaking the figures down into more basic geometric
objects.
2.3 Compute the length of the perimeter, the surface area of the faces, and the
volume of a three-dimensional object built from rectangular solids. Understand that when the lengths of all dimensions are multiplied by a scale factor,
the surface area is multiplied by the square of the scale factor and the volume
is multiplied by the cube of the scale factor.
2.4 Relate the changes in measurement with a change of scale to the units used
(e.g., square inches, cubic feet) and to conversions between units (1 square
foot = 144 square inches or [1 ft2] = [144 in2], 1 cubic inch is approximately
16.38 cubic centimeters or [1 in3] = [16.38 cm3]).
3.0 Students know the Pythagorean theorem and deepen their understanding of
plane and solid geometric shapes by constructing figures that meet given
conditions and by identifying attributes of figures:
3.1 Identify and construct basic elements of geometric figures (e.g., altitudes,
midpoints, diagonals, angle bisectors, and perpendicular bisectors; central
angles, radii, diameters, and chords of circles) by using a compass and
straightedge.
3.2 Understand and use coordinate graphs to plot simple figures, determine
lengths and areas related to them, and determine their image under translations
and reflections.
3.3 Know and understand the Pythagorean theorem and its converse and use it to
find the length of the missing side of a right triangle and the lengths of other
line segments and, in some situations, empirically verify the Pythagorean
theorem by direct measurement.
3.4 Demonstrate an understanding of conditions that indicate two geometrical
figures are congruent and what congruence means about the relationships
between the sides and angles of the two figures.
3.5 Construct two-dimensional patterns for three-dimensional models, such as
cylinders, prisms, and cones.
3.6 Identify elements of three-dimensional geometric objects (e.g., diagonals of
rectangular solids) and describe how two or more objects are related in space
(e.g., skew lines, the possible ways three planes might intersect).
Geometry
1.0 Students demonstrate understanding by identifying and giving examples of
undefined terms, axioms, theorems, and inductive and deductive reasoning.
2.0 Students write geometric proofs, including proofs by contradiction.
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Math 580
Geometry for Middle Grade Teachers
3.0 Students construct and judge the validity of a logical argument and give
counterexamples to disprove a statement.
4.0 Students prove basic theorems involving congruence and similarity.
5.0 Students prove that triangles are congruent or similar, and they are able to use the
concept of corresponding parts of congruent triangles.
6.0 Students know and are able to use the triangle inequality theorem.
7.0 Students prove and use theorems involving the properties of parallel lines cut by a
transversal, the properties of quadrilaterals, and the properties of circles.
8.0 Students know, derive, and solve problems involving the perimeter, circumference,
area, volume, lateral area, and surface area of common geometric figures.
9.0 Students compute the volumes and surface areas of prisms, pyramids, cylinders,
cones, and spheres; and students commit to memory the formulas for prisms,
pyramids, and cylinders.
10.0 Students compute areas of polygons, including rectangles, scalene triangles,
equilateral triangles, rhombi, parallelograms, and trapezoids.
11.0 Students determine how changes in dimensions affect the perimeter, area, and
volume of common geometric figures and solids.
12.0 Students find and use measures of sides and of interior and exterior angles of
triangles and polygons to classify figures and solve problems.
13.0 Students prove relationships between angles in polygons by using properties of
complementary, supplementary, vertical, and exterior angles.
14.0 Students prove the Pythagorean theorem.
15.0 Students use the Pythagorean theorem to determine distance and find missing
lengths of sides of right triangles.
16.0 Students perform basic constructions with a straightedge and compass, such as
angle bisectors, perpendicular bisectors, and the line parallel to a given line through
a point off the line.
17.0 Students prove theorems by using coordinate geometry, including the midpoint of a
line segment, the distance formula, and various forms of equations of lines and
circles.
18.0 Students know the definitions of the basic trigonometric functions defined by the
angles of a right triangle. They also know and are able to use elementary
relationships between them. For example, tan(x) = sin(x)/cos(x), (sin(x))2 +(cos(x))2
= 1.
19.0 Students use trigonometric functions to solve for an unknown length of a side of a
right triangle, given an angle and a length of a side.
