Download Course Title: Intuitive Geometry

SJSU Mathematics Department Course Syllabus
Fall 2005
Course Title: Intuitive Geometry
SJSU Number: Math 106
Prerequisites: Math 12 and Math 105 (with grades of “C-“ or better); two years of high school
algebra; one year of high school geometry
Course Description
Mathematics 106 is the third course designed for prospective elementary and middle school
teachers. Students explore and develop understanding of mathematical concepts and processes
taught at those levels. In particular, students study two- and three-dimensional geometric
objects; analyze characteristics and properties of two- and three-dimensional geometric shapes;
develop mathematical arguments about geometric relationships; apply transformations and use
symmetry to analyze mathematical situations; represent geometric objects using representational
systems such as concrete models, drawings and coordinate geometry; and use techniques, tools
and formulas for determining measurements.
In general students are encouraged to think about geometry as the study of objects in a plane or
in space. They are asked to investigate situations in which they need to identify the geometric
objects, the conditions that vary, the relationships between the given conditions and the outcome,
state conjectures, and provide explanations that support their conjectures. They are encouraged
to use dynamic geometry systems such as Cabri or Geometer’s Sketchpad as a tool for visual
investigations. Throughout the course, students experience mathematics learning in the way that
we want their future students to experience mathematics learning. In addition, students analyze
their learning experiences from the perspective of a future teacher.
Note: This is the third course in a three-course sequence of mathematics courses for future
elementary and middle school teachers. For an overview of the three-course sequence, see p. 11.
Bibliography – Knowledge Base
Textbook
Musser, Burger, & Peterson’s Mathematics for Elementary School Teachers, 7th edition. (Note
that Math 12, Number Systems, and Math 105 Concepts in Mathematics, Probability, and
Statistics, use the same textbook.)
Required Topics and Suggested Schedule
Chapter 12
(3 weeks)
Chapter 13
(3 weeks)
Geometric Shapes
 Section 12.1 Recognizing Geometric Shapes
 Section 12.2 Analyzing Shapes
 Section 12.3 Properties of Geometric Shapes: Lines and Angles
 Section 12.4 Regular Polygons and Tessellations
 Section 12.5 Describing Three-Dimensional Shapes
Measurement
 Section 13.1 Measurement with Nonstandard and Standard Units
 Section 13.2 Length and Area
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 Section 13.3. Surface Area
 Section 13.4 Volume
Chapter 14
Geometry Using Triangle Congruence and Similarity
(3 weeks)
 Section 14.1 Congruence of Triangles
 Section 14.2 Similarity of Triangles
 Section 14.3 Basic Euclidean Constructions
 Section 14.4 Additional Euclidean Constructions
 Section 14.5 Geometric Problem Solving Using Triangle Congruence
Geometry Using Coordinates (last section)
Chapter 15
(0.5 weeks)
 Section 15.3 Geometric Problem Solving Using Coordinates
Chapter 16
Geometry Using Transformations
(3 weeks)
 Section 16.1 Transformations
 Section 16.2 Congruence and Similarity Using Transformations
 Section 16.3 Geometric Problem Solving Using Transformations
Miscellaneous Review, quizzes, exams (2.5 weeks)
Total Time
15 weeks
Allocation
Journal Articles
1. J. Michael Shaughnessy and William F. Burger, “Spadework Prior to Deduction in Geometry”
Mathematics Teacher 6(September 1985):419-427.
2. Glenda Lappan, “Geometry: The Forgotten Strand” NCTM News Bulletin, December 1999.
3. National Council of Teachers of Mathematics, “Navigating through Geometry, Introduction”
Navigating through Geometry in Grades 6 – 8, 1-8.
4. Wheatley, Grayson H., “Spatial Sense and the Construction of Abstract Units in Tiling”
Arithmetic Teacher (April 1992), reprinted in Chambers, Donald L. (Ed.) Putting Research into
Practice in the Elementary Grades, NCTM, Washington, DC.
