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Beverley Taylor Sorenson College of Education and Human Development 351 W. University Blvd. – OM 311 Cedar City, UT 84720 Information: (435) 865-8320 Advisement: (435) 865-8083 FAX: (435) 865-8046 www.suu.edu/ed EDUC 5525 - Mathematics for Elementary Teachers K-8: Geometry and Measurement Co-Facilitator: Email: Credit Hours: 3 Co-Facilitator: E-Mail: Semester: Winter 2011-12 Course Description: To provide practicing teachers a deeper understanding of the geometry and measurement content that exists in the state core and instructional strategies to facilitate the instruction of this content. This course aligns with the Utah State Office of Education’s Elementary Math Endorsement Syllabus Framework - 01/2010. Course Objectives NCTM Standards 1. Increase geometry and measurement content knowledge. 2. Design worthwhile geometric and measurement tasks for elementary students. 3. Orchestrate discourse related to geometric and measurement topics in an elementary classroom. 4. Model and represent geometric and measurement concepts and relationships. 5. Improve problem-solving skills using geometry and measurement. 6. Reason, justify, and analyze about geometric and measurement relationships. 7. Communicate geometric and measurement ideas orally, visually, and in writing. 8. Use a variety of tools including technology to enhance classroom instruction and increase student understanding of geometry and measurement. 9. Address developmental issues associated with learning geometry and measurement. 11.1, 11.2, 11.3, 11.4 11.1, 11.2 Course Outline: Subject to change as necessary Class Topics 11.1 1.1, 1.3 11.3, 11.4 3.1, 3.2, 3.3, 11.1, 11.2, 11.3, 11.4 7.6, 8.1, 8.2, 8.7, 8.9 7.4, 8.1, 8.7 Assignments Session 1 11.1, 11.2, 11.3 Welcome / Introductions & Class Norms Review assignments: [1] Overview of the course Course Goals and Outcomes Assignments Measurement Understanding and using measurement concepts Units of measurement including length, weight, capacity, and angles. Include metric, customary, and non-standard units of measurement. Conversion of units within and between different systems. 2 3 4 Line and Angle Relationships Parallel and perpendicular lines. Finding angles when parallel lines are cut by a transversal and showing lines are parallel given information about angles including corresponding angles, same side interior/exterior angles, and alternate interior/exterior angles. Angles that sum to a straight angle. Vertical angles Attributes of Geometric Figures Intersection properties of lines and circles with themselves and each other. Attributes of triangles including the sum of angles, the sum of the non-right angles of a right triangle, the exterior angles of a triangle, the base angles of an isosceles triangle, and the angles of an equilateral triangle. Classifying triangles and symmetry of triangles. Attributes of quadrilaterals including the opposite angles in a parallelogram and the interior angles between two parallel sides in a trapezoid. The definition of a (regular) polygon and attributes of polygons including the sum of the angles of a polygon and the sum of exterior angles of a polygon. Definitions and Axiomatic Nature of Geometry Mathematical problem solving task Differentiated Lesson Math Journal Administer Pre-test Administer M-CAP & MCOMP Math Journal (group A) Homework Math Journal (group B) Homework Math Journal (group A) Homework Undefined and defined terms. Axioms for plane geometry including the parallel lines Similarity and Congruence Congruence tests for triangles, proving facts about side-lengths and angles using congruent triangles. Math Journal (group B) Homework o Similarity and congruence tests for quadrilaterals and how they differ from those for triangles. 5 Constructions Using Appropriate tools to perform constructions to: 1. Copy line segments and angles 2. Copy triangles given three sides, two sides and an included angle, or two angles and the included side. 3. Construct a line parallel or perpendicular to a given line through a given point when possible [2] Math Journal (group A) Homework Construct a circle that is circumscribed about a triangle or that is 6 inscribed in a triangle by the use of perpendicular bisectors and angle bisectors. Transformations and Tessellations 7 Rotations, translations, reflections and their composites. Transformation of geometric figures including deconstructing and recombining geometric figures Tessellations of the plane by regular polygons and irregular shapes Definition of circle, radius, diameter and circumference The number π defined as the ratio of the circumference of a circle to the diameter and why it is the same for every circle. Central and inscribed angles, arc, and sector Areas of circles and sectors, and π as the ratio of the area circle to the square of the radius. Demonstrations of the area formula for a circle and areas of sectors. Perimeter and Area 9 Square units, rectangles with whole number side lengths Intersections and unions of regions and properties of area including finding area by deconstructing and recombining geometric figures. The relationship between area and perimeter in rectangles; i.e., among all rectangles, squares have the largest area with the fixed perimeter and the smallest perimeter with a fixed area. Derive area formulas for triangles, parallelograms, and trapezoids. Convert area units and show how scaling affects area. Surface Area and Volume 10 Homework Circles 8 Math Journal (group B) Derive the surface area and volume formulas for a rectangular prism. Define prisms and cylinders including oblique prisms and cylinders and derive surface area and volume formulas for right prisms and cylinders. Use the slice (Cavalieri) principle for comparing areas and volumes Define pyramids and cones and derive volume and surface area formulas. Convert volume units and show how scaling effects area and volume Pythagorean Theorem Proof of the Pythagorean Theorem and its converse Square roots and Pythagorean Triples Special right triangles- isosceles right triangles and 30-60-90 triangles Applications of Geometry to Algebra Use the Pythagorean Theorem to find a formula for the distance [3] Math Journal (group A) Homework Math Journal (group B) Homework Differentiated Lesson Due Math Journal (group A) Homework Math Journal (group B) Homework between two points in the coordinate plane 11 Review Tasks Class registration Post-Test Course Evaluation Post-Test Assignments: Assignment Homework Pre-test (participation score) Differentiated