20.0 Students know and are able to use angle and side relationships in problems with
special right triangles, such as 30°, 60°, and 90° triangles and 45°, 45°, and 90°
triangles.
21.0 Students prove and solve problems regarding relationships among chords, secants,
tangents, inscribed angles, and inscribed and circumscribed polygons of circles.
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22.0 Students know the effect of rigid motions on figures in the coordinate plane and
space, including rotations, translations, and reflections.
An “advance perspective”:
What does it mean to build an advanced perspective on the Geometry standards in middle
grades?
1. Gaining an advanced viewpoint of geometric concepts means that you are consider
these concepts in the context of higher order thinking skills.
2. Gaining an advanced viewpoint of geometric concepts means that you are able to
place these concepts in a broader context or mathematical system.
Let’s expand these two points below:
Higher order thinking skills in Geometry – The van Hiele Levels of
Geometric Thinking.1
Dina and Pierre van Hiele are two Dutch educators who were concerned about the
difficulties that their students were having in geometry. This concern motivated their
research aimed at understanding students’ levels of geometric thinking to determine the
kinds of instruction that can best help students.
The five levels that are described below are not age-dependent, but, instead, are related
more to the experiences students have had. The levels are sequential; that is, students
must pass through the levels in order as their understanding increases. The descriptions
of the levels are in terms of “students” – and remember that we are all students in some
sense.
Level 0 – Visualization
Students recognize shapes by their global, holistic appearance.
Students at level 0 think about shapes in terms of what they resemble and are able to sort
shapes into groups that “seem to be alike.” For example, a student at this level might
describe a triangle as a “clown’s hat.” The student, however, might not recognize the
same triangle if it is rotated so that it “stands on its point.”
Level 1 – Analysis
1
Cathcart, et al. Learning Mathematics in Elementary and Middle Schools. p.282-3.
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Math 580
Geometry for Middle Grade Teachers
Students observe the component parts of figures (e.g., a parallelogram has
opposite sides that are parallel) but are unable to explain the relationships
between properties within a shape or among shapes.
Student at level 1 are able to understand that all shapes in a group such as parallelograms
have the same properties, and they can describe those properties.
Level 2 – Informal deduction (relationships)
Students deduce properties of figures and express interrelationships both within
and between figures.
Students at level 2 are able to notice relationships between properties and to understand
informal deductive discussions about shapes and their properties.
Level 3 – Formal deduction
Students can create formal deductive proofs.
Students at level 3 think about relationships between properties of shapes and also
understand relationships between axions, definitions, theorems, corollaries, and
postulates. At this level, students are able to “work with abstract statements about
geometric properties and make conclusions based more on logic than intuition” (Van de
Walle).
Level 4 – Rigor
Students rigorously compare different axiomatic systems.
Students at this level think about deductive axiomatic systems of geometry. This is the
level that college mathematics majors think about Geometry.
In general, most elementary school students are at levels 0 or 1; some middle school
students are at level 2. The CA standards are written to begin the transition from levels 0
and 1 to level 2 as early as 5th grade “Students identify, describe, draw and classify
properties of, and
relationships between, plane and solid geometric figures.” (5th grade, standard 2 under
Geometry and Measurement) This emphasis on relationships is magnified in the 6th and
7th grade standards.
Interestingly, the sixth National Assessment of Educational Progress report (1997)
reported that “most of the students at all three grade levels (fourth, eight, and twelfth)
appear to be performing at the ‘holistic’ level (level 0) of the van Heile levels of
geometric thought.”
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In this course, our goal is increase your experiences with geometry at and above level 2.
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Math 580
Geometry for Middle Grade Teachers
Broader mathematics Context – Mathematical (Geometric) Systems
We have already begun building our “big picture” of geometry. To do so requires that
we look at
1. The process of classification
2. The nature of postulates, definitions, propositions, and proof
3. The consequences of alternative assumptions
As we do so, systemic views of geometry will emerge:
1. Euclidean Geometry in 2 and 3 dimensions (one emphasis)
2. Hyperbolic (“Pringel’s Potato chip” surface) and Elliptic (Sphere surface)
geometry
3. Transformational Geometry
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