Bemidji State University
M3066/ GEOMETRY AND TECHNOLOGY IN THE MIDDLE SCHOOL MATHEMATICS CLASSROOM(4)
Summer 2016
MTWR, 8-11:30 am

Instructor: Dr. Glen Richgels

Email: -- grichgels@bemidjistate.edu

Office Phone: 755- 2824

Office Hours:

Professional Education Department Mission Statement:

“The Bemidji State University Professional Education program is preparing today's teachers for tomorrow, through effective, inquisitive, and reflective practice. Our students are proficient, self-reliant, and thoughtful practitioners, developed in a viable and growing program, who can teach effectively in various settings with diverse learners."

Course Description

This course helps meet the licensure rule with respect to concepts of patterns, shape and space; spatial sense; plane, solid, and coordinate geometry systems; generalizing geometric principals; limits, derivatives and integrals; and appropriate use of technology in the classroom. 

Prerequisites
MATH 1011 and MATH 1100 or MATH 1110 or equivalent

Required Text
No text required – course taught with handouts

Resources:

This course is taught with handouts from a variety of sources and using several computer application / instructional programs. Richgels, G.W., Frauenholtz, T., Hansen, H., Severson, A.R., Rypkema, C. Computer Activities for Teachers, Bemidji State University. Mathematics for Elementary Teachers a Contempory Approach, Musser, Burger & Peterson Papert, S. MINDSTORMS Children, Computers, and Powerful Ideas. Basic Books, Inc., Publishers, 1980. Software: · Cinderella · CoreLite FTP · FUGU · Geometer’s Sketchpad 5 · Macromedia Dreamweaver · Microsoft Excel · Microsoft Word · Microworlds EX Robotics (LOGO) · Mindstorms NXT Internet Browsers: · Internet Explorer · Mozilla FireFox · Safari

Technology:

A computer or calculator

STANDARDS OF EFFECTIVE PRACTICE FOR TEACHERS

"K" indicates standard is taught at the KNOWLEDGE level. "A" indicates standard is taught and ASSESSED.