Lesson Math Journal (biweekly homework) Post-test Total Points Possible 10 Points for each task 25 Points 50 Points 10 points each: 50 Points Total Possible 100 Points 25 Points 50 Points 50 Points 50 points 50 Points 275 Points Assignment Descriptions: Homework: Selected mathematical tasks focused on developing problem solving strategies and exposing important ideas about geometry and measurement. These are completed outside of class and turned in the following week. Sometimes a reading assignment will be given as part of the homework. Differentiated lesson: Select one of the course objectives. Develop a lesson to meet the objective for whole group instruction and additional differentiation activities for three populations: high-achieving students, English Language Learners, and Special Education. (Due Class #9) Reflections on your own mathematical understanding: Keep a biweekly reflection log that discusses: 1. a mathematical idea from the session that is new or important for you, 2. a question you have, or 3. an application of the mathematical idea for your classroom. You will be assigned to be a member of either “Group A” or “Group B” for the purpose of collecting and scoring math journal entries on a biweekly basis. Pre and Post Exams: The pre- and post-tests are based upon key concepts and principles that are taught during the course. Content of the test may include multiple choice, short answer, computational, and application questions. Study guides will not be made for the post-test, but students are encouraged to use the pre-test, class notes, and math tasks as resources for preparation. The posttest is open book, open notes, but must be completed individually without assistance from peers or the instructors. Course Texts: Parker, T. H. & Baldridge, S. J. (2004). Elementary mathematics for teachers. USA: Ash Publishing Beckman, S. (2008) Mathematic for Elementary Teachers Van de Walle, J. A. (2010). Elementary and middle school mathematics: Teaching developmentally (7th edition). Boston: Allyn & Bacon. [4] Sefton- Grading Scale: All assignments must be submitted in order to receive a grade. Attendance and participation are essential components of the course. If a student misses more than one class, the final grade will be deducted one full letter grade. If a student is often late for class, the instructors may deduct accumulated missed class time and dock the final grade accordingly. Attending one of the other cohort sessions of the same class and content can make up missed classes. Percentages for grades are as follows: 100-95 94-90 89-87 86-83 - A AB+ B 82-80 79-77 76-73 72-70 - BC+ C C- Academic Integrity: Scholastic dishonesty will not be tolerated and will be prosecuted to the fullest extent. You are expected to have read and understood the current issue of the student handbook (published by Student Services) regarding student responsibilities and rights, and the intellectual property policy, for information about procedures and about what constitutes acceptable on-campus behavior. Pedagogy (infused into the mathematics topics, not taught in isolation) (Standards for Elementary Mathematics Specialists, Association of Mathematics Teacher Educators) Knowledge of mathematics learners: Employ prior knowledge, skills, dispositions, beliefs, conceptions, misconceptions and confusions around particular topics; Employ knowledge of cognition, child development, and learning theory; Use knowledge of learners culturally and socially to know what motivates and inspires them. Knowledge of effective mathematics teaching practices for student and teacher learning: Design, select and/or adapt worthwhile tasks and sequences of examples that support a particular learning goal, including tasks to be solved by teachers themselves, and others for which they try to anticipate children's solutions, and design responses. Develop sensitivity with the careful use of technical language, attending to both mathematical integrity, and usability by learners. Construct and evaluate multiple representations of a mathematical idea or process, and establish correspondences between representations. Use questions to effectively probe mathematical understanding and make productive use of responses. Develop learners' skills at clear and coherent public mathematical communication in a classroom setting, using a variety of resources and media. Model mathematical practices – questioning, representing, communicating, problem solving, inquiring and conjecturing, making connections, reasoning and proving, self-assessing and cultivate the learning of such practices. Use various instructional applications of technology and be able to make judicious, mathematically and pedagogically grounded use of them. [5] Construct and mathematically evaluate contexts or story problems for various mathematical situations and processes. Analyze and evaluate student work, and design appropriate responses. Develop skillful and flexible use of different instructional formats – whole group, small group, partner, and individual – in support of learning goals. Know the resources for managing the diversities – cultural, linguistic, gender, socioeconomic, developmental – of the classroom, and ways to skillfully deploy them in support of the learning of all students. Understand and address diversity in planning and implementing instruction and in helping other teachers foster mathematics classroom environments that support the learning of all students. Understand and utilize various cultures' approaches to learning mathematics. Knowledge of mathematics curriculum and assessment: Know common developmental paths (learning trajectories) related to foundational mathematical topics (e.g., place value, rational numbers and equivalence); understand these developmental paths, and use them to sequence activities to build learning environments that are developmentally appropriate and effective. Select, use, adapt and determine the suitability of mathematics curricula and teaching materials (e.g., textbooks, technology, manipulatives) for particular learning goals. Be cognizant of a formative assessment cycle (administering a formative assessment task, analyzing student work from the task, using that analysis to enhance teacher knowledge, and designing and teaching reengagement lessons) and resources. Provide appropriate interpretations of assessment results, and communicate results (in context) to specific individuals and groups (e.g., students, parents, caregivers, colleagues, administrators, policymakers, community members). [6]