5. Lappan, G., & Even, R. “Similarity in the Middle Grades” Arithmetic Teacher (May 1988),
reprinted in Chambers, Donald L. (Ed.) Putting Research into Practice in the Elementary
Grades, NCTM, Washington, DC.
Instructors’ References
Principles and Standards for School Mathematics, National Council of Teachers of
Mathematics, 2000.
Mathematics Framework for California Public Schools, California Department of Education,
1999.
Navigating through Geometry Series, National Council of Teacher of Mathematics, 2002
Harel, G and L. Sowder, Proof Schemes, Unpublished Manuscript, 1995
Embse, C and A. Engebretsen, Explorations: Geometric Investigations for the Classroom, Texas
Instruments, 1996.
Lappan, G, Fey, J, Fitzgerald, W, Friel, S and E. Phillips, Ruins of Montareck, Connected
Mathematics, 1998
Clements, D. Tierney, C. Murray, M. Akers, J. and J. Sarama, Picturing Polygons, TERC, 1998
Supplementary Activities Packet, SJSU Math Education Committee
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Goals and Objectives – Mathematical Content*
1. Two- and Three-dimensional Geometric Objects – Two- and three-dimensional objects can
be visualized, described, classified, characterized and analyzed relative to component parts and
relationships across these components.
Key ideas include:
• point-line-plane relationships including undefined terms and conditions such as collinear,
concurrent, coplanar, parallel, perpendicular and skew
• angles including types of angles acute, right and obtuse and relationships complementary
and supplementary
• triangles including classification schemes, lines with triangles (altitudes, medians,
perpendicular bisectors, angle bisectors) and relationship (orthocenter, centroid,
circumcenter, incenter, Euler’s line)
• quadrilaterals including descriptions and relationships (diagonals and midsegments)
• polygons
• circles including basic description and relationships (central angles, inscribed angles, etc,
inscribed quadrilaterals).
• three-dimensional figures including regular polyhedra, prisms, pyramids, cylinders, cones,
and spheres.
2. Transformations – Geometric objects can be visualized and analyzed according to what
varies and what remains the same under geometric transformations.
Key ideas include:
• transformations including translations, rotations, reflections and dilations
• isometry/congruence
• similarity
• tessellation
• combining and dissecting
3. Representational Systems, Including Concrete Models, Drawings, and Coordinate
Geometry –Representation of geometric objects can take on many forms and include the use
of many tools.
Key ideas include:
• representations, including physical models, manipulatives (power polygons, tangrams,
geoboards, etc.), free-hand drawings, isometric drawings, coordinates, and computer
generated drawings
• tools used to realize the representations include tracing, paper folding, compass and
straight edge, mira, dynamic geometry environments
4. Techniques, Tools, and Formulas for Determining Measurements – Measurement of
geometric objects quantifies geometric objects.
Key ideas include:
• techniques for estimating, measuring and representing time, length, angles, weight/mass
and temperature
*
Goals and objections in italic are directly quoted from the mathematics content specifications described in The
California Commission on Teacher Credentialing document Standards of Program Quality and Effectiveness for the
Subject Matter Requirement for the Multiple Subject Teaching Credential. Goals and objectives in italic must be
covered in the course in order for the course to remain compliant with this document. See pp. 8-10 for the full text
of these mathematics content specifications.
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•
•
•
•
•
•
importance of unit
rates and proportional reasoning
different measurement systems including the metric system
tools for measurement (ruler, protractor, calculator based labs, dynamic geometry tools)
formulas such as the Pythagorean theorem
calculations of perimeter, area, surface area, volume
Goals and Objectives – Mathematical Processes
One of the ways in which we weave together the three mathematics courses for prospective
teachers is to have well-defined, long-term goals, which guide the implementation of the syllabi
throughout the sequence. These are goals that go above and beyond the learning of specific
content topics, and which take students longer than one semester to achieve. These goals
correspond roughly to the National Council of Teachers of Mathematics’ process standards –
they “highlight ways of acquiring and using content knowledge” (NCTM, 2000, p. 29). Each
course builds upon these goals in a progressive fashion.