Board of Teaching Standards

8710.3320 MIDDLE LEVEL ENDORSEMENT LICENSE FOR TEACHERS OF MATHEMATICS.

Department of Mathematics and Computer Science
8710.3320 MIDDLE LEVEL ENDORSEMENT LICENSE FOR TEACHERS OF MATHEMATICS		In this syllabus you will find the word TEACH. This will mean to: Launch: This is where the teacher sets the context of the problem or activity being worked on this day. This involves making sure the students clearly understand the mathematical context and the mathematical challenge of the day’s activities. Explore: This is the time where the students get to work in pairs, individually, or as a class to solve problems presented by the lesson. Share: This occurs when most of the students have made sufficient progress toward solving the problem presented with today’s lesson. It is during this phase that the students learn how others approached the problem and possible solution routes. Helps students deepen their understanding of the mathematical ideas presented in the day’s lesson. Summarize: During this phase the teacher concludes the lesson by clearly stating what the main idea was in the lesson, being sure to clear up any confusion that may arise during the “share” segment. Helps students focus their understanding of the mathematical ideas presented in the lesson.
Standard	K/A	Activity	Assessment
(1) concepts of patterns, relations, and functions:
(e) apply properties of boundedness and limits to investigate problems involving sequences and series; and	K A	TEACH: Limit or bound as a number from a sequence - fibonacci/lucas numbers; seq/ratios; n/n+1, n+1/n, golden ratio Limit or bound as a point or asymptote in the graph of functions Limit or bound as a number from a series – Area and lower Riemann sum series, upper Riemann sum series, other Riemann sum series. The integral is the limit of a Riemann sum. Assignment: 22, 24	Assessment: Teacher observation -Students will construct a spreadsheet to find a bound or limit of a sequence. -Students will find the area between the x-axis and a function. They will use the limit of sequences and series that bound the area above and below to find the area as a limit of a Riemann sequence or series.
(f) apply concepts of derivatives to investigate problems involving rates of change;	K A	TEACH: Derive the concept of rate of change, derivative of a curve from the concept of slope of a line. Demonstrate that the derivative is the limit of a difference quotient. Find the derivative of a function at a point, find the equation of the tangent line, graph the function and equation of the tangent line. Prove the power rule for polynomial functions. Use the power rule to find the derivative of a function at a point, find the equation of the tangent line, graph the function and equation of the tangent line. Assignment: 23	Assessment: Teacher observation -Students will calculate the derivative or rate of change of a polynomial function at a point using the difference quotient. They will then write the equation of the tangent line, then graph the function and the tangent line to verify their calculations. -Students will calculate the derivative, or rate of change, of a polynomial function at a point using the power rule for polynomials. They will then write the equation of the tangent line, then graph the function and the tangent line to verify their calculations. They will check their equations visually and use Geometers Sketchpad to confirm their calculation.
(4) concepts of shape and space:
(a) shapes and the ways in which shape and space can be derived and described in terms of dimension, direction, orientation, perspective, and relationships among these properties;	K A	TEACH: Locate a point in different dimensions with different reference systems; What are the applications of different dimensions and different reference systems. Assignment: 14 TEACH: What is a fractal? Definition includes self similarity, simple rule, iterative / recursive growth Investigate fractals, fractal structures and fractal dimension. Fractals as a more “real” model for the “real” world than Euclidean and Platonic structures. Fractals in nature and fractals as link between mathematics and biology. Assignment: 14 TEACH: Tessellations are a covering / tiling of a plane surface. Triangles and quadrilaterals tessellate all planes. Regular and semi-regular tessellations are finite – > 3 and 8. Escher merged art and mathematics into examples of orientation, perspective, and dynamic tessellations. Mathematical rules, geometric transformations or the geometry of motion consists of translations, rotations, and reflections. Escher used translations, translation reflections, mid-point rotations, and vertex rotations to create his art and tessellations. Look at the polygons that allow for the use of single or multiple transformations in a tessellation. Assignment: 20	Assessment: - Teacher observation will verify that students can specify locations in space using different reference systems that include, dimension, direction, orientation and relationships between these properties. Assessment: -Students will construct fractals in a computer / Microworlds environment using logo. Students will construct fractals with different fractal dimensions. Assessment: -Students will create at least two tessellations of a plane utilizing at least two different transformations. The tessellations will be derived from shapes whose relationships can be described in terms of direction, orientation, perspective, and relationships among these properties;
(b) spatial sense and the ways in which shapes can be visualized, combined, subdivided, and changed to illustrate concepts, properties, and relationships;	K A	TEACH: - LOGO fundamentals of turtle movement, turtle orientation, heading and programming procedures. Use graph paper to model construction of two-dimensional shapes that can be scaled using sliders to assign parameter-values. Construct circle and arc procedures by approximation with n-gon, n>=30. Help students visualize a linear geometric object as a combination of simple geometric shapes. Students should decompose the geometric object into simple shapes that can be combined to create the whole object. Make project constructions using simple component tools such as square, triangle, right triangle, rectangle, … to create a two dimensional linear object. Assignment: 7 TEACH: - LOGO fundamentals of turtle movement, turtle orientation, heading and programming procedures. Use graph paper to model construction of two-dimensional shapes that can be scaled using sliders to assign parameter values. Construct circle and arc procedures by approximation with n-gon, n>=30. Demonstrate how to visualize a set of movements within a background. Animate the “turtle” to carry out the movements through geometric commands. Assignment: 7 TEACH: - LOGO fundamentals of turtle movement, turtle orientation, heading and programming procedures. Use graph paper to model construction of two-dimensional shapes that can be scaled using sliders to assign parameter values. Construct circle and arc procedures by approximation with n-gon, n>=30. Help students visualize geometric objects as a combination of simple geometric shapes including circles or arcs. They should decompose the geometric object into simple shapes that can be combined to create the whole object. Make project constructions using simple