# Transformational Geometry

This 3 credit course will be offered during the summer of 2016, from June 13 – July 22. It will be offered online through University of Idaho using BbLearn.

Three groups of students are recruited:

- In-service high school teachers seeking further education or renewal credits
- MAT students
- Undergraduate Mathematics Education Students

There are three different course numbers for the course: Math 427/MTHE 527/PD MATH 505. Expectations will be different depending on which course number a student registers for. Information is given below in the course description area.

Common Core calls for transformational geometry to be a primary component of high school and Grade 8 geometry. In transformational geometry, the investigative mode through which students recognize a property/theorem is very closely related to the theorems and axioms they appeal to when it comes time to prove the property/theorem. In Euclidean geometry this is often not the case—a theorem that is discovered by investigation with transformations is proven with other things like the SAS theorem or postulate.

## Course Description

This course develops the geometry concepts of congruence, parallelism, and similarity using transformations (e.g., rigid motions or isometries) and the corresponding matrix representations with respect to a chosen basis. Students will learn how shape geometry (including congruence, parallelism, and similarity) can be developed using transformations and how this can be done with middle school and secondary geometry students.

In addition, graduate students (MTHE 527) will learn how transformational geometry approaches can be used to more rigorously develop undergraduate second and third semester calculus strategies such as change of variable strategies for integration problems.

An objective for all three courses is to engage students in an authentic mathematical experience, where they will take advantage of technology (geometry software), prior knowledge, intuition, heuristics, and formal mathematical systemization as they answer mathematical questions such as how must the definitions, axioms, and postulates from traditional Euclidean geometry be modified to develop an axiomatic approach to congruence, parallelism, and similarity using a transformational approach?

This course does not count towards a MS or PhD in mathematics.

#### Course Expectations

Expectations will be different for the three courses. Professional development students (PD Math 505) will enroll as pass/fail and will not be formally assessed and assigned letter grades. Adequate participation in online discussions and adequately posting on discussion boards will constitute a pass. Math 427 and MTHE 527 students will be expected to participate in asynchronous online discussions, attend several live online collaborative sessions, develop collective write-ups, and complete a formal assessment. MTHE 527 students will be expected to complete an additional assessment on transformational approaches for the undergraduate calculus topics covered in this course.

### Prerequisite

Linear Algebra (330) or equivalent

### Course Fees

During the summer, there are only per-credit fees, with no additional in-state, out-of-state fees.

Summer Session: Tuition & Fees

The cost for Math 427 is $1,158.

The cost for Math 527 is $1,476.

The cost for PD Math 505 is $393.

## High School Teacher Information

Common Core State Standards for Mathematics (CCSSM-M, 2010) asserts that students will experiment with transformations in the plane and understand congruence in terms of rigid motions. Similar statements are made for similarity. These standards have been adopted by the State of Idaho and are now the required content for school mathematics. Traditional secondary teacher preparation has focused on Euclidean geometry, with systematic treatment of analytic geometry occurring primarily in traditional calculus. This training is insufficient for middle school and secondary teachers, who must navigate transformational geometry as prescribed by Common Core. Transformational geometry depends heavily on several fields of mathematics traditionally viewed as separate from school geometry including matrix theory and abstract algebra.

## Community College Teachers Information

Community colleges often teach second and third semester calculus and linear algebra. This course will strengthen future teachers’ preparation to teach calculus by giving careful attention to transformational geometry and its applications to change of variables in integration problems. This course will also strengthen future teachers’ preparation for teaching linear algebra by giving careful attention to uses of matrices (e.g., applications of matrix theory and matrix representations with respect to the basis chosen).

## Contact Information

For more information about these courses, contact the instructor: Dr. David Yopp, dyopp@uidaho.edu .