1. Understand understanding
 Recognize the validity of different approaches
 Recognize the equivalence of different answers
 Analyze errors to identify misunderstandings
 Analyze levels of understanding
 Explain multiple ways of understanding the same idea
 Recognize when language use is ambiguous, well-defined, or meaningless
 Recognize examples and non-examples
2. Utilize representations and connections
 Identify situations that can be modeled using mathematics
 Represent situations appropriately using mathematics
 Translate from one representation to one another
 Explain how representations are connected to one another
3. Develop new reasoning and problem-solving skills
 Experiment, conjecture, verify
 Recognize patterns, recognize recurring ideas in different settings
 Reason by analogy, infer in situations of uncertainty
 Synthesize, deduce
4. Communicate mathematical ideas
 Use mathematical terminology, notation, and language effectively and accurately
 Express ideas logically and clearly
 Model English with mathematics, interpret mathematics into English
 Illustrate and support ideas graphically, numerically, symbolically, or verbally as needed
5. Develop positive attitudes and beliefs about mathematics
 Learn and articulate how mathematics is useful outside of school
 Identify potential sources of negative attitudes and beliefs
 Model helpful attitudes and beliefs while working with fellow students
6. Use technology appropriately
 Appreciate the role of technology as a tool for learning and problem solving
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

Recognize the capabilities and limitations of computational aids
See how elementary mathematics and technology interact
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California Commission on Teacher Credentialing Standards
The California Commission on Teacher Credentialing requires all approved subject matter
programs to meet certain standards of program quality and effectiveness, as described in the
September 2001 document Standards of Program Quality and Effectiveness for the Subject
Matter Requirement for the Multiple Subject Teaching Credential. Here, we describe how
SJSU’s Math 106 contributes to meeting the content specifications in mathematics. (See pp. 8-10
for the mathematics content specifications.)
Part I: Content Domains for Subject Matter Understanding and Skill in Mathematics
Course Topics
Math 106
2D & 3D Objects
Transformations
Representational
Systems
Measurements
Domain 1
Number
Sense
Domain 2
Algebra &
Functions
1.1
2.1
1.2
x
x
x
Domains
Domain 3
Measurement &
Geometry
2.2
3.1
3.2
x
x
x
x
x
x
x
3.3
Domain 4
Statistics, Data
Analysis, &
Probability
4.1
4.2
4.3
x
Domain 1: Number Sense
Within the area of measurement, students use numbers and computational algorithms for the
numbers. For example, when students find the length of sides of a right triangle, they need to be
able to have a general number sense relative to irrational numbers and to operate on irrational
numbers.
Domain 2: Algebra and Functions
Transformations, especially dilations require that students use proportional reasoning. Similar
triangles are also explored in detail within the section on transformations. Concepts in
coordinate geometry apply algebraic reasoning. In fact, the chapter is called connecting algebra
and geometry.
Domain 3.1: Two-and Three-dimensional Geometric Objects
The elements of this domain form an important component of Math 106, the third course of our
three-semester sequence of courses for future elementary and middle school teachers. Through
activities and projects involving various representations and tools, our future teachers are
encouraged to characterize and compare geometric objects. Transformations including
isometries, dilations and sheers help support the explorations of what changes and what remains
the same under variations in the plane or in space. Through dynamic geometry environments
such as CABRI or Sketchpad, students are encouraged to develop dynamic visualization of
geometric figures and then explore mentally what changes and what does not under prescribed
variations. This visual exploration acts as a bridge between intuitive geometric reasoning and
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more formal geometric proofs. Measurement of also contributes to issues in this domain such as
the Pythagorean theorem.