component tools such as square, triangle, right triangle, rectangle, circles, arcs … to create a two dimensional object. Assignment: 7 TEACH: Regular polygons have properties including number of sides, central angles, interior and exterior angles, side length of sides and perimeter. Assignment: 7 TEACH: Standard form of families of mathematical functions includes parameters such as this quadratic form: . The use of sliders allows for the investigation of the role each parameter plays in the graph of the function. Assignments: 11	Assessment: Students will create a changeable, scalable, linear two-dimensional figure construction. The figure will be one that can be subdivided into simpler figures and constructed as the combination of these figures. Students will use a turtle and component tools or computer procedures. Assessment: Students will animate at least one turtle to carry out a student- designed path. This project should be posted to the WWW. Assessment: Students will create a changeable, scalable, two-dimensional construction with a turtle and using component tools. This object must contain circles or arcs as components in the object. Assessment: Students will create procedures that will make regular polygonal shapes. The number of sides and side length will be changeable and dynamic to illustrate concepts, properties, and relationships of the parts of the geometric shapes polygons and to connect polygons to circles and its parts, radius, diameter, and circumference. Assessment: Students will graph the standard forms of families of functions and be able to use sliders to dynamically change the value of parameters and to illustrate concepts, properties, and relationships of the parameters.
(c) spatial reasoning and the use of geometric models to represent, visualize, and solve problems;	K A	TEACH: Review linear equations and slope as a way to characterize the shape and orientation of a line. Utilize spatial reasoning to conclude that the slope of a non-linear function is constantly changing. Use the geometric model of a tangent line to model the slope of a curve at a point. Use spatial reasoning to view that the limit of a chord through A and B, as B approaches A, is the line tangent to the function at A. Formulate the derivative definition and derive the power rule for polynomials. Students will practice finding the derivative of a polynomial, writing the equation of the tangent line, and graphing the function and tangent line to check their work. Review area formulas for simple polygonal shapes: cube, rectangle, triangle, parallelogram, trapezoid. Develop notation for finding the area between a function and the x-axis in an interval. Use the mean value theorem to derive the fundamental theorem of calculus. Students will find the area of simple polygons and polynomial functions using the anti-derivative. Develop the Riemann sum approach to visualizing the area of a region. Use the geometric model of rectangles, trapezoids, or parabolas to approximate the area under the function. Demonstrate that the limit as n approaches of a Riemann sum model for a two dimensional region is the area and is equal to the value for the area that can be calculated through the anti-derivative (integral) approach. Model a geometric solid of revolution as the sum of many, thinly sliced pieces of the whole. Each slice can be modeled as a thin or short cylinder. Model the volume of a solid of revolution, sphere, cylinder, cone, as a Riemann sum of many thin cylinders and calculate the volume formula. Assignment: 22-25	Assessment: Teacher observation -Students will calculate the derivative or rate of change of a polynomial function at a point using the difference quotient. They will then write the equation of the tangent line. They will then graph the function and tangent line to verify their calculations. -Students will calculate the derivative or rate of change of a polynomial function at a point using the power rule for polynomials. They will then write the equation of the tangent line. They will then graph the function and tangent line to verify their calculations. -Students will calculate the area between a function and the x-axis on an interval for elementary geometric shapes and for polynomial functions.
(d) motion and the ways in which rotation, reflection, and translation of shapes can illustrate concepts, properties, and relationships;	K A	TEACH: Tessellations are a covering / tiling of a plane surface. Triangles and quadrilaterals tessellate all planes. Regular and semi-regular tessellations are finite –> 3 and 8. Escher merged art and mathematics into examples of orientation, perspective, and dynamic tessellations. Mathematical rules, geometric transformations or the geometry of motion consists of translations, rotations, and reflections. Escher used translations, translation reflections, mid-point rotations, and vertex rotations to create his art and tessellations. Look at the polygons that allow for the use of single or multiple transformations in a tessellation. Assignment: 20	Assessment: -Students will create at least two tessellations of a plane utilizing at least two different transformations from: · translation, · rotation, · reflection, · translation and rotation, · translation and reflection, · and rotation and reflection.
(e) formal and informal argument, including the processes of making assumptions; formulating, testing, and reformulating conjectures; justifying arguments based on geometric figures; and evaluating the arguments of others;	K A	TEACH: Demonstrate the creation of a model mathematical system. A mathematical system consists of undefined terms, defined terms, assumptions, axioms or postulates, lemmas, theorems, and corollaries. Study the system used to find the measure of arcs and angles with respect to circles: central, inscribed, interior, and exterior angles. Begin with the undefined terms, point and line. Review the definintion of angle, circle, and major and minor arcs. Begin with the definition of a central angle and its intercepted arc. Begin with an inscribed angle definition. Use geometric figures to model the three cases for an inscribed angle. Have students utilize the given conditions, their assumptions and a dynamic geometry tool to formulate hypotheses or conjectures about the relationship between an inscribed angle and its intercepted arc. Use the geometric figure constructed in the dynamic geometry environment to investigate their conjecture and to establish an informal proof or conclusion about their conjecture. Then construct the formal arguments necessary to justify the conjecture to prove a lemma and the first theorem of the mathematical system. Next define an interior angle. Model an interior angle with static and dynamic geometric figures. Utilize previous knowledge to augment the existing geometric figure, utilize the figure to formulate a conjecture, make informal arguments and finally prove an interior angle measurement theorem. Finally define an exterior angle. Model an exterior angle with static and dynamic geometric figures. Utilize previous knowledge to augment the existing geometric figure, utilize the figure to formulate a conjecture, make informal arguments and finally prove an interior angle measurement theorem. Assignment: 12	Assessment: -Students will solve a puzzle involving unknown central, interior, inscribed, and exterior angles and their associated arcs. They will use the properties that were derived from formal arguments. The formal arguments are the culmination of the process that includes the processes of making assumptions; formulating, testing, and reformulating conjectures; justifying arguments based on geometric figures; and evaluating the arguments of others. Informal arguments can be made based upon dynamic investigation of the objects that lead to the formal arguments. All answers need to be correct before proceeding. Instructor corrected. -Students will construct a puzzle to apply the results of the informal and formal arguments. The puzzle will have at least one unknown central, interior, inscribed and exterior angle and associated arcs.
