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 Geometric Shapes (3 weeks)  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 Chapter 13 Measurement (3 weeks)  Section 13.1 Measurement with Nonstandard and Standard Units  Section 13.2 Length and Area

1  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 Chapter 15 Geometry Using Coordinates (last section) (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 Miscellaneou Review, quizzes, exams (2.5 weeks) s 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.

2 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

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

* 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.

3 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 • 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

4  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  Recognize the capabilities and limitations of computational aids  See how elementary mathematics and technology interact

5 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

Domains Domain 1 Domain 2 Domain 3 Domain 4 Number Algebra & Measurement & Statistics, Data Sense Functions Geometry Analysis, & Probability Course Topics 1.1 1.2 2.1 2.2 3.1 3.2 3.3 4.1 4.2 4.3 Math 106 2D & 3D Objects x Transformations x x Representational x x x x Systems Measurements x x x 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

6 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.

7 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.

8 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.

9 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.

10 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

11 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.