Domain 3.2 Representational Systems, Including Concrete Models, Drawings, and Coordinate
Geometry
The elements of this domain are integrated across the Math 106 course as students use
manipulatives, drawings and coordinate geometry to represent geometric figures and geometric
relationships. Construction of drawings involves a number of approaches including paper
folding, the MIRA, traditional compass and straight edge and dynamic geometry environments.
Regardless of the representational system, however, students are encouraged to generalize
drawings to figures and through variations, supported within the representational system, focus
on properties of the figure that remain invariant.
Domain 3.3 Techniques, Tools and Formulas for Determining Measurements
The elements of this domain are closely linked to ideas included in the previous two courses.
Within this domain, students quantify geometric objects with a careful consideration of the unit
of analysis. Applications, such as rates and scale drawings, connect to work done in previous
courses through the ideas of ratio and proportional reasoning. Lengths, areas, and volumes
require application of the real number system including irrational number. Although the role of
the estimate or measure is the focus, investigations in this section continue to support the
understandings of geometric relationships.
Part II: Subject Matter Skills and Abilities Applicable to the Content Domain in
Mathematics
Subject matter skills and abilities in mathematics are developed in a progressive fashion over the
three-semester sequence of mathematics courses for future elementary school teachers. Broadly,
our goals are for our students to (1) understand understanding, (2) utilize representations and
connections, (3) develop new reasoning and problem-solving skills, (4) communicate
mathematical ideas, (5) develop positive attitudes and beliefs about mathematics, and (6) use
technology appropriately. Math 106 allows a unique opportunity to examine these goals with the
environment of geometric objects.
The dynamic visualization theme in Math 106 requires students to examine explicitly the nature
and importance of geometric relationships and make use of tools that can help in exploring these
relationships. Through varied experiences with both manipulatives and computer environments,
students learn to create multiple images of geometric objects and through investigations of these
images understand, at least intuitively, geometric relationships. Connections to formal reasoning
are supported through discussions of invariance.
Subject Matter Competency in Mathematics for Multiple-Subjects Credential Candidates
All future multiple-subjects credential candidates will have to pass the CSET exam in multiple
subjects prior to entering a credential program in California. Subtest II covers mathematics and
science. Sample questions can be viewed at the web site http://www.cset.nesinc.com/.
Instructors might wish to go over sample mathematics questions from the CSET on occasion.
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Content Specifications in Mathematics*
Part I: Content Domains for
Subject Matter Understanding and Skill in
Mathematics
Domain 1: Number Sense
1.1
Numbers, Relationships Among Numbers, and Number Systems. Candidates for
Multiple Subject Teaching Credentials understand base ten place value, number theory
concepts (e.g., greatest common factor), and the structure of the whole, integer, rational,
and real number systems. They order integers, mixed numbers, rational numbers
(including fractions, decimals, and percents) and real numbers. They represent numbers in
exponential and scientific notation. They describe the relationships between the algorithms
for addition, subtraction, multiplication, and division. They understand properties of
number systems and their relationship to the algorithms, [e.g., 1 is the multiplicative
identity; 27 + 34 = 2 X 10 + 7 + 3 X 10 + 4 = (2 + 3) X 10 + (7 + 4)]. Candidates perform
operations with positive, negative, and fractional exponents, as they apply to whole
numbers and fractions.
1.2
Computational Tools, Procedures, and Strategies. Candidates demonstrate fluency in
standard algorithms for computation and evaluate the correctness of nonstandard
algorithms. They demonstrate an understanding of the order of operations. They round
numbers, estimate the results of calculations, and place numbers accurately on a number
line. They demonstrate the ability to use technology, such as calculators or software, for
complex calculations.
Domain 2: Algebra and Functions
2.1
Patterns and Functional Relationships. Candidates represent patterns, including
relations and functions, through tables, graphs, verbal rules, or symbolic rules. They use
proportional reasoning such as ratios, equivalent fractions, and similar triangles, to solve
numerical, algebraic, and geometric problems.