(f) plane, solid, and coordinate geometry systems, including relations between coordinate and synthetic geometry and generalizing geometric principles from a two-dimensional system to a three-dimensional system;	K A	TEACH: Construct a table of the 8 measurement concepts to be taught in grades K-8: length, area, volume, capacity, mass, time, temperature, angle measure. These measurements would be in column 1, column 2 would contain English system examples of the concept, column 3 would contain metric or SI system examples of the concept. Include the defining relationship in metric between volume, mass, and capacity. Discuss the evolution or design of the two systems. Assignment: 17 TEACH: Students will be introduced to different geometric systems during the examination of the van Heile levels. The different geometries occur at level four. Assignment: 8 TEACH: The students will investigate the theorem: the lengths of the diagonals of a rectangle are equal. The students will develop a sketch that demonstrates the theorem dynamically using geometer’s sketchpad. This approach supports student learning at level two of the van Hiele levels. Students at this level need to see informal proofs through dynamic examples. The students will then construct coordinate and synthetic proofs of the theorem. These approaches will help students see what is needed to help public school learners in middle school transition to high school. Assignment: 10 TEACH: Students will be asked to locate a point in different referent systems. One dimensional system: label, number line coordinate Two dimensional system: label, Cartesian coordinates, polar coordinates. Three dimensional system: label, Cartesian coordinates, spherical coordinates, cylindrical coordinates. Cartesian coordinates : most people use this system Spherical coordinates : physicists and astronomers use this system Cylindrical coordinates : computer scientists use this Conversion between coordinate systems. Distance formula between points p and q in n dimensions is: Assignment: 14, 15, 16	Assessment: -Students will construct a table showing the eight measurement concepts and examples from the SI and English measurement systems. Assessment: Students will investigate the following problem: What is the sum of the angles in a triangle? They will use the software Cinderella to investigate the problem in plane, hyperbolic, and spherical geometries. They will show triangles that support their answer to the instructor. Assessment: -Instructor observation of the sketch, the synthetic and coordinate geometry proofs. Assessment: -Students will identify points in the different reference systems and will calculate the distance between two points in each system. Student work will be checked by instructor observation.
(g) attributes of shapes and objects that can be measured, including length, area, volume, capacity, size of angles, weight, and mass;	K A	TEACH: Students will order geometric solids consisting of cylinders, spheres, hemi-spheres, cones, prisms, and pyramids based upon volume. They will use conservation of volume as they compare volumes using a medium such as rice or water. Students will be able to use standard naming conventions to name the different solids as they order them from least volume to greatest volume. Assignment: 18 TEACH: Construct a table of the 8 measurement concepts to be taught in grades K-8: length, area, volume, capacity, mass, time, temperature, angle measure. These measurements would be in column 1, column 2 would contain English system examples of the concept, column 3 would contain metric or SI system examples of the concept. Include the defining relationship in metric between volume, mass, and capacity. Discuss the evolution or design of the two systems. Assignment: 17 TEACH: Examine the formula that are used for computation of different geometric objects. Two dimensional geometric objects: square, rectangle, triangle, parallelogram, trapezoid, regular n-gon. Formula to compute perimeter and area. Three dimensional geometric objects: prism, pyramid, cylinder, cone, sphere. Formula to compute surface area and volume. Assignment: 19	Assessment: - Students will put their orderings of geometric solids on a common display and then resolve all differences until an ordering has been agreed upon. Assessment: -Students will construct a table showing the eight measurement concepts, length, area, volume, capacity, size of angles, weight, degree measure and mass, and examples from the SI and English measurement systems. This table will be posted on their homepage. Assessment: -Students will construct a table of the formula used to calculate perimeter, area, surface area, and volume and post the table on their homepage.
(h) the structure of systems of measurement, including the development and use of measurement systems and the relationships among different systems;	K A	TEACH: Construct a table of the 8 measurement concepts to be taught in grades K-8: length, area, volume, capacity, mass, time, temperature, angle measure. These measurements would be in column 1, column 2 would contain English system examples of the concept, column 3 would contain metric or SI system examples of the concept. Include the defining relationship in metric between volume, mass, and capacity. Discuss the evolution or design of the two systems. Assignment: 17	Assessment: -Students will construct a table showing the eight measurement concepts, length, area, volume, capacity, size of angles, weight, degree measure and mass, and examples from the SI and English measurement systems. This table will be posted on their homepage.
(i) measuring, estimating, and using measurements to describe and compare geometric phenomena;	K A	TEACH: Students will construct an inclinometer from simple classroom materials: straws, folders, paper, centimeter graph paper, string, paper clips, and tape. Students will measure / determine their stride length in English or metric units – instructor choice. Students will gather data on at least two tall objects outside of the classroom, e.g. trees, flag poles, light poles, multiple story buildings, etc. The class will develop how to take the data and use it to measure / estimate the heights of the objects that they measured. Assignment: 21	Assessment: - Students will construct an inclinometer from simple classroom materials: straws, folders, paper, centimeter graph paper, string, paper clips, and tape, measure different geometric phenomena, post their measurements to the board, make an oral presentation given to the class and instructor for comparison with the all of the class.

Technology Requirements and Expectations
Students will use internet browsers to access information and answer questions posed in class. Students will may use graphing calculators, Geometer’s Sketchpad, or data programs such as Excel, Tinkerplots, Fathom 2, or Minitab as needed. Written assignments for class will be composed using a word processor such as Microsoft Word.

Students in M3066 will be expected to construct a homepage and post it to the WWW.
Teaching Methodology
Polya’s problem solving steps

1. Understand the problem

2. Devise a plan
3. Carry out the plan
4. Reflect

Lesson Sequencing

Intuitions Þ Concrete Û Semi-Concrete Û Abstract

Glen’s Teaching/Learning Principles

1. Teach the way students learn

2. Use group work, heterogeneous, 3-4, change monthly

3. Communication student Û student

4. Communication teacher Û student

5. Multiple solution paths

6. Use contextual settings / problem solving

7. Assessment

1. Grading
2. To inform instruction

University Policies and Procedures
http://www.bemidjistate.edu/students/handbook/policies/

Academic Integrity
BSU students are expected to practice the highest standards of ethics, honesty and integrity in all of their academic work. Any form of academic dishonesty (e.g., plagiarism, cheating and misrepresentation) may result in disciplinary action. Possible disciplinary actions may include failure for part of all of a course as well as suspension from the University.

Students with Special Needs
Upon request this document can be made available in alternate formats. Please contact Kathi Hagen at Disabilities Services at (218) 755-3883 for assistance or the AUC Office at 262-6753 or (800) 369-4970.

Student Rights and Responsibilities

Student Code of Ethics

http://www.bemidjistate.edu/academics/catalog/10catalog/GradCatalog/Frontpages/sectionIV/rights.html

Student Academic Rights and Responsibilities

http://www.bemidjistate.edu/students/handbook/policies/academic_integrity/rights_responsibilities.cfm

Instructor Rights and Responsibilities
- I work with all students and expect success from all students. It is my expectation for those students who attend class regularly and complete assignments that they will earn an A or B.