2.2
Linear and Quadratic Equations and Inequalities. Candidates are able to find
equivalent expressions for equalities and inequalities, explain the meaning of symbolic
expressions (e.g., relating an expression to a situation and vice versa), find the solutions,
and represent them on graphs. They recognize and create equivalent algebraic expressions
(e.g., 2(a+3) = 2a + 6), and represent geometric problems algebraically (e.g., the area of a
triangle). Candidates have a basic understanding of linear equations and their properties
(e.g., slope, perpendicularity); the multiplication, division, and factoring of polynomials;
and graphing and solving quadratic equations through factoring and completing the square.
*
Reprinted from California State Standards of Program Quality and Effectiveness for the Subject Matter Requirement for the Multiple Subject
Teaching Credential, September, 2001.
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They interpret graphs of linear and quadratic equations and inequalities, including solutions
to systems of equations.
Domain 3: Measurement and Geometry
3.1
Two- and Three-dimensional Geometric Objects. Candidates for Multiple Subject
Teaching Credentials understand characteristics of common two- and three-dimensional
figures, such as triangles (e.g., isosceles and right triangles), quadrilaterals, and spheres.
They are able to draw conclusions based on the congruence, similarity, or lack thereof, of
two figures. They identify different forms of symmetry, translations, rotations, and
reflections. They understand the Pythagorean theorem and its converse. They are able to
work with properties of parallel lines.
3.2
Representational Systems, Including Concrete Models, Drawings, and Coordinate
Geometry. Candidates use concrete representations, such as manipulatives, drawings, and
coordinate geometry to represent geometric objects. They construct basic geometric
figures using a compass and straightedge, and represent three-dimensional objects through
two-dimensional drawings. They combine and dissect two- and three-dimensional figures
into familiar shapes, such as dissecting a parallelogram and rearranging the pieces to form a
rectangle of equal area.
3.3
Techniques, Tools, and Formulas for Determining Measurements. Candidates estimate
and measure time, length, angles, perimeter, area, surface area, volume, weight/mass, and
temperature through appropriate units and scales. They identify relationships between
different measures within the metric or customary systems of measurements and estimate
an equivalent measurement across the two systems. They calculate perimeters and areas of
two-dimensional objects and surface areas and volumes of three-dimensional objects. They
relate proportional reasoning to the construction of scale drawings or models. They use
measures such as miles per hour to analyze and solve problems.
Domain 4: Statistics, Data Analysis, and Probability
4.1
Collection, Organization, and Representation of Data. Candidates represent a collection
of data through graphs, tables, or charts. They understand the mean, median, mode, and
range of a collection of data. They have a basic understanding of the design of surveys,
such as the role of a random sample.
4.2
Inferences, Predictions, and Arguments Based on Data. Candidates interpret a graph,
table, or chart representing a data set. They draw conclusions about a population from a
random sample, and identify potential sources and effects of bias.
4.3
Basic Notions of Chance and Probability. Candidates can define the concept of
probability in terms of a sample space of equally likely outcomes. They use their
understanding of complementary, mutually exclusive, dependent, and independent events
to calculate probabilities of simple events. They can express probabilities in a variety of
ways, including ratios, proportions, decimals, and percents.
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Part II: Subject Matter Skills and Abilities
Applicable to the Content Domains in Mathematics
Candidates for Multiple Subject Teaching Credentials identify and prioritize relevant and
missing information in mathematical problems. They analyze complex problems to identify
similar simple problems that might suggest solution strategies. They represent a problem in
alternate ways, such as words, symbols, concrete models, and diagrams, to gain greater insight.
They consider examples and patterns as means to formulating a conjecture.