- I am available for help whenever I am in my office. I encourage students to do homework at a table outside of my office so that I can help them whenever they have difficulties. Help is also available through email and at my home, if prior arrangements have been made.

- I will try to give grade status reports at least every three weeks.

Course Grades
A: 100 – 90% B: 89 – 80% C: 79 – 70% D: 69 – 60%

Course Policies
Attendance: Daily attendance is expected
Participation: Class participation and group work is expected

Tentative Course Calendar

Assign 1	Conceptual compuer; Evolution of von Neuman machine and current state of computer architecture and devices. Two computers review-- Features and prices
Assign 2	Instructional software. Software review
Assign 3	Internet resources for the classroom. WWW Review
Assign 4	Internet resources for computers. Java Applets
Assign 5	Internet communication E-mail game
Assign 6	WWW Homepage construction. Home Page
Assign 7	The use of LOGO for learning geometric fundamentals. Microworlds EX project / LOGO
Assign 8	Van Hiele levels and geometric learning.
Assign 9	The relationships between different types of triangles and different types of quadrilaterals. Use GSP to construct Venn diagrams. Geometer's Sketchpad -- Geometry relationships
Assign 10	Develop geometric theorems from observation. Geometric plausibility through dynamic examples. Geometer's Sketchpad -- Geometry theorem
Assign 11	Geometric visualization of standardized equations. The use of dynamic parameters for exploration of standard forms. Geometer's Sketchpad
Assign 12	The development of a mathematical system: undefined terms, defined terms, axioms, postulates, lemmas, theorems, corollaries. Proof, application, and synthesis. Circles and angles
Assign 13	Construct and program a simple robotic car to follow a predetermined course. Lego robotics
Assign 14	Reference systems: Number Line, Cartesian, Polar, Spherical, Cylindrical. What are they and who uses them. Fractals: dust, lines, area. Point, logo fractals, Euclidean and Fractal Dimension
Assign 15	Distance between two points in different reference systems. Distance between two points in 2,3,n Euclidean dimensions
Assign 16	Synthetic, Coordinate and dynamic geometry proofs. Prove diagonals of a rectangle are equal, bisect each other
Assign 17	Measurement systems: evolution, design, units. English and metric (SI) systems
Assign 18	Conservation of volume, analyze and compare geometric solids
Assign 19	Geomeric measurment of polygons and solids, perimeter, area, surface area, and volume. Platonic and Archimedean Solids
Assign 20	Reflections (Flips), Translations (Slides), Rotations (Turns), Escher tessellations
Assign 21	Similarity, congruence, constant of proportionality. Construct an inclinometer. Apply similar triangles (trigonometry) to find unknown heights / distances.
Assign 22	Examine limits: movement, sequences, functions Characterize a function by slope, rate of change or derivative. The derivative is the slope of the line tangent to a curve at a point. Calculate the derivative of polynomials from the difference quotient.
Assign 23	Derive the power rule for the derivative of a polynomial function. Calculate the derivative rule for a function, find the derivative at a specified point, write the equation the tangent line at the point, and demonstrate that the line is tangent to the function at the point with a graphing tool.
Assign 24	List area formulae for elementary polygonal shapes. Consider the area between a function and the x-axis between two points. Derive the fundamental theorem of the Calculus, i.e. integral. Compare polygonal formulae to integration to find the area. Use integral to find area under curves.
Assign 25	Develop Riemann sums as a model to find area. Demonstrate that an n-partition model yields the area under a curve as n increases without bound. Model a sphere, cylinder, cone as a volume of revolution. Derive the formula for a sphere, cylinder, or cone.
Assign 26	Examine Euclidean, Spherical and Hyperbolic geometries with Cinderella. Explore the sum of the measure of the angles in a triangle using a dynamic geometry tool such as Cinderella.
Assign 27	Explore the non-Euclidean geometry – Taxi-cab geometry. Investigate segments, perpendicular bisectors, midpoints, and circles in Euclidean and Taxi-cab geometry.
Assign GS	Instructional Lesson and Task (Graduate Students only) – Improve at least 10 days of instruction by including technology and concepts from class. Include assessment to evaluate students and the effectiveness of the changes in the instructional unit.
Assign Final	Final Paper – synthesize the information, concepts and technology from class into a proposal for technology for your classroom.