Candidates apply logical reasoning and techniques from arithmetic, algebra, geometry, and
probability/statistics to solve mathematical problems. They analyze problems to identify
alternative solution strategies. They evaluate the truth of mathematical statements (i.e., whether
a given statement is always, sometimes, or never true). They apply different solution strategies
(e.g., estimation) to check the reasonableness of a solution. They demonstrate that a solution is
correct.
Candidates explain their mathematical reasoning through a variety of methods, such as words,
numbers, symbols, charts, graphs, tables, diagrams, and concrete models. They use appropriate
mathematical notation with clear and accurate language. They explain how to derive a result
based on previously developed ideas, and explain how a result is related to other ideas.
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Mathematics Course Sequence at SJSU for Prospective Elementary and Middle School Teachers
Math 12, Math 105, Math 106
The San Jose State Mathematics Department offers a three-semester sequence of courses designed for prospective
elementary and middle school teachers. In these courses, students explore and develop understanding of
mathematical concepts and processes taught at those levels. Throughout the three-course sequence, students
experience mathematics learning in the way that we want their future students to experience mathematics learning,
using technology, as appropriate. In addition, students analyze their own learning experiences from the perspective
of a future teacher. Students are expected to grow in mathematical sophistication, scholarly responsibility, and
pedagogical perspectives over the three-course sequence.
Math 12 Number Systems
In Math 12 is the first course in the three-course sequence. Several local community colleges offer equivalent
courses. In this course, students study problem solving techniques, numeration systems, the structure of the real
number system, and elementary number theory.
Course Prerequisites (reprinted from the SJSU 2004-2006 catalog, p. 303)
Two years of high school algebra, one year of high school geometry, satisfaction of ELM requirement
Math 105 Concepts in Mathematics, Probability, & Statistics
Math 105 is the second course in the three-course sequence. This is an upper division class that cannot be taken at a
local community college. Students study problem-solving techniques, functions and algebraic reasoning, ratio and
proportions, probability, data, graphs, and statistics.
Course Prerequisites (reprinted from the SJSU 2004-2006 catalog, p. 304)
Two years of high school algebra, one year of high school geometry, SJSU’s Math 12 with a C- or better
Math 106 Intuitive Geometry
Mathematics 106 is the third course in the three-course sequence. This is an upper division class that cannot be
taken at a local community college. Students analyze characteristics and properties of two- and three-dimensional
geometric shapes; develop mathematical arguments about geometric relationships; apply transformations and use
symmetry to analyze mathematical situations; represent geometric objects using representational systems such as
concrete models, drawings, and coordinate geometry; and use techniques, tools, and formulas for determining
measurements. In general, students are encouraged to think about geometry as the study of objects in a plane or in
space. They are asked to investigate situations involving geometric objects, state conjectures, and provide
explanations that support their conjectures. Technology is integrated extensively. In particular, students will use a
dynamic geometry system, such as Geometer’s Sketchpad or Cabri, as a tool for visual investigations.
Course Prerequisites (reprinted from the SJSU 2004-2006 catalog, p. 304)
Two years of high school algebra, one year of high school geometry, SJSU’s Math 12 and Math 105 with grades of
C- or better



General Notes
There are no exceptions to the prerequisites. This means that students must take the three courses
sequentially. (The level of mathematical sophistication, extent of scholarly expectations, and breadth of
pedagogical perspectives increase substantially from one course to the next in this sequence. The
prerequisites are designed to provide students with the greatest opportunity of success in the three-course
sequence as well as the best possible preparation for teaching mathematics at the elementary and middle
school levels.)
Grades of C or better MAY be required by some students’ majors in order for them to graduate. Students
are expected to determine this in consultation with their major advisors.
The topics in the three-course sequence include all of the topics covered in the mathematics portion of
Subtest II of the California Subject Examination for Teachers: Multiple Subjects. Students preparing for
teaching careers in California are strongly encouraged to take this three-course mathematics sequence in
preparation for this exam, even if their majors do not require all or any of the